2011 Caruntu Constantin

UNIVERSITATEA TEHNICĂ “GHEORGHE ASACHI” DIN IAŞI

Şcoala Doctorală a Facultăţii de

Automatică şi Calculatoare

NETWORKED PREDICTIVE CONTROL FOR FAST PROCESSES

(Controlul predictiv în reţea al proceselor rapide) - TEZĂ DE DOCTORAT -

Conducător de doctorat: Prof. univ. dr. ing. Corneliu Lazăr

Doctorand: Ing. Constantin Florin Căruntu

IAŞI - 2011

UNIUNEA EUROPEANĂ GUVERNUL ROMÂNIEI

MINISTERUL MUNCII, FAMILIEI ŞI PROTECŢIEI SOCIALE

AMPOSDRU

Fondul Social European POSDRU 2007-2013

Instrumente Structurale 2007-2013

OIPOSDRU UNIVERSITATEA TEHNICĂ “GHEORGHE ASACHI”

DIN IAŞI

UNIVERSITATEA TEHNICĂ “GHEORGHE ASACHI” DIN IAŞI

Şcoala Doctorală a Facultăţii de Automatică şi Calculatoare

NETWORKED PREDICTIVE CONTROL FOR FAST PROCESSES

(Controlul predictiv în reţea al proceselor rapide) - TEZĂ DE DOCTORAT -

Conducător de doctorat: Prof. univ. dr. ing. Corneliu Lazăr

Doctorand: Ing. Constantin Florin Căruntu

IAŞI - 2011



AMPOSDRU




DIN IAŞI

Teza de doctorat a fost realizată cu sprijinul financiar al

proiectului „Burse Doctorale - O Investiţie în Inteligenţă (BRAIN)”.

Proiectul „Burse Doctorale - O Investiţie în Inteligenţă (BRAIN)”,

POSDRU/6/1.5/S/9, ID 6681, este un proiect strategic care are ca

obiectiv general „Îmbunătățirea formării viitorilor cercetători în cadrul

ciclului 3 al învățământului superior - studiile universitare de doctorat

- cu impact asupra creșterii atractivității şi motivației pentru cariera în

cercetare”.

Proiect finanţat în perioada 2008 - 2011.

Finanţare proiect: 14.424.856,15 RON

Beneficiar: Universitatea Tehnică “Gheorghe Asachi” din Iaşi

Partener: Universitatea “Vasile Alecsandri” din Bacău

Director proiect: Prof. univ. dr. ing. Carmen TEODOSIU

Responsabil proiect partener: Prof. univ. dr. ing. Gabriel LAZĂR



AMPOSDRU




DIN IAŞI

familiei mele to my family

Acknowledgements

I would like to thank all people who have helped and inspired me during my

doctoral study.

Foremost, I would like to express my sincere gratitude to my advisor Prof. Dr.

Eng. Corneliu Lazar, Head of the Department of Automatic Control and Applied

Informatics, “Gheorghe Asachi” Technical University of Iasi, for the continuous

support of my Ph.D study and research, for his detailed and constructive comments,

for his patience, motivation, enthusiasm, and immense knowledge. His guidance

helped me in all the time of research and writing of this thesis.

My sincere thanks also goes to Prof. Dr. Ir. Paul P. J. van den Bosch and Dr.

Mircea Lazar, for giving me the opportunity to work with them for three months in

the Department of Control Systems, Faculty of Electrical Engineering at Eindhoven

University of Technology in the Netherlands and gave me untiring help during my

difficult moments. Dr. Mircea Lazar sets an example of a world-class researcher

for his rigor and passion on research. I thank him for his insights and suggestions

that helped to increase my research skills.

I am also grateful to the staff members of the Department of Automatic Control

and Applied Informatics, “Gheorghe Asachi” Technical University of Iasi, for their

help during the three years of Ph.D study and research. I would like to mention

some of them: Prof. Dr. Eng. Octavian pastravanu, Prof. Dr. Eng. Alexandru

Onea, Prof. Dr. Eng. Mihaela Hanako Matcovschi, Conf. Dr. Eng. Mihai Posto-

lache, Conf. Dr. Eng. Lavinia Ferariu.

I would like to thank Michael Kiener, director of the R&D Center in Iasi of Con-

tinental Automotive Romania, Eng. George Costeniuc and Eng. Sorin Carari, for

giving me the opportunity to work in their laboratories. The experimental results,

which are a part of this Ph.D thesis, couldn’t be obtained without their help.

During this work I have collaborated with many colleagues for whom I have

great regard, and I wish to extend my warmest thanks to all those who have helped

me with my work: Eng. Andreea Elena Balau, Eng. Catalin Braescu (TUIASI), Ir.

Rob H. Gielen (TUE), Dr. Eng. Stefano Di Cairano (Ford Research and Advanced

Engineering, US), Eng. Daniel Ionut Patrascu, Eng. Doina Onu (Continental Au-

tomotive Romania) for their constructive cooperation on our papers.

I thank my fellow colleagues in the Department of Automatic Control and Ap-

plied Informatics: Andreea, Cosmin, Alina, Carlos, Bogdan, Adrian, Simona, Mar-

ius, Cristina, Alex, Mircea, for the stimulating discussions, for the sleepless nights

we were working together before deadlines, and for all the fun we have had in the

last three years.

Last but not the least, I would like to thank my family: my parents Iuliea and

Constantin, and my brother Cristi, for all their support and love.

Contents

Contents v

List of Figures ix

1 Introduction 11.1 Networked control systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Direct structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.2 Hierarchical structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Network-induced imperfections . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Model predictive control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Delay modeling and compensation strategies . . . . . . . . . . . . . . . . . . . 5

1.5 Thesis summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.6 Summary of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.7 Basic mathematical notation and definitions . . . . . . . . . . . . . . . . . . . 11

2 Networked control systems 132.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 Networked control systems modeling . . . . . . . . . . . . . . . . . . . . . . 16

2.3.1 NCSs addings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.2 Modeling NCSs with network-induced delay . . . . . . . . . . . . . . 20

2.3.2.1 Delays smaller than the sampling period . . . . . . . . . . . 20

2.3.2.2 Delays larger than the sampling period . . . . . . . . . . . . 21

2.4 Time-varying delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4.1 Delays modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4.1.1 Delays modeled as polytopic inclusions . . . . . . . . . . . . 25

2.4.1.2 Delays modeled as disturbances . . . . . . . . . . . . . . . . 27

v

2.4.2 Delay compensation strategies . . . . . . . . . . . . . . . . . . . . . . 27

2.4.2.1 Adaptive Smith Predictor . . . . . . . . . . . . . . . . . . . 32

2.4.2.2 Explicit Model Predictive Control . . . . . . . . . . . . . . . 33

2.4.2.3 Lyapunov-based Model Predictive Control . . . . . . . . . . 34

2.4.2.4 Robust Lyapunov-based Model Predictive Control . . . . . . 36

2.4.2.5 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . 38

2.5 Automotive networked control systems . . . . . . . . . . . . . . . . . . . . . . 39

2.5.1 Automotive control systems . . . . . . . . . . . . . . . . . . . . . . . 39

2.5.2 Drivetrain control systems . . . . . . . . . . . . . . . . . . . . . . . . 41

2.5.3 CAN-based networked control systems . . . . . . . . . . . . . . . . . 43

2.5.3.1 In-vehicle communication networks for control . . . . . . . 44

2.5.3.2 Controller Area Network . . . . . . . . . . . . . . . . . . . 46

2.5.3.3 CAN-induced delays in automotive applications . . . . . . . 47

2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3 Linear networked predictive control 513.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.2 Modeling the plant including the delays . . . . . . . . . . . . . . . . . . . . . 52

3.2.1 Average method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.2.2 Identification method . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.2.3 Adaptation method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.3 Linear networked predictive controller design . . . . . . . . . . . . . . . . . . 54

3.3.1 Prediction model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.3.2 λ-scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.4 Linear networked predictive control of automotive fast systems . . . . . . . . . 59

3.4.1 Average delay modeling and LNPC applied to a DC motor . . . . . . . 59

3.4.2 Identification modeling and LNPC of an ICE . . . . . . . . . . . . . . 62

3.4.3 λ-scheduling control applied to a valve-clutch subsystem . . . . . . . . 66

3.4.3.1 Valve-clutch system model . . . . . . . . . . . . . . . . . . 68

3.4.3.2 Simulation results . . . . . . . . . . . . . . . . . . . . . . . 70

3.5 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.5.1 Real-time test-bench . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.5.2 Drivetrain model including the clutch flexibility . . . . . . . . . . . . . 77

3.5.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

vi

4 Nonlinear networked predictive control 894.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.2 Delays modeled as polytopic inclusions . . . . . . . . . . . . . . . . . . . . . 90

4.3 Lyapunov-based Model Predictive Control . . . . . . . . . . . . . . . . . . . . 92

4.3.1 Lyapunov-based predictive controller . . . . . . . . . . . . . . . . . . 92

4.3.2 Implementing Lyapunov-based predictive controller as a single linear

program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.4 LMPC applied to a two masses drivetrain controlled through CAN . . . . . . . 97

4.4.1 Control Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.4.2 Plant model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.4.3 Matlab/Simulink results . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.4.4 TrueTime results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.4.4.1 Simulation environment . . . . . . . . . . . . . . . . . . . . 108

4.4.4.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . 109

4.4.4.3 Acceleration test . . . . . . . . . . . . . . . . . . . . . . . . 110

4.4.4.4 Tip-in tip-out maneuvers . . . . . . . . . . . . . . . . . . . 111

4.4.5 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

4.4.5.1 HIL simulation . . . . . . . . . . . . . . . . . . . . . . . . . 114

4.4.5.2 Control architecture description . . . . . . . . . . . . . . . . 115


4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5 Robust nonlinear networked predictive control 1235.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.2 Modeling input delays as disturbances . . . . . . . . . . . . . . . . . . . . . . 124

5.3 Modeling input and output delays as disturbances . . . . . . . . . . . . . . . . 125

5.3.1 Forward channel delays . . . . . . . . . . . . . . . . . . . . . . . . . 126

5.3.2 Feedback channel delays . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.4 Robust Lyapunov-based Model Predictive Control . . . . . . . . . . . . . . . . 128

5.4.1 Robust Lyapunov-based predictive controller . . . . . . . . . . . . . . 129

5.4.2 Implementation issues . . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.5 Robust LMPC applied to control a DC motor through Ethernet . . . . . . . . . 135

5.5.1 Networked control architecture . . . . . . . . . . . . . . . . . . . . . . 136

5.5.2 DC motor modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 137


vii


5.5.4.1 Simulation environment . . . . . . . . . . . . . . . . . . . . 143


5.6 Robust LMPC for a CAN-controlled drivetrain including the clutch . . . . . . . 147

5.6.1 Discrete-time model . . . . . . . . . . . . . . . . . . . . . . . . . . . 147


5.6.2.1 Scenario 1: Acceleration test . . . . . . . . . . . . . . . . . 153

5.6.2.2 Scenario 2: Deceleration test . . . . . . . . . . . . . . . . . 154

5.6.2.3 Scenario 3: Tip-in tip-out maneuvers . . . . . . . . . . . . . 154

5.6.2.4 Scenario 4: Stress test . . . . . . . . . . . . . . . . . . . . . 155


5.6.4 Real-time results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

6 Conclusions 1636.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

6.1.1 Delay modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

6.1.2 Linear networked predictive control . . . . . . . . . . . . . . . . . . . 164

6.1.3 Nonlinear and Robust nonlinear networked predictive control . . . . . 165

6.1.4 Automotive networked applications . . . . . . . . . . . . . . . . . . . 166

6.2 Future research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

Appendix A 169

References 173

viii

List of Figures

1.1 NCS with direct structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 NCS with hierarchical structure. . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Control system with network-induced time delay. . . . . . . . . . . . . . . . . 16

2.2 Control system with grouped network-induced time delay. . . . . . . . . . . . 16

2.3 NCS model introduced by [Luck and Ray, 1990]. . . . . . . . . . . . . . . . . 17

2.4 NCS model introduced by [Krtolica et al., 1994]. . . . . . . . . . . . . . . . . 18

2.5 NCS model introduced by [Chan and Ozguner, 1995]. . . . . . . . . . . . . . . 18

2.6 Delay distribution considered in [Nilsson et al., 1998]. . . . . . . . . . . . . . 19

2.7 NCS Markov chain modeling introduced by [Nilsson et al., 1998]. . . . . . . . 19

2.8 Network-induced delays time diagram. . . . . . . . . . . . . . . . . . . . . . . 23

2.9 Concept of network disturbance in bilateral teleoperation systems. (a) Bilateral

teleoperation system with time delay. (b) Bilateral teleoperation system with

network disturbance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.10 NCS with two predictive controllers. . . . . . . . . . . . . . . . . . . . . . . . 32

2.11 Time-delay compensation in bilateral teleoperation systems. . . . . . . . . . . 32

2.12 NCS with adaptive Smith Predictor. . . . . . . . . . . . . . . . . . . . . . . . 33

2.13 Schematic vehicle structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.14 Automotive networked control systems interconnected through gateways. . . . 44

2.15 Vehicle networks technologies. . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.1 System output - different delays. . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.2 Delay time distribution (DC motor control). . . . . . . . . . . . . . . . . . . . 60

3.3 DC motor output for d = 3 (step reference). . . . . . . . . . . . . . . . . . . . 61



3.6 DC motor output for d = 3 (sequence of pulses reference). . . . . . . . . . . . 62

ix



3.9 Delay time distribution (ICE control). . . . . . . . . . . . . . . . . . . . . . . 64

3.10 Engine output (PI & Smith predictor). . . . . . . . . . . . . . . . . . . . . . . 65

3.11 Throttle valve angle (PI & Smith predictor). . . . . . . . . . . . . . . . . . . . 66

3.12 Engine output (LNPC). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.13 Throttle valve angle (LNPC). . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.14 Charging phase of the valve clutch system (dashed - discharging phase). . . . . 68

3.15 Valve-clutch and simulation models responses. . . . . . . . . . . . . . . . . . 70

3.16 Clutch displacements (without delays). . . . . . . . . . . . . . . . . . . . . . . 71

3.17 Schematic representation of the valve-clutch networked control system. . . . . 71

3.18 Time distribution of communication delay (valve-clutch system). . . . . . . . . 72

3.19 Clutch displacements for different lambdas. . . . . . . . . . . . . . . . . . . . 72

3.20 Clutch displacements (PI & Smith predictor). . . . . . . . . . . . . . . . . . . 73

3.21 Clutch displacements (LNPC). . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.22 Voltage signals (PI & Smith predictor). . . . . . . . . . . . . . . . . . . . . . . 74

3.23 Voltage signals (LNPC). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.24 Control signal increments (PI & Smith predictor). . . . . . . . . . . . . . . . . 74

3.25 Control signal increments (LNPC). . . . . . . . . . . . . . . . . . . . . . . . . 75

3.26 Average communication delay. . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.27 Real-time test-bench. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.28 Three inertias drivetrain model including the clutch. . . . . . . . . . . . . . . . 78

3.29 Clutch functionality a) stiffness characteristic; b) clutch springs. . . . . . . . . 78

3.30 ControlDesk console. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.31 Vehicle velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.32 Clutch mode of operation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.33 Engine torque (control signal). . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.34 Driveshaft torque. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.35 Speed difference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.36 Speed difference - detail. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.1 Simplified drivetrain schematic representation and control architecture. . . . . . 98

4.2 Delays induced by CAN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.3 History of λ∗k throughout the simulation. . . . . . . . . . . . . . . . . . . . . . 105

4.4 Vehicle velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

x

4.5 Speed difference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.6 Engine torque (control signal). . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.7 Driveshaft torque. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.8 TrueTime Simulink diagram - two inertias drivetrain model. . . . . . . . . . . 109

4.9 Acceleration test: History of λ∗k throughout the simulation. . . . . . . . . . . . 110

4.10 Acceleration test: TrueTime simulation results. . . . . . . . . . . . . . . . . . 111

4.11 Acceleration test: Delays induced by CAN. . . . . . . . . . . . . . . . . . . . 112

4.12 Tip-in tip-out maneuvers: Simulation results. . . . . . . . . . . . . . . . . . . 112

4.13 Tip-in tip-out maneuvers: History of λ∗k throughout the simulation. . . . . . . . 113

4.14 Tip-in tip-out maneuvers: Delays induced by CAN. . . . . . . . . . . . . . . . 113

4.15 HIL simulation setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4.16 Matlab/Simulink model of the drivetrain HIL setup. . . . . . . . . . . . . . . . 115

4.17 CAN communication between ECU and plant. . . . . . . . . . . . . . . . . . . 116

4.18 Delays management. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

4.19 HIL: Vehicle velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4.20 HIL: Engine torque (control signal). . . . . . . . . . . . . . . . . . . . . . . . 119

4.21 HIL: History of λ∗k throughout the simulation. . . . . . . . . . . . . . . . . . . 119

4.22 HIL: Speed difference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.1 Control system with network disturbance. . . . . . . . . . . . . . . . . . . . . 124

5.2 Control system with network-induced time delay as disturbance. . . . . . . . . 126

5.3 DC motor networked control system. . . . . . . . . . . . . . . . . . . . . . . . 136

5.4 Delays introduced by the communication network. . . . . . . . . . . . . . . . 140

5.5 History of λ∗k. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

5.6 Rotor angular velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

5.7 Armature voltage (control signal). . . . . . . . . . . . . . . . . . . . . . . . . 142

5.8 Armature current. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

5.9 TrueTime Simulink diagram - DC motor. . . . . . . . . . . . . . . . . . . . . . 144

5.10 Simulation results: 10% network bandwidth occupied by the interfering node. . 145



5.13 Clutch active mode and velocity histories. . . . . . . . . . . . . . . . . . . . . 148

5.14 Scenario 1: Acceleration test. . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

5.15 Scenario 1: Clutch mode of operation. . . . . . . . . . . . . . . . . . . . . . . 153

5.16 Scenario 2: Deceleration test. . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

xi

5.17 Scenario 3: Tip-in, tip-out maneuver simulation. . . . . . . . . . . . . . . . . . 155

5.18 Scenario 4: Stress test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

5.19 TrueTime Simulink diagram - three inertias drivetrain model. . . . . . . . . . . 156

5.20 TrueTime Scenario 1: Acceleration test. . . . . . . . . . . . . . . . . . . . . . 157

5.21 TrueTime Scenario 1: Clutch mode of operation. . . . . . . . . . . . . . . . . 157

5.22 TrueTime Scenario 2: Deceleration test. . . . . . . . . . . . . . . . . . . . . . 158

5.23 TrueTime Scenario 3: Tip-in, tip-out maneuver simulation. . . . . . . . . . . . 158

5.24 TrueTime Scenario 4: Stress test. . . . . . . . . . . . . . . . . . . . . . . . . . 159

5.25 Real-time results: Tip-in, tip-out maneuver simulation. . . . . . . . . . . . . . 160

xii

Abbreviations

ABS Anti-lock Braking System

ARMA Auto-Regressive Moving Average

ARX AutoRegressive with eXogenous

ASR Anti-Slip Regulation

CAN Controller Area Network

CARIMA Controlled Auto-Regressive Integrated Moving Average

CLF Control Lyapunov Function

CLRF Control Lyapunov-Razumikhin Function

DC Direct Current

DSC Dynamic Stability Control

ECU Electronic Control Unit

ESC Electronic Stability Control

ESP Electronic Stability Program

FCLF Flexible Control Lyapunov Function

GPC Generalized Predictive Control

HIL Hardware-In-the-Loop

ICE Internal Combustion Engine

ISS Input-to-State Stability

xiii

LIN Local Interconnect Network

LMPC Lyapunov-based Model Predictive Control

LNPC Linear Networked Predictive Controller

LQG Linear Quadratic Gaussian

LQR Linear Quadratic Regulator

MOST Media Oriented Systems Transport

MPC Model Predictive Control

NCS Networked Control System

PI Proportional-Integral (controller)

PID Proportional-Integral-Derivative (controller)

PRBS PseudoRandom Binary Sequence

PWA PieceWise Affine

TCS Traction Control System

V DC Vehicle Dynamics Control

ZOH Zero-Order Hold

xiv

Chapter 1

Introduction

Feedback control systems over real-time communication networks, also called networked con-

trol systems, are now widely used in different industries, ranging from automated manufac-

turing plants to automotive and aero-spatial applications. This evolution of standalone con-

trol systems to networked control systems brought many attractive advantages, which include

low cost, simple installation and maintenance, increased system agility, higher reliability and

greater flexibility [Johansson et al., 2005, Jiangang et al., 2007], but the use of communica-

tion networks makes it necessary to deal with the effects of the network-induced imperfec-

tions and constraints: time-varying delays, packet dropouts, variable sampling periods. The

delays may be unknown and time-varying and may deteriorate the performances of the control

systems designed without considering them and even destabilize the closed-loop control sys-

tem [Jiangang et al., 2007]. Existing constant time-delay control methodologies may not be

directly suitable for controlling a system over a communication network since network delays

are usually unpredictable and time-varying. Therefore, to handle network delays in a closed-

loop control system over a communication network, an advanced methodology is required, a

significant emphasis being on developing control methodologies to handle the network delay

effect in NCSs [Tipsuwan and Chow, 2003].

1.1 Networked control systems

In a networked control system, the sensors have the task of measuring the outputs of the plant

and sending the samples to the controller through the network. The controller receives the

measurements from the sensors, calculates the control command and sends the values through

the network to the actuators. The actuators have the task of applying the control commands

1

Introduction

Controller

Actuator

Physical plant

Sensor

Network

Control signal

Sensor measurement

Sensor

Physical

plant

Actuator

Main

controller Network

Reference signal

Sensor measurement

Remote

controller

+

-

Remote system

∆

+

∆

∆ …

0β

2Dβ

ky

kw

Figure 1.1: NCS with direct structure.

received through the network to the physical plant [Onat and Parlakay, 2007].

There are two general configurations to design networked control systems: direct struc-

ture and hierarchical structure, but many of the existing NCSs are realized as hybrid structures

between these two.

1.1.1 Direct structure

This structure is composed of a controller and a remote system which contains a physical plant,

sensors and actuators [Chow and Tipsuwan, 2001]. The controller and the physical plant are

at different spatial locations and connected through a communication network to form a closed

control loop, as it is represented in Fig. 1.1 [Tipsuwan and Chow, 2003, Yang et al., 2006].

The systems works very simple: the controller receives the measurements from the sensors and

calculates the control command which is sent to the remote system through the communication

network. The remote system receives the control command from the controller, acts on the

physical plant, the sensors measure the outputs of the system and then sends the samples to the

controller through the communication network.

An advantage of this structure is that the data is exchanged directly between the controller

and the other components of the NCS (sensors and actuators).

1.1.2 Hierarchical structure

This structure is composed of several subsystems organized in a hierarchical structure, in which

every subsystem contains (Fig. 1.2): a set of sensors, a set of actuators and a controller [Tip-

suwan and Chow, 2003]. These components are attached to one physical plant. In this case, a

controller from a subsystem receives reference signals from a central controller through a com-

munication network [Gupta and Chow, 2008]. At the same time, the measurements from the

sensors of each subsystem are transmitted to the local controller and to the central controller

through the same communication network.

Periodically, the central controller calculates and sends the reference signal to the controllers

from each subsystem. In this case, the sampling period of the external control loop must be

2

Network-induced imperfections

Controller

Actuator

Physical plant

Sensor

Network

Control signal

Sensor measurement

Sensor

Physical

plant

Actuator

Main

controller Network

Reference signal

Sensor measurement

Remote

controller

+

-

Remote system

∆

+

∆

∆ …

0β

2Dβ

ky

kw

Figure 1.2: NCS with hierarchical structure.

greater than the sampling period of the internal control loop [Chow and Tipsuwan, 2001].

1.2 Network-induced imperfections

In most cases, introducing a communication network does not significantly affect the con-

trol system performances [Vatanski et al., 2007]. Nevertheless, for certain physical plants

which are time-restricted, implementing a networked control system must be realized tak-

ing into account the fact that a communication network is not perfect. For such systems,

implementing a communication network for closed-loop control leads to introducing an ad-

ditional delay, either constant, either time-varying, which makes system analysis and con-

troller design become of higher complexity. Another consequence of introducing communi-

cation networks in the control loop is represented by the possibility of data packet dropout.

These two problems can seriously affect the control performances and can even destabilize

the closed-loop control system [Halevi and Ray, 1990, Zhang et al., 2001, Chow and Tip-

suwan, 2001, Juanole and Mouney, 2005, Jiangang et al., 2007, Gupta and Chow, 2008].

There are many network-induced imperfections and constraints which are categorized in

five types in [Heemels et al., 2010], but, usually, the available literature on NCS considers only

some of them in the analysis of NCS: (i) quantization errors in the signals transmitted over the

network due to the finite word length of the packets; (ii) packet dropouts caused by the unreli-

ability of the network are considered in [Onat et al., 2008, de la Pena and Cristofides, 2008,

Sheng and Pan, 2010, Li et al., 2008a, Cloosterman et al., 2010, Shen et al., 2010, Onat et

al., 2010]; (iii) variable sampling/transmission intervals are taken into account in [Clooster-

man et al., 2010, Bernardini et al., 2010]; (iv) time-varying network-induced delays, which can

be smaller than the sampling period [Zhang et al., 2005,Salehi et al., 2008,Gielen and M.Lazar, 2009a,

Wang et al., 2010,Huang et al., 2010] or larger than the sampling period [Li et al., 2008a,Cloost-

erman et al., 2010, Shen et al., 2010, Cloosterman et al., 2009]; (v) communication constraints

caused by the sharing of the network by multiple nodes and the fact that only one node is al-

lowed to transmit its packet per transmission are addressed in [Liu et al., 2010]. Moreover, an

analysis of different aspects of the limitations imposed by the use of communication channels

3

Introduction

to connect the elements of NCSs is given in [Hespanha et al., 2007].

1.3 Model predictive control

The predictive control strategies were initially utilized for slow processes: oil refineries, petro-

chemicals, pulp and paper, primary metal industries, gas plant [Tran and Vlacic, 2006], but

starting with the evolution of hardware components and algorithms, the possibility to imple-

ment these types of control algorithms to fast processes, which have reduced sampling periods,

appeared: vehicle engine and traction control, aero-spatial applications, autonomous vehicles,

power generation and distribution [Gu et al., 2009].

Predictive control techniques have been introduced mainly in order to deal with plants that

have complex dynamics (unstable inverse systems, time-varying delay, etc.) and plant model

mismatch. They are of a particular interest from the point of view of both broad applicability

and implementation simplicity, being applied on large scale in industry processes, having good

performances and being robust at the same time.

Model predictive control is a popular control strategy based on using a model to predict at

each sampling period the future evolution of the system from the current state along a given

prediction horizon, see, e.g., [Maciejowski, 2002], [Camacho and Bordons, 2004] and [Rawl-

ings and Mayne, 2009] for a review of results in the area of MPC. The optimal control sequence

of control inputs is obtained by minimizing an optimization criterion; the first control signal

is applied to the process and the procedure is repeated at the next sampling instant. This fea-

ture makes the MPC approach very appropriate to incorporate the input/output constraints into

the on-line optimization problem as well as to compensate time-delays, which increases the

possibility of its application in the synthesis and analysis of NCSs.

Traditional control methods such as PID or LQR control cannot explicitly enforce hard con-

straints. On the other hand, the capability of dealing with constraints is one of the main reason

why MPC has become successful. In MPC, the control action is usually computed by solving a

finite-horizon open-loop optimization problem on-line, at each control sampling instant, using

the measured state as initial condition for predicting the plant’s behavior, and while explicitly

taking input and state constraints into account. Unfortunately, standard MPC techniques require

a long prediction horizon to guarantee stability, which makes the corresponding optimization

problem too complex to solve.

There are many research results reported on MPC for NCS. Among these, the segmented

time-stamped dynamic matrix control is proposed in [Zhang et al., 2005] and an adaptive pre-

4

Delay modeling and compensation strategies

dictive functional control strategy is described in [Wang et al., 2010]. In [Onat et al., 2008] a

plant model is kept inside the controller node, which is used to predict the plant states into the

future to generate corresponding control outputs. Furthermore, a state machine is used in the

actuator to select the proper output to reduce the difference between the actual plant states and

those predicted by the model. In [Li et al., 2008a] two predictive observer-based controllers are

designed, which are used when the full state vector is not available. A Lyapunov-based model

predictive controller is designed in [de la Pena and Cristofides, 2008] for nonlinear uncertain

systems, which takes data losses explicitly into account, both in the optimization problem and

in the controller implementation.

MPC is increasingly seen as an attractive methodology for the fast processes from auto-

motive applications due to its capability to directly handle various specifications including the

optimization of a cost function while enforcing constrains on state and control variables. This

fits the problems considered in this thesis, i.e., controlling different automotive subsystems,

which have fast dynamics, through a communication network, i.e., CAN. However, due to the

complexity of the models and the stringent time limitations for finding a solution on-line, stan-

dard “sufficiently long horizon” MPC approaches [Maciejowski, 2002] do not represent a viable

solution, leading to solutions that are too complex and, at the same time, too conservative to be

implemented in real-time control applications, keeping in mind that the physical plants from

automotive applications have fast dynamics, requiring short sampling periods.

1.4 Delay modeling and compensation strategies

The delays introduced by the communication network are time-varying and unpredictable and

may deteriorate the closed-loop system performance and even reduce the system’s stability re-

gion [Jiangang et al., 2007]. In [Wu et al., 2009] the network-induced delays are modeled by

Markovian chains, the closed-loop systems resulting as jump linear systems with two modes.

The time-varying bounded delays are modeled as random processes described by continuous

probability density functions in [Bernardini et al., 2010]. In [Yue et al., 2009] it is considered

that the probability distribution of delays is known a priori. The networked control system is

formulated using Jordan canonical form in [Salehi et al., 2008,Cloosterman et al., 2009,Bernar-

dini et al., 2010, Cloosterman et al., 2010] or based on the Cayley-Hamilton theorem in [Gie-

len and M.Lazar, 2009a, Gielen et al., 2010] of continuous-time matrices as a mode dependent

polytopic uncertain system and, based on these uncertainty functions, a convex overapproxi-

mation of the discrete-time model that explicitly takes into account the bounds of the time-

5

Introduction

varying delays is used for analysis. Another recent approach is to consider the communication

delay as a disturbance and to use a communication delay observer (CDOB) as in [Natori et

al., 2008, Gadamsetty et al., 2009], which observes the condition of the network and compen-

sates the effect of the variable delays in the control algorithm.

Various research has been conducted on compensating the time-varying delays induced in

NCSs and numerous control strategies were reported in the literature: multiple-delay Smith pre-

dictor based controller for systems with bounded uncertain delay [Ibeas et al., 2007], variable-

period sampling scheme for NCSs with random time delay based on BP neural network pre-

diction [Jiangang et al., 2007], Smith dynamic predictor combined with fuzzy immune PID

control [Du and Qian, 2008], predictive networked controller based on Smith predictor that in-

cludes an adaptation loop to decrease the influence of the communication delay on the control

performance [Velagic, 2008] and even model predictive control [Gielen and M.Lazar, 2009b].

1.5 Thesis summary

The remainder of this thesis is structured as follows.

Chapter 2, entitled Networked control systems, provides general information regarding the

problems that arise and the solutions that were proposed in the available literature due to the in-

troduction of communication networks in the closed-loop control systems. Firstly, the network-

induced imperfections and constraints are presented and then different mathematical models

which describe the functioning of a NCS are depicted. Secondly, several delay models and

compensation strategies are illustrated. Thirdly, the control systems for the fast processes that

are in nowadays vehicles are presented and then the most utilized communication network in

automotive closed-loop control systems, i.e., CAN, is briefly described and a method to find the

bounds of the delays that appear on CAN in automotive applications is presented.

Chapter 3, entitled Linear networked predictive control, focuses on decreasing the influence

of the time-varying delay induced in a NCS on the control system performance, by proposing

a networked controller based on a new predictive strategy, the delays that appear in the NCS

being taken into account by the presented strategy. It is assumed that the delays in the com-

munication network are bounded and three methods of considering the delays by the predictive

control algorithm are proposed: an average delay method, an identification method, which uses

an estimated model that includes the delays in the plant model, and an adaptation method, which

adapts the control algorithm to the time-varying delays in the communication network. More-

over, a new method by which the controller is adapted to the difference between the desired

6

Thesis summary

reference and the plant output is proposed. It is shown that the proposed control algorithm can

be analytically determined off-line, meaning that the actual control command can be computed

very fast, so this algorithm is capable of satisfying the timing constraints associated to the fast

processes that require short sampling periods.

Another contribution of this chapter is the development of a PWA model for an automotive

driveline, which brings several improvements with respect to the models available in the litera-

ture, by considering the driving load given by the airdrag torque, gravity and rolling resistance,

and four modes to describe the clutch dynamics. Taking into account all these factors yields a

more accurate model of the driveline dynamics.

The strategy is then tested on several physical plants (fast processes) from automotive ap-

plications (DC motor, ICE, valve-clutch system, drivetrain) and comparisons were made with

two different networked controllers: a PI controller and a Smith-like predictive controller with

adaptation to communication delay developed in [Velagic, 2008], to illustrate the performance

of the proposed method. Moreover, the proposed networked predictive control strategy was

tested on a real-time simulation test-bench and the obtained results illustrate that the strategy

can be implemented to control the actual physical plant.

Chapter 4, entitled Nonlinear networked predictive control, addresses the problem of mod-

eling a CAN-based network control system that accounts for time-varying delays within a pro-

vided bound based on polypotic approximations [Gielen et al., 2010] for controller design pur-

poses. Then, a one step ahead predictive controller based on flexible control Lyapunov func-

tions [M.Lazar 2009] is designed for the resulting model with polytopic uncertainty and hard

constraints. The developed algorithms have the potential to satisfy the timing requirements, due

to the short horizon, while it can still offer a non-conservative solution to stabilization due to

the flexibility of the Lyapunov function. Moreover, it is shown that for an appropriately chosen

Lyapunov function, the MPC algorithms amount to solving a single, low-complexity linear pro-

gram each sampling instant, meaning that the proposed LMPC algorithm can be applied for the

class of fast processes, which require short sampling periods.

The strategy was then tested on a validated simplified drivetrain model. In this chapter,

TrueTime toolbox (for Matlab/Simulink), which allows the simulation of distributed real-time

control systems, taking into account the effects of the execution of the control tasks and the

data transmission on the controlled system dynamics was used as the simulation environment.

Several TrueTime simulation experiments defined in collaboration with Ford Research and Ad-

vanced Engineering, US validate the proposed approach, indicating that the developed scheme

has the potential to meet the required real-time control specifications. The simulation results

also indicate that the proposed scheme can outperform other types of controllers, such as PID

7

Introduction

or explicit MPC.

Moreover, the proposed approach was experimentally verified using an industrial HIL test-

bench setup consisting of a Freescale based electronic control unit (which implements the con-

troller) linked via a CAN bus with a dSPACE MicroAutoBox plant simulator.

Chapter 5, entitled Robust nonlinear networked predictive control, focuses on the design of

robustly stabilizing MPC algorithms for delay-perturbed systems. The problem addressed in

this chapter is to control a system through a communication network, while compensating the

time-varying delays introduced by the network. The proposed solution consists of two steps:

firstly, the error caused by the time-varying network-induced delays is considered as a distur-

bance and two novel methods of finding the bounds of the disturbance are proposed. Secondly,

an one step ahead MPC scheme is designed using the concept of flexible control Lyapunov

functions [M.Lazar 2009] that explicitly accounts for rejection of disturbances introduced by

the time-varying delay in the communication network and do not require a precise model of

time delay. The main idea of the proposed LMPC is to formulate appropriate constraints in the

controller optimization problem, which are based on an existing Lyapunov-based controller, in

such a way that the LMPC controller inherits the stability properties and the robustness of the

Lyapunov-based controller, in a non-conservative way. Standard MPC strategies, which typi-

cally require a sufficiently long prediction horizon to assure stability and performance, would

lead to solutions that are too complex and, at the same time, too conservative to be imple-

mented in real-time control applications, for physical plants with fast dynamics, which require

short sampling periods.

Firstly, a TrueTime simulation of an Ethernet-based controlled DC motor, which is one

of the most popular devices for actuation and propulsion systems, is used in this chapter to

illustrate the effectiveness and robustness of the proposed delay modeling and control strategy.

Then, the proposed control solution was tested on a validated vehicle drivetrain model in-

tegrated with CAN and the simulation scenarios indicate that the proposed scheme, besides

yielding a feasible algorithm, outperforms controllers obtained using classical approaches, such

as explicit MPC and PID control. Moreover, the performed tests indicate that the proposed

design methodology can handle both the performance/physical constraints and the strict limi-

tations on the computational complexity corresponding to vehicle drivetrain oscillations damp-

ing. Furthermore, the proposed robust one step ahead predictive control strategy was tested

on a real-time simulation test-bench and the obtained results illustrate that the strategy can be

implemented to control the actual physical plant.

8

Summary of publications

1.6 Summary of publications

This thesis is mostly based on published or submitted articles. The results obtained during the

three years were published in 2 ISI indexed papers with impact factor (MSSP - 1.762, IET

CT& A - 1.283), 1 ISI indexed paper (CEAI), 3 ISI Proceedings papers, 2 SCOPUS papers, 1

INSPEC paper, 1 ZBMATH paper, 4 IEEE Proceedings papers, 2 IFAC Proceedings papers, 4

papers in International Conferences volumes and 1 paper submitted to a journal (IEEE TII).

Chapter 3 contains results presented in:

• [Caruntu and Lazar, 2009a]: C. F. Caruntu and C. Lazar. Network-Induced Variable Time De-

lay Compensation Technique Based on Predictive Control. In 17th International Conference on

Control Systems and Computer Science, pages 65–71, 2009.

• [Patrascu et al., 2009]: D. I. Patrascu, A. E. Balau, C. F. Caruntu, C. Lazar, M. H. Matcovschi

and O. Pastravanu. Modelling of a Solenoid Valve Actuator for Automotive Control Systems. In

17th International Conference on Control Systems and Computer Science, pages 541–546, 2009.

• [Balau, Caruntu et al., 2009b]: A. E. Balau, C. F. Caruntu, D. I. Patrascu, C. Lazar, M. H.

Matcovschi and O. Pastravanu. Modeling of a Pressure Reducing Valve Actuator for Automo-

tive Applications. In 18th IEEE International Conference on Control Applications, Part of 2009

IEEE Multi-conference on Systems and Control, Saint Petersburg, Russia, 2009. (indexata ISIProceedings)

• [Caruntu et al., 2009c]: C. F. Caruntu, M. H. Matcovschi, A. E. Balau, D. I. Patrascu, C. Lazar and

O. Pastravanu. Modeling of an Electromagnetic Valve Actuator. Buletinul Institutului Politehnic

Iasi, vol. LV (LIX), pages 9–28, 2009. (indexata ZBMATH)

• [Caruntu and Lazar, 2009b]: C. F. Caruntu and C. Lazar. Predictive Control for Time-Varying

Delay in Networked Control Systems. In 8th IFAC Workshop on Time Delay Systems, pages

49–54, Sinaia, Romania, 2009. (indexata SCOPUS)

• [Balau, Caruntu et al., 2009a]: A. E. Balau, C. F. Caruntu, C. Lazar and D. I. Patrascu. New Model

for Predictive Control of an Electro-Hydraulic Actuated Clutch. In 8th International Conference

Fuel Economy, Safety and Reliability of Motor Vehicles, volume 1, pages 463–472, Bucuresti,

Romania, 2009.

• [C.Lazar, Caruntu and Balau, 2010]: C. Lazar, C. F. Caruntu and A. E. Balau. Modelling and

Predictive Control of an Electro-Hydraulic Actuated Wet Clutch for Automatic Transmission. In

IEEE Symposium on Industrial Electronics, Bari, Italy, 2010. (indexata SCOPUS)

9

Introduction

• [Caruntu et al., 2010b]: C. F. Caruntu, A. E. Balau and C. Lazar. Networked Predictive Con-

trol Strategy for an Electro-Hydraulic Actuated Wet Clutch. In IFAC Symposium Advances in

Automotive Control, pages 419–424, Munchen, Germany, 2010.

• [Balau, Caruntu and C.Lazar, 2011b]: A. E. Balau, C. F. Caruntu and C. Lazar. Simulation and

Control of an Electro-Hydraulic Actuated Clutch. Mechanical Systems and Signal Processing,

vol. 25, pages 1911–1922, 2011. (cotata ISI, Impact factor = 1.726)

• [Caruntu and C.Lazar, 2011b]: C. F. Caruntu and C. Lazar. Networked Predictive Control for

Time-varying Delay Compensation with an Application to Automotive Mechatronic Systems. ac-

cepted for Journal of Control Engineering and Applied Informatics, 2011. (indexata ISI)

• [Caruntu et al., 2011h]: C. F. Caruntu, D. Onu, F. C. Braescu and C. Lazar. Model predictive con-

trol for real-time simulation of a network-controlled vehicle drivetrain. In 2nd Eastern European

Regional Conference on the Engineering of Computer Based Systems, pages 115–123, Bratislava,

Slovakia, 2011.

The results presented in Chapter 4 are published in:

• [Caruntu et al., 2011f]: C. F. Caruntu, M. Lazar, S. Di Cairano, R. H. Gielen and P. P. J. van den

Bosch. Horizon-1 predictive control of networked controlled vehicle drivetrains. In 18th IFAC

World Congress, pages 3824–3830, Milano, Italy, 2011.

• [Braescu, Caruntu et al., 2011]: F. C. Braescu, C. F. Caruntu, L. Ferariu and C. Lazar. OSEK

Based Embedded Networked Controller Designed to Handle Communication Delays. In 2nd East-

ern European Regional Conference on the Engineering of Computer Based Systems, pages 71–77,

Bratislava, Slovakia, 2011.

Chapter 5 is based on:

• [Caruntu et al., 2010a]: C. F. Caruntu, A. E. Balau and C. Lazar. Cascade based Control of

a Drivetrain with Backlash. In 12th International Conference on Optimization of Electrical and

Electronic Equipment, Brasov, Romania, 2010. (indexata ISI Proceedings)

• [Caruntu et al., 2011a]: C. F. Caruntu, A. E. Balau, M. Lazar, P. P. J. van den Bosch and S. Di

Cairano. A Predictive Control Solution for Driveline Oscillations Damping. In Hybrid Systems:

Computation and Control, pages 181–190, Chicago, USA, 2011. (indexata ISI Proceedings)

• [Caruntu and C.Lazar, 2011e]: C. F. Caruntu and C. Lazar. Stabilizing MPC for network-

controlled systems with an application to DC motors. In IEEE International Conference on

Mechatronics, pages 973–978, Istanbul, Turkey, 2011. (indexata INSPEC)

10

Basic mathematical notation and definitions

• [Caruntu and C.Lazar, 2011d]: C. F. Caruntu and C. Lazar. Robustly stabilizing MPC design for

networked control systems with an application to DC motors. accepted for IET Control Theory

and Applications, 2011. (cotata ISI, Impact factor = 1.283)

• [Balau, Caruntu and C.Lazar, 2011a]: A. E. Balau, C. F. Caruntu and C. Lazar. Driveline oscilla-

tions modeling and control. In 18th International Conference on Control Systems and Computer

Science, pages 332–338, Bucuresti, Romania, 2011.

• [Caruntu and C.Lazar, 2011c]: C. F. Caruntu and C. Lazar. Robust MPC for TrueTime Simula-

tion of a Vehicle Drivetrain Controlled through CAN. In 16th IEEE International Conference on

Emerging Technologies and Factory Automation, Toulouse, France, 2011.

• [Caruntu et al., 2011i]: C. F. Caruntu, D. Onu and C. Lazar. Real-time Simulation of a Vehicle

Drivetrain Controlled through CAN using a Robust MPC Strategy. In 15th International Confer-

ence on System Theory, Control and Computing, Sinaia, Romania, 2011.

• [Caruntu and C.Lazar, 2011j]: C. F. Caruntu and C. Lazar. Network-induced time-varying delay

modeling and predictive compensation with stability guarantee. submitted to IEEE Transactions

on Industrial Informatics, 2011.

1.7 Basic mathematical notation and definitions

In this section, some basic mathematical notation and standard definitions are recalled to make

the manuscript self-contained.

• R, R+, Z and Z+ denote the field of real numbers, the set of non-negative reals, the set of

integers and the set of non-negative integers, respectively;

• Z≥c1 and Z(c1,c2] denote the sets k ∈ Z+ | k ≥ c1 and k ∈ Z+ | c1 < k ≤ c2,respectively, for some c1, c2 ∈ Z+;

• For a set P ⊆ Rn, int(P) denotes the interior of P, cl(P) denotes the closure of P and

Co(P) denotes the convex hull of P;

• A convex and compact set in Rn that contains the origin in its interior is called a C-set;

• A polyhedron (or a polyhedral set) in Rn is a set obtained as the intersection of a finite

number of open and/or closed half-spaces;

• A polytope is a compact (closed and bounded) polyhedron.

11

Introduction

• A piecewise polyhedral set is a set obtained as the union of a finite number of polyhedral

sets.

• For a real number a ∈ R, |a| denotes its absolute value and dae denotes the smallest

integer larger than a;

• For a sequence zjj∈Z+ with zj ∈ Rl, z[k] denotes the truncation of zjj∈Z+ at time

k ∈ Z+, i.e. z[k] = zjj∈Z[0,k], and z[k1,k2] denotes the truncation of zjj∈Z+ at times

k1 ∈ Z≥1 and k2 ∈ Z≥k1 , i.e. z[k1,k2] = zjj∈Z[k1,k2];

• For a vector x ∈ Rn let [x]i, i ∈ Z[1,n] denote the i-th component of x.

• The Holder p-norm of a vector x ∈ Rn is defined as:

‖x‖p ,

(|x1|p + . . .+ |xn|p)1p , p ∈ Z[1,∞)

maxi=1,...,n |xi|, p =∞,

where xi, i = 1, . . . , n is the i-th component of x, ‖x‖2 is also called the Euclidean norm

and ‖x‖∞ is also called the infinity (or the maximum) norm;

• Let ‖ · ‖ denote an arbitrary Holder p-norm. For a sequence zjj∈Z+ with zj ∈ Rn,

‖zjj∈Z+‖ , sup‖zj‖ | j ∈ Z+;

• In denotes the identity matrix of dimension n× n;

• For a matrixZ ∈ Rm×n, Z> denotes its transpose andZ−1 denotes its inverse (if it exists);

• A function ϕ : R+ → R+ belongs to class K if it is continuous, strictly increasing and

ϕ(0) = 0;

• A functionϕ : R+ → R+ is said to belong to class K∞ if it is of class K and lims→∞ ϕ(s) =

∞.

12

Chapter 2

Networked control systems

In this chapter the networked control systems and the problems that arise in their usage (time-

varying delays, data-packet dropout) are presented. Moreover, different modeling methods and

compensation strategies for the network-induced time-varying delays are described. Further-

more, choosing the automotive applications as the fast systems on which this thesis experiments

are based on, some of the control systems that are implemented in nowadays vehicles are briefly

described and the delays that are introduced by CAN are modeled.

2.1 Introduction

A networked control system is a system for which the control loop is closed through a real-time

communication network. These NCSs are composed of the following components: physical

plant, sensors, actuators and controllers which are coordinated through the communication net-

work. In a networked control system, the sensors have the task of measuring the outputs of the

plant and sending the samples to the controller through the network. The controller receives the

measurements from the sensors, calculates the control command and sends the values through

the network to the actuators. The actuators have the task of applying the control commands

received through the network to the physical plant [Onat and Parlakay, 2007].

Feedback control systems over real-time communication networks are now widely used

in different industries, where, in a remarkable evolution over the last few years, the embed-

ded control systems have grown from standalone control systems to highly integrated and net-

worked control systems [Johansson et al., 2005]. Their evolution raised numerous questions

in the control, communication and information processing fields of study regarding the rela-

tions between the components and the quality of the whole system. NCSs have many attrac-

13


tive advantages which include low cost, simple installation and maintenance, increased system

agility, higher reliability and greater flexibility [Jiangang et al., 2007]. Moreover, NCSs can

be easily modified and upgraded by adding new sensors, actuators and/or controllers. Further-

more, due to efficient exchange of information between the controllers, these systems have the

possibility to combine the global informations from the system in order to take the best deci-

sions [Gupta and Chow, 2008]. But, the main disadvantages of using NCSs are represented by

the effects of the network-induced delays in the control loop and the data packet dropouts that

can appear in sending the data [Tipsuwan and Chow, 2003].

In most cases, introducing a communication network does not significantly affect the con-

trol system performances [Vatanski et al., 2007]. Nevertheless, for certain physical plants

which are time-restricted, implementing a networked control system must be realized tak-

ing into account the fact that a communication network is not perfect. For such systems,

implementing a communication network for closed-loop control leads to introducing an ad-

ditional delay, either constant, either time-varying, which makes system analysis and con-

troller design become of higher complexity. Another consequence of introducing communi-

cation networks in the control loop is represented by the possibility of data packet dropout.

These two problems can seriously affect the control performances and can even destabilize

the closed-loop control system [Halevi and Ray, 1990, Zhang et al., 2001, Chow and Tip-

suwan, 2001, Juanole and Mouney, 2005, Jiangang et al., 2007, Gupta and Chow, 2008].

There are many network-induced imperfections and constraints which are categorized in

five types in [Heemels et al., 2010], but, usually, the available literature on NCS considers only

some of them in the analysis of NCS: (i) quantization errors in the signals transmitted over the

network due to the finite word length of the packets; (ii) packet dropouts caused by the unreli-

ability of the network are considered in [Onat et al., 2008, de la Pena and Cristofides, 2008,

Sheng and Pan, 2010, Li et al., 2008a, Cloosterman et al., 2010, Shen et al., 2010, Onat et

al., 2010]; (iii) variable sampling/transmission intervals are taken into account in [Clooster-

man et al., 2010, Bernardini et al., 2010]; (iv) time-varying network-induced delays, which can

be smaller than the sampling period [Zhang et al., 2005,Salehi et al., 2008,Gielen and M.Lazar, 2009a,

Wang et al., 2010,Huang et al., 2010] or larger than the sampling period [Li et al., 2008a,Cloost-

erman et al., 2010, Shen et al., 2010, Cloosterman et al., 2009]; (v) communication constraints

caused by the sharing of the network by multiple nodes and the fact that only one node is al-

lowed to transmit its packet per transmission are addressed in [Liu et al., 2010]. Moreover, an

analysis of different aspects of the limitations imposed by the use of communication channels

to connect the elements of NCSs is given in [Hespanha et al., 2007].

14

Preliminaries

2.2 Preliminaries

Consider the standard networked control system illustrated in Fig. 2.1, which is composed

of five parts: a communication network, where it is assumed that delay occurs completely

randomly, a physical plant, one sensor node (S), one controller node and one actuator node (A).

The delays introduced by the communication network are represented as τ ca for the time delay

in the forward channel (from controller to actuator) and as τ sc for the time delay in the feedback

channel (from sensor to controller). In the same figure, u and y represent the control signal and

the output of the system, respectively.

Remark 2.2.1 The sensor is clock-driven and it samples the plant outputs periodically at sam-

pling instants. The controller is event-driven and it calculates the control signal as soon as the

data from the sensor arrives. The actuator is event-driven and it acts on the plant as soon as the

control data becomes available.

Remark 2.2.2 A priori knowledge of the reference signal is not assumed as this is not possible

with most systems.

Remark 2.2.3 The delays τ sc and τ ca can be both smaller and larger than a sampling period.

Notation: Let ak and bk denote the delay in the forward channel and in the feedback channel,

respectively, at time instant k, expressed as a number of sampling periods

ak = dτ cak /Tse,bk = dτ sck /Tse,

(2.1)

where Ts is the sampling period of the system. Moreover, let a and b denote the maximum

delays in the forward channel and in the feedback channel, respectively, expressed as a number

of sampling periods

a = dτ camax/Tse,b = dτ scmax/Tse.

(2.2)

The time delay generated in the communication line makes feedback signal delayed relative

to control input, so, from the controller point of view, Fig. 2.1 and Fig. 2.2 are equivalent if at

least one of the following holds:

15


caτ

scτ

r

kyku kk ay

−

k kk a by− −

kk au−Discrete-time

Controller

ca scτ τ+

k kk a b ku u− −

−

++

Remote process Network

Network

Network

r

kyku k kk a by

− −

k kk a by− −

k kk a bu− −Discrete-time

Controller

Remote process

r

kyku k kk a by

− −

k kk a by− −


Controller

Remote process

Continuous-

time Plant ZOH

A Ts

S

Continuous-

time Plant ZOH

A Ts

S

Continuous-

time Plant ZOH

A Ts

S

Figure 2.1: Control system with network-induced time delay.

caτ

scτ

r

kyku kk ay

−

k kk a by− −


Controller

ca scτ τ+


−

++


Network

Network

r

kyku k kk a by

− −

k kk a by− −


Controller

Remote process

r

kyku k kk a by

− −

k kk a by− −


Controller

Remote process

Continuous-

time Plant ZOH

A Ts

S

Continuous-

time Plant ZOH

A Ts

S

Continuous-

time Plant ZOH

A Ts

S

Figure 2.2: Control system with grouped network-induced time delay.

• the time-varying delays are smaller than the sampling period [Zhang et al., 2005,Salehi et

al., 2008, Gielen and M.Lazar, 2009a, Wang et al., 2010, Huang et al., 2010]: τmax ≤ Ts,

where τ = τ ca + τ sc, τmax ∈ R>0 is the maximum delay that can be introduced by the

communication network;

• a buffer is introduced before the controller and another one before the actuator [Luck and Ray, 1990,

Richard, 2003, Lin and Antsaklis, 2004, Lopez et al., 2006] with length b and a, respec-

tively; by using this method, the time-varying delay may be kept constant at the expense

of making it larger;

• a low-pass filter is implemented in the controller or before the controller, similar to the

one in [Mu et al., 2005], with cutoff frequency ft = ωt/2π; the filter is introduced to

reduce the influence of the fast time-varying delays on the signal that is transmitted from

the sensor to the controller as the plant filters the effects of the fast time-varying delays

on the control signal that is transmitted from the controller through the network; ωt is the

angular cutoff frequency of the plant;

• the delays do not increase as fast as the time [Witrant et al., 2007]: τ(t) < 1, ∀t ∈ R+.

2.3 Networked control systems modeling

Over the years, networked control systems were modeled based on necessities and the commu-

nication network in case.

16

Networked control systems modeling

.

.

.

.

.

.

Actuatornode

ProcessSensornode

Controllernode

Network

Buffer

Buffer

τcak τ

sck

u(t) y(t)h

wk

Ts

Figure 2.3: NCS model introduced by [Luck and Ray, 1990].

2.3.1 NCSs addings

In [Luck and Ray, 1990, Luck and Ray, 1994] the closed-loop control system is made time-

invariant by introducing buffers before the controller and actuator (see Fig. 2.3).

All nodes in the communication network are clocked and act synchronized. In this case, the

buffers lengths must be greater than the largest value of the delays that can appear due to the

communication network, so the discrete-time model of the process can be written as

xk+1 = Axk +Buk−∆1 ,

yk = Cxk,(2.3)

where ∆1 is the length in samples of the buffer at the actuator node. If the buffer at the controller

node is assumed to have he length ∆2 samples, the process output available for the controller

at time k is wk = yk−∆2 . The design problem is now formulated as a standard sampled data

control problem. The information set available for calculation on uk is

Wk = wk, wk−1, . . .. (2.4)

The most important disadvantage of this method is that it makes the control delay longer

than necessary in most cases.

The model utilized in [Krtolica et al., 1994] implies delaying the output of the system with

a certain amount of sampling periods, corresponding to the time-varying delays introduced by

the communication network.

The discretized signal yk is delayed with a number of sampling periods equal to the delay

17


Controller

Actuator

Physical plant

Sensor

Network

Control signal

Sensor measurement

Sensor

Physical

plant

Actuator

Main

controller Network

Reference signal

Sensor measurement

Remote

controller

+

-

Remote system

∆

+

∆

∆ …

0β

2Dβ

ky

kw

Figure 2.4: NCS model introduced by [Krtolica et al., 1994].

Chapter 2. Problem Formulation

node. It is shown that by this method the controller node can reduce itsuncertainty about which sensor reading it is using down to two possiblecases. By knowing the probability for the two possible cases of delay, astate estimator is constructed. The sensor node and the controller node areboth time-driven with a skew of ∆sp. It is also shown how pole placementcan be done for the described setup.

yk

ω k

Queue

Single register

Communication link

Figure 2.7 Block diagram of the transmission from the sensor node to the con-troller node in Chan and Özgüner (1995). The sampled signal yk is delayed duringthe transmission to the controller node. The controller reads the sensor value ω kfrom a register in the controller node. A simple form of timestamping is done byappending every message with the size of the queue when the message was sent.

Jump Linear Systems

Jump systems in continuous time was introduced and studied in the 1960’sby Krasovskii and Lidskii (1961). Jump linear systems can in discrete timebe written as

xk+1 A(rk)xk + B(rk)uk, (2.20)

where A(rk) and B(rk) are real-valued matrix functions of the randomprocess rk, see Ji et al. (1991). An interesting special case of rk-processis to let rk be a time homogeneous Markov chain taking values in a finiteset 1, . . . , s. The Markov chain has the transition probabilities

P(rk+1 j t rk i) qij ≥ 0, (2.21)

where

s∑

j1

qij 1. (2.22)

26

Figure 2.5: NCS model introduced by [Chan and Ozguner, 1995].

introduced by the communication network, the controller receiving the signal wk. Only one of

the coefficients βi is equal to 1 at the discrete time instant k, the other coefficients being equal

to 0. Notice that the delay must be a multiple of the sampling period (all nodes are time-driven).

In [Chan and Ozguner, 1995] a queue is introduced after the sensor and a single shifting

register between the sensor and the controller (Fig. 2.5). The sampled signal yk is delayed

during the transmission to the controller node. The controller reads the sensor value ωk from

a register in the controller node. At the same time, a simple way to count the amount of time

data is delayed is introduced by adding to each message the queue dimension at the moment the

message is sent.

In [Nilsson et al., 1998] a model that captures different network loads is utilized, having

three states: one for low load, one for medium load and one for high load as it is represented in

Fig. 2.6 (L is the state for low network load, M is the state for medium network load and H is

the state for high network load). The transitions between the different states of the communica-

tion network are modeled by using a Markov chain (Fig. 2.7). The transition probabilities are

indicated on the arcs.

18


Chapter 3. Modeling of Network Delays

L M H

qLL qMM qH H

qLM qM H

qM L qH M

qLH

qH L

Figure 3.1 An example of a Markov chain modeling the state in a communicationnetwork. L is the state for low network load, M is the state for medium networkload, and H is the state for high network load. The arrows show possible transitionsin the system.

L

M

H

Delay

Delay

Delay

Figure 3.2 The delay distributions corresponding to the states of the Markovchain in Figure 3.1. L is the state for low network load, M is the state for mediumnetwork load, and H is the state for high network load.

lated by sampling of the continuous-time process. Let the controlled pro-cess be

dxdt Ax(t) + Bu(t) + v(t), (3.2)

where x(t) ∈R n, u(t) ∈R m and v(t) ∈R n. A and B are matrices of appro-priate sizes, u(t) is the controlled input and v(t) is white noise with zero

32

Figure 2.6: Delay distribution considered in [Nilsson et al., 1998].

Chapter 3. Modeling of Network Delays

L M H

qLL qMM qH H

qLM qM H

qM L qH M

qLH

qH L

Figure 3.1 An example of a Markov chain modeling the state in a communicationnetwork. L is the state for low network load, M is the state for medium networkload, and H is the state for high network load. The arrows show possible transitionsin the system.

L

M

H

Delay

Delay

Delay

Figure 3.2 The delay distributions corresponding to the states of the Markovchain in Figure 3.1. L is the state for low network load, M is the state for mediumnetwork load, and H is the state for high network load.

lated by sampling of the continuous-time process. Let the controlled pro-cess be

dxdt Ax(t) + Bu(t) + v(t), (3.2)

where x(t) ∈R n, u(t) ∈R m and v(t) ∈R n. A and B are matrices of appro-priate sizes, u(t) is the controlled input and v(t) is white noise with zero

32

Figure 2.7: NCS Markov chain modeling introduced by [Nilsson et al., 1998].

19


2.3.2 Modeling NCSs with network-induced delay

The NCS model considering network-induced delay is represented in Fig. 2.1. The model

consists of a continuous-time plant

x(t) = Ax(t) +Bu(t)

y(t) = Cx(t)(2.5)

and a discrete-time controller

u(kTs) = −Kx(kTs), k = 0, 1, 2, . . . . (2.6)

Here, x ∈ Rn, u ∈ Rm, y ∈ Rp and A,B,C,K are of compatible dimensions.

2.3.2.1 Delays smaller than the sampling period

In this case, the constraint τk ≤ Ts, k = 0, 1, 2, . . . means that at most two control samples

u((k − 1)Ts) and u(kTs) need to be applied during the kth sampling period. The system equa-

tions can be written as in ( [Zhang et al., 2001])

x(t) = Ax(t) +Bu(t), t ∈ [kTs + τk, (k + 1)Ts + τk+1),

y(t) = Cx(t),

u(t+) = −Kx(t− τk), t ∈ kTs + τk, k = 0, 1, 2, . . . ,(2.7)

where u(t+) is piecewise continuous and only changes its value at kTs + τk. Sampling the

system with period Ts it is obtained

x((k + 1)Ts) = Φx(kTs) + Γ0(τk)u(kTs) + Γ1(τk)u((k − 1)Ts),

y(kTs) = Cx(kTs),(2.8)

20


where

Φ = eATs ,

Γ0(τk) =

∫ Ts−τk

0

eAθBdθ,

Γ1(τk) =

∫ Ts

Ts−τkeAθBdθ.

(2.9)

Defining z(kTs) = [x>(kTs), u>((k−1)Ts)]

> as the augmented state vector, the augmented

closed-loop systems is

z((k + 1)Ts) = Φ(k)z(kTs), (2.10)

where

Φ(k) =

[Φ− Γ0(τk)K Γ1(τk)

−K 0

]. (2.11)

If the delay is constant, the system is still time-invariant, which simplifies the system anal-

ysis.

2.3.2.2 Delays larger than the sampling period

If the delays can be longer than the sampling period (e.g., 0 ≤ τk ≤ lTs), the actuator may

receive zero, one or more than one control samples (maximum l) in a single sampling period.

In the special case where ((l−1)Ts) ≤ τk ≤ lTs for all k ∈ Z>1, one control sample is received

every sampling period. In this case (see, e.g., [Zhang et al., 2001]), it results that

Φ(k) =

Φ Γ1(τ′

k) Γ0(τ′

k) . . . 0

0 0 I . . . 0...

...... . . . ...

−K 0 0 . . . 0

, (2.12)

where τ ′k = τk − (l − 1)Ts and the augmented state vector is z(kTs) = [x>(kTs), u>((k −

l)Ts), . . . , u>((k − 1)Ts)]

>.

21


2.4 Time-varying delays

When sensors, actuators and controllers exchange data across the network, various delays with

variable length occur due to sharing the common network medium. These delays are called

network-induced delays and can vary widely according to the transmission time of messages

and the overhead of the network. Usually, these network delays are randomly time-varying

[Rodriguez and Menendez, 2007] and can be categorized from the direction of data transfers

as the sensor-to-controller delay τ sc and the controller-to-actuator delay τ ca, the delays being

computed as [Tipsuwan and Chow, 2003]:

τ sc = tcs − tse

τ ca = trs − tce,(2.13)

where tse is the time instant that the sensor encapsulates the measurement to a frame or a packet

to be sent, tcs is the time instant that the controller starts processing the measurement in the

delivered frame or packet, tce is the time instant that the main controller encapsulates the control

signal to a packet to be sent, and trs is the time instant that the system starts processing the

control signal. Fig. 2.8 ( [Tipsuwan and Chow, 2003,Ge et al., 2007]) shows the corresponding

timing diagram of network delay propagations, where u is the control signal and y the output

signal.

In fact, both network delays can be longer or shorter than the sampling time Ts. The con-

troller processing delay τ c and both network delays can be lumped together as the control delay

τ for ease of analysis. This approach has been used in some networked control methodologies.

Although the controller processing delay τ c always exists, this delay is usually small compared

to the network delays, and could be neglected. In addition, the sampling periods of the main

controller and of the remote system may be different in some cases.

Generally, both the controller and the actuator are event-driven and the sensor is time-driven,

sampling the plant output every period. When one sampling begins, the data sampled by the

sensor y (k) will be sent to the controller via the communication network, the delay τ sck oc-

curring during this interval. Then, the controller computes the control data u (k) according to

the sampling data and then the control data will be sent to the actuator via the communication

network, the delay τ cak occurring in this interval. The actuator acts on the plant in no time once

it receives the control data.

The time delays τ sc and τ ca are composed of at least the following parts [Tipsuwan and Chow, 2003]:

waiting time delay, frame time delay and propagation delay. These three delay parts are fun-

22

Time-varying delays

Output signal

Control signal

Actual output

signal

Delayed output

signal (by τsc)

Control signal with

respect to y(k)

Delayed control

signal (by τca)

y(k) y(kTs- τsc)

u(k) u(kTs- τca)

t se

τcaτsc τc

t cs t

ce t

rs

τ

τW τF τP kTs (k+1)Ts

Ts

Figure 2.8: Network-induced delays time diagram.

damental delays that occur on a local area network, i.e., CAN, Ethernet. There can be also

additional delays when the control or sensory data travel across networks, such as the queuing

delay at a switch or a router and the propagation delay between network hops. The delays τ sc

and τ ca also depend on other factors, being functions of network variables such as type and

number of transmitted signals, network traffic congestion condition, network throughput, net-

work protocol used and network management/policy used. In fact, both network delays can be

longer or shorter than the sampling time Ts.

The existence of time-varying and unpredictable communication delay on the communica-

tion line is the most serious problem in the research area of networked control systems. The

communication delay in the communication system can be regarded as time delay in the con-

trol system, which introduces phase-delay and, in the end, it can destabilize the system and

deteriorate the performances of the closed-loop control system [Tipsuwan and Chow, 2003].

2.4.1 Delays modeling

The sensor to actuator delays were modeled using simple models that consider the delays:

23


• constant (simplest model of the network-induced delays: converts time-varying to time-

invariant delays): τk = τmax [Luck and Ray, 1990] or τk = Ts/2 [Nilsson et al., 1998],

where k is the discrete time instant at which the delay is calculated, τmax is the upper

bound of the delays that can appear in the communication network and Ts is the sampling

period of the system;

• average: τk = τksc + 1

w

n∑i=n−w+1

τica, where w is the number of memorized delays from

controller to actuator, τ sck is the current delay from sensor to controller , τ cai is the delay

from controller to actuator at time instant i and n ≤ k − 1 [Zhang et al., 2006] or τk =

2 1N

k−1∑i=k−N

τisc, where N is the number of past delays of which the average is calculated

and τ sci is the delay from sensor to controller at time instant i [Velagic, 2008];

• random, uniform distributed in the interval [0, Ts] or [0, αTs], where 0 ≤ α ≤ 1 [Nils-

son et al., 1998],

but these models do not take into account the communication network dynamics.

Also, AutoRegressive Moving-Average (ARMA) models were utilized to model the delays

introduced by the communication network

τk = M + ek + g∞∑

j=1

ek−j, (2.14)

where τk is the delay at the discrete time instant k, M is a constant representing the mean

value of the delays, ei, i = −∞, ..., k − 1, k is a sequence of random variables, indepen-

dent, identically distributed, having the mean value equal to 0 and g ∈ (0, 1) is a constant

[Li and Mills, 2001].

Markov chains were utilized to model random delays in [Krtolica et al., 1994], as it is

illustrated in Fig. 2.4.

Markov chains and Hidden Markov Models (HMM) were also utilized to model the delays,

which take into account the load of the network and are formulate based on:

• probability distributions (the network’s delay characteristics were incorporated as one of

the control design parameters), the delay at the discrete time instant k being given by the

state rk of a Markov chain

τk =

0, if rk = 1,

rect (d− α, d+ α) , if rk = 2,(2.15)

24

Time-varying delays

for a chain with two states, state 1 corresponding to the system without delays and state

2 corresponding to a delay in the interval [d− α, d+ α], where d and α depend on the

sampling period, with d − α > 0 and d + α < T and rect (x, y) represents an uniform

distribution on the interval [x, y] [Nilsson et al., 1998] or on

• symbols to represent the delays:

yt =

M, dt =∞⌈dt−dmin

b

⌉, dmin ≤ dt ≤ dmax

u, dt = u

, (2.16)

where M is the number of symbols utilized to represent the delays, dtTst=1 is the obser-

vation sequence, dt = ∞ represents that the packet generated at time instant t was lost,

dt = u represent the missing of t moment symbol from the observation sequence, dmin

and dmax are the minimum and the maximum delays that can appear in the communication

network and b = dmax−dmin

M[Wei et al., 2002].

These delay modeling methods are utilized to design and to evaluate the controllers used

in NCSs regarding the stability and the performances of the system. A model which precisely

characterizes the delays introduced by the communication networks is hard to obtain, but, in

general, the proposed models are reliable.

2.4.1.1 Delays modeled as polytopic inclusions

Consider the continuous-time system with input delay [Gielen and M.Lazar, 2009b]

x(t) = Acx(t) +Bcu(t),

u(t) = uk,∀t ∈ [tk + τk, tk+1 + τk+1),(2.17)

where x(t) is the system state, u(t) is the system input, Ac and Bc are the continuous-time

system matrices, τ is the time delay introduced by the communication network, tk = kTs, k ∈Z+, Ts ∈ R+ denotes the sampling period and assume that u(t) = u0 for all t ∈ [0, τ0] with

u0 ∈ Rm some predetermined constant vector. uk ∈ R is the control action generated at time

t = tk. τk ∈ R[0,τmax] denotes the delay induced by the network at time k ∈ Z+ and τmax ∈ R+

is the maximal possible delay.

Consider the maximum delay as τmax = (Υ + υ)Ts, where Υ ∈ Z≥1 and υ ∈ R[0,1).

Assuming that uk = ψk for all k ∈ Z[−Υ−1,−1] with ψ[−Υ−1,−1] some predetermined vector, the

25


discretized model is

xk+1 = Adxk +Bduk + ∆0(τk)(uk−1 − uk)++ ∆1(τk)(uk−2 − uk−1) + . . .

. . .+ ∆Υ(τk)(uk−Υ−1 − uk−Υ),

(2.18)

where Ad = eAcTs , Bd =∫ Ts

0eAc(Ts−θ)dθBc,

∆i(τk) :=

0, τk−i − iTs ≤ 0∫ τk−i−iTs

0eAc(Ts−θ)dθBc, 0 < τk−i − iTs < Ts

∫ Ts0eAc(Ts−θ)dθBc, Ts ≤ τk−i − iTs

(2.19)

for all k ∈ Z+ and i ∈ Z[0,Υ]. Furthermore, letting j ∈ Z[0,Υ]

∆i(τk) := 0 if τk−j ≤ τk−i − (i− j)Ts, (2.20)

for all j < i, i.e., a newer control update arrives before uk−i arrives and thus uk−i is ignored.

A polytopic over-approximation of the nonlinear functions ∆i(τk) can be found using the

Cayley-Hamilton theorem as presented in [Gielen et al., 2010]. Therefore, the following poly-

topic set is defined

∆τ := Co(∆ll∈Z[1,L]), (2.21)

with τ ∈ R[0,Ts], ∆l ∈ Rn×m such that ∆i(τk) ∈ ∆τ for all τk ∈ R[0,τ ] and L ∈ Z≥1 is

finite. In [Gielen et al., 2010] several methods to create the polytope (2.21) were assessed. This

reference is referred for further details and assume for the remainder of this subsection that the

polytopic set (2.21) is known.

As ∆i(τk) ∈ ∆Ts for all i ∈ Z[0,Υ−1] and ∆Υ(τk) ∈ ∆υTs , it is obtained that (2.18) is

contained in

xk+1 ∈ φ(xk, uk, uk−1), k ∈ Z+, (2.22)

where

φ(xk, uk, uk−1) :=Adxk +Bduk

+E∆(uk−i−1 − uk−i) + υ | E∆ ∈ ∆Ts , υ ∈ Sk(2.23)

26

Time-varying delays

and

Sk := v ∈ Rn | v ∈Υ−1⊕

i=1

∆Ts(uk−1−i − uk−i)

⊕∆υTs(uk−1−Υ − uk−Υ).(2.24)

Observe that the input vectors uk−i are known at time k ∈ Z+ for all i ∈ Z≤1.

This method of modeling the delays as polytopic inclusions will be used in Chapter 4 to

model the CAN-induced delays.

2.4.1.2 Delays modeled as disturbances

Fig. 2.9 (a) shows a simple example of a bilateral teleoperation system. F means a control input

for slave (torque dimension), and sXeTs is an output of slave (angular velocity dimension). J

means slaves moment of inertia. T1 denotes a time delay from master to slave, and T2 is a time

delay from slave to master (T = T1 + T2). In the case of bilateral teleoperation under time-

varying delay, T1 and T2 are time varying. Fig. 2.9 (a) shows that the output of slave sXeTs

is delayed relative to the input for slave F . Considering the concept of network disturbance

described in [Natori et al., 2008, Natori et al., 2010], Fig. 2.9 (a) changes, as shown in Fig. 2.9

(b). In Fig. 2.9 (b), there is no time-delay element. Instead of the time-delay element, there

exists network disturbance described as follows:

Dnet(s) = F (1− e−Ts). (2.25)

Starting from this method of modeling the delays as disturbances, in Chapter 5, a new

method of modeling separately the forward channel delays and the feedback channel delays

as disturbances is proposed. Moreover, two new methods of bounding the disturbances are

proposed, bounds which will be used in the controllers design phase.

2.4.2 Delay compensation strategies

Existing constant time-delay compensation methodologies may not be directly suitable for con-

trolling a system over a communication network; therefore, to handle network delays in a

closed-loop control system over a network, an advanced methodology is required, a signifi-

cant emphasis being on developing control methodologies to handle the network delay effect in

NCSs [Tipsuwan and Chow, 2003].

27

Networked control systems1054 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 57, NO. 3, MARCH 2010

Fig. 5. Concept of ND in bilateral teleoperation systems. (a) Bilateral teleop-eration system with time delay. (b) Bilateral teleoperation system with ND.

Fig. 6. Time-delay compensation in bilateral teleoperation systems.

the slave system. After introduction of the four-channel time-delayed bilateral control system, the bilateral teleoperationsystem used in the experiment is presented.

A. Time-Delay Compensation

Fig. 5(a) shows a simple example of a bilateral teleoperationsystem. F means a control input for slave (torque dimension),and sXe−Ts is an output of slave (angular velocity dimension).J means slave’s moment of inertia. T1 denotes a time delayfrom master to slave, and T2 is a time delay from slave tomaster (T = T1 + T2). In the case of bilateral teleoperationunder time-varying delay, T1 and T2 are time varying. Fig. 5(a)shows that the output of slave sXe−Ts is delayed relative to theinput for slave F . Considering the concept of ND describedin Section II, Fig. 5(a) changes, as shown in Fig. 5(b). InFig. 5(b), there is no time-delay element. Instead of the time-delay element, there exists ND described as follows:

Dnet(s) = F (1 − e−Ts). (31)

The ND is estimated by CDOB and the estimated ND isutilized for time-delay compensation, as shown in Fig. 6. (In

Fig. 7. Internal structure of CDOB in bilateral teleoperation systems.

Fig. 6, for easy understanding, it is assumed that the cutofffrequency of LPF in CDOB is ideally infinity.) Although theoutput of the slave sXe−Ts is delayed by the round trip timedelay T , the time-delay effect is compensated by the time-delay compensation method with CDOB and then the inputfor master sX is no longer delayed. Therefore, it turns outthat time delay is sufficiently compensated. The effectivenessis the same as Smith predictor. Furthermore, the time-delaycompensation method works without time-delay model (theinternal structure of CDOB (Fig. 7) does not include time-delaymodel) in contrast to model-based or predictive control methodslike Smith predictor which needs accurate time-delay model.Consequently, the time-delay compensation method is expectedto work even in the case of unknown and time-varying delaylike that of IP networks.

B. Design of Cutoff Frequency of LPF in CDOB

Fig. 7 shows internal structure of CDOB in a simple exampleof a bilateral teleoperation system shown in Fig. 6. Describedas (26), ND is estimated with limited bandwidth in actualimplementation (gnet is the cutoff frequency of CDOB)

Dnet(s) =gnet

s + gnetDnet(s)

=gnet

s + gnetF (1 − e−Ts). (32)

Then, the cutoff frequency of LPF in CDOB gnet should bedesigned appropriately. In addition, since DOB [27], [28] isimplemented at slave side for realization of robust accelerationcontrol, we should also consider the cutoff frequency of LPFin DOB at slave side gd, as shown in Fig. 8 (Fdis means adisturbance exerted on slave). In Fig. 8, the estimated valueby CDOB is affected by cutoff frequencies of CDOB (gnet)and DOB at slave side (gd). Then, considering Fig. 3(b), theestimated value by CDOB Fdisnet is described as follows:

Fdisnet =gnet

s + gnet

(F (1 − e−Ts) +

s

s + gdFdis

)

=gnet

s + gnetF (1 − e−Ts) +

gnet

s + gnet

s

s + gdFdis.

(33)

Figure 2.9: Concept of network disturbance in bilateral teleoperation systems. (a) Bilateral tele-operation system with time delay. (b) Bilateral teleoperation system with network disturbance.

Over the last years, the problem of compensating network-induced time-varying delays was

extensively studied and different control strategies were proposed:

• an LQG-optimal controller uk = ζ(Wk), with Wk from (2.4), that can handle control

delays that are longer than the sampling period [Luck and Ray, 1990];

• a deterministic method for delay compensation (see Fig. 2.3) - it utilizes an estimator

for the system states and a predictor to calculate the control command based on the past

outputs of the system [Luck and Ray, 1994] as follows:

- using the set of past measurements z(k) = y(k−φ), y(k−φ−1), . . . in the controller

buffer, where φ is the number of packets in the buffer, the observer estimates the plant

states x(k − θ + 1), where θ is the length of the controller buffer;

- the predictor uses x(k−θ+1) to predict the future states x(k+µ), where µ is the length

of the actuator buffer;

- the controller computes the predictive control u(k + µ) from x(k + µ) and then sends

u(k + µ) to be stored in the actuator buffer.

• a probabilistic method for delay compensation (see Fig. 2.5) - it utilizes probabilistic

information along with the number of packets in a queue to improve state prediction

28

Time-varying delays

[Chan and Ozguner, 1995]; the predictor estimates the current state x(k) by

x(k) = P0(Φφ−1w(k) +Wφ) + P1(Φφw(k) +Wφ+1),

Wi =

0, φ = 1,

[Γ,ΦΓ, . . . ,Φφ−2Γ]··[u>(k − 1), u>(k − 2), . . . , u>(k − φ− 1)]>, i 6= 1,

(2.26)

where P0 and P1 are weighting matrices, which are computed from the probabilities of the

occurrences of y(k− φ) and y(k− φ+ 1). These equations require full state information

(y(k) = x(k)).

• an optimal stochastic control method for a process controlled through a network with ran-

dom delays; this method considers that the delays introduced by the communication net-

work are known and smaller than the sampling period of the system [Nilsson et al., 1998];

The dynamics of the remote plant is described by (2.8), (2.9) and the goal of the optimal

stochastic control methodology is to minimize the following cost function in the case that

the full state information is available

J(k) = E[x>(N)QNx(N)] + EN−1∑

k=0

[x(k)

u(k)

]>Q

[x(k)

u(k)

], (2.27)

where E[·] is the expected value and QN and Q are weighting matrices. The control law

for the optimal state feedback is derived by using dynamic programming and is described

as

u(k) = −K(k, τk)

[x(k)

u(k − 1)

], (2.28)

where K is the optimal gain matrix after solving the formulated LQG problem.

• a method of estimating the undelayed plant state using time domain solutions of the plant

state equations [Zhang et al., 2001]; the estimation of the plant state x(kTs + τ sck ) is

based on the sensor measurement at time kTs and the sensor to controller delay τ sck . It is

assumed that the delay is less than one sampling period (τ sc < Ts).

With full state feedback, the only task of the estimator is to compensate for the delay τ sc

in order to achieve a more accurate plant state at the time the control signal is calculated.

29


Assuming the plant and controller models are given by (2.5) and (2.6), this can be done

by using

x(t) = eAtx(0) +

∫ t

0

eA(t−s)Bu(s)ds. (2.29)

The plant state estimate x(kTs + τ sck ) can be calculated by

x(kTs + τ sck ) = x(kTs + τ sck ) =

= eAτsck x(kTs) +

∫ kTs+τsck

kTs

eA(kTs+τsck −s)Bu(s)ds(2.30)

and the control law is computed by

u(kTs + τ sck ) = −Kx(kTs + τ sck ). (2.31)

Using this control law, the closed-loop system becomes

x((k + 1)Ts + τk+1) = Φ(δk)x(kTs + τk), (2.32)

where

δk = Ts + τ sck+1 − τ sck ,Φ(δk) = Φ(δk)− Γ(δk)K,

Φ(δk) = eAδk ,

Γ(δk) =

∫ δk

0

eAsBds.

(2.33)

• a predictive control strategy to compensate for random delays that appear between the

controller and the actuator, but the delays that appear between the sensor and controller

are not taken into account [Liu et al., 2004]. The control structure (Fig. 2.10) is composed

of a conventional predictive controller, designed based on the GPC strategy, which assures

the closed-loop performances and a predictive controller (Networked Control Predictor),

which compensates the delays introduced by the network. Since the network can transmit

a set of data at the same time, it is assumed that the all predictive control sequence at time

30

Time-varying delays

t is packed and sent to the plant. The networked control predictor takes the latest control

value from the predictive control sequence available on the plant side.

• a method which applies Fuzzy Logic to a PID controller [Cao and Zhang, 2006]; the

classic PID controller is modified in order to introduce high-order information, which can

accurately reflect the delays; the parameters in modified Fuzzy controller can dynamically

adapt the change of delays because of the adaption of the Fuzzy Logic. In order to get the

control signal, firstly the fuzzy relationship matrix R is determined as

Ri = ei × ecj × uij,

R =n∑

i=1

Ri,(2.34)

where ei, ecj and uij are the error, the error change and the corresponding control in ith

sample. Then, the control signal is calculated using

U = [e(k)× ec(k)] R, (2.35)

where is the composition operator.

To attain the real control signal, the Fuzzy control signal is transfered using a Fuzzy

Decision

uc =

∑i u(Zi) Zi∑

i u(Zi), (2.36)

where u(Zi) = Ui is the ith value of the fuzzy control and uc is the control signal that can

be directly used.

• a communication disturbance observer for bilateral teleoperation under time-varying de-

lay; the effects of the time-varying delays are considered as network disturbances (see

subsection 2.4.1.2) and this network disturbance is estimated by a communication distur-

bance observer (CDOB) as shown in Fig. 2.11 [Natori et al., 2010]. Although the output

of the slave sXeTs is delayed by the round trip time delay T , the time-delay effect is

compensated by the time-delay compensation method with CDOB and then the input for

master sX is no longer delayed. Therefore, it turns out that time delay is sufficiently com-

pensated. Furthermore, the time-delay compensation method works without time-delay

model.

31


Conventional

Predictive

Controller

Networked

Control

Predictor

Network

Plant

( )r t

( )y t

( )u t

( )( )

|

1|

u t t

u t t

+ ⋮

Networked predictive controller

Sensor

Physical

plant

Actuator

Network

Model of

process

Adaptation

of delay

Filter

Controller +

-

e

r

u

y

my

py

pe

+

-

+

+

ay

Adaptive Predictor

Figure 2.10: NCS with two predictive controllers.

1054 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 57, NO. 3, MARCH 2010

Fig. 5. Concept of ND in bilateral teleoperation systems. (a) Bilateral teleop-eration system with time delay. (b) Bilateral teleoperation system with ND.

Fig. 6. Time-delay compensation in bilateral teleoperation systems.

the slave system. After introduction of the four-channel time-delayed bilateral control system, the bilateral teleoperationsystem used in the experiment is presented.

A. Time-Delay Compensation

Fig. 5(a) shows a simple example of a bilateral teleoperationsystem. F means a control input for slave (torque dimension),and sXe−Ts is an output of slave (angular velocity dimension).J means slave’s moment of inertia. T1 denotes a time delayfrom master to slave, and T2 is a time delay from slave tomaster (T = T1 + T2). In the case of bilateral teleoperationunder time-varying delay, T1 and T2 are time varying. Fig. 5(a)shows that the output of slave sXe−Ts is delayed relative to theinput for slave F . Considering the concept of ND describedin Section II, Fig. 5(a) changes, as shown in Fig. 5(b). InFig. 5(b), there is no time-delay element. Instead of the time-delay element, there exists ND described as follows:

Dnet(s) = F (1 − e−Ts). (31)

The ND is estimated by CDOB and the estimated ND isutilized for time-delay compensation, as shown in Fig. 6. (In

Fig. 7. Internal structure of CDOB in bilateral teleoperation systems.

Fig. 6, for easy understanding, it is assumed that the cutofffrequency of LPF in CDOB is ideally infinity.) Although theoutput of the slave sXe−Ts is delayed by the round trip timedelay T , the time-delay effect is compensated by the time-delay compensation method with CDOB and then the inputfor master sX is no longer delayed. Therefore, it turns outthat time delay is sufficiently compensated. The effectivenessis the same as Smith predictor. Furthermore, the time-delaycompensation method works without time-delay model (theinternal structure of CDOB (Fig. 7) does not include time-delaymodel) in contrast to model-based or predictive control methodslike Smith predictor which needs accurate time-delay model.Consequently, the time-delay compensation method is expectedto work even in the case of unknown and time-varying delaylike that of IP networks.

B. Design of Cutoff Frequency of LPF in CDOB

Fig. 7 shows internal structure of CDOB in a simple exampleof a bilateral teleoperation system shown in Fig. 6. Describedas (26), ND is estimated with limited bandwidth in actualimplementation (gnet is the cutoff frequency of CDOB)

Dnet(s) =gnet

s + gnetDnet(s)

=gnet

s + gnetF (1 − e−Ts). (32)

Then, the cutoff frequency of LPF in CDOB gnet should bedesigned appropriately. In addition, since DOB [27], [28] isimplemented at slave side for realization of robust accelerationcontrol, we should also consider the cutoff frequency of LPFin DOB at slave side gd, as shown in Fig. 8 (Fdis means adisturbance exerted on slave). In Fig. 8, the estimated valueby CDOB is affected by cutoff frequencies of CDOB (gnet)and DOB at slave side (gd). Then, considering Fig. 3(b), theestimated value by CDOB Fdisnet is described as follows:

Fdisnet =gnet

s + gnet

(F (1 − e−Ts) +

s

s + gdFdis

)

=gnet

s + gnetF (1 − e−Ts) +

gnet

s + gnet

s

s + gdFdis.

(33)

Figure 2.11: Time-delay compensation in bilateral teleoperation systems.

2.4.2.1 Adaptive Smith Predictor

A Smith predictor with an adaptation loop to the time-varying sensor to actuator delays that

are introduced by the communication network (Fig. 2.12) [Velagic, 2008]; the PI controller

is designed using technical optimum technique and its discrete-time form is described by the

following equation

Gr(z) =KrTsz

Tiz − Ti, (2.37)

whereKr is the proportional gain and Ti is the integral time constant. The process model, which

predicts the future behavior of the process, is given in discrete form as follows

Gm(z) =b1z−1 + b2z

−2 + ...+ bmz−m

a0 + a1z−1 + a2z−2 + ...+ anz−n, (2.38)

where m and n are the polynomials grades.

The adaptation algorithm is derived with assumption that average communication delays

from sensor to controller and from controller to actuator are equal, both delays being variables

and uniformly distributed.

In the adaptation loop, the communication delay in the Smith predictor is calculated based

32

Time-varying delays

Conventional

Predictive

Controller

Networked

Control

Predictor

Network

Plant

( )r t

( )y t

( )u t

( )( )

|

1|

u t t

u t t

+ ⋮

Networked predictive controller

Sensor

Physical

plant

Actuator

Network

Model of

process

Adaptation

of delay

Filter

Controller +

-

e

r

u

y

my

py

pe

+

-

+

+

ay

Adaptive Predictor

Figure 2.12: NCS with adaptive Smith Predictor.

on N previous delays in the network from sensor to controller

d = 2

N∑i=1

τ sci

N, (2.39)

where τ sci is the communication delay from sensor-to-controller in i-th step. In this way, the

average value of N previous delays is calculated.

Moreover, to improve the robustness of the closed-loop control system, a first-oder filter is

introduced, its transfer function being given by

Gf (z) =1

1− Tfz−1, (2.40)

where Tf is a time constant of the filter, which is recommended to be set as follows

Tf =Tcα, (2.41)

where Tc is a whole communication delay and the parameter α influences the robustness of the

control signal (smaller α => better robustness; greater α => faster delay compensation).

2.4.2.2 Explicit Model Predictive Control

Optimal control of constrained linear and piecewise affine systems has garnered great interest

in the research community due to the ease with which complex problems can be stated and

solved. The Multi-Parametric Toolbox (MPT) provides efficient computational means to obtain

feedback controllers for these types of constrained optimal control problems in a Matlab pro-

33


gramming environment. By multi-parametric programming, a linear or quadratic optimization

problem is solved off-line. The associated solution takes the form of a PWA state feedback law.

In particular, the state-space is partitioned into polyhedral sets and for each of those sets the

optimal control law is given as one affine function of the state. In the online implementation

of such controllers, computation of the controller action reduces to a simple set-membership

test, which is one of the reason why this method has attracted so much interest in the research

community [Kvasnica et al., 2006].

PWA systems are models for describing hybrid systems and the dynamical behavior of such

systems is capture by relations of the following form:

xk+1 = Aixk + Biuk + fi

yk = Cixk + Diuk + gi, (2.42)

subject to constraints on outputs, control input, and control input slew rate:

ymin ≤ yk ≤ ymax

umin ≤ uk ≤ umax

∆umin ≤ uk − uk−1 ≤ ∆umax

. (2.43)

The cost function used for the explicit MPC scheme is

minukk∈Z[0,N−1]

(‖PNxN‖p +

N−1∑

k=0

‖Qxxk‖p + ‖Ruuk‖p), (2.44)

where u is the vector of manipulated variables over which the optimization is performed, N is

the prediction horizon, p is the linear norm and can be 1 or∞ for 1- and Infinity-norm, respec-

tively. Also, Qx, Ru and PN represents the weighting matrices imposed on states, manipulated

variables and terminal states, respectively.

2.4.2.3 Lyapunov-based Model Predictive Control

Consider the non-autonomous system

xk+1 ∈ φ(x[k−h,k], uk), k ∈ Z+, (2.45)

34

Time-varying delays

where x[k−h,k] ∈ Xh+1, h ∈ Z+ is the maximum delay and uk ∈ U ⊆ Rm is the control input

at the discrete time instant k.The mapping φ : Rn × . . . × Rn × Rm ⇒ Rn is an arbitrary

set-valued function with φ(0[−h,0], 0) = 0. It is assumed that 0 ∈ int(X) and 0 ∈ int(U).

Next, let α1, α2 ∈ K∞ and let ρ ∈ R[0,1).

Definition 2.4.1 A function V : Rn → R+ that satisfies

α1(‖x‖) ≤ V (x) ≤ α2(‖x‖), ∀x ∈ Rn, (2.46)

and for which there exists a control law, possibly set-valued, π : Rn × . . .×Rn ⇒ U such that

V (x+) ≤ maxi∈Z[−h,0]

ρV (xk+i), ∀x[−h,0] ∈ Xh+1 (2.47)

for all u ∈ π(x[−h,0]) and all x+ ∈ φ(x[−h,0], u) is called a CLRF for the difference inclusion

corresponding to system (2.45). 2

In [Gielen and M.Lazar, 2009b,Gielen et al., 2010] it is assumed that the delays are smaller

than or equal to the sampling period and the main results are derived with respect to this as-

sumption.

Consider the following inequality corresponding to (2.47)

V (x+) ≤ maxi∈Z[−1,0]

ρV (xk+i) + λk, ∀k ∈ Z+, (2.48)

for all x+ ∈ φ(xk, uk, uk−1). Here λk is a variable which allows for additional freedom in

the evolution of the CLRF, i.e., it can increase if (2.47) is too conservative at time instant k ∈Z+, possibly due to active state/input constraints. Based on (2.48) the following optimization

problem is formulated.

Let α3, α4 ∈ K∞ and J : R → R+ be an arbitrary function such that α3(|λ|) ≤ J(λ) ≤α4(|λ|) for all λ ∈ R. Let V (·) be a candidate CLRF for system (2.22).

Problem 2.4.2 Assume that at time k ∈ Z+, xk, xk−1 and uk−1 are known and minimize the

cost J(λk) over λk subject to

uk ∈ U, φ(xk, uk, uk−1) ⊆ X, λk ≥ 0, (2.49a)

V (x+) ≤ maxi∈Z[−1,0]

ρV (xk+i) + λk, (2.49b)

for all x+ ∈ φ(xk, uk, uk−1). 2

35


Let π(xk, uk−1) := uk ∈ Rm | ∃λk ∈ R s.t. (2.49) holds and let φcl(xk, π(xk, uk−1), uk−1) :=

φ(xk, u, uk−1) | u ∈ π(xk, uk−1). Furthermore, let VΓ := x ∈ Rn | V (x) ≤ Γ for any

Γ ∈ R+. Let λ∗k denote the optimum in Problem 2.4.2 for all k ∈ Z+.

Theorem 2.4.3 Suppose that V (·) is a function that satisfies (2.46) and that X and U are

bounded. Furthermore, suppose that Problem 2.4.2 is feasible for all (x[−1,0], u−1) ∈ X2 × Uand that limk→∞ λ

∗k = 0. Then, the origin of the difference inclusion

xk+1 ∈ φcl(xk, π(xk, uk−1), uk−1), k ∈ Z+ (2.50)

is attractive. Moreover, if ∃Γ ∈ R>0 such that V (·) is a CLRF for initial conditions in VΓ for

system (2.22), then system (2.50) is asymptotically stable in X. 2

The proof of Theorem 2.4.3 follows from standard arguments employed in proving input-

to-state stability and Lyapunov stability and consideration of the time-delay setting, and is there-

fore omitted here. The interested reader is referred to [M.Lazar 2009] and [Gielen and M.Lazar, 2009b]

for more details. The key of the stability proof is the limiting condition limk→∞ λ∗k = 0.

2.4.2.4 Robust Lyapunov-based Model Predictive Control

Consider the perturbed discrete-time nonlinear system

xk+1 = ψ(xk, wk), k ∈ Z+, (2.51)

where xk ∈ Rn is the state, wk ∈ Rl is an unknown disturbance input and ψ : Rn×Rl → Rn is

an arbitrary nonlinear, possibly discontinuous, set-valued function. For simplicity of notation,

it is assumed that the origin is an equilibrium for (2.51) for zero disturbance, i.e., ψ(0, 0) = 0.

Consider the case when wk takes values at all times k ∈ Z+ in a bounded set W ⊂ Rl.

Definition 2.4.4 A set P ⊆ Rn is robustly positively invariant (RPI) for system (2.51) with

respect to W if for all x ∈ P it holds that ψ(x,w) ⊆ P for all disturbances w ∈W.

Now consider the perturbed discrete-time constrained nonlinear system of the form

xk+1 = φ(xk, uk, wk) := φ(xk, uk) + wk

:= f(xk) + g(xk)uk + wk, k ∈ Z+,(2.52)

where xk ∈ X ⊆ Rn is the state, uk ∈ U ⊆ Rm is the control input and wk ∈ W ⊆ Rn is an

unknown disturbance at the discrete-time instant k. φ : Rn ×Rm ×Rn → Rn, φ : Rn ×Rm →

36

Time-varying delays

Rn, f : Rn → Rn and g : Rn → Rn×m are arbitrary nonlinear, possibly discontinuous,

functions with φ(0, 0, 0) = 0, φ(0, 0) = 0, f(0) = 0 and g(0) = 0. Naturally, it is assumed that

the set of feasible states X, the set of feasible inputs U and the disturbance set W are bounded

polyhedra with non-empty interiors containing the origin. Next, let α1, α2, α3 ∈ K∞ and let

σ ∈ K.


α1(‖x‖) ≤ V (x) ≤ α2(‖x‖), ∀x ∈ Rn (2.53)

and for which there exists a possibly set-valued control law π : Rn ⇒ U such that

V (φ(x, u, w))− V (x) ≤ −α3(‖x‖) + σ(‖w‖),∀x ∈ X,∀u ∈ π(x),∀w ∈W,

(2.54)

is called an input-to-state stability control Lyapunov function (ISS-CLF) in X for system (2.52)

and disturbances in W.

ISS theory (see [Jiang and Wang, 2001]) can be used to derive an input-to-state stabilizing

predictive control scheme with improved disturbance rejection, as done in [M.Lazar and Heemels, 2008],

where this property is refereed to as optimized ISS.

As such, let W be a convex hull of the vertices we, e = 1, . . . , E, and let λek, k ∈ Z+, be

optimization variables associated with each vertex we. Let J(λ1, . . . , λE) : RE+ → R+ be a

strictly convex, radially unbounded function (i.e. J(·) tends to infinity when its arguments tend

to infinity) and let J(λ1, . . . , λE)→ 0⇒ λe → 0 for all e = 1, . . . , E and J(0, . . . , 0, ) = 0.

Choose off-line a CLF V (·) for system (2.52) without disturbances and let α3 ∈ K∞ and

x ∈ X be given. At each control sampling instant k ∈ Z+ the one step ahead ISS MPC

controller solves the following problem.

Problem 2.4.6 At time k ∈ Z+ measure the state xk and minimize the cost J(λ1k, . . . , λ

Ek ) over

ukandλ1k, . . . , λ

Ek , subject to the constraints

uk ∈ U, φ(xk, uk) ∈ X, λek ≥ 0, (2.55a)

V (φ(xk, uk, 0))− V (xk) + α3(‖xk‖) ≤ 0, (2.55b)

V (φ(xk, uk, we))− V (xk) + α3(‖xk‖)− λek ≤ 0, (2.55c)

for all e = 1, . . . , E. 2

37


Let π(xk) := uk ∈ Rm | ∃λek, e ∈ Z[1,E] s.t. (2.55) holds and let φcl(xk, π(xk), wk) :=

φ(xk, uk, wk) | uk ∈ π(xk) denote the difference inclusion corresponding to system (2.52)

in closed-loop with the set of feasible solutions obtained by solving Problem 2.4.6 at each

sampling instant k ∈ Z+.

Next, the main robust stability result in terms of ISS is stated.

Theorem 2.4.7 Let α1, α2, α3 ∈ K∞, a continuous and convex CLF V (·) and a cost J(·) be

given. Suppose that Problem 2.4.6 is feasible for all states x in X. Then, the trajectories gener-

ated by the difference inclusion

xk+1 ∈ φcl(xk, π(xk), wk), k ∈ Z+, (2.56)

with initial state x0 ∈ X converge in finite time to a RPI subset of X, in which the difference

inclusion is ISS for disturbances in W.


to-state stability and Lyapunov stability and is therefore omitted here. The interested reader is

referred to [M.Lazar and Heemels, 2008] and [M.Lazar 2009] for more details. Advantageous

properties of the proposed robust controller are that ISS is guaranteed for any (feasible) solution

of the optimization problem, state and input constraints can be explicitly accounted for, and

feedback to disturbances is provided actively, on-line.

2.4.2.5 Stability analysis

New methods for stability analysis of NCSs are needed and this is motivated by the fact that a

control system can be destabilized as a consequence of constant time delays. However, the sit-

uation can become even more interesting as it is shown in [Cloosterman et al., 2006]: a system,

which is stable for the best- and worst-case constant delays (and all constant values in between),

can become unstable if the delay is time-varying within these bounds. Several methods have

been proposed to improve stability of NCSs with time-varying delays. In [Li et al., 2008a] the

stability conditions are derived via both network-condition-dependent Lyapunov function and

common Lyapunov function. Stability is guaranteed by giving sufficient conditions in terms of

linear matrix inequalities (LMIs) in [Sheng and Pan, 2010,Liu et al., 2010,Wu et al., 2010] or bi-

linear matrix inequalities (BMIs) in [Salehi et al., 2008], using Lyapunov-Krasovskii methods.

A Lyapunov-based stability criterion in terms of LMIs for discrete-time NCS models is given

in [Cloosterman et al., 2009, Huang et al., 2010] and for continuous-time models in [Yue et

al., 2009]. Moreover, in [Cloosterman et al., 2010] a parameter-dependent Lyapunov function

38

Automotive networked control systems

is compared with a Lyapunov-Krasovskii function for stability analysis. In [Zhao et al., 2010]

a stabilized controller design method obtained using time delay switched theory is proposed.

2.5 Automotive networked control systems

From the wide variety of fast systems, in this thesis, the focus is on the subclass of automotive

systems, which have fast dynamics, requiring small sampling periods.

A modern vehicle is composed of mechanical systems (engine, clutch, transmission, pro-

peller shaft, final drive, driveshafts, wheels), electronic and digital components (ECUs, sensors,

actuators, digital communication networks) and software applications (embedded operating sys-

tems, multimedia applications). In these types of vehicles, the communication networks have

an important role, because the vehicle is a distributed system, so its well functioning depends on

the collaboration of different components located at different spatial locations. Before the elec-

tronic components appearance, the communication between the different elements in a vehicle

was made with mechanical or hydraulic systems [Li et al., 2008a]. The modern vehicle systems

are hierarchically organized and distributed in a communication network with decentralized

control.

2.5.1 Automotive control systems

Recent studies in automotive engineering explore various engine, transmission and chassis mod-

els and advanced control methods to increase overall vehicle performance, fuel economy, safety

and comfort. Moreover, nowadays, the driver does not have direct access to the mechanical

components of a vehicle, instead, its commands pass through ECUs that control the main com-

ponents of a vehicle: engine, transmission, brakes, final drive and even the steering system.

The throttle and the brake pedals are not directly connected to the throttle valve and the brake

pressure valve; the two signals pass through an ECU which decides if the drivers commands

should be followed, or, based on the sensor measurements, it should apply different commands

to maintain the vehicle stability in case of emergency. Also, in automatic transmission equipped

vehicles, the gear ratio is changed by a different ECU and in some 4x4 vehicles, the final drive

can be locked or unlocked by another ECU.

Furthermore, modern vehicles are generally equipped with passive and/or active safety sys-

tems. The main objective of passive safety systems (safety belts, airbags) is to diminish the

consequences of a car crash. In return, the active safety systems, which make use of the above

described control systems, help to prevent the accidents by partially taking the control of the

39


vehicle until the unwanted behavior concerning the vehicle dynamics is corrected [Cerone et

al., 2007].

Lately, there has been an increased effort and different researches were carried out in the

automotive industry towards developing new safety systems meant to increase the vehicles be-

havior in critical situations, the passengers comfort and to lessen the drivers tasks. Some of the

developed systems are already installed on new vehicles, e.g. ABS, TCS (ASR), ESC (ESP,

VDC, DSC), but these systems are not yet optimized and their are still under research activities

in both industry and academics programs [Oudghiri et al., 2007].

Various research has been conducted on understanding and damping the driveline oscilla-

tions of conventional vehicles and numerous control strategies were reported in the literature:

robust pole placement [Richard et al., 1999,Stewart et al., 2005], H∞ optimization [Lefebvre et

al., 2003], linear quadratic gaussian control design with loop transfer recovery (LQG/LTR) [Pet-

tersson, 1997, Fredriksson et al., 2002, Berriri et al., 2008] and even model predictive con-

trol [Rostalski et al., 2007].

Also, different control strategies were proposed to control the brakes by using the anti-

lock breaking system: PD control [Choi, 2008], hybrid control [Johansen et al., 2001], sliding

mode control [Schinkel and Hunt, 2002, Hong et al., 2006], logic control [Lv et al., 2007],

predictive control [Yoo and Wang, 2007], fuzzy control [Lee and Zak, 2001, Yu et al., 2002,

Lin and Hsu, 2003, Keshmiri and Shahri, 2007].

Different control algorithms to maintain the stability of a vehicle in emergency situations

were proposed in the available literature: sliding mode control [Drakunov et al., 2000, Hong et

al., 2006,Wang and Longoria, 2006,Canale et al., 2008a,Canale et al., 2008b], nonlinear alloca-

tion [Tondel and Johansen, 2005], feedforward-feedback control [Canale et al., 2006,Cerone et

al., 2007], predictive control [Anwar, 2007], robust Fuzzy control [Oudghiri et al., 2007].

Although the majority of the control strategies that are implemented on real vehicles are

based on heuristics and look-up tables, it was shown that MPC has a large potential for control

of automotive subsystems, e.g., ABS [Yoo and Wang, 2007], vehicle dynamics [Anwar, 2007],

mechatronic actuators [Di Cairano et al., 2007], driveline [Saerens et al., 2008], engine [Di

Cairano et al., 2010]. This is supported by the fast proliferation of powerful embedded compo-

nents that enable complex real-time control algorithms.

All of the above control solutions assume that the sensors, controllers and actuators are di-

rectly connected, which is not realistic. Rather, in modern vehicles, the control signals from

the controllers and the measurements from the sensors are exchanged using a communication

network, e.g., CAN or Flexray, among control system components. This brings up a new chal-

lenge on how to deal with the effects of the network-induced delays and packet losses in the

40


Transmission

Engine

Wheel

Final drive

Driveshaft

Clutch Driveshaft

Propeller

shaft

Figure 2.13: Schematic vehicle structure.

control loop. The delays may be unknown and time-varying and may degrade the performances

of control systems designed without considering them and can even destabilize the closed-loop

system.

2.5.2 Drivetrain control systems

An automotive drivetrain is a system that includes the mechanical components (engine, clutch,

transmission, propeller shaft, final reduction gear, driveshafts and wheels) which have the func-

tion of transmitting the engine torque to the driving wheels, as it can be seen in Fig. 2.13. In

order to transmit the torque in an efficient way, an accurate model of the drivetrain is needed

for controller design purposes with the aim of increasing comfort by reducing the vibrational

oscillations.

Recently, the greater demand for passenger comfort regarding the noise and vibration char-

acteristics of vehicles has led to the design of proper models for vehicle drivetrains and to the

development of different control strategies to minimize the effects of drivetrain oscillations.

When a vehicle is subjected to acceleration, the elasticity of the various components in the

driveline may cause torsional vibrations or disturbances. The torsional vibrations can result in

driveline or vehicle speed oscillations, also known as shuffle mode, which are low-frequency

oscillations corresponding to the first resonance frequency of the driveline. The oscillations

give rise, apart from material stress, to noticeable reduced driveability.

Driveability, the ability to quickly and appropriately respond to driver commands and a high

degree of driving comfort are expected in a modern vehicle. Due to the elasticity of the driveline

components, mechanical resonance may occur. This phenomenon is known as driveline oscil-

lations or “shuffles”. When driveline oscillations are induced, the driveability of the vehicle is

reduced, as the oscillations are transmitted via the chassis to the driver who experiences longi-

tudinal jerking. The control design objective is to increase the passenger comfort by reducing

41


the oscillations that occur during gearshift, while traversing backlash or when tip-in and tip-out

maneuvers are performed.

In recent years the driveline oscillation problem has received an increasing interest due to

the introduction of dual-clutch powershift (or shortly, DPS) automatic transmissions. These

dry clutch transmissions offer improved fuel economy, easier packaging and reduced weight

with respect to the standard wet-clutch planetary gear transmissions. Also the torque converter,

which provides a smooth hydrodynamic coupling between the engine and the transmission and

which is present in standard automatic transmissions, can be removed. However, the absence

of the torque converter makes the torque transfer path from the engine to the wheels entirely

mechanical, which means that disturbances, including the inherent reciprocating behavior of

the engine, have more impact on the driveline. In manual transmissions the driver avoids or

reduces the shuffle mode by modulating the clutch pedal. In automatic DPS transmissions the

engine torque is modulated by a control algorithm, since torque actuators (airflow and/or spark

timing) attain a higher precision and higher bandwidth than clutch actuators. In this thesis a

specific design for this control set-up is proposed.

The automotive driveline is an essential part of the vehicle and its dynamics have been

modeled differently, according to the driving necessities. The complexity of the numerous

models reported in the literature varies [Hrovat et al., 2000], but the two masses models are

more commonly used, and this fact is justified in [Pettersson, 1997], where it is shown that

this model is able to capture the first torsional vibrational mode. There are also more com-

plex three-masses models reported in different research papers, as it will be indicated next.

In [Templin and Egardt, 2009] a simple driveline model with two inertias, one for the engine

and the transmission, and one for the wheel and the vehicle mass, was presented. A more com-

plex two-masses model, including a nonlinearity introduced by the backlash, was presented

in [Templin, 2008]. A mathematical model of a drivetrain was introduced in [Baumann et

al., 2006] and [Bruce et al., 2005] in the form of a third order linear state-space model. A sim-

ple model with the pressure in the engine manifold and the engine speed as state variables and

the throttle valve angle as control input was presented in [Saerens et al., 2008]. A piecewise

linear (PWL) two-masses model, with one inertia representing the engine and the other inertia

representing the vehicle (including the clutch, main-shaft and the powertrain), was presented

in [Bemporad et al., 2001]. [Rostalski et al., 2007] presents a PWA two masses model for a

driveline including a backlash nonlinearity. In [Grotjahn et al., 2006] a two masses model was

presented with the driveline main flexibility represented by the driveshafts, as well as a three

mass model to reproduce the behavior of a vehicle with a dual-mass flywheel. Linear and non-

linear three masses models, in which the clutch flexibility was also considered, were presented

42


in [Kiencke and Nielsen, 2005]. A complex three masses model that includes certain nonlinear

aspects of the clutch was presented in [Van Der Heijden et al., 2007].

Concerning the applied control strategies, different approaches to damp driveline oscilla-

tions have also been proposed in literature. In [Templin and Egardt, 2009] a linear quadratic

regulator (LQR) design that damps driveline oscillations by compensating the driver’s en-

gine torque demand was presented. Other linear quadratic gaussian controllers designed with

loop transfer recovery were presented in [Pettersson, 1997, Fredriksson et al., 2002, Berriri et

al., 2007,Berriri et al., 2008]. Furthermore, [Bruce et al., 2005] proposed the usage of a feedfor-

ward controller in combination with a LQR controller and considering the engine as an actuator

to damp powertrain oscillations. A robust pole placement strategy was employed in [Richard et

al., 1999, Stewart et al., 2005], an H∞ optimization approach was presented in [Lefebvre et

al., 2003], while MPC strategies were proposed in [Lagerberg and Egardt, 2005, Rostalski et

al., 2007, Baumann et al., 2006]. In [Baumann et al., 2006], a model-based approach for anti-

jerk control of passenger cars that minimizes driveline oscillations while retaining fast acceler-

ation was introduced. The controller was designed with the help of the root locus method and

an analogy to a classical PI-controller was drawn. In [Rostalski et al., 2007], a constraint was

imposed on the difference between the motor speed and the load speed to minimize the driveline

oscillations, while reducing the impact of forces between the mechanical parts.

2.5.3 CAN-based networked control systems

Until recently, the in-vehicle communications between different components, as switches and

actuators, were realized using point-to-point connections, resulting in a sizable, expensive and

complicated wiring network, which was difficult to realize and install. With the continuously

increase of in-vehicle communication possibilities, the connections number increased so much

that their volume, their reliability and their weight became a true problem [Leen et al., 1999].

Nowadays, the vehicles electronic systems architecture is composed of electronic control

units, sensors and actuators. In an exact domain, e.g., powetrain, chassis, body, these units

are interconnected through a communication network. Furthermore, gateway nodes enable the

information exchange between different domains [Stroop and Stolpe, 2006] as it can be seen

in Fig. 2.14. Current control systems (powertrain, chassis, body) are individual systems with

a limited interdependency. Integrating these systems would bring two great benefits [Lauer et

al., 2006]:

• All informations from different domains can be combined to offer a better view of vehi-

cles states and surroundings; in this way better decisions can be made.

43


Power EngineAutomatic

Bluetooth

Gateway

Audio

SystemGPS

Video

SystemHMITelematics

GW

Infotainment

CAN

MOST

26 Mbps

Introducere

Power EngineAutomatic

Fig. 1. Reţele de sisteme de control interconectate prin gateway-uri (GW)

Body

Control

Door

Control

Parking

Assistant

Climate

Control

Seat

Control

Instrument

Panel

Stability

Control

Roll

Control

Brake

Control

Power

Steering

Engine

Control

Automatic

Transmission

GW

Body System

Powertrain

CAN

Low Speed

125 Kbps

CAN

High Speed

500 Kbps

Stability

Control

Roll

Control

Brake

Control

Power

Steering

Engine

Control

Automatic

Transmission

Powertrain

Figure 2.14: Automotive networked control systems interconnected through gateways.

Table 2.1: Automotive networks grouping

Group Subbus Event-triggered Time-triggeredLIN CAN FlexRay

Representative K-Line VAN TTPI2C PLC TTCAN

• The vehicle can be controlled in an integrated way, by coordinating the actions from

different domains.

2.5.3.1 In-vehicle communication networks for control

The rising number of sensors, actuators and ECUs led to an increased complexity of commu-

nication networks in automotive applications, from LIN, to CAN, to MOST and even FlexRay,

which can be seen in Fig. 2.15 [Li and Zhao, 2008b] along other existing or in development

technologies. As shown in Table 2.1 [Wolf et al., 2004], the vehicle networks are distributed in

five different groups according to their essential technical properties and application areas.

CAN protocol is utilized in the automotive industry as the basis communication network,

which allows the information exchange between the different components sharing the network.

However, the introduction of advanced control systems implemented in several ECUs, which

use informations from different sensors and send control signals to different actuators, requires

a protocol with a higher transfer rate, i.e., FlexRay, which is a new protocol that will be used

for the development of future in-vehicle communication networks.

44


WiMAX

GSM (2G) GSM (3G)

Coverage Scope

Wide-Area

Data Rate

Around-Car

50M

802.11g

25M

DSRC

TTP/C

MOSTD2B

5M

Byteflight

FlexRayTTP/C

TTP/A CAN

J1850 LIN

Bluetooth

20M10M1M

In-Car

Technologies for future use Technologies in use

Figure 1．Communication Technologies in Automotive Network Systems

2. Challenges and Approaches

Digital communication plays a critical role in current and future vehicle systems. However, a requirement for communication that takes place in an automotive system is significantly different from traditional Internet-like systems. Specifically, the following are important properties that should exist within an automotive system’s communication network:

1) Fault Isolation. Refers to the fact that an error of transmitting one signal will not affect the transmission of other signals. In conventional mechanical automotive systems, different components use dedicated mechanical or hydraulic systems to transmit control information (e.g., steering and brake). Therefore, one component’s failure will not affect the normal operations of others. Hence, when X-by-wire in the future vehicle system is deployed for function X, this property of fault isolation must be preserved in order to ensure the safety of the vehicle.

2) Timing Correctness. After a vehicle’s careful design and implementation, its components will function in a coherent and synchronized manner to realize correct functionalities. Hence, in communication, timing correctness must be guaranteed. Note that fault isolation can only guarantee that different type of signal transmission will not affect each other, while the timing correctness ensures the correct functionalities.

3) Heterogeneity. Many current vehicle systems have already had multiple sub-networks. For example [8], it is typical that in a vehicle, a low-speed CAN manages “comfort electronics”, e.g., seat and window control, while a high speed network runs “mission critical” functions, e.g., engine control and brake system. While in many current vehicles these sub-networks have not been connected, it is expected that for future vehicle systems, these sub-networks will be inter-connected in order to achieve more functionality and better reliability. Thus, in future vehicle systems, communication network will be heterogeneous.

These three properties are equally important. Compounded together, they ensure the safety and utility of the underlying vehicle systems. They present significant challenges in the design and

Position Paper V 3.0 2 3/11/2008, 2:46:14 PM

Figure 2.15: Vehicle networks technologies.

The communication networks from automotive applications were formally classified by

SAE (Society of Automotive Engineers) based on their transfer rate, Table 2.2 showing the

categories [Leen et al., 1999].

LIN is a low cost serial communication network used to control the systems in the comfort

domain: automatic door locking, electrical windows and mirrors, trunk opening. It is also used

to communicate with different smart sensors that detect, e.g., rain or darkness. CAN is an event-

triggered communication protocol used for soft real-time control applications, e.g., ABS, engine

management, VDC. Time-triggered CAN (TT-CAN) is a communication protocol based on

CAN and it is used for hard real-time control applications, being capable to transmit also event-

Table 2.2: Vehicle networks classification

Category Transfer rate Application domainClass A < 10 kb/s

Low transfer rateComfort systems: trunk opening,electrical mirrors

Class B 10 - 125 kb/sMedium transfer rate

General information: board in-struments, electrical windows

Class C 125 kb/s - 1 Mb/sHigh transfer rate

Real-time control: powertrain,dynamics control

Class D > 1 Mb/s Multimedia applications: Inter-net, TV. Critical functions: ”X-by-Wire”

45


triggered messages in certain ”arbitrating” time windows. FlexRay is also a time-triggered

communication protocol used for highly safety X-by-Wire systems, being very flexible and

with an increased network throughput.

2.5.3.2 Controller Area Network

CAN is a serial, event-triggered, communication protocol that was developed in the mid 1980s

by Robert Bosch GmbH. CAN is usually utilized in real-time applications, defining a standard

for efficient and reliable communication between sensors, controllers, actuators and other dif-

ferent nodes. Although it was initially designed as a bus system for automotive applications,

nowadays, it is also utilized for other applications in different domains, e.g., robotics, factory

automation, medical equipment, building automation.

Any node may start transmitting a message when the bus is free (multi-master capability)

and all the nodes in the network synchronize at the beginning of a message transmission by

using the SOF (Start of Frame) bit. In other words, every node which has a message to transmit,

starts transmitting it, bit by bit, quasi-simultaneous. The nodes that do not have anything to

transmit, start receiving the message that is sent on the line. All the nodes may receive (multi-

cast capability) and accept or reject a message.

The arbitration is determined by the arbitration field, i.e., identifier, which is located just

after the SOF bit. There are two types of identifiers depending on the message frame format: 11

bits identifiers for the standard CAN and 29 bits identifiers for the extended CAN. Both these

frames can coexist on the same communication bus. Every node listens to its own transmission

and when it detects that the transmitted bit is different than the one that appears on the line, it

gives up on transmitting the rest of the message (it had lost the access) and receives further the

transmitted message on the bus.

After sending the last bit in the arbitration field, a single node must remain to continue

transmitting the message (it had won the arbitration). In order to be a single node that keeps

on transmitting its message, every message type must have an unique identifier (the identifier

automatically defines the message priority).

The protocol uses a bitwise arbitration scheme for medium access, i.e., Carrier Sense Mul-

tiple Access/Bitwise Arbitration (CSMA/BA) or Carrier Sense Multiple Access/Arbitration

on Message Priority (CSMA/AMP). Possible conflicts are resolved by using a nondestructive

priority-based arbitration process: if two or more nodes start transmitting at the same time, the

highest priority frame will win the conflict and it will be sent over the network. A retransmission

of the interrupted frame will be triggered automatically.

46


The CAN communication protocol has the following advantages:

• higher flexibility (allows incremental system design); nodes can be added or removed

without affecting the other nodes;

• safety; assures the detection and correction of errors;

• low cost;

• low worst-case latency for the highest priority messages;

• high average performance;

and disadvantages:

• non-deterministic behavior; all the nodes that have a message to transmit must participate

in the bus access arbitration process;

• limited throughput compared to other networks used for control;

• not suitable for large data sized messages;

• the transmission of all safety messages is not guaranteed;

• there is no possibility to determine the point in time when the message will be transmitted

(with high precision);

• it does not support fault-tolerance by network redundancy;

• the transmission latencies and jitter are not bounded;

• being event-triggered, it is difficult to analyze.

2.5.3.3 CAN-induced delays in automotive applications

The safe operation of a vehicle depends not only on the performance and access mechanism

of the communication system, but also requires reliable data transfer with respect to transmis-

sion errors and retransmissions, robustness of the bus system and control units. To assure the

consistent operation of a vehicle, hard real-time transmission deadlines have to be met, so there

appears the need to determine reasonable bounds for delays of safety critical data transmission.

The result presented in [Klehmet et al., 2008, Herpel et al., 2009], which aims to provide

a method for calculating the worst-case response time of each message sent on CAN in auto-

motive applications, is used to determine the upper bound of the delays that are induced by

47


the communication network. The result is based on Network Calculus [Le Boudec and Thi-

ran, 2001], which is a system theory for deterministic queuing systems, based on min-plus

algebra. Its main focus is on determination of bounds on worst case performance. One aim is to

determine lower and upper bounds for end-to-end delays of nodes or collections of nodes within

a network, for traffic backlog and for output limitations. The Network Calculus technique re-

quires only the statically assigned CAN identifiers and the specific cycle times at which each

message is sent as input data. There is no need to know the global bus-wide schedule of traffic

to obtain reasonable upper delay bounds for each priority class.

Let xj(t) denote the input data and yj(t) the output data at time t per node of priority

j = 0, ..., N . The procedure starts with input x0(t) and output y0(t). Then, the delay for the

first priority class yields:

d0 ≤ supt≥0infτ : α0(t) ≤ β0(t+ τ), (2.57)

where α0(t) is a step-function-type arrival curve of input x0(t) and the rate-latency service curve

β0(t) = βR,T (t) = 500 kbps·[t− 0.000272s]+, with [t−T ]+ := max0, t−T. The maximum

delay d1 for frames of priority 1 is

d1 ≤ supt≥0infτ : α1(t) ≤ β1(t+ τ) = β0(t+ τ)− α1(t+ τ), (2.58)

where β1(t) is the service curve for the nodes of priority 1 and α1(t) is the upper bound for the

cumulative higher priority arrivals. Going on in this way, lastly, it yields:

dj ≤ supt≥0infτ ≥ 0 : αj(t) ≤ βj(t+ τ), (2.59)

where αj(t) and βj(t) are the arrival curve and, respectively, the service curve for the nodes of

priority j. The last inequality can be solved geometrically, resulting that the upper bound for

the delays of the j-th priority node is calculated using

dj ≤(j + 2) · l

R−∑j−1i=0 (l/ci)

, (2.60)

where l = 136 bits denotes the maximum frame length including the 6 bit CS time, R = 500

kbps is the rate of a high-speed CAN and ci is the cycle length of the i-th priority message. A

cycle length cn, corresponding to a message of priority n, represents the period after which the

message is repeated.

48

Conclusions

2.6 Conclusions

In this chapter, the state-of-the-art in NCS modeling and delay compensation is presented. Gen-

eral information regarding the problems that arise and the solutions that were proposed in the

available literature due to the introduction of communication networks in the closed-loop con-

trol systems is provided.

NCSs have many attractive advantages (low cost, simple installation and maintenance, higher

reliability and greater flexibility), but there are also some disadvantages represented by the

network-induced imperfections and constraints (quantization errors, data-packet dropouts, vari-

able sampling periods, time-varying network-induced delays, communication constraints), which

are firstly presented.

Secondly, different models which describe the functioning of a NCS are depicted, starting

with the early modeling methods of [Luck and Ray, 1990, Krtolica et al., 1994, Chan and Oz-

guner, 1995], which introduce buffers and queues in the closed-loop control system, to the

detailed mathematical model derived by [Zhang et al., 2001] concerning the time-varying de-

lays that can be smaller or larger than the sampling period. Then, several delay models and

compensation strategies are illustrated.

Thirdly, the control systems that are in nowadays vehicles, including the different strategies

of controlling these systems proposed in the available literature, are presented and then the

most utilized communication network in automotive closed-loop control systems, i.e., CAN, is

briefly described.

Finally, a method of finding the upper bound of the communication delays that appear on

CAN in automotive applications is illustrated. The method will be used in the following chap-

ters for controller design purposes.

49


50

Chapter 3

Linear networked predictive control

In this chapter, three new methods of modeling the network-induced time-varying delays for

controller design purposes are proposed. In addition, a linear networked predictive control strat-

egy, which makes use of the three modeling methods, is proposed with the aim of compensating

the time-varying delays which appear in NCSs.

3.1 Introduction

The variable-time delay introduced by communication networks is the main factor that dete-

riorates the performance of networked control systems. In this chapter, it is assumed that the

delays in the communication network are bounded and three methods of modeling the time-

varying delays introduced by the communication network are proposed: average method, iden-

tification method and adaptation method. The average delay modeling method considers the

mean value of the delays that can appear in the communication network. The identification

modeling method considers that the delay is equal to the minimum delay that can appear in the

communication network and uses a different model of the physical plant that is identified in

order to capture the dynamics of the system including the delays between the minimum and the

maximum delays that can appear in the communication network. The adaptation method is de-

signed in order to adapt the control algorithm to the network-induced time-varying delays. The

adaptation algorithm is derived with assumption that the average communication delays from

sensor-to-controller and from controller-to-actuator are equal, both being variables. Unlike for

the first two modeling methods, in the case of the adaptation method there is no need to know

the upper bound of the delays.

Moreover, a new networked predictive control strategy is proposed with the aim of control-

51


ling the output of a physical plant, while compensating the effects of the time-varying delay by

using the proposed modeling methods. Furthermore, a new method (λ-scheduling) by which

the controller is adapted to the difference between the desired reference and the plant output

is proposed. Then, the proposed strategy is applied in order to control several physical plants

(fast processes) from automotive applications (DC motor, ICE, valve-clutch system, drivetrain)

and to decrease the influence of the variable-time delay induced in the NCS on the control

performance.

The performance of the proposed strategy is demonstrated by simulation results and corre-

sponding comparisons with other networked controllers (PID, adaptive Smith Predictor) prove

the significance of the proposed modeling methods and control strategy. Also, the linear net-

worked predictive control strategy was tested on a real-time test-bench and the obtained results

prove that the proposed strategy can be implemented to control the actual physical plant.

3.2 Modeling the plant including the delays

Consider the plant described by the CARIMA (Controlled AutoRegressive Integrated Moving

Average) model [Camacho and Bordons, 2004]

A(z−1)yk = z−dB

(z−1)uk−1 +

ekC (z−1)

D (z−1), (3.1)

where d = ak+bk is the delay introduced by the communication network at time instant k ∈ Z+

to be considered by the predictive strategy and ek is white noise with zero mean value.

A (z−1) and B (z−1) are the system polynomials

A(z−1)

= 1 + a1z−1 + ...+ anAz

−nA ,

B(z−1)

= b0 + b1z−1 + ...+ bnBz

−nB ,(3.2)

where nA and nB represent the polynomials degrees and C (z−1) and D (z−1) are the distur-

bances polynomials equal to

C(z−1)

= 1,

D(z−1)

= 1− z−1,(3.3)

for obtaining a zero steady-state error.

Now, consider that the delay introduced by the communication network dc is time-varying,

52

Modeling the plant including the delays

but bounded

dm ≤ dc ≤ dM , (3.4)

where dm is the minimum delay and dM is the maximum delay that appear in the communication

network.

In the following subsections, three methods of considering the communication delay [Caruntu and Lazar, 2009b,

Caruntu et al., 2010b, Balau, Caruntu and C.Lazar, 2011b, Caruntu and C.Lazar, 2011b] pro-

posed to be used by the predictive algorithm are discussed.

3.2.1 Average method

In this approach, the delay to be considered by the predictive strategy is calculated using the

mean value of the delays that can appear in the communication network given by the following

relation

d =dm + dM

2. (3.5)

This is a very simple method of considering the network-induced time-varying delays by the

LNPC, but, as it will be shown in Section 3.4, the performances of the closed-loop control

systems are indeed improved w.r.t. other networked controllers.

3.2.2 Identification method

For the second approach, the delay considered by the predictive strategy is equal to the minimum

delay that can appear in the communication network

d = dm (3.6)

and instead of the polynomial B, another polynomial B, identified in order to model the system

including the delays between dm and dM is introduced

B(z−1)

= b0 + b1z−1 + ...+ bnBz

−nB , (3.7)

with

nB = nB + dM − dm,

b0 = b1 = ... = bnB =b0 + b1 + ...+ bnB

nB + 1,

B(1) = B(1),

(3.8)

53


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

2

4

6

8

Time [s]

Tor

que

[Nm

]

d=dm=0d=dM=12Td=(dm+dM)/2=6TARX model

Figure 3.1: System output - different delays.

In Fig. 3.1 the difference between the model of the plant with average delay and the model

obtained with the identification method can be observed. It can be seen that the identified model

tries to capture the whole interval between dm and dM .

3.2.3 Adaptation method

In the third approach, at each sampling instant the delay used by the predictive strategy is

calculated using

d = 2

N∑i=1

τ sci

N, (3.9)

where τ sci is the communication delay from sensor-to-controller in i-th step. In this way, the

average value of N previous delays is calculated.

This method is designed in order to adapt the control algorithm to the variable-time delays

that appear in the communication network. The adaptation algorithm is derived with assump-

tion that the average communication delays from sensor-to-controller and from controller-to-

actuator are equal, both being variables.

Unlike for the other two modeling methods, in the case of the adaptation method the upper

bound of the delays can be unknown.

3.3 Linear networked predictive controller design

Consider the plant described by the CARIMA model (3.1) and the three methods of considering

the communication delays, which are subjected to (3.4).

54

Linear networked predictive controller design

3.3.1 Prediction model

The prediction model is given by

yk+j|k = Gj−d(z−1)D(z−1)z−d−1uk+j+

+Hj−d (z−1)D (z−1)

C (z−1)uk−1 +

Fj−d (z−1)

C (z−1)yk

(3.10)

with j = hi, hp, where hi is the minimum prediction horizon and hp is the prediction horizon.

uk+j−1|k, j = 1, hc is the future control, computed at time k and yk+j|k are the predicted values

of the output, hc being the control horizon.

Using the three methods of considering the communication delays by the LNPC, d from

(3.10) is given by (3.5) for the average delay modeling method, (3.6) for the identification

modeling method and (3.9) for the adaptation method.

For determining the polynomials Fj−d (z−1), Gj−d (z−1) and Hj−d (z−1) two Diophantine

equations are used. The first is:

C (z−1)

A (z−1)D (z−1)= Ej−d

(z−1)

+ z−(j−d) Fj−d (z−1)

A (z−1)D (z−1), (3.11)

where

Ej−d(z−1)

= 1 + e1z−1 + ...+ enEz

−nE ,

Fj−d(z−1)

= f0 + f1z−1 + ...+ fnF z

−nF ,(3.12)

with

nE = j − d− 1,

nF = max (nA + nD − 1, nC − (j − d)) .(3.13)

For the average and adaptation methods, the second Diophantine equation is

Ej−d(z−1)B(z−1)

= C(z−1)Gj−d

(z−1)

+ z−(j−d)Hj−d(z−1), (3.14)

where

Gj−d(z−1)

= g0 + g1z−1 + ...+ gnGz

−nG

Hj−d(z−1)

= h0 + h1z−1 + ...+ hnHz

−nH ,(3.15)

55


with:

nG = j − d− 1,

nH = max (nC , nB + d)− 1.(3.16)

For the identification methodB is replaced by B, so the second Diophantine equation (3.14)

becomes

Ej−d(z−1)B(z−1)

= C(z−1)Gj−d

(z−1)

+ z−(j−d)Hj−d(z−1), (3.17)

where Gj−d and Hj−d are given by (3.15) with:

nG = j − d− 1,

nH = max (nC , nB + d)− 1.(3.18)

The prediction model yields as

yk+j|k = Gj−d(z−1)D(z−1)z−d−1uk+j + y0,k+j|k (3.19)

where

y0,k+j|k =Hj−d (z−1)D (z−1)

C (z−1)uk−1 +

Fj−d (z−1)

C (z−1)yk (3.20)

represents the free response.

Considering as inputs D(z−1)uk and collecting the j-step predictors in a matrix notation,

the prediction model can be written as

y = Gud + y0 (3.21)

where

y = [yk+hi|k, yk+hi+1|k, . . . , yk+hp|k]>, (3.22)

G =

ghi−d−1 . . . g0 0 . . . 0

ghi−d . . . g1 g0 . . . 0

. . . . . . . . . . . . . . . . . .

ghc−1 . . . . . . . . . . . . g0

ghp−d−1 . . . . . . . . . . . . ghp−hc−1

, (3.23)

56

Linear networked predictive controller design

ud =[D(z−1)uk, . . . , D(z−1)uk+hc−1

]>, (3.24)

y0 = [y0,k+hi|k, y0,k+hi+1|k, . . . , y0,k+hp|k]>. (3.25)

The objective function is based on the minimization of the tracking error and on the min-

imization of the controller output, the control weighting factor λ being introduced in order to

make a trade-off between these objectives

J = (Gud + y0 −w)T (Gud + y0 −w) + λudTud, (3.26)

subject to D(z−1)uk+i = 0 for i ∈ [hc, hp− d− 1], where w is the reference trajectory vector

with the components wk+j|k, j = hi, hp. Minimizing the objective function (∂J/∂ud = 0), the

optimal control sequence yields as

u∗d =(GTG + λIhc

)−1GT [w − y0] . (3.27)

Using the receding horizon principle and considering that γj, j = hi, hp are the elements of

the first row of the matrix(GTG + λIhc

)−1GT, the following control algorithm results:

D(z−1)uk =

hp∑

j=hi

γj[wk+j|k − y0,k+j|k

]. (3.28)

With y0,k+j|k from (3.20), the control algorithm can be rewritten as

C(z−1)D(z−1)uk =

−hp∑

j=hi

γjHj−d(z−1)D(z−1)uk−1−

−hp∑

j=hi

γjFj−d(z−1)yk +

hp∑

j=hi

γjC(z−1)wk+j,

(3.29)

The LNPC can be analytically determined off-line using (3.29), which means that the con-

trol command can be computed very fast, so this algorithm is capable of satisfying the timing

constraints associated to the fast processes.

57


3.3.2 λ-scheduling

The aim of this method is to adapt the control algorithm to the difference between the de-

sired reference and the process output. Being known that λ is the parameter used to make

the trade-off between the minimization of the tracking error and the minimization of the con-

troller output, it is easy to observe that by increasing λ, the process output tracks the reference

more slowly, and by decreasing λ, the process output tracks the reference faster, but with an

overshoot, as it will be shown in Subsection 3.4.3.2. So, the proposed solution is to modify

the parameter λ on-line depending on the error between the desired reference and the process

output [Caruntu and C.Lazar, 2011b].

Consider the following normalized error

ek =wk − ykwk

, (3.30)

which takes values in the interval [1, 0].

Then, λ can be calculated using

- linear functions:

λk = a1ek + b1, (3.31)

- nonlinear functions:

λk = a1e2k + b1ek + c, (3.32)

- PWA functions:

λk =

a1ek + b1, ek ∈ [e1, e2],

a2ek + b2, ek ∈ (e2, e3],

· · · ,aNek + bN , ek ∈ (eN , eN+1],

(3.33)

where ai, bi and c are suitable chosen constants, for i = 1, N .

Now, considering that λmin and λmax are two values sufficiently small and large, respec-

tively, for λ, it is easy to make a connection between ek and λk such that

λk = (1− ek)(λmax − λmin) + λmin =

= −(λmax − λmin)ek + λmax,(3.34)

so, as the error ek decreases, the parameter λk increases. In this case, a linear function of the

form (3.31) results.

58

Linear networked predictive control of automotive fast systems

3.4 Linear networked predictive control of automotive fast

systems

In this section, several LNPCs are designed for different physical plants from automotive ap-

plications. Firstly, a LNPC is designed based on the average delay modeling method for a DC

motor used as actuator in different automotive subsystems; secondly, based on the identifica-

tion modeling method, a LNPC is designed to control an internal combustion engine; thirdly,

two LNPCs based on the adaptation modeling and λ-scheduling methods are tested on a valve-

clutch subsystem of an automatic transmission. CAN is used as the communication medium for

all case studies as it is the most used communication network from automotive applications.

3.4.1 Average delay modeling and LNPC applied to a DC motor

A LNPC based on the average delay modeling method was tested to control a DC motor through

a CAN and the results are presented in this subsection. Comparisons were made with a PI

controller and the adaptive Smith predictor described in Subsection 2.4.2.1 and proposed in

[Velagic, 2008].

The control of a DC motor was chosen because is the most distributed application in au-

tomotive systems. The applications that they are used to are: seat adjustment, window lift,

wipers, mirror adjustment, trunk lock, sunroof control, ABS-motor, water pump, engine cool-

ing fan, electronic throttle plate [Gerling, 2001], fuel pump, engine management [Angleviel et

al., 2006], steering system [Gao et al., 2008], active suspensions [Zhang et al., 2008] etc.

Considering the control of a DC motor through a network, the plant can be modelled in the

s-domain as follows [Velagic, 2008]

G (s) =1

0.1s2 + 0.7s+ 1(3.35)

and its discrete model is given by

Gm

(z−1)

=0.0042z−1 + 0.0039z−2

1− 1.8025z−1 + 0.8106z−2=B (z−1)

A (z−1), (3.36)

where a system sampling time of 0.03s it is used.

In this section delays larger than a sample time as in [Velagic, 2008] are considered, as-

suming that communication delays from sensor to controller and from controller to actuator are

variables (random), have the same values and they are uniform distributed. In Fig. 3.2 a time

distribution of the communication delay under interval [3Ts, 9Ts] (with average 6Ts), generated

59


0 5 10 150.05

0.1

0.15

0.2

0.25

0.3

Time [s]

Co

mm

un

ica

tion

de

lay

[s]

Figure 3.2: Delay time distribution (DC motor control).

using a Matlab/Simulink model from [Velagic, 2008], it is shown, where Ts = 0.03s is the

sample period of the system.

The networked predictive control strategy was applied using the average delay modeling

method and the performances of the closed-loop system were analyzed considering different

communication delays having as average values 3Ts, 6Ts and 9Ts as in [Velagic, 2008]. It was

considered that d = 3, 6, 9 and the predictive control strategy was applied using the following

parameters: hc = na + 1 = 3, hi = d + 1, hp = hc + d. Applying the linear networked

predictive control strategy for d = 6 and λ = 3.2 resulted in the following control law

uk = 0.9993uk−1 − 0.0004uk−2 − 0.0004uk−3

− 0.0005uk−4 − 0.0005uk−5 − 0.0005uk−6

+ 0.0014uk−7 + 0.0015uk−8 − 0.4498yk

+ 0.7484yk−1 − 0.3142yk−2 + 0.0155,

(3.37)

when it is considered that w(k) = 1, k = 1, 2, ....

The results obtained with the proposed LNPC based on the average delay modeling method

are compared with the ones obtained in [Velagic, 2008]. First, a step signal was applied to the

system and it was desired that the system tracks the reference signal as fast as possible (Figs. 3.3

- 3.5). In these figures five signals are represented: the reference value (blue), the response of

the system using a PI controller without communication delay (green) and with communication

delay (red), the response of the system with delay when the Smith-like predictive controller is

applied (cyan) and the response of the system with communication delay controlled using the

LNPC (magenta). In Figs. 3.3 - 3.5 it can be seen that the performances of the LNPC are better

than the performances of the Smith predictor used in [Velagic, 2008], the output of the system

when using the LNPC asymptotically tracking the desired reference signal.

60


0 5 10 150

0.5

1

1.5

Time [s]

Vol

tage

[V]

referencePI without delayPI with delaySmith Predictor (Velagic, 2008)LNPC with delay

Figure 3.3: DC motor output for d = 3 (step reference).

0 5 10 150

0.5

1

1.5

Time [s]

Vol

tage

[V]



0 5 10 150

0.5

1

1.5

Time [s]

Vol

tage

[V]



61


0 10 20 30 40 50 60−0.5

0

0.5

1

1.5

Time [s]

Vo

ltag

e [V

]

Figure 3.6: DC motor output for d = 3 (sequence of pulses reference).

0 10 20 30 40 50 60−0.5

0

0.5

1

1.5

Time [s]

Vo

ltag

e [V

]


The responses of the system when the LNPC was applied are clearly different from those

obtained with the PI controller and the Smith predictor. The set-point responses for the PI and

the Smith-like predictive controllers have an obvious overshoot, while the set-point curve for

the LNPC has a lower rise time and an overshoot less than 0.2%.

Second, a sequence of pulses was applied, the results being represented in Figs. 3.6 - 3.8,

using the same colors as for the step responses. The performances of the proposed method are

still better; at the same time it can be seen that the rise time for the output of the system is

smaller than the ones of the other methods.

Even if the networked induced delays are time-variant and large compared to one sampling

period, the system performance is improved and the influence of the time delay on networked

control systems is alleviated using the proposed GPC method.

3.4.2 Identification modeling and LNPC of an ICE

In this subsection, the results of a LNPC based on the identification modeling method tested on

an ICE controlled through CAN are presented. Comparisons were made with the LNPC based

on the average modeling method, a PI controller and the adaptive Smith predictor described in

62


0 10 20 30 40 50 60−0.5

0

0.5

1

1.5

Time [s]

Vo

ltag

e [V

]


Subsection 2.4.2.1 to illustrate the performances obtained by this method.

In order to develop a driver model, in [Kiencke and Nielsen, 2005] two different controllers

were designed: one for the longitudinal dynamics and one for the lateral dynamics of a vehicle.

For the design of the longitudinal controller, which calculates the control variables for the gas

or brake pedals, a simplified longitudinal closed-loop system was identified. The identification

of the longitudinal control plant was divided into drive and brake control, two models being

identified: one for the drive system, which has the throttle valve angle as the input and the drive

torque as the output, and one for the brake system, which has the brake pedal force as the input

and the brake torque as the output.

For the drive system, a linear second order ARMA transfer function is assumed [Kiencke and Nielsen, 2005].

For the identification of the coefficients, a test drive on a straight dry road was carried out,

whereby the initial velocity corresponds to 24m/s and the transmission ratio is igear = 1.41.

The test signals were a minimum throttle angle αt,sim,1 = 1 and a maximum αt,sim,2 = 10.

The transfer function for the drive system was estimated as

GD

(z−1)

=0.112295 + 0.112314z−1

1− 1.7713z−1 + 0.810704z−2, (3.38)

where a system sampling time of 0.01s it is used.

The relationship between the accelerator pedal position and the opening angle of the throt-

tle valve is not fixed, instead an ECU determines how much torque the engine is required to

produce and then opens the throttle valve to the appropriate angle. The chosen throttle opening

is based on the difference between the engines instantaneous torque output and the required

torque output, as requested not only by the driver, but also by in-car systems.

In this section, the equation (2.60) developed in [Klehmet et al., 2008] it is used to determine

the upper bound of the communication delays that appear on CAN in automotive applications. It

is assumed that communication delays from sensor to controller and from controller to actuator

63


0 0.5 1 1.50

0.02

0.04

0.06

0.08

0.1

0.12

Time [s]

Co

mm

un

ica

tion

de

lay

[s]

Figure 3.9: Delay time distribution (ICE control).

have the same values and they are uniform distributed. In Fig. 3.9, a time distribution of the

communication delay under interval [0, 12Ts], generated using a Matlab/Simulink model, it is

shown.

The CARIMA model for the drive system (3.38) it is used by the predictive control method,

where:

A(z−1)

= 1− 1.7713z−1 + 0.810704z−2;

B(z−1)

= 0.112295 + 0.112314z−1;

C(z−1) = 1;D(z−1)

= 1− z−1 = ∆.

(3.39)

For the average delay modeling method it was considered that dm = 0 and dM = 12 from

(2.60) and after applying the designed control algorithm for hc = na + 1 = 3, hi = d + 1,

hp = hc+ d, λ = 110 and using d = 6 from (3.5), the following control law resulted:

uk = 0.9847uk−1 − 0.0076uk−2 − 0.0081uk−3−− 0.0082uk−4 − 0.008uk−5 − 0.0075uk−6+

+ 0.0255uk−7 + 0.0291uk−8 − 0.2878yk+

+ 0.4853yk−1 − 0.2103yk−2 + 0.001wk+7+

+ 0.0038wk+8 + 0.0079wk+9.

(3.40)

The CARIMA model for the identification modeling method including the delays it is used

by the predictive control strategy, where

B(z−1)

= 0.016043 + 0.016043z−1+...+0.016043z−13 (3.41)

64


0 0.5 1 1.5 20

100

200

300

400

500

600

Time [s]

Tor

que

[Nm

]

referencePI without delayPI with delaySmith predictor with delay

Figure 3.10: Engine output (PI & Smith predictor).

and after applying the predictive control algorithm for hc = na + 1 = 3, d = dm = 0,

hi = d+ 1, hp = hc+ d and λ = 24 the following control action resulted

uk = 0.98988uk−1 − 0.0001uk−2 − 0.0001uk−12−− 0.0003uk−13 − 0.0009uk−14 − 0.20608yk+

+ 0.0823yk−1 − 0.0307yk−2 + 0.0007wk+1+

+ 0.0025wk+2 + 0.0059wk+3.

(3.42)

The results obtained with the LNPC based on the average modeling and identification mod-

eling methods are compared with two different controllers: a PI controller and a Smith-like

predictive controller with adaptation to communication delay developed in [Velagic, 2008].

The PI controller has the following tuning parameters: Kr = 1.65 and Ti = 1.

A step signal was applied to the system and it was desired that the system tracks the refer-

ence signal as fast as possible, the following figures showing the controlled outputs and the ref-

erence signal. In Fig. 3.10, four signals were represented: the reference drive torque value, the

response of the system using the PI controller without communication delay and with commu-

nication delay, the response of the system with delay when the Smith-like predictive controller

is applied.

In Fig. 3.11, the angles of the throttle valve for the PI controllers and for the Smith predictor

were represented.

Fig. 3.12 illustrates the reference drive torque value and the responses of the system with

communication delay when the predictive average value method is applied and when the pre-

dictive identification method is applied. The responses are clearly different from those obtained

with the PI controller and the Smith predictor. The set-point response for the PI and the Smith-

like predictive controllers have an obvious overshoot, while the set-point curves for the two

65


0 0.5 1 1.5 20

20

40

60

80

Time [s]

Thr

ottle

val

ve a

ngle

[deg

]

PI without delayPI with delaySmith predictor with delay

Figure 3.11: Throttle valve angle (PI & Smith predictor).

0 0.5 1 1.5 20

100

200

300

400

500

600

Time [s]

Tor

que

[Nm

]

referenceaverage predictivemethod with delay

identification predictivemethod with delay

Figure 3.12: Engine output (LNPC).

predictive methods are similar except that the rise time for the average value method is a little

longer than that for the identification method and their overshoot is less than 0.5%.

It can be seen that the performances of the predictive control methods are better than the

performances of the Smith predictor used in [Velagic, 2008], the output of the system when

using predictive controllers asymptotically tracking the desired reference signal.

In Fig. 3.13, the angles of the throttle valve (the control signals) for the predictive methods

were represented. The throttle valve angle was limited to between 0 and 90, only this range

being physically realizable. The results illustrate that the variations of the throttle valve angle

are much smaller for the proposed identification method.

3.4.3 λ-scheduling control applied to a valve-clutch subsystem

In this subsection, two LNPCs based on the adaptation modeling and λ-scheduling methods

were designed to control a valve-clutch subsystem of an automatic transmission through CAN.

Comparisons were mare with LNPCs based on the average and identification modeling methods

and also with a PI controller and the adaptive Smith predictor described in Subsection 2.4.2.1.

The clutch is the main component of an automatic transmission, having the aim of cou-

66


0 0.5 1 1.5 20

20

40

60

80

Time [s]

Thr

ottle

val

ve a

ngle

[deg

]

average predictivemethod with delay

identification predictivemethod with delay

Figure 3.13: Throttle valve angle (LNPC).

pling/decoupling the engines output shaft to the transmission shaft. Clutch control is seen

more and more as an important enabling technology for the automotive industry, being cen-

tral for automatic gear shifting and traction control for improved safety, drivability, comfort

and fuel economy. During the last years this topic has been actively researched and the at-

tention has focused on modelling and developing control methods for automated clutch ac-

tuators [Morselli et al., 2003, Van Der Heijden et al., 2007, Langjord et al., 2008, Neelekan-

tan, 2008]. The automatic transmission clutch is controlled by using a hydraulic 3-way pressure

reducing valve. For a valve of this type, which is used in the automatic transmissions of Volk-

swagen vehicles, two models were developed: a linearized input-output model [Patrascu et

al., 2009] and a state-space model [Balau, Caruntu et al., 2009b, Caruntu et al., 2009c]. In

order to design a predictive controller to control the valve-clutch system (Fig. 3.14), starting

from the equations presented in [Merritt, 1967], an input-output model for the valve-clutch

system was also developed. The model was validated by comparing the simulation results

with the experimental ones obtained at Continental Automotive Romania [Balau, Caruntu et

al., 2009a, C.Lazar, Caruntu and Balau, 2010].

All of the above control solutions assume that the sensors, controllers and actuators are di-

rectly connected, which is not realistic. Rather, in modern vehicles, the control signals from the

controllers and the measurements from the sensors are exchanged using a communication net-

work, e.g., CAN or Flexray, among control system components. This brings up a new challenge

on how to deal with the effects of the network-induced delays and packet losses in the control

loop.

In this section, a network-controlled wet clutch actuated by an electro-hydraulic valve is

used to illustrate the effectiveness of the proposed delay modeling methodologies proposed

in Section 3.2 and the performance of the proposed predictive control strategy presented in

Section 3.3. The goal of the control algorithm is to achieve zero steady-state error and fast

67


Figure 1: Actuator and clutch system: a) Charging phase; b) Discharging phase

Line

pressure

DQCQ

RP

TP

1K 2K

CPA,

DPD,

CF DF

SP

magF

Output axle

Output cooling

oil to tank

Input

cooling

oil

0>x

Valve

plunger

3KL

Q

L

L

A

P

,C

C P

,LA

LP

0<x

Solenoid

i u

Reset spring

Piston stop

Input axle

Figure 3.14: Charging phase of the valve clutch system (dashed - discharging phase).

response of the valve-clutch system to step references for the clutch piston displacement, while

compensating the time-varying delays introduced by CAN.

3.4.3.1 Valve-clutch system model

The model of the valve-clutch system, which is illustrated in Fig. 3.14, designed based on phys-

ical principles for flow and fluid dynamics, is briefly described below by the following equations

[Balau, Caruntu et al., 2009b,C.Lazar, Caruntu and Balau, 2010,Balau, Caruntu and C.Lazar, 2011b]:

- force balance for the valve plunger:

Fmag − ACPC + ADPD = Mvs2x+Kex (3.43)

with

Fmag =kai

2

2(kb + x)2 , LSdi

dt+RSi = u, (3.44)

where Fmag is the electromagnetic force that acts on the plunger, PC represents the oil pressure

in chamber C exerted on area AC on the left end of the plunger, PD represents the oil pressure

in chamber D exerted on area AD on the right end of the plunger, Mv is the plunger mass, Ke =

0.43w(PS0 −PR0) is the flow force spring gradient, Ps is the supply pressure, PR is the reduced

pressure, w represents the area gradient of the main orifice, x represents the displacement of the

plunger, s is the Laplace operator, ka and kb are constants, Ls the solenoid induction and Rs the

68


resistance, i is the current in the solenoid and u is the input voltage;

- flow continuity in the chambers C, D, R and L:

QC = K1 (PR − PC) =VCβesPC − ACsx, (3.45)

QD = K2 (PR − PD) =VDβesPD + ADsx, (3.46)

KC (PS − PR)−QL − klPR −K1 (PR − PC)−

−K2 (PR − PD) +Kqx =VtβesPR,

QL +K1 (PC − PR) +K2 (PD − PR)−

−KD (PR − PT )− klPR +Kqx =VtβesPR,

(3.47)

QL = K3 (PR − PL) =VLβesPL + ALsxp, (3.48)

where K1, K2, K3 are the flow-pressure coefficients of the restrictors, KC is the flow-pressure

coefficient corresponding to the main orifice, PL represents the oil pressure in the clutch cham-

ber exerted on area AL, VC , VD, VL represent the chambers volumes, Kq is the flow gain of

the main orifice, kl is the leakage coefficient, Vt represents the total volume of the chamber

where the pressure is being controlled, βe is the bulk modulus and xp is the clutch displacement

(output of the system);

- force balance for the clutch piston:

ALPL = Mps2xp +Kxp, (3.49)

where K is the force flow spring rate for the clutch and Mp is the mass of the piston.

Using the parameters estimated experimentally or already given by Continental Automotive

Romania or the manufacturer (see Table 1 in Appendix A), the model of the valve-clutch system

was validated by comparing the simulation results with data obtained on a real test-bench at

Continental Automotive Romania [C.Lazar, Caruntu and Balau, 2010].

In order to apply the predictive control strategy, a CARIMA model for the valve-clutch sys-

tem was developed [Caruntu et al., 2010b,Balau, Caruntu and C.Lazar, 2011b,Caruntu and C.Lazar, 2011b],

using as input the supply voltage u and as output the clutch piston displacement xp. The system

69


0 0.5 1 1.5 2

0

2

4

6

x 10−3

Time [s]

Clu

tch

dis

pla

cem

en

t [m

]

PRBSvalve−clutch systemARX

Figure 3.15: Valve-clutch and simulation models responses.

was identified with an ARX equivalent model employing the design model, utilizing as input

a PRBS signal. These sequences are successions of squared pulses, width modulated, that ap-

proximate a discrete white noise and their richness in frequencies helps capturing the dynamical

behavior of the system.

The ARX model is given by the following system polynomials

A(z−1)

= 1− 1.781z−1 + 0.8039z−2,

B(z−1)

= 0.00003312z−1 + 0.0001122z−2.(3.50)

The comparison results are illustrated in Fig. 3.15 , where it can be seen that the valve-clutch

model and the identified design model produce similar results to a given PRBS input signal.

3.4.3.2 Simulation results

This section presents the validations of the proposed linear networked predictive control strategy

investigated on the valve-clutch system model using the Matlab/Simulink program.

Firstly, the predictive control strategy presented in Section 3.3 was tested on the valve-clutch

system without considering that the system is controlled through a communication network,

so no delays appear in the control loop. The obtained results [Balau, Caruntu et al., 2009a,

C.Lazar, Caruntu and Balau, 2010] are illustrated in Fig. 3.16.

It can be seen that the output of the system tracks the reference signal, having no steady

state error.

Being a subsystem of the automatic transmission of a Volkswagen vehicle, the valve-clutch

system is controlled through a communication network [Caruntu et al., 2010b,Balau, Caruntu and C.Lazar, 2011b].

The control commands from the controller, which is implemented in an ECU, are transmitted to

the actuator (valve) through CAN; also, the measurements from the sensors are sent to the con-

70


0 0.2 0.4 0.6 0.8 1

0

1

2

3

4x 10

−3

Time [s]

Dis

plac

emen

t [m

]

clutch displacementreference

Figure 3.16: Clutch displacements (without delays).

Controlul în reţea al proceselor rapide

44

sunt ilustrate în detaliu în Fig. 24. Valva primeşte la intrare mărimea de control

(tensiunea u ) transmisă de regulatorul predictiv prin intermediul reţelei de comunicaţii şi

are ca ieşire presiunea reglată RP , care reprezintă intrarea blocului Wet clutch. Ieşirile

ambreiajului sunt presiunea uleiului din camera acestuia LP şi deplasarea pistonului px , a

cărui valoare este trimisă ca semnal de măsură către regulator prin intermediul aceleiaşi

reţele. În blocul Predictive controller este implementat algoritmul de reglare descris în

Secţiunea 3.1. În Fig. 27 sunt reprezentate, de asemenea, întârzierile introduse de reţeaua

de comunicaţii de la senzor la regulator sc şi de la regulator la elementul de execuţie ca .

Fig. 27. Reprezentare schematică a structurii de de control în reţea

Fiind componentele aceluiaşi subsistem din cadrul lanţului de transmisie a puterii,

se poate considera că întârzierile de la senzor la regulator şi de la regulator la elementul

de execuţie au aceleaşi valori. În Fig. 28 este reprezentată o distribuţie a întârzierilor de la

senzor la regulator în intervalul 0, 6T , generată utilizând un model Simulink.

0 0.05 0.1 0.15 0.20

1

2

3

4

5

6x 10

-3

Time [s]

Com

mun

icat

ion

dela

y [s

]

Fig. 28. Distribuţia întârzierilor în intervalul 0, 6T

Predictive controller

Valve actuator

Wet clutch C

A N

Reference generator

ECU

r k r kT u k u kT ca

ku kT

y k y kT scky kT

Figure 3.17: Schematic representation of the valve-clutch networked control system.

trollers through the same communication network, resulting a networked control system, which

is represented in Fig. 3.17. The Reference generator block calculates and sends the reference

trajectory to the predictive controller, which has to control the clutch plates to follow the desired

reference. The Valve actuator and Wet clutch blocks represent the actuator and the controlled

system. The valve receives the control command (voltage u) sent by the controller through the

communication network and outputs the controlled pressurePR, which is the input of the Wet

clutch block. The outputs of the system are represented by the pressure in the clutch chamber

PL and by the clutch piston displacement xp. The measured value xp is sent to the controller

through the same communication network.

In this section, the equation (2.60) developed in [Klehmet et al., 2008] it is used to determine

the upper bound of the communication delays that appear on CAN in automotive applications.

Using typical values for the parameters in (2.60), it yields that the sum of the delays (τ sc +

τ ca = τmax) is randomly distributed in the interval [0, 12Ts], where Ts = 1ms is the sampling

period of the system. Furthermore, being components of the same powertrain subsystem it can

be considered that the communication delays from sensor to controller and from controller to

actuator have the same values and they are uniformly distributed (see Fig. 3.18).

The predictive control algorithm described in Section 3.3 was designed using the CARIMA

model of the electro-hydraulic actuated clutch with the system polynomials from (3.50). It was

considered that dm = 0 and dM = 12 from (2.60) and the predictive control strategy with

71


0 0.05 0.1 0.15 0.2 0.25 0.30

0.002

0.004

0.006

0.008

0.01

0.012

Time [s]

Co

mm

un

ica

tion

de

lay

[s]

Figure 3.18: Time distribution of communication delay (valve-clutch system).

0 0.05 0.1 0.15 0.2 0.25 0.30

2

4

6x 10

−3

Time [s]

Clu

tch

disp

lace

men

t [m

]

λ1

λ2

λ3

Figure 3.19: Clutch displacements for different lambdas.

the three modeling methods was applied using the following parameters: hc = na + 1 = 3,

hi = d + 1, hp = hc + d. The control action was then applied to the initial model of the

valve-clutch system.

The results obtained with the LNPCs based on the three modeling methods are compared

with two different controllers: a PI controller and a Smith-like predictive controller with adap-

tation to communication delay developed in [Velagic, 2008].

The λ-scheduling method was designed starting from the average delay modeling method.

As it can be seen in Fig. 3.19, for λ1 = 3 · 10−5, the response of the system has a big overshoot,

but it reaches the steady-state faster. For λ2 = 9.5 · 10−5, the response of the system has no

overshoot and it reaches the steady-state in a reasonable period of time, while for λ3 = 12·10−5,

the response of the system is really slow and for sure it has no overshoot. So, for λmin = λ1

and for λmax = λ3, the relation (3.34) becomes:

λk = −9 · 10−5ek + 12 · 10−5. (3.51)

A step signal was applied as the reference for the clutch piston displacement and it was

desired that the system tracks the reference signal as fast as possible, the following figures

72


0 0.05 0.1 0.15 0.2 0.25 0.30

1

2

3

4

5x 10

−3

Time [s]

Clu

tch

disp

lace

men

t [m

]

referencePI without delayPI with delaySmith predictor with delay

Figure 3.20: Clutch displacements (PI & Smith predictor).

0 0.05 0.1 0.15 0.2 0.25 0.30

1

2

3

4

5x 10

−3

Time [s]

Clu

tch

disp

lace

men

t [m

]

referenceaverageidentificationadaptationλ−scheduling

Figure 3.21: Clutch displacements (LNPC).

showing the controlled outputs and the reference signal. In Fig. 3.20 the clutch displacements

obtained using the PI controllers and the Smith predictor are represented.

Fig. 3.21 illustrates the reference clutch displacement value and the responses of the system

with communication delay when the predictive strategy is applied. It can be seen that the system

tracks the reference signal, having no steady state error and it has a rise time in accordance with

the needs in this kind of automotive applications.

The responses are clearly different from those obtained with the PI controller and the Smith

predictor. The set-point response for the PI and the Smith-like predictive controllers have an

obvious overshoot, while the set-point curves for the predictive methods are similar except that

the rise time for the adaptation and λ-scheduling methods is much smaller than those for the

other methods and all the responses have almost no overshoot.

In Fig. 3.22 and Fig. 3.24, the voltage signals (control signals) and the one step ahead

control signal increments for the PI controllers and for the Smith predictor were represented.

Fig. 3.23 and Fig. 3.25 illustrate the voltage signals (control signals) and the one step ahead

control signal increments for the predictive methods. It can be seen that the variations of the

voltage signal are much smaller for the proposed identification method, while the variations are

73


0 0.05 0.1 0.15 0.2 0.25 0.30

0.2

0.4

0.6

0.8

1

Time [s]

Vol

tage

[V]


Figure 3.22: Voltage signals (PI & Smith predictor).

0 0.05 0.1 0.15 0.2 0.25 0.30

0.2

0.4

0.6

0.8

1

Time [s]

Vol

tage

[V]

averageidentificationadaptationλ−scheduling

Figure 3.23: Voltage signals (LNPC).

much bigger for the adaptation method.

Also, in Fig. 3.26, the delay used by the average method is represented and compared with

the value of the delay used by the adaptation method for N = 20 from (3.9).

It can be concluded that the performances of the predictive control methods are better than

the performances of the Smith predictor proposed in [Velagic, 2008].

0 0.05 0.1 0.15 0.2 0.25 0.3−0.02

−0.01

0

0.01

0.02

Time [s]

Vo

ltag

e [

V]


Figure 3.24: Control signal increments (PI & Smith predictor).

74

Experimental results

0 0.05 0.1 0.15 0.2 0.25 0.3−0.2

−0.1

0

0.1

0.2

Time [s]

Vo

ltag

e [

V]

averageidentificationadaptationλ−scheduling

Figure 3.25: Control signal increments (LNPC).

0 0.05 0.1 0.15 0.2 0.25 0.30

2

4

6

8

Time [s]

Ave

rage c

om

munic

atio

n d

ela

y [S

am

plin

g p

eriods]

averageadaptation

Figure 3.26: Average communication delay.

3.5 Experimental results

In this section, the performances of the linear networked predictive control algorithm developed

with the aim of damping driveline oscillations, while decreasing the influence of the communi-

cation time-varying delays on the closed-loop control performances over the CAN network, are

evaluated on the results obtained on a real-time simulation test-bench.

The testing of control algorithms in their real environment represents an important phase in

the development of control units for drivetrains. The major feature of a real-time simulation is

that the simulation can be carried out as quickly as the real system would actually run, thereby

allowing to combine the simulation and the real CAN network. The absence of the real device

in the simulation particularly simplifies the development, since it allows systematically and

effectively validation of the closed-loop control system, even if the controller is not hardware

implemented.

The use of real-time simulation can replace significantly the construction of expensive pro-

totypes to test drivetrain systems. Due to the utilization of the model in the closed loop, the

simulation is cheaper, easier and less time consuming than experiments on the real device.

Moreover, the risks to damage the real device are eliminated.

75


Ethernet

MicroAutoBoxHost PC

C A N

Figure 3.27: Real-time test-bench.

The computing power required by real-time simulation highly depends on the characteristics

of the simulated model: if it contains very demanding calculations you have to provide a lot of

computing power because the timing cannot be satisfied otherwise. A simulation performed in

Simulink takes as much computer time as needed to compute the control command and then

to calculate the behavior of the system. If the model is complex, much more time is needed to

carry out the necessary calculations. Because a strict time requirement for Simulink simulation

does not need to be fulfilled, the complexity of the model does not need to be reduced.

3.5.1 Real-time test-bench

The test-bench setup for real-time simulation of the closed-loop drivetrain control system,

which is designed to validate the control algorithm, consists of: dSPACE MicroAutoBox, Host

PC, CAN network, power supply, a Simulink model of the drivetrain and controller and a Graph-

ical User Interface (GUI) as it can be seen in Fig. 3.27.

MicroAutoBox consists of the base board DS1401 and it performs the drivetrain plant-

model and controller calculations. It has an Ethernet interface for direct connection to the host

PC, for tasks such as loading models and reading or adjustig parameters in GUI. The HostPC

also runs the simulator’s console software (GUI), designed in ControlDesk, to communicate

with the real-time Simulator (MicroAutoBox). The application allows to download and control

the real time process such as download, start, stop, variable observation and data collection.

Real-Time Interface (RTI) is the link between dSPACE MicroAutoBox hardware and the

MATLAB/Simulink models. Real-Time Workshop (RTW) generates the model code which is

compiled and downloaded in the real-time hardware. The RTI CAN MultiMessage Blockset

76


is an extension for Real-Time Interface and it is used for configuring the CAN network. CAN

configurations are read from a database container (DBC) file.

Note that the Simulink model of the drivetrain and controller exchange data via a CAN bus.

Also, the Host PC has access to the CAN bus through the developed ControlDesk application.

3.5.2 Drivetrain model including the clutch flexibility

This section deals with the problem of damping driveline oscillations, which is crucial for im-

proving driveability and passenger comfort. The design of a controller that damps driveline

oscillations can be a challenging problem considering that vehicle drivetrains are characterized

by fast dynamics that are subject to physical and control constraints. Furthermore, current im-

plementations use a networked controlled environment, which introduces time-varying delays.

Recently, the driveline oscillations damping has received an increased interest due to the

introduction in several production vehicles of the dual-clutch powershift automatic transmis-

sion with dry clutches. This type of transmission improves fuel economy, but it results in a

challenging control problem, due to driveline oscillations. These oscillations, also called “shuf-

fles”, occur during gear-shift, while traversing backlash or when tip-in and tip-out maneuvers

are performed.

Since in this section the focus is on damping the drivetrain oscillations, there is no need to

model dynamics that are much faster than the longitudinal dynamics of the vehicle, so the de-

veloped model is simple enough for controller design, also knowing that in the designing phase

the time-varying delays have to be considered, but complex enough to capture the essential dy-

namics of the drivetrain. In this case the oscillations originating from driveline flexibilities are

of interest.

A PWA three inertias model, which takes into consideration the clutch flexibility together

with the driveshaft flexibility, was derived from the laws of motion [Kiencke and Nielsen, 2005,

Grotjahn et al., 2006,Van Der Heijden et al., 2007], in which the first inertia corresponds to the

engine, the second one includes the inertia of the gearbox and the inertia of the final reduction

gear, and the last inertia corresponds to the wheel and vehicle mass, as it is represented in Fig.

3.28.

When studying a clutch in detail, it is seen that its torsional flexibility is a result of an

arrangement with smaller stiffness springs in series with springs with higher stiffness, like it

can be seen in Fig. 3.29. Fig. 3.29.a) illustrates the clutch stiffness characteristic and the

spring arrangement of the clutch is presented in Fig. 3.29.b). The reason for this arrangement is

vibration insulation. Unlike in the previously mentioned modeling approaches (see Subsection

77


1 eJ J=

2

2 /t f fJ J J i= +

2

3 w COG statJ J m r= +

1 ed d=

2

2 /t f fd d d i= +

3 2w r statd d c r= +

Engine

eJ eT

fJ

Shaft Gearbox

Final drive Driveshaft

Wheels

+

Vehicle

wJ

dk

dd

tiL

fd

tJ

td

fi

ck

cd

wd

,e eω θ

,t tω θ

ed

,w wω θ

Engine

eJ eT

fJ

Clutch Gearbox

Final drive Driveshaft

Wheels

+

Vehicle

wJ

dk

dd

loadT

fd

tJ

td

fg

wd

,e eω θ

,t tω θ

ed

,w wω θ ck

cd

tg

Figure 3.28: Three inertias drivetrain model including the clutch.

a) b)

3ck 2ck

1θ 2θ

1θ− 2θ−

Torque

Clutch torsion

2ck

3ck

4ck

4ck

3ck

2ck

Mechanical stop

Mechanical stop

Figure 3.29: Clutch functionality a) stiffness characteristic; b) clutch springs.

2.5.2), where only two or three working modes are considered for the clutch dynamics, in this

section four working modes are introduced, i.e., open, closing, closed and locked, which yields

a more accurate model of the clutch.

In the open mode, there is no mechanical connection between the engine and the rest of

the driveline, so no torque is transmitted towards the wheels. In the closing mode, the smaller

springs in the clutch are compressed, which means that the engine torque is gradually transmit-

ted to the driveline. In the closed mode, the stiffer springs in the clutch are compressed and

the gradual transmission of torque to the driveline continues. The locked mode corresponds to

the phase when the springs in the clutch cannot be compressed any further, i.e., the clutch hits

a mechanical stop, and the maximum amount of torque is transmitted from the engine to the

wheels through the driveline.

Continuous-time model

78


Considering as state variables the torsional angle between engine and transmission, the tor-

sional angle between transmission and wheel, the angular speed of the engine, the angular speed

of the transmission and the angular speed of the wheel, and as control input of the system, the

engine torque, i.e.,

x1 = θe − θtgt, x2 =θtgf− θw,

x3 = ωe, x4 = ωt, x5 = ωw, u = Te,

(3.52)

the following PWA state-space model is obtained:

x(t) = Acix(t) + bcu(t) + fc if x(t) ∈ Ωi, (3.53)

where x := (x1, . . . , x5)> ∈ R5 and i ∈ I := Z[1,4]. Here i denotes the active mode at time

t ∈ R+, Aci ∈ R5×5, bc ∈ R5×1 are the system matrices and fc ∈ R5×1 is the affine term. Notice

that although there are 3 angles, only two states are introduced as only the angle difference is

relevant. The collection of sets Ωi | i ∈ I defines a partition of the state-space X ⊆ R5as

follows:

Ω1 := x ∈ R5 | x3 ≤ ωclosinge , - open

Ω2 := x ∈ R5 | x3 > ωclosinge & |x1| ≤ θ1, - closing

Ω3 := x ∈ R5 | x3 > ωclosinge & θ1 < |x1| ≤ θ2, - closed

Ω4 := x ∈ R5 | x3 > ωclosinge & θ2 < |x1|, - locked

(3.54)

where ωclosinge is the engine closing speed and θ1, θ2 are threshold values for the torsional angle

between the engine and the transmission, which determine the working mode of the clutch.

Note that when a transition from the open mode to the closing mode occurs, the following reset

condition must be imposed:

∀t1 ∈ R+,∀t2 ∈ R>t1 , if x(τ) ∈ Ω1,∀τ ∈ R[t1,t2)

and x(t2) ∈ Ω2, set x1(t2) := 0.(3.55)

As the engine angle θe tends to infinity in the open mode, so the state x1 tends to infinity,

synchronization of the engine angle and the transmission angle must be attained at the moment

the clutch switches from the open mode to the closing mode. Notice that the reset condition

does not apply to initial conditions, as the model cannot be initialized in the closing mode. The

matrices Aci, bc and fc can be obtained by deriving the whole dynamical model of the drivetrain

using the generalized Newton’s second law of motion. The equation of motion for the first

79


rotational mass yields

Jeωe = Te − Tc − deωe, (3.56)

where the engine is described as an ideal torque source Te with a mass moment of inertia Jeand a viscous friction coefficient de; the engine speed is represented by ωe. Although this will

not be pursued here, a dynamical model of torque production (i.e., manifold dynamics and

combustion delay) can also be included. On the other hand, instantaneous torque production

can be achieved by creating and maintaining a torque reserve via airflow control. The torque

reserve is then exploited by actuating the spark timing which provides fast torque production

changes [Di Cairano et al., 2010]. The clutch torque Tc can be modeled as

Tc = kci(θe − θtgt) + dci(ωe − ωtgt), (3.57)

where the clutch stiffness kci and damping dci have different values for the four modes of op-

eration. In (3.57), gt is the transmission ratio, while ωt stands for the transmission speed. The

corresponding angles of the engine and the transmission are given by θe and θt.

The equation of motion of the second body, the transmission and the shafts, can be derived

as

J2ωt = gtTc − d2ωt −1

gfTd, (3.58)

where J2 = Jt +Jfg2f

is the second inertia with damping d2 = dt +dfg2f

composed by the trans-

mission and final drive damping and gf is the final reduction gear ratio. The torque in the

driveshafts can be expressed as

Td = kd

(θtgf− θw

)+ dd

(ωtgf− ωw

), (3.59)

where ωw is the wheel speed, θw is the wheel angle, and kd, dd are the driveshaft stiffness and

damping, respectively.

Remark 3.5.1 In this thesis, a parallel structure that is equivalent to the series structure adopted

in [Hrovat et al., 2000] is used to model the stiffness and damping of the driveshafts. Testing

how the series structure would affect the performance of the designed control strategy makes

the object of future research. 2

The final law of motion corresponds to the wheels and vehicle body and can be written as

J3ωw = Td − dwωw − Tload, (3.60)

80


where J3 = Jw + mCOGr2w is the wheel and vehicle inertia with damping dw, mCOG is the

vehicle mass and rw is the wheel radius. The load torque is modeled as

Tload = Troll + Tgrade + Tairdrag, (3.61)

where Troll is the rolling resistance torque, Tgrade is the torque due to the road slope and Tairdrag

is the aerodynamic drag torque of the vehicle body,

Troll = cr1mCOGg cos(χroad)rw,

Tgrade = mCOGg sin(χroad)rw,

Tairdrag = 0.5ρairAfcdv2vrw,

(3.62)

where cr1 is the rolling resistance coefficient, g is the gravitational acceleration, χroad is the road

slope gradient, ρair is the air mass density, Af is the frontal area of the vehicle, cd is the airdrag

coefficient and vv = rwωw is the vehicle velocity. To avoid unnecessary complications, only the

terms given by the rolling torque Troll and the aerodynamic drag torque Tairdrag are considered,

while the road slope gradient is assumed to be zero. Furthermore, instead of the nonlinear

function that describes the aerodynamic drag torque given in (3.62), a linear approximation will

be used with da as an approximation parameter, i.e.,

Tairdrag = daωw. (3.63)

Rewriting the equations in a matrix form, yields the model matrices

Aci =

0 0 1 −gt 0

0 0 0 1gf

−1

−kciJe

0 −Dsum1

Je

dcigtJe

0kcigtJ2

− kdgfJ2

dcigtJ2

−Dsum2

J2

ddgfJ2

0 kdJ3

0 ddgfJ3

−dwheel

J3

, (3.64)

with Dsum1 = dci + de, Dsum2 = dcigt2 + d2 + dd

gf 2 , dwheel = dw + dd + da and

bc =(

0 0 1Je

0 0)>

, fc =(

0 0 0 0 −Troll

J3

)>. (3.65)

The engine torque (i.e., the control input) is restricted by lower and upper bounds and by a

81


Figure 3.30: ControlDesk console.

torque rate constraint as follows:

0 ≤ u(t) ≤ Tmaxe , ∀t ∈ R+,

Tme ≤ u(t) ≤ TM

e , ∀t ∈ R+,(3.66)

where Tmaxe is the maximum torque that can be generated by the internal combustion engine

and Tme , TM

e are torque rate bounds. Furthermore, the engine and wheel speeds are bounded,

i.e.,

ωmine ≤ x3(t) ≤ ωmax

e , ∀t ∈ R+, (3.67)

ωminw ≤ x5(t) ≤ ωmax

w , ∀t ∈ R+, (3.68)

where ωmine and ωmax

e are the idle speed and the engine limit speed, respectively, and ωminw and

ωmaxw are the minimum and the maximum speed of the wheels.

3.5.3 Numerical results

This subsection presents the validations of the proposed linear networked predictive control

strategy investigated on the vehicle drivetrain model using the designed real-time simulation

test-bench. The simulator’s console software (GUI), designed in ControlDesk, is represented in

Fig. 3.30, including the messages sent on CAN.

Using typical values for the parameters in (2.60), it yields that the sum of the delays (τ sc +

τ ca) is randomly distributed in the interval [0, 4Ts], where Ts = 5ms is the sampling period of the

system. Furthermore, being components of the same powertrain subsystem it can be considered

that the communication delays from sensor to controller and from controller to actuator have

82


the same values and they are uniformly distributed.

In order to apply the predictive control strategy, a CARIMA model for the drivetrain model

was developed, using as input the engine torque Te and as output the wheel speed ωw. The

system was identified with an ARX equivalent model employing the design model, which is

given by the following system polynomials

A(z−1)

= 1− 1.962z−1 + 0.96203z−2,

B(z−1)

= −0.77 · 10−6z−1 + 0.143 · 10−4z−2.(3.69)

Using the identification modeling method, it was considered that dm = 0 and dM = 4 from

(2.60) and the polynomial B was identified using (3.7), which yielded

B(z−1)

= 0.3383 · 10−5z−1 + 0.3383 · 10−5z−2+

+ 0.3383 · 10−5z−3 + 0.3383 · 10−5z−4(3.70)

The predictive control strategy was designed using the following parameters: hc = na +

nd = 3, hi = d+ 1, hp = hc+ d, with d from (3.6). The control action was then applied to the

initial model of the vehicle drivetrain.

Also, a PI controller was designed using [O’Dwyer, 2006] and it was tuned to have a fast

response, which yielded the proportional and integral terms: KR = 0.01 and Ki = 0.001.

A sequence of stairs was applied as the reference for the vehicle velocity and it was desired

that the system tracks the reference signal as fast as possible, the following figures showing the

results obtained in the simulations.

Fig. 3.31 illustrates the reference vehicle velocity value and the response of the system

with communication delay when both control strategies are applied. It can be seen that the

closed-loop control system tracks the reference signal, having no steady state error.

The working modes of the clutch are presented in Fig. 3.32, where it can be seen that the

clutch starts from open mode and after it reaches the engine closing speed ωclosinge , the clutch

enters in closing mode. In order to put the vehicle in motion, the load torque has to be defeated,

so the engine speed varies around the threshold value, which results in the switching between

the open and closing mode. Note that when the clutch springs are being compressed, the clutch

enters the closed mode.

As it can be seen in Fig. 3.33 the engine torque (control signal) for both control method

have smooth behaviors so the stiffer springs do not fully compress. As a result, the first state x1

does not pass the θ2 threshold so the clutch can not reach the looked mode.

83


Fig. 3.34 illustrates the driveshaft torque in which it can be seen that the switching between

the clutch modes results in very high increases/decreases in the driveshaft torque. Note that

when the vehicle velocity reference is switched from 30 km/h to 10 km/h, the driveshaft torque

reaches negative values corresponding to the engine breaking.

The axle wrap angular speed (ωd = ωe/itot−ωw) is represented in Fig. 3.35 (and a detail in

Fig. 3.36) as a measure of driveline oscillations that appears when the clutch switches through

the operating modes.

In the figures it can be seen that, due to the undershoot of the PI response (represented in

Fig. 3.31), the clutch opens (see Fig. 3.32) , which results in an increased driveshaft torque (see

Fig. 3.34) and driveline oscillations (see Fig. 3.35 and 3.36).

This section considered the design of a real-time simulation test-bench in order to validate

and to test the performances of the control algorithm developed with the aim of damping driv-

eline oscillations, while decreasing the influence of the communication time-varying delays on

the closed-loop control performances over the CAN network. The designed predictive strategy,

which uses an identification method to model the physical plant (vehicle drivetrain) including

the delays, were tested on the real-time simulation test-bench and the experiments designed

based on realistic scenarios verify the better performances of the proposed method when com-

paring the results with the ones obtained with classical controllers (PI).

3.6 Conclusions

In order to overcome the influences of communication delays on the NCS performance, a new

linear networked predictive control strategy, which makes use of three time-varying delays mod-

eling methods (average, identification and adaptation), is proposed in this chapter. The average

delay modeling method considers the mean value of the delays that can appear in the commu-

0 50 100 150 2000

10

20

30

40

Time [s]

v v[k

m/h]

referencePIDLNPC

Figure 3.31: Vehicle velocity.

84

Conclusions

0 50 100 150 2001

2

3

4

Time [s]

Clu

tch

mod

e

PIDLNPC

Figure 3.32: Clutch mode of operation.

0 50 100 150 2000

50

100

Time [s]

Te

[Nm

]

PIDLNPC

Figure 3.33: Engine torque (control signal).

0 50 100 150 200−1000

0

1000

2000

3000

Time [s]

Tf

[Nm

]

PIDLNPC

Figure 3.34: Driveshaft torque.

0 50 100 150 200−100

−50

0

50

100

Time [s]

ωd

[rpm

]

PIDLNPC

Figure 3.35: Speed difference.

85


0 50 100 150 200−2

−1

0

1

2

Time [s]

ωd

[rpm

]

PIDLNPC

Figure 3.36: Speed difference - detail.

nication network. The identification modeling method considers that the delay is equal to the

minimum delay that can appear in the communication network and uses a different model of

the physical plant that is identified in order to capture the dynamics of the system including

the delays between the minimum and the maximum delays that can appear in the communica-

tion network. The adaptation method is designed in order to adapt the control algorithm to the

network-induced time-varying delays. The adaptation algorithm is derived with assumption that

the average communication delays from sensor-to-controller and from controller-to-actuator are

equal, both being variables. Unlike for the first two modeling methods, in the case of the adap-

tation method there is no need to know the upper bound of the delays. Moreover, the proposed

λ-scheduling algorithm is a new method by which the LNPC is adapted to the difference be-

tween the desired reference and the plant output.

One of the advantages of proposed algorithm is that the LNPC can be analytically deter-

mined off-line using (3.29), which yields control laws of the form (3.37), (3.40) or (3.42). This

means that the actual control command can be computed very fast, so this algorithm is capable

of satisfying the timing constraints associated to the fast processes that require short sampling

periods, which are used as case studies to prove the performances of the proposed delay mod-

eling methods and predictive control strategy.

The last contribution of this chapter is the development of a PWA model for an automotive

driveline, which brings several improvements with respect to the models available in the litera-

ture, by considering the driving load given by the airdrag torque, gravity and rolling resistance,

and four modes to describe the clutch dynamics. Taking into account all these factors yields a

more accurate model of the driveline dynamics.

Several simulation examples are presented to illustrate the performances obtained and the

effectiveness of the new predictive algorithm. The strategy was applied to control physical

plants (fast processes) from automotive applications (DC motor, ICE, valve-clutch system, driv-

86

Conclusions

etrain) with the aim of decreasing the influence of the communication time-varying delays on

the closed-loop control performances over CAN. Comparisons were made with a PI controller

and a Smith-like predictive controller with adaptation to communication delay developed in [Ve-

lagic, 2008], in order to illustrate the performance of the proposed approaches. The simulations

designed to test the strategies developed in this chapter verify the better performances of the

proposed methods.

Furthermore, a case study on the real-time control of a vehicle drivetrain is included to

illustrate the potential of the proposed linear networked predictive control strategy for real-time

applications. The LNPC was tested on a real-time test-bench and the obtained results prove that

it can be implemented to control the actual physical plant (drivetrain).

The results presented in this chapter were published in:

• [Caruntu and Lazar, 2009a]: C. F. Caruntu and C. Lazar. Network-Induced Variable Time De-

lay Compensation Technique Based on Predictive Control. In 17th International Conference on

Control Systems and Computer Science, pages 65–71, 2009.

• [Patrascu et al., 2009]: D. I. Patrascu, A. E. Balau, C. F. Caruntu, C. Lazar, M. H. Matcovschi

and O. Pastravanu. Modelling of a Solenoid Valve Actuator for Automotive Control Systems. In

17th International Conference on Control Systems and Computer Science, pages 541–546, 2009.

• [Balau, Caruntu et al., 2009b]: A. E. Balau, C. F. Caruntu, D. I. Patrascu, C. Lazar, M. H.

Matcovschi and O. Pastravanu. Modeling of a Pressure Reducing Valve Actuator for Automotive

Applications. In 18th IEEE International Conference on Control Applications, Part of 2009 IEEE

Multi-conference on Systems and Control, Saint Petersburg, Russia, 2009.

• [Caruntu et al., 2009c]: C. F. Caruntu, M. H. Matcovschi, A. E. Balau, D. I. Patrascu, C. Lazar and

O. Pastravanu. Modeling of an Electromagnetic Valve Actuator. Buletinul Institutului Politehnic

Iasi, vol. LV (LIX), pages 9–28, 2009.

• [Caruntu and Lazar, 2009b]: C. F. Caruntu and C. Lazar. Predictive Control for Time-Varying

Delay in Networked Control Systems. In 8th IFAC Workshop on Time Delay Systems, Sinaia,

Romania, 2009.

• [Balau, Caruntu et al., 2009a]: A. E. Balau, C. F. Caruntu, C. Lazar and D. I. Patrascu. New Model

for Predictive Control of an Electro-Hydraulic Actuated Clutch. In 8th International Conference

Fuel Economy, Safety and Reliability of Motor Vehicles, volume 1, pages 463–472, Bucuresti,

Romania, 2009.

• [C.Lazar, Caruntu and Balau, 2010]: C. Lazar, C. F. Caruntu and A. E. Balau. Modelling and

Predictive Control of an Electro-Hydraulic Actuated Wet Clutch for Automatic Transmission. In

IEEE Symposium on Industrial Electronics, Bari, Italy, 2010.

87


• [Caruntu et al., 2010b]: C. F. Caruntu, A. E. Balau and C. Lazar. Networked Predictive Con-

trol Strategy for an Electro-Hydraulic Actuated Wet Clutch. In IFAC Symposium Advances in

Automotive Control, pages 419–424, Munchen, Germany, 2010.

• [Balau, Caruntu and C.Lazar, 2011b]: A. E. Balau, C. F. Caruntu and C. Lazar. Simulation and

Control of an Electro-Hydraulic Actuated Clutch. Mechanical Systems and Signal Processing,

vol. 25, pages 1911–1922, 2011.

• [Caruntu and C.Lazar, 2011b]: C. F. Caruntu and C. Lazar. Networked Predictive Control for

Time-varying Delay Compensation with an Application to Automotive Mechatronic Systems. Jour-

nal of Control Engineering and Applied Informatics, 2011.

• [Caruntu et al., 2011h]: C. F. Caruntu, D. Onu, F. C. Braescu and C. Lazar. Model predictive con-

trol for real-time simulation of a network-controlled vehicle drivetrain. In 2nd Eastern European

Regional Conference on the Engineering of Computer Based Systems, Bratislava, Slovakia, 2011.

88

Chapter 4

Nonlinear networked predictive control

In this chapter, a systematic approach to modeling NCSs, which accounts for time-varying

delays, and controller design with robustness to delays are proposed. A polytopic approximation

technique is applied to obtain a discrete-time model of the closed-loop CAN system and a one

step ahead predictive controller based on flexible control Lyapunov functions is designed for

the resulting model with polytopic uncertainty and hard constraints.

4.1 Introduction

The design of a controller for systems that are characterized by fast dynamics and subjected to

physical and control constraints can be a challenging problem. Furthermore, in current imple-

mentations, the connections between the controller and the physical plant are realized using a

communication network, which introduces time-varying delays.

MPC is increasingly seen as an attractive methodology for fast-dynamics applications due to

its capability to directly handle various specifications including the optimization of a cost func-

tion while enforcing constrains on state and control variables. However, due to the complexity

of the models and the stringent time limitations for finding a solution on-line, standard MPC

strategies [Maciejowski, 2002], which typically require a sufficiently long prediction horizon to

assure stability and performance, would lead to solutions that are too complex and, at the same

time, too conservative to be implemented in real-time control applications, keeping in mind that

the physical plants have fast dynamics that require short sampling periods.

The goal of this chapter is to provide a control design methodology that can cope with

all these challenges and limitations and still yield an effective solution. To this end, firstly, a

polytopic approximation technique is applied to obtain a discrete-time model of the closed-loop

89


CAN system [Gielen et al., 2010]. Secondly, a one step ahead predictive controller based on

flexible control Lyapunov functions [M.Lazar 2009] is designed for the resulting model with

polytopic uncertainty and hard constraints. The algorithm therein has the potential to satisfy the

timing requirements, due to the short horizon, while it can still offer a non-conservative solution

to stabilization due to the flexibility of the Lyapunov function. An optimization problem (to be

solved on-line) is constructed, that enforces stability by an explicit CLRF-type constraint and

includes state/input constraints as well. To handle the latter constraints, a solution for relaxing

the CLRF-type condition is provided, by introducing an auxiliary optimization variable that

adds flexibility to the CLRF. Convergence to the equilibrium is still guaranteed via a particular

new limiting condition on the auxiliary optimization variable.

To illustrate the potential for applications to fast processes, with short sampling periods, the

proposed controller is applied to control a vehicle drivetrain. As such, the case study consid-

ered in this chapter is to minimize the oscillations of a vehicle drivetrain while compensating

the time-varying delays introduced by CAN. TrueTime toolbox (for Matlab/Simulink), which

allows the simulation of distributed real-time control systems, taking into account the effects

of the execution of the control tasks and the data transmission on the controlled system dy-

namics, was used as the simulation environment. Several TrueTime simulation experiments

defined in collaboration with Ford Research and Advanced Engineering, US validate the pro-

posed approach, indicating that the developed scheme has the potential to meet the required

real-time control specifications. The simulation results also indicate that the proposed scheme

can outperform other types of controllers, such as PID or explicit MPC.

The proposed approach was also experimentally verified using a HIL test-bench consisting

of a Freescale based electronic control unit (which implements the controller) linked via a CAN

bus with a dSPACE MicroAutoBox plant simulator.

4.2 Delays modeled as polytopic inclusions

Given the explicit upper bound derived previously for the CAN-induced time-varying delays

(Subsection 2.5.3.3), the task is now to obtain a discrete-time model of the network-controlled

architecture that accounts for time-varying delays within the provided bound [Caruntu et al., 2011f].

To this end, starting from the modeling method described in Subsection 2.4.1.1, consider the

continuous-time system with input delay induced by CAN, i.e.,

x(t) = Acx(t) +Bcu(t− τ),

u(t) = uk,∀t ∈ [tk + τk, tk+1 + τk+1),(4.1)

90

Delays modeled as polytopic inclusions

where x(t) is the system state, u(t) is the system input, Ac and Bc are the continuous-time

system matrices, τ = τ sc + τ ca is the time delay introduced by the communication network,

tk = kTs, k ∈ Z+, Ts ∈ R+ denotes the sampling period and assume that u(t) = u0 for all

t ∈ [0, τ0] with u0 ∈ Rm some predetermined constant vector. uk ∈ R is the control action

generated at time t = tk. τk ∈ R[0,τmax] denotes the delay induced by the network at time

k ∈ Z+ and τmax ∈ R+ is the maximal possible delay.

Consider the maximum delay determined using (2.60) as τmax = (Υ + υ)Ts, where Υ ∈Z≥1 and υ ∈ R[0,1). Assuming that uk = ψk for all k ∈ Z[−Υ−1,−1] with ψ[−Υ−1,−1] some

predetermined vector, the discretized model is

xk+1 = Adxk +Bduk + ∆0(τk)(uk−1 − uk)++ ∆1(τk)(uk−2 − uk−1) + . . .

. . .+ ∆Υ(τk)(uk−Υ−1 − uk−Υ),

(4.2)

where Ad = eAcTs , Bd =∫ Ts

0eAc(Ts−θ)dθBc,

∆i(τk) :=

0, τk−i − iTs ≤ 0∫ τk−i−iTs

0eAc(Ts−θ)dθBc, 0 < τk−i − iTs < Ts

∫ Ts0eAc(Ts−θ)dθBc, Ts ≤ τk−i − iTs

(4.3)

for all k ∈ Z+ and i ∈ Z[0,Υ] with

τk ≥ τk−1 − Ts. (4.4)

The inequality (4.4) is due to the fact that within the same message ID group, the messages

sent on CAN are queued and they are resent until they are received, i.e., they can not be lost in

the network. Thus, a message transmitted at a certain time can never arrive before a message

that was transmitted at a previous time. Notice that this feature of CAN is different from NCS

set-ups considered in the control literature, see, e.g, [Gielen and M.Lazar, 2009b], where it is

assumed that a newer control update can arrive before an older one and thus the latter can be

ignored.

A polytopic over-approximation of the nonlinear functions ∆i(τk) can be found using the

Cayley-Hamilton theorem as presented in [Gielen et al., 2010]. Therefore, the following set is

defined

∆τ := Co(∆ll∈Z[1,L]), (4.5)

with τ ∈ R[0,Ts], ∆l ∈ Rn×m such that ∆i(τk) ∈ ∆τ for all τk ∈ R[0,τ ] and L ∈ Z≥1 is

91


finite. In [Gielen et al., 2010] several methods to create the polytope (4.5) were assessed. This

reference is referred for further details and assume for the remainder of this chapter that the

polytopic set (4.5) is known.

As ∆i(τk) ∈ ∆Ts for all i ∈ Z[0,Υ−1] and ∆Υ(τk) ∈ ∆υTs , it is obtained that (4.2) is

contained in

xk+1 ∈ φ(xk,u[k−Υ−1,k]), k ∈ Z+, (4.6)

where the inputs u[−Υ−1,0] are equal to some fixed, predetermined values and

φ(xk,u[k−Υ−1,k]) := Adxk +Bduk

+Υ∑

i=0

∆i(uk−i−1 − uk−i) |

∆i ∈ ∆Ts , i ∈ Z[0,Υ−1], ∆Υ ∈ ∆υTs.

(4.7)

Observe that the input vectors u[k−Υ−1,k] are known at time k ∈ Z+. The above model will be

used in the next chapter for controller design purposes.

4.3 Lyapunov-based Model Predictive Control

In this section, a one step ahead MPC scheme that employs a flexible Lyapunov function

[M.Lazar 2009] to attain stability and performance, while considering the network-induced

time-varying delays, via the usage of the model described in Subsection 4.2, is proposed. The

scheme is an adaption of the one presented in Subsection 2.4.2.3, to compensate the time-

varying delays induced by CAN in automotive applications modeled as polytopic inclusions.

4.3.1 Lyapunov-based predictive controller

Consider the non-autonomous system

xk+1 ∈ φ(xk,u[k−Υ−1,k]), k ∈ Z+, (4.8)

where xk ∈ X ⊆ Rn is the state at the discrete time instant k and u[k−Υ−1,k] ∈ UΥ+2 :=

U × . . . × U ⊆ Rm × . . . × Rm, are the control inputs starting with discrete time instant

k −Υ− 1 up to and including k. The mapping φ : Rn × Rm × . . .× Rm ⇒ Rn is an arbitrary

set-valued function with φ(0,0[−Υ−1,0]) = 0. It is assumed that 0 ∈ int(X) and 0 ∈ int(U).

Next, let α1, α2 ∈ K∞ and let ρ ∈ R[0,1).

92

Lyapunov-based Model Predictive Control


α1(‖x‖) ≤ V (x) ≤ α2(‖x‖), ∀x ∈ Rn, (4.9)

and for which there exists a control law, possibly set-valued, π : Rn ⇒ U such that x+ ∈ X and

that

V (x+) ≤ ρ maxθ∈Z[−Υ−1,0]

V (xk+θ), (4.10)

for all xθ ∈ X, θ ∈ Z[−Υ−1,0], uθ ∈ π(xθ) and all x+ ∈ φ(x,u[−Υ−1,0]) is called a CLRF for the

difference inclusion corresponding to system (4.8). 2

Consider the following inequality corresponding to (4.10)

V (xk+1) ≤ ρ maxθ∈Z[−Υ−1,0]

V (xk+θ) + λk, (4.11)

for all k ∈ Z+ and all xk+1 ∈ φ(xk,u[k−Υ−1,k]). Here λk is a variable which allows for addi-

tional freedom in the evolution of the CLRF, i.e., it can increase if (4.10) is too conservative

at time instant k ∈ Z+, possibly due to active state/input constraints. Based on (4.11) the

following optimization problem is formulated.

Let α3, α4 ∈ K∞ and J : R → R+ be an arbitrary function such that α3(|λ|) ≤ J(λ) ≤α4(|λ|) for all λ ∈ R. Let V (·) be a candidate CLRF for system (4.6).

Problem 4.3.2 Assume that at time k ∈ Z+, xk and u[k−Υ−1,k−1] are known and minimize the

cost J(λk) over uk and λk subject to

uk ∈ U, φ(xk,u[k−Υ−1,k]) ⊆ X, λk ≥ 0, (4.12a)

V (xk+1) ≤ ρ maxθ∈Z[−Υ−1,0]

V (xk+θ) + λk, (4.12b)

for all xk+1 ∈ φ(xk,u[k−Υ−1,k]), with ρ ∈ R[0,1). 2

Let π(xk) := uk ∈ Rm | ∃λk ∈ R s.t. (4.12) holds and let φcl(xk, π(xk),u[k−Υ−1,k−1]) :=

φ(xk, u,u[k−Υ−1,k−1]) | u ∈ π(xk). Furthermore, let VΓ := x ∈ Rn | V (x) ≤ Γ for any

Γ ∈ R+. Let λ∗k denote the optimum in Problem 4.3.2 for all k ∈ Z+.

Next, the main stability result is stated.

Theorem 4.3.3 Suppose that V (·) is a function that satisfies (4.9) and that X and U are bounded.

Furthermore, suppose that Problem 4.3.2 is feasible for all (x0,u[−Υ−1,0]) ∈ X×UΥ+2 and that

93


limk→∞ λ∗k = 0. Then, the origin of the difference inclusion

xk+1 ∈ φcl(xk, π(xk),u[k−Υ−1,k−1]), k ∈ Z+ (4.13)

is attractive. Moreover, if ∃Γ ∈ R>0 such that V (·) is a CLRF for initial conditions in VΓ for

system (4.6), then system (4.13) is asymptotically stable in X. 2


to-state stability and Lyapunov stability and consideration of the time-delay setting, and is there-

fore omitted here. The interested reader is referred to [M.Lazar 2009] and [Gielen and M.Lazar, 2009b]

for more details. The key of the stability proof is the limiting condition limk→∞ λ∗k = 0. In what

follows a new non-conservative solution for guaranteeing this condition is provided, such that

λk can be small or even 0 for some time after which it is allowed to increase again, as long as

this does not violate the upper bound.

Lemma 4.3.4 Let Ω ∈ R+ be a fixed constant to be chosen a priori and let ρ ∈ R[0,1). If

0 ≤ λk ≤ ρ(λ∗k−1 + ρk−1Ω), ∀k ∈ Z+, (4.14)

then limk→∞ λk = 0.

Proof: Since λ∗k is bounded for all k ∈ Z+, there exists an Nλ ∈ R+ such that λ∗0 ≤ Nλ.

Applying (4.14) recursively, the following inequality is obtained:

λk ≤ ρk(Nλ + kΩ), ∀k ∈ Z+. (4.15)

Since ρ ∈ R[0,1) it holds that limk→∞ ρk = 0. Moreover,

limk→∞

kρk = limk→∞

k(ρ−1)−k = limk→∞

k

(ρ−1)k(4.16)

and applying l’Hospital’s rule [Stewart, 2003] it follows that

limk→∞

k

(ρ−1)k= lim

k→∞

ddkk

ddk

(ρ−1)k= lim

k→∞

1

(ρ−1)k ln(ρ−1)= 0. (4.17)

The fact that limk→∞ kρk = 0 completes the proof.

By augmenting Problem 4.3.2 with constraint (4.14) the property limk→∞ λ∗k = 0 is thus

guaranteed, which is sufficient for asymptotic stability. Note that, in constraint (4.14), ρk−1Ω

94

Lyapunov-based Model Predictive Control

gives a decreasing ramp and λ∗k−1 dictates with what value the ramp is allowed to be violated in

order to maintain the feasibility of the closed-loop control system. It is worth to mention that the

proposed limiting condition is less conservative than the solution presented in [M.Lazar 2009]

and [Gielen and M.Lazar, 2009b], which corresponds to setting Ω = 0.

4.3.2 Implementing Lyapunov-based predictive controller as a single lin-ear program

In this subsection it is shown how a CLRF of the form (4.10) defined using infinity norms

and corresponding control law can be obtained. Moreover, it is indicated how the optimization

problem derived previously can be formulated as a single low-complexity linear program (LP),

while strictly enforcing the constraints and the time-varying delays.

Consider the following cost function to be minimized

J1(xk,u[k−Υ−1,k], λk) :=JMPC(xk,u[k−Υ−1,k]) + J(λk)

:=‖Qxk+1‖∞ + ‖Ruk‖∞ + ‖Gλk‖∞,(4.18)

where xk+1 ∈ φ(xk,u[k−Υ−1,k]) and the matrices Q and R are known full-column rank matrices

of appropriate dimensions.

The constraints (4.40) can be specified as

umin ≤uk ≤ umax,

−u∆ ≤ ∆uk ≤ u∆,

xmin ≤ Adxk + bduk+Υ∑

i=0

∆i(τk)(uk−i−1 − uk−i) ≤ xmax

(4.19)

where xmin and xmax are appropriate vectors with values that bound the states of the system and

∆i(τk) ∈ ∆Ts for i ∈ Z[0,Υ−1] and ∆Υ(τk) ∈ ∆υTs .

Note that constraint satisfaction of the difference inclusion (4.6) can be guaranteed by guar-

anteeing constraint satisfaction for all combinations of the vertices of the sets ∆Ts and ∆υTs .

Now, consider the following infinity-norm based candidate CLRF, i.e.,

V (x) = ‖Px‖∞, (4.20)

where P ∈ Rp×n, p ≥ n, is a full-column rank matrix to be determined. This function satisfies

(4.9) with α1(s) = σ√ps, where σ is the smallest singular value of P , and with α2(s) = ‖P‖∞s.

95


Substituting (4.6) and (4.20) in (4.12) yields

‖P (Adxk + bduk +Υ∑

i=0

∆i(τk)(uk−i−1 − uk−i))‖∞ ≤ ρ maxθ∈Z[−Υ−1,0]

‖Pxk+θ‖∞ + λk, (4.21)

where x[k−Υ−1,k], P , u[k−Υ−1,k−1] and ρ ∈ R[0,1) are known. Furthermore, ∆i(τk) ∈ ∆Ts for

i ∈ Z[0,Υ−1] and ∆Υ(τk) ∈ ∆υTs .

Although (4.18) and (4.21) appear to be nonlinear in the optimization variables, due to

using only horizon 1, the optimization problem can be recast as a LP via an appropriate set of

equivalent linear inequalities. By definition of the infinity norm, for ‖x‖∞ ≤ c, x ∈ Rn, to be

satisfied, it is necessary and sufficient to require that ±[x]j ≤ c for all j ∈ 1, 2, ..., n. So, for

(4.21) to be satisfied, it is necessary and sufficient to require that

±[P (Adxk+bduk+Υ∑

i=0

∆i(τk)(uk−i−1−uk−i))]j ≤ ρ maxθ∈Z[−Υ−1,0]

‖Pxk+θ‖∞+λk, j ∈ 1, 2, 3,

(4.22)

where ∆i(τk) ∈ ∆Ts for i ∈ Z[0,Υ−1] and ∆Υ(τk) ∈ ∆υTs .

Solving Problem 4.3.2, which includes minimizing the cost function (4.18), can be refor-

mulated as the following problem by introducing three auxiliary optimization variables ε1k, ε2kand ε3k.

Problem 4.3.5minuk,λk

(ε1k + ε2k + ε3k) (4.23)

subject to (4.14), (4.19), (4.22), and

±[Q(Adxk + bduk +Υ∑

i=0

∆i(τk)× (uk−i−1 − uk−i))]j ≤ ε1k, j ∈ 1, 2, 3, (4.24a)

±Ruk ≤ ε2k, (4.24b)

Gλk ≤ ε3k, (4.24c)

where ∆i(τk) ∈ ∆Ts for i ∈ Z[0,Υ−1] and ∆Υ(τk) ∈ ∆υTs .

When (4.24) is imposed for all combinations of the vertices of the sets ∆Ts and ∆υTs Problem

4.3.5 is a LP. Indeed, xk, λ∗k−1 and all u[k−Υ−1,k−1] are known at time k ∈ Z+ and thus, all

constraints are linear in the unknowns uk, λk and ε1,2,3k .

Next, the one step ahead MPC algorithm can be summarized as follows:

96

LMPC applied to a two masses drivetrain controlled through CAN

Algorithm 4.3.6 At each instant k ∈ Z+:

Step 1: Measure or estimate the current local state xk;

Step 2: Solve LP (4.23) and pick any feasible solution u∗(xk);

Step 3: Use uk = u∗(xk) as control action.

Using the technique of rewriting inequalities that include∞-norms as linear inequalities in

combination with the above optimization problem, one can formulate a linear program whose

solution minimizes the cost J(λ) for a horizon equal to one and it satisfies all constraints in

(4.12).

Notice that in Step 3 of Algorithm 4.3.6, only a feasible solution of (4.23) is required, instead

of the optimal one. This implies the control algorithm can be executed in a shorter time than

standard MPC schemes, which typically require an optimal solution to be found. This means

that the proposed controller can be applied for the class of fast processes, with short sampling

periods, implying that the controller can be used in real-time applications.

4.4 LMPC applied to a two masses drivetrain controlled through

CAN

The two mass model illustrated in Fig. 4.1, which is a schematic representation of a simplified

drivetrain, has two inertias, one for the engine, gearbox and final reduction gear, i.e., Jegf ,

and the other one representing the contributions from the vehicle and from the driving wheels,

i.e., Jv, connected through a flexible driveshaft. The engine subsystem is also characterized

by a damping coefficient, i.e., de. The engine generates a torque, i.e., Te, which is transmitted

towards the wheels through the driveline. The engine torque is used as the control signal and

assumed to be available on demand, e.g., engine torque is requested from the engine controller

unit and obtained by modulating spark timing and airflow. A fundamental requirement for

model-based control is a model that captures the main characteristics of the plant. In this case

the oscillations originating from driveline flexibilities are of interest. The drivetrain flexibility is

given by the driveshaft which transmits the received torque from the gearbox to the wheels and

it is characterized by an elasticity factor kd and a damping coefficient dd. The driving wheels

are the final components of the drivetrain, which have the purpose of moving the vehicle by

defeating the friction forces with the road surface and the aerodynamic drag, which are given

by the load torque Tload.

97


Engine

totieT

Flexible

driveshaft

Wheels,

Vehicle

loadTdT

dk

dd

Gearbox,

FRG

dT tot

dT

iFh

egfJ

Fh

vJ

Engine

ECU

r

eT

Sensor 1

eω

Sensor 2

wω

r

vvTransmission

ECU

tot

e

w

e wi

ω

ω

θ θ−

r

eT

, ,e e edω θ ,w wω θ

Sensor 3

tote wiθ θ−

CAN 011100001011110001010101 CAN

Figure 4.1: Simplified drivetrain schematic representation and control architecture.

4.4.1 Control Architecture

The different subsystems of modern vehicles, e.g., engine, transmission, brakes, suspensions,

steering, are controlled by specific ECUs, e.g., engine control unit, transmission control unit,

located at different spatial locations than the physical plants, through communication networks.

The resulting closed-loop control systems is called a NCS, meaning that the sensors measure-

ments are sent to the controllers, implemented in ECUs, through the same communication net-

work as the control signals from the controllers to the actuators. These so called NCSs have

many attractive advantages which include low cost, simple installation and maintenance, higher

reliability and greater flexibility.

As it can be seen in Fig. 4.1 the components of the closed-loop vehicle drivetrain control

system, e.g., sensors, controller, actuator, communicate through a communication network, i.e.,

the engine torque (control signal), and also the engine speed and the wheel speed are sent

through CAN. Many of the messages transmitted on CAN, have real-time constraints associated

with them.

The complete networked controlled architecture considered in this section, which is graph-

ically depicted in Fig. 4.1, consists of the following operations:

• The time-driven sensors, which contain periodic tasks, sample the outputs of the system

98


and send the measurements to the controller via CAN;

• The transmission controller (Transmission ECU), which contains an event-driven task that

is triggered each time a measurement data arrives on CAN, receives the measurements

from the sensors and the desired velocity reference vrv = rwωrw and computes the required

torque, while handling the physical/control constraints and the delays;

• The control signal, i.e., the torque computed by the transmission controller, is sent to the

engine controller (Engine ECU) via CAN;

• The engine controller, which contains an event-driven task that is triggered each time

a measurement data arrives on CAN, receives the control signal and actuates the spark

timing and airflow as requested for driveline control.

In Fig. 4.1 the dashed arrows represent the direction of the messages sent to and from the

controllers.

In this case study, the actuator is the internal combustion engine and it is assumed that

the request is accomplished instantaneously. For the engine torque this can be achieved by

creating a torque reserve by changing the spark timing in order to have quickly available torque

modifications [Di Cairano et al., 2010]. The torque reserve can be maintained by appropriately

actuating the engine airflow.

4.4.2 Plant model

The whole dynamical model can be obtained by applying the equilibrium torque condition at

the different nodes of the structure presented in Fig. 4.1. The engine speed dynamics, modeled

as a single inertia system, are given as:

Jegf ωe = Te − deωe − Td/itot, (4.25)

where ωe is the engine speed and Jegf := Je +Jgfi2tot

is the equivalent inertia of the engine-

gearbox-final reduction gear subsystem, with Jgf the gearbox and final drive inertia and itot is

the overall transmission ratio from the gearbox and the final reduction gear. From the gearbox

the driveline consists of a propeller shaft, final reduction gear and driveshafts. The propeller

shaft is assumed to be stiff, which means that the final reduction gear can be included in the

transmission, i.e., itot := itif .

99


The driveshafts can not be considered as stiff as the propeller shaft [Kiencke and Nielsen, 2005]

and they are modeled as a spring and damper system. The torque in the driveshaft can be ex-

pressed as

Td = dd (ωe/itot − ωw) + kd (θe/itot − θw) , (4.26)

where θw and ωw are the wheel angle and speed, respectively. Since the vehicle jerk performance

depends primarily on the driveshaft torque oscillations, this part will be used as a criterion to

determine the driver’s and passenger’s comfort.

Analogously, the wheel and vehicle body dynamics are modeled as a single inertia with

Jvωw = Td − Tload, (4.27)

and the tire was accounted as a rolling element without slip, i.e., vv = rwωw, where vv is the

vehicle speed and rw is the effective wheel radius. The vehicle inertia can be obtained by adding

the wheel inertia to the equivalent inertia of the vehicle mass, i.e.,

Jv = Jw +mvr2w, (4.28)

where mv is the mass of the vehicle and Jw is the wheel inertia.

The load torque is modeled as:

Tload = Tairdrag + Troll + Tgrade, (4.29)

where Tairdrag is the aerodynamic drag torque of the vehicle body, Troll is the rolling torque of

the tires and Tgrade is the torque due to road grade, which are defined as

Tairdrag = 0.5ρairAfcdv2vrw,

Troll = crmvg cos(χroad)rw,

Tgrade = mvg sin(χroad)rw,

(4.30)

where ρair is the air density, Af is the frontal area of the vehicle, cd is the airdrag coefficient,

cr is the coefficient of rolling resistance, g is the gravitational acceleration and χroad is the road

grade. Note that vv = rwωw, which means that Tairdrag is a nonlinear function of ωw. Instead of

this nonlinear function, a linear approximation will be used, i.e.,

Tairdrag = daωw, (4.31)

100


where da is an approximation parameter.

By introducing the engine speed, the wheel speed and the torsion in the driveshaft, also

called axle wrap, as state variables, i.e.,

xm1 = ωe, xm2 = ωw, xm3 =θeitot− θw, (4.32)

the drivetrain model can be written in a state space form as

xm(t) = Amc xm(t) + bmc u

m(t) + fmc , (4.33)

where xm =(xm1 xm2 xm3

)>is the system state, um ∈ R is the system input, fmc =

(0

−Troll+Tgrade

Jv0

), bmc =

(1Jeg

00

)and Amc =

− deJegf

− ddi2totJegf

dditotJegf

− kditotJegf

dditotJv

− da+ddJv

kdJv

1itot

−1 0

.

The engine torque (control input) is restricted by lower and upper bounds and by a torque

rate constraint, i.e.,

0 ≤um(t) ≤ Tmaxe , (4.34a)

Tme ≤um(t) ≤ TMe , (4.34b)

where Tmaxe is the maximum torque that can be generated by the internal combustion engine and

Tme , TMe are torque rate bounds.

The engine and wheel speeds are bounded, i.e.,

ωmine ≤ xm1 (t) ≤ ωmax

e , (4.35a)

ωminw ≤ xm2 (t) ≤ ωmax

w , (4.35b)

where ωmine , ωmax

e are the idle and the maximum speed of the engine, respectively, and ωminw , ωmax

w

are the minimum and the maximum speed of the wheel, respectively.

The control objective is to reach a desired value of the wheel speed, i.e., xss2 , as fast as

possible and with minimum overshoot. Note that for a desired wheel speed value xss2 , one can

obtain the steady-state engine speed xss1 , axle wrap xss3 and engine torque uss as

xss1 = itotxss2 , x

ss3 =

bakfxss2 +

Troll + Tgrade

kf,

uss = beitotxss2 +

baitotxss2 +

Troll + Tgrade

itot.

(4.36)

101


To facilitate the implementation of the one step ahead MPC algorithm of [M.Lazar 2009], a

coordinate transformation is performed on (4.33), i.e.,

x(t) = xm(t)− xss(t), u(t) = um(t)− uss(t), (4.37)

where xss =(xss1 xss2 xss3

)>, which yields the following continuous-time system descrip-

tion

x(t) = Acx(t) + bcu(t), (4.38)

where Ac and bc are the transformed system matrices.

Discretizing model (4.38) yields

xk+1 = Adxk + bduk, (4.39)

for all k ∈ Z+, where Ad and bd are the transformed and discretized system matrices and xkand uk are the state and the input of the system at time instant k ∈ Z+.

Using (4.37) and (4.39), the constraints given in (4.34) and (4.35) can be converted to:

x1,k ∈ [bx1 , bx1 ], x2,k ∈ [bx2 , bx2 ],

uk ∈ [bu, bu],∆uk ∈ [b∆u, b∆u],(4.40)

for suitably defined constants.

The control objective can now be formulated as stabilization of the zero equilibrium of

system (4.39), while fulfilling the constraints (4.40).

4.4.3 Matlab/Simulink results

In this subsection, the results obtained when applying the Lyapunov-based predictive controller

proposed in Section 4.3 on the drivetrain model described in Subsection 4.4.2 are presented.

The proposed control structure was implemented in Matlab/Simulink and the delays introduced

by the communication network were simulated using Variable Time Delay blocks.

102


0 2 4 6 8 100

0.005

0.01

0.015

0.02

Time [s]

Com

mun

icat

ion

dela

ys [s

]

Figure 4.2: Delays induced by CAN.

The discrete-time model (4.7), with

Ad =

0.9806 0.1594 −22.7747

0.0003 0.9954 0.6539

0.0007 −0.0100 1.0000

and bd =

0.0526

0

0

(4.41)

from (4.39), was implemented in Matlab and three different control strategies have been applied

in order to damp out driveline oscillations: a PID controller, an explicit MPC controller and

the one step ahead networked predictive controller (Section 4.3) proposed in this thesis. The

sampling period of the system was chosen as Ts = 0.01s and the values of the parameters used

in simulations were obtained with the help of Ford Research and Advanced Engineering, US,

and are given in Table 4 in Appendix A.

The upper bound of the delays that are induced by CAN was calculated using the method-

ology described in Subsection 2.5.3.3, resulting that τmax = 1.7Ts = 0.017s. The delays

are time-varying and uniformly distributed in the interval [0, τmax], as shown in Fig. 4.2. The

delays were considered in the design phase only for the one step ahead predictive controller

proposed in this thesis, but they were introduced in simulations for all the strategies.

The control objective is to reach a desired speed reference in a short time, but, at the same

time, to increase the passenger comfort by reducing the oscillations that occur when tip-in and

tip-out maneuvers are performed. All control strategies were tuned to have the same response

regarding the wheel speed, so the results could be discussed w.r.t. the drivetrain oscillations.

A PID controller was designed based on [O’Dwyer, 2006] and it was tuned to have a fast

response, which yielded the proportional, integral and derivative terms KR = 13, Ti = 0.45

and Td = 0.1, respectively. Also, an explicit MPC controller was designed for the delay-

free affine model and the same operating constraints using the MultiParametric Toolbox for

Matlab. The cost function (2.44) with PN = 11I3, Qx = 0.1I3 and Ru = 0.5 was employed

103


in the corresponding finite horizon optimization problem. It should be noted that many tests

were performed to obtain the explicit MPC controller, including using the CLRF weight matrix

(see below) as PN . The above cost led to the best performance for the considered application

and several simulation scenarios. A feasible solution to the corresponding multiparametric

LP problem could be obtained for a maximum prediction horizon of two sampling periods

(N = 2), which resulted in a controller defined by 614 regions. For a prediction horizon larger

than 2, despite using a powerful workstation and several robust solvers, a solution could not be

obtained, which reflects the non-trivial nature of the considered case study.

The one step ahead predictive controller was designed using the following weight matri-

ces of the cost (4.18): Q = 11I3, R = 0.5 and G = 1. The technique presented in [Gie-

len and M.Lazar, 2009b] was used for the off-line computation of the infinity norm based

CLRF V (x) = ‖Px‖∞ for ρ = 0.99 and the affine model of the drivetrain in closed-loop

with uk := Kxk. The following matrices were obtained:

P =

2.2669 32.1093 946.2197

2.5314 −71.6993 −363.0408

2.4062 81.7867 741.4772

,

K =(−6.3471 45.7387 −229.6221

).

(4.42)

The bounds for the constraints in (4.19) are defined as umin = 0− uss, umax = Tmaxe − uss,

u∆ = T∆e , xmin =

(ωmine −xss1ωminw −xss2−∞

)and xmax =

(ωmaxe −xss1ωmaxw −xss2∞

).

Note that the control law uk := Kxk was only employed off-line, to calculate the weight

matrix P of the local CLRF V (·), and it was never used for controlling the system. The evolution

of the CLRF relaxation variable λ∗k and the corresponding upper bound defined by (4.14) for

ρ = 0.99, λ∗0 ≈ 50 and Ω = 350 is shown in Fig. 4.3, for this particular simulation. Notice that

this leads to an initial upper bound on λk at time k = 1 approximately equal to 400, as it can

be seen in Fig. 4.3. Therein, it can also be observed that λ∗k may be small or even 0 for some

time after which it is allowed to increase again, as long as this does not violate the upper bound.

However as k →∞, λ∗k is forced to converge to 0, which in turn implies asymptotic stability as

guaranteed by Theorem 4.3.3.

The following paragraph is dedicated to the stability of the resulting closed-loop systems

for each technique. Clearly, no stability guarantee can be obtained for the affine system in

closed-loop with the PID controller. The stability of the closed-loop system that corresponds

to the explicit MPC scheme was analyzed a posteriori, but the explicit MPC controller failed

all the stability tests that can be performed with the Multi Parametric Toolbox (e.g., piecewise

104


0 2 4 6 8 100

100

200

300

400

Time [s]

λ∗ k

actual valueupper bound

Figure 4.3: History of λ∗k throughout the simulation.

0 2 4 6 8 105

10

15

20

25

30

35

Time [s]

v v[k

m/h

]

referencePIDExplicit MPCFLCF MPC

Figure 4.4: Vehicle velocity.

quadratic, linear and even polynomial Lyapunov functions were searched for). For the pro-

posed one step ahead MPC scheme, recursive feasibility implies asymptotic stability. However,

recursive feasibility is not a priori guaranteed and hinges mainly on the constraint (4.14) on the

future evolution of λ∗k. For the considered case study, through extensive simulations, the em-

ployed Ω = 350 proved to be large enough to guarantee recursive feasibility for a wide range

of operating scenarios.

In what follows the performance of the resulting closed-loop systems for each technique is

analyzed using the trajectories plotted in Fig. 4.4, Fig. 4.5, Fig. 4.6 and Fig. 4.7. Also, the

time needed for computation of the control input is given in Table 4.1. Note that, although the

proposed one step ahead controller takes into account in the designing phase the time-varying

delays that appear on CAN, the worst case time needed for computation of a control input is less

than a sampling period. Simulations were performed using the CDD Dual Simplex LP solver in

Matlab R2010a on Windows 7 running on a HP EliteBook 8740w 4x Intel Core i5 2.53 GHz,

4GB RAM mobile workstation.

In Fig. 4.4, the vehicle velocity is represented, as obtained by the three different controllers,

and it can be seen that in terms of rising time the controllers have almost the same response.

105


0 2 4 6 8 10−10

−5

0

5

10

Time [s]

ωe/i

tot−

ωw

[rpm

]

PIDExplicit MPCFLCF MPC

Figure 4.5: Speed difference.

0 2 4 6 8 100

50

100

150

Time [s]

Te

[Nm

]

Upper boundPIDExplicit MPCFLCF MPC

Figure 4.6: Engine torque (control signal).

In Fig. 4.5 one can see that for the first 2.5 seconds the difference between the engine speed

divided by the total gear ratio itot and the wheel speed is almost the same for the three controllers.

After that, the driveline oscillations go to 0 for the one step ahead predictive controller proposed

in this chapter, but the amplitude of the oscillations obtained with the explicit MPC controller

increases by approximately a factor 4 w.r.t. the ones obtained with the one step ahead predictive

controller and this difference in speeds keeps on oscillating further for both the the PID and the

explicit MPC controller. These increased oscillations amplitude and frequency are undesirable

because they can produce excessive wear to the driveline components and reduced comfort for

the driver.

The engine torque is represented in Fig. 4.6, where one can see that the constraint on the

upper bound of the control signal is satisfied by all tested control strategies. Note that, although

the PID controller does not enforce constraints on control command, its output was saturated in

order to enforce the engine limitations, i.e., the torque limit Tmaxe .

The oscillations of the drivetrain obtained using the PID and the explicit MPC controllers

can also be seen in Fig. 4.7, where the driveshaft torque is presented; the longitudinal oscilla-

tions produced by the PID controller and especially the ones produced with the explicit MPC

106


0 2 4 6 8 10−500

0

500

1000

1500

Time [s]

Te

[Nm

]


Figure 4.7: Driveshaft torque.

Table 4.1: Worst case computational times

Controller CPU time [s]PID 0.0001

Explicit MPC 0.009FCLF MPC 0.008

controller are quite severe and they can be easily felt by the passengers of the vehicle.

4.4.4 TrueTime results

In this subsection, the results obtained when applying the Lyapunov-based predictive controller

proposed in Section 4.3 on the drivetrain model described in Subsection 4.4.2 are presented. The

proposed control structure was implemented in Matlab/Simulink using the TrueTime Toolbox.

The continuous-time model (4.33), with

Amc = 103 ·

−0.0019 0.0159 −2.2775

0.0000 −0.0005 0.0654

0.0001 −0.0010 0

,

bmc =

5.2587

0

0

and fmc =

0

−0.3350

0

(4.43)

was implemented in Matlab/Simulink and the three different control strategies designed in the

previous Subsection have been applied n order to damp out driveline oscillations. The sampling

period of the system was chosen as Ts = 0.01s and the values of the parameters used in simu-

107


lations were obtained with the help of Ford Research and Advanced Engineering, US, and are

given in Table 4 in Appendix A.


ology described in Subsection 2.5.3.3, resulting that τmax = 1.7Ts = 0.017s. The delays are

time-varying and uniformly distributed in the interval [0, τmax]. The delays were considered in

the design phase only for the one step ahead predictive controller proposed in this chapter, but

they were introduced in simulations for all the strategies.


time, to increase the passenger comfort by reducing the oscillations that occur when tip-in

and tip-out maneuvers are performed. All control strategies were tuned to have almost the

same response regarding the wheel speed, so the results could be discussed w.r.t. the drivetrain

oscillations.

4.4.4.1 Simulation environment

The control strategies were simulated using TrueTime simulator [Henriksson et al., 2003],

which is a very powerful MATLAB-based network simulation toolbox that can effectively sim-

ulate real-time networked control systems. There are two primary Simulink blocks in the True-

Time package: the computer block and the network block, both being easy to customize in

order to obtain a practical NCS. The computer block simulates an ECU with a flexible real-

time kernel, A/D and D/A converters, a network interface, and external interrupt channels. The

user has to write code functions, which define the evolution of the control tasks and the in-

terrupt handlers. The network block, which allows nodes to communicate over the simulated

network, is event-driven, so it executes every time a message enters or leaves the network. To

simulate a required network condition, different parameters such as: medium access control

(MAC) protocol (Ethernet, CAN, etc.), transmission rate, minimum frame size, etc., can be

customized [Henriksson et al., 2003].

In the designed NCS simulation platform (see Fig. 4.8), the sensors (Node 3, 4, 5), con-

troller (Node 2) and actuator (Node 1) are implemented using computer blocks and the CAN

communication network is realized using a network block in which the MAC protocol is speci-

fied as CSMA/AMP. The sensors and the actuator are connected to the plant through their A/D

converters and D/A converter, respectively.

Since the propagation delays are not modeled by TrueTime toolbox, these delays were in-

troduced in the simulation platform by using uniform random number (URN) generators and

variable time-delay (VTD) blocks, which are also represented in Fig. 4.8.

108


Network

(CAN)

Vehicle velocity

reference [km/h]

VTD4 To

VTD3 To

VTD2 To

VTD1

To

URN4URN3URN2

URN1

Node 5

(Sensor3)

A/D

Snd

Node 4

(Sensor2)

A/D

Snd

Node 3

(Sensor1)

A/D

Snd

Node 2

(Controller - Transmission ECU )

Rcv

r

Snd

Node 1

(Actuator -

Engine ECU +

Engine )RcvD/A

snd2

snd3

snd4

snd5

rcv1

rcv2

Drivetrain model

u x

Figure 4.8: TrueTime Simulink diagram - two inertias drivetrain model.

4.4.4.2 Numerical results

The PID controller designed based on [O’Dwyer, 2006] was re-tuned to have a fast response,

which yielded the proportional, integral and derivative terms KR = 17, Ti = 0.45 and Td =

0.005, respectively. Also, the explicit MPC controller was redesigned for the delay-free affine

model and the same operating constraints using the MultiParametric Toolbox for Matlab. The

cost function (2.44) with PN = 15I3, Qx = 0.1I3 and Ru = 0.5 was employed in the corre-

sponding finite horizon optimization problem.

The one step ahead predictive controller was designed using the following weight matrices

of the cost (4.18): Q = 11I3, R = 0.05 and G = 1. The same P matrix as in (4.42) was used in

simulations.

The time needed for computation of the control input is given in Table 4.1. Note that,

although the proposed one step ahead controller takes into account in the designing phase the

time-varying delays that appear on CAN, the worst case time needed for computation of a

control input is less than a sampling period. Simulations were performed using the CDD Dual

Simplex LP solver in Matlab R2010a on Windows 7 running on a HP EliteBook 8740w 4x Intel

Core i5 2.53 GHz, 4GB RAM mobile workstation.

Different simulations were conducted, to evaluate the vehicle behavior in response to ac-

109


0 2 4 6 8 100

100

200

300

400

Time [s]

λ


Figure 4.9: Acceleration test: History of λ∗k throughout the simulation.

celeration and tip-in and tip-out maneuvers, which are presented in the following subsections.

Note that, although the PID controller does not enforce constraints on control command, its

output was saturated in order to enforce the engine limitations, i.e., the torque limit Tmaxe , and

the limitation on the control input increment.

In what follows the performance of the resulting closed-loop systems for each technique is

analyzed using the trajectories plotted in Fig. 4.9 - 4.14, but only the PID and the one step ahead

predictive controller proposed in this chapter were simulated using Matlab TrueTime toolbox.

4.4.4.3 Acceleration test

A first simulation test is performed on an acceleration scenario where the vehicle has to ac-

celerate from 7 km/h to 30 km/h. The evolution of the CLRF relaxation variable λ∗k and the

corresponding upper bound defined by (4.14) for ρ = 0.99, λ∗0 ≈ 100 and Ω = 550 is shown

in Fig. 4.9, for this particular simulation. Notice that this leads to an initial upper bound on λkat time k = 1 approximately equal to 650, as it can be seen in Fig. 4.9. Therein, it can also

be observed that λ∗k may be small or even 0 for some time after which it is allowed to increase

again, as long as this does not violate the upper bound. However as k → ∞, λ∗k is forced to

converge to 0, which in turn implies asymptotic stability as guaranteed by Theorem 4.3.3.

In Fig. 4.10 a), the vehicle velocity is represented, as obtained by the three different con-

trollers, and it can be seen that in terms of rising time the explicit MPC and the one step ahead

predictive controller proposed in this chapter have almost the same response, while the PID

controller has a slight overshoot.

In Fig. 4.10 b) one can see that for the first 2.5 seconds the difference between the engine

speed divided by the total gear ratio itot and the wheel speed is the same for the three controllers.

After that, the driveline oscillations go to 0 for the one step ahead predictive controller proposed

in this chapter and for the explicit MPC controller and this difference in speeds keeps on os-

110


0 2 4 6 8 100

10

20

30

40

Time [s]

v v[k

m/h]

a) Vehicle velocity.

referencePIDExplicit MPCFLCF MPC

0 2 4 6 8 10−10

−5

0

5

10

15

Time [s]

ωe/i t

ot−

ωw

[rpm

]

b) Speed difference.


0 2 4 6 8 100

50

100

150

Time [s]

Te

[Nm

]

c) Engine torque (control signal).


0 2 4 6 8 10−500

0

500

1000

1500

Time [s]

Tf

[Nm

]

d) Driveshaft torque.


Figure 4.10: Acceleration test: TrueTime simulation results.

cillating further for the PID controller. These increased oscillations frequency are undesirable

because they can produce excessive wear to the driveline components and reduced comfort for

the driver.

The engine torque is represented in Fig. 4.10 c), where one can see that the constraint on

the upper bound of the control signal is satisfied by all tested control strategies. Note that,

although the PID controller does not enforce constraints on control command, its output was

saturated in order to enforce the engine limitations, i.e., the torque limit Tmaxe , and the limitation

on the control input increment. It can also be seen that the engine torque for the explicit MPC

controller starts from the steady-state value and that makes it have a faster response regarding

the vehicle velocity.

The oscillations of the drivetrain obtained using the PID and the explicit MPC controllers

can also be seen in Fig. 4.10 d), where the driveshaft torque is presented; the longitudinal

oscillations produced by the PID controller and especially the ones produced with the explicit

MPC controller are quite severe and they can be easily felt by the passengers of the vehicle.

Also, the delays introduced by CAN were represented in Fig. 4.11, where it can be seen that

they are time-varying and uniformly distributed in the interval [0, τmax].

4.4.4.4 Tip-in tip-out maneuvers

The results of a tip-in, tip-out maneuver simulation, in which the reference vehicle velocity goes

from 30 km/h to 10 km/h and back to 30 km/h, are presented in Fig. 4.12. Because the explicit

MPC control scheme could not be simulated using TrueTime toolbox, for this simulation sce-

111


0 2 4 6 8 100

0.005

0.01

0.015

0.02

Time [s]

Com

mun

icat

ion

dela

ys [s

]

Figure 4.11: Acceleration test: Delays induced by CAN.

0 5 10 15 20 25 300

10

20

30

40

Time [s]

v v[k

m/h]

a) Vehicle velocity.

0 5 10 15 20 25 30−10

−5

0

5

10

Time [s]

ωe/i t

ot−

ωw

[rpm

]

b) Speed difference.

0 5 10 15 20 25 300

50

100

Time [s]

Te

[Nm

]

c) Engine torque (control signal).

0 5 10 15 20 25 30−1000

−500

0

500

1000

1500

Time [s]

Tf

[Nm

]

d) Driveshaft torque.

Figure 4.12: Tip-in tip-out maneuvers: Simulation results.

nario the results are illustrated only for the PID controller and the proposed one step ahead MPC

controller, using the same colors. It can be seen that the one step ahead predictive controller has

a slightly faster response with no overshoot when it approaches the reference velocity. More-

over, the oscillations of the axle wrap are damped much faster in the acceleration phase. In the

deceleration phase, again, the PID controller produces undesired oscillations of the axle wrap.

Note that the controller performance during deceleration is limited by the actuator authority.

For this experiment the evolution of the CLF relaxation variable λ∗k and the corresponding

upper bound defined by (4.14) are shown in Fig. 4.13. Due to changing the reference vehicle

velocity, the upper bound of the CLF relaxation variable defined by (4.14) may become unfeasi-

ble, so whenever a change in the reference vehicle velocity occurs, the value of the upper bound

was re-initialized. However as k →∞, λ∗k is forced to converge to 0.

The delays introduced by CAN for this simulation scenario were represented in Fig. 4.14,

where it can be seen that they are time-varying and uniformly distributed in the interval [0, τmax].

112


0 5 10 15 20 25 300

200

400

600

Time [s]

λ


Figure 4.13: Tip-in tip-out maneuvers: History of λ∗k throughout the simulation.

0 5 10 15 20 25 300

0.005

0.01

0.015

0.02

Time [s]

Com

mun

icat

ion

dela

ys [s

]

Figure 4.14: Tip-in tip-out maneuvers: Delays induced by CAN.

4.4.5 Experimental results

The Lyapunov-based predictive controller proposed in Section 4.3 was also tested on an indus-

trial HIL test-bench consisting of a Freescale based electronic control unit (which implements

the controller) linked via a CAN bus with a dSPACE MicroAutoBox plant simulator.

Real-time networked embedded systems represent a viable alternative for complex engi-

neering applications. The main advantage of distributed architectures lies in improved overall

performances and higher flexibility resulted by means of an efficient exploitation of any avail-

able information communicated between the entities of the system. One relevant example is the

distributed control application - in this case the controller can compute the control inputs using

process measurements acquired from distinct local nodes. Unlike the local approach, when the

design methodology has to deal with nonlinearities, disturbances, time variance, etc., the dis-

tribute control is confronted with significantly increased difficulties due to potential occurrence

of communication errors and delays [Zurawski, 2006, Li and Cao, 2003, Buttazzo, 2006].

113


I. EXTENDED ABSTRACT To optimize the development time and costs, the hardware-in-the-loop (HIL) simulation was adopted by the automotive

industry and not only. This concept offers the possibility to investigate new control structures for automotive drivetrains without expensive experiments on the real system. Using the HILs test benches the future cars can be developed increasing the quality of the final products. New Hybrid Electric Vehicles (HEVs), zero emission Electric Vehicles (EV) or classical conventional vehicles are developed and tested using HIL simulation. In this paper the HIL concept is explained, considering a MicroAutoBox from dSPACE as plant emulator. Then, the HIL concept has been exemplified using the drivetrain of a conventional vehicle as a plant.

In recent years, with the development of computer technology, HIL simulation technology became widely used in automotive, aeronautics and astronautics industries by design and test engineers to evaluate and validate components during the development process. HIL simulation is in fact a form of real-time simulation, which should be adopted whenever possible to validate together both the hardware and the software under realistic conditions [1]. HIL simulation is a modern technique that is used in the development and test of complex real-time embedded systems. Rather than testing components in complete system setups, HIL simulation allows the testing of new components and prototypes while communicating with software models that simulate the rest of the system [2], [3]. The purpose of a HIL system is to provide all of the electrical stimuli needed to fully exercise the Electronic Control Unit (ECU). One application of the HIL system in automotive industry is to test the Internal Combustion Engine (ICE) control structure. This subject is presented in detail in [4] and [5]. The use of HIL system in developing process of an Automatic Transmission Control Unit (ATCU) is presented in [6]. With the increasing interest in the Electric and Hybrid Electric Vehicles, the HIL system was used for development, testing and validation of the new prototype-cars. A research report on this topic is presented in [7], to simulate test conditions and vehicle behavior of a hybrid vehicle at FORD Company. The use of HIL systems in electric drives is also presented in [8]. The subject regarding conventional powertrain control testing and developed on HIL system is discussed in [9]. Other automotive applications of the HIL system can be found in [10], [11], [12], [13], [14] and [15]. The current industry definition of a HIL system shows that the plant is simulated and the ECU is real, as it is shown in Fig. 1.

Fig. 1. HIL simulation setup

The typical HIL system is comprised of the following components: the ECU with inputs/outputs (I/O) for running the

controller software, the plant emulator for emulating the behavior of the real system (usually is described by a mathematical model or a set of lookup tables), sensor models, real or simulated loads, fault insertion relay matrix (for diagnostic and safety checks), a host PC which runs a Graphical User Interface (GUI) for the user, with communications link to target computer and diagnostic link to ECU, a Supervisory Control and Data Acquisition (SCADA) application to download and control the

Hardware-In-the-Loop Simulation for Automotive Applications Daniel I. Patrascu*, Constantin F. Caruntu**, George Costeniuc*, Theodor Paulus*, Doina Onu*,

Catalin F. Braescu**, Lavinia Ferariu**, Corneliu Lazar** *Continental Automotive Romania

Daniel.Patrascu, George.Costeniuc, Theodor.Paulus, [email protected] **Department of Automatic Control and Applied Informatics,

”Gheorghe Asachi” Technical University of Iasi, Romania caruntuc, cbraescu, lferaru, [email protected]

Figure 4.15: HIL simulation setup.

4.4.5.1 HIL simulation

To optimize the development time and costs, the HIL simulation was adopted by the automotive

industry and not only. This concept offers the possibility to investigate new control structures

for automotive drivetrains without expensive experiments on the real system. Using the HIL

test benches, the future cars can be developed increasing the quality of the final products. New

Hybrid Electric Vehicles (HEVs), zero emission Electric Vehicles (EV) or classical conventional

vehicles are developed and tested using HIL simulation. In this subsection the HIL concept

is explained, considering a MicroAutoBox from dSPACE as plant emulator. Then, the HIL

concept has been exemplified using the drivetrain of a conventional vehicle as a plant.

In recent years, with the development of computer technology, HIL simulation technol-

ogy became widely used in automotive, aeronautics and astronautics industries by design and

test engineers to evaluate and validate components during the development process. HIL sim-

ulation is in fact a form of real-time simulation, which should be adopted whenever possi-

ble to validate together both the hardware and the software under realistic conditions [John-

son and Fontaine, 2001]. HIL simulation is a modern technique that is used in the devel-

opment and test of complex real-time embedded systems. Rather than testing components

in complete system setups, HIL simulation allows the testing of new components and proto-

types while communicating with software models that simulate the rest of the system [Mc-

Neal and Belkhayat, 2007, Lin and Zhang 2008]. The purpose of a HIL system is to provide all

of the electrical stimuli needed to fully exercise the ECU.

One application of the HIL system in automotive industry is to test the internal combustion

engine control structure. This subject is presented in detail in [Rabbath et al., 2001] and [Wu et

al., 2008]. The use of HIL system in developing process of an Automatic Transmission Control

Unit (ATCU) is presented in [Matsura et al., 2004].

The current industry definition of a HIL system shows that the plant is simulated and the

ECU is real, as it is shown in Fig. 4.15.

114


real-time process and a test automation application to automate all aspects of the test, to develop/change models and tests,

and to collect, store, and report test results.

MicroAutoBox is developed by dSPACE and can be used for many different rapid control prototyping (RCP) applications

[15], for example: Chassis control, Powertrain control, Body control, X-by-wire applications.

The automotive drivetrain is composed from the following components: engine, transmission, final reduction gear, flexible

drive-shaft and driving wheel. Fig. 2 presents the drivetrain of a conventional vehicle.

Fig. 2. Conventional vehicle drivetrain

In a HIL system the drivetrain is modeled based on mathematical equations and is emulated on MicroAutoBox. The

Matlab/Simulink model of the drivetrain is presented in Fig. 3.

RTI CAN Demo: Receive (RX) and Transmit (TX) messages

theta _trimis _pe CAN

Ww_trimis_pe_CAN

We_trimis_pe_CAN

Variation @RTICANMM

ControllerSetup _

1

Variation @RTICANMM

ControllerSetup

1

Mapping

to RTICANMMTerminator 3

Saturation 2

Saturation 1

Saturation

Reset@RTICANMM

ControllerSetup _

1

Reset@RTICANMM

ControllerSetup

1

RTICANMM __

ControllerSetupController1: DS1401_B1_C1

RTI CAN

MultiMessage

ControllerSetup

Variation

Reset

RTICANMM _

GeneralSetup

RTI CAN

MultiMessage

GeneralSetup

RTICANMM

MainBlock

Controller1[CAN ]: DS1401_B1_C1_V1

RTI CAN

MultiMessage

MainSetup

TX Data RX Data

RTICANMM

ControllerSetup

Controller2: DS1401_B1_C2

RTI CAN

MultiMessage

ControllerSetup

Variation

Reset

RTI Data

RS1_C5

Convert

Drivetrain model 1

u x

Display 1_primit

Fig. 3. Matlab/Simulink model of the drivetrain HIL setup

The HIL system signal flow is presented in Fig. 4. It can be seen the model of the plant emulated on MicroAutoBox. The

communication between process and controller is done by Controller Area Network (CAN).

Fig. 4. HIL system signal flow

This paper presents an overview of the HIL systems and its components, considering a MicroAutoBox developed by

dSPACE as plant emulator. MicroAutoBox is a real-time system for performing fast function prototyping from scratch. It

operates without user intervention, just like an ECU. The presented concept was then exemplified using the drivetrain of a

conventional vehicle as a plant. The drivetrain model was developed and an appropiate control structure was designed and

tested using HIL simulation.

Power

supply

Host PC

Drivetrain model

Control algorithm C

AN

dSPACE

MicroAutoBox

ECU

Figure 4.16: Matlab/Simulink model of the drivetrain HIL setup.

The typical HIL system is comprised of the following components: the ECU with in-

puts/outputs (I/O) for running the controller software, the plant emulator for emulating the

behavior of the real system (usually is described by a mathematical model or a set of lookup

tables), sensor models, real or simulated loads, fault insertion relay matrix (for diagnostic and

safety checks), a host PC which runs a Graphical User Interface (GUI) for the user, with com-

munications link to target computer and diagnostic link to ECU, a Supervisory Control and

Data Acquisition (SCADA) application to download and control the real-time process and a

test automation application to automate all aspects of the test, to develop/change models and

tests, and to collect, store, and report test results. MicroAutoBox is developed by dSPACE and

can be used for many different rapid control prototyping (RCP) applications [Ping et al., 2010],

for example: Chassis control, Powertrain control, Body control, X-by-wire applications.

In a HIL system the drivetrain is modeled based on mathematical equations and is emulated

on MicroAutoBox. The Matlab/Simulink model of the drivetrain is presented in Fig. 4.16.

4.4.5.2 Control architecture description

The proposed approach is experimentally verified using an industrial test-bench setup consisting

of a Freescale based electronic control unit (which implements the controller) linked via a CAN

bus with a dSPACE MicroAutoBox plant simulator.

The OSEK-based networked controller requirements are defined having in mind the dy-

namic properties of the plant and the HIL system configuration. Based on the most recent

values of the state variables (xk) received encapsulated within a CAN message, the controller

computes and sends the command uk by means of another CAN message (Fig. 4.17). In fact,

the controller implements a single linear program, obtained as a result of the one step ahead pre-

115


Figure 4.17: CAN communication between ECU and plant.

dictive controller design. To ensure increased reliability, safety and predictability OSEK/VDX

has been adopted as real-time operating system.

A key issue for this software implementation is to handle the command computation and

the potential delays occurred during communication, without demanding increased computa-

tional resources (time and memory). In addition, the design has to take into consideration the

characteristics of the adopted real-time operating system.

A dedicated periodic task (Task CAN Drv) is in charge with messages processing (via CAN

driver functions). The activation of this task is controlled by an alarm, according to the adopted

sampling period. The CAN driver includes all functions necessary for handling message trans-

mission and reception. The messages are standard type ones with 11 bits IDs. They are managed

in two distinct queues (one for reception and one for sending), using different identifiers for

the messages transmitted by the ECU (0x205) and messages transmitted by the HIL simulator

(0x105).

Because the communication delays could affect the time interval between successive mes-

sages arriving to the ECU, the command computation cannot be performed periodically. As

soon as a CAN message is received, Task CAN Drv saves it in the reception queue and acti-

vates the task Task Ctrl which is responsible with command update and command transmission

(to the HIL simulator). This task accepts multiple activations, in order to deal with the case

when an additional message is received before finalizing the update of the current command

value.

It should be noted that the control algorithm determines the optimal command by means of

dual simplex method. The management of delays translates in the need of processing huge data

arrays, therefore special attention has to be paid both to temporal and memory performances of

the application. Based on OSEK debugging facilities, the designer has to set a proper balance

between the precision used for constraints verification, the minimum accepted sampling period

and the compliant maximum delays. As indicated in upper sequence depicted in Fig. 4.18, the

116


communication delays. By means of multiple instances of Task_Ctrl, one can treat successive messages occurred at time intervals less than the execution time of Task_Ctrl (see lower sequence in Fig. 2). The limitation is given by the fact that the resulted command could be available too late, leading to improper behavior of the automatic system.

Ts 2Ts 3Ts

Ts 2Ts 3Ts

Message detected by Task_CAN_Drv

Task_Ctrl

Figure 2. Delays management.

III. ONE STEP AHEAD PREDICTIVE CONTROLLER DESIGN This section describes the proposed horizon-1 MPC

scheme based on a flexible Lyapunov function [10], which is employed to provide the real-time control of a vehicle drivetrain system. As indicated below, for a particular choice of the Lyapunov function candidate, the suggested design leads to a low-complexity linear program that can be efficiently solved within the required time range, while strictly enforcing the constraints and considering the time-varying delays, via the usage of the model described in the first Subsection.

A. Drivetrain model Introducing the engine speed, the wheel speed and the

torsion in the driveshaft, also called axle wrap, as state variables, i.e.,

1 2 3, /,m m me tow e wtx x x iω ω θ θ= = = − , (1)

the drivetrain model from [16] can be written in a state space form as

( ) ( ) ( )m m m m m mc c cx t A x t b u t f= + + , (2)

where ( )1 2 3m m m mx x x x= is the system state, mu ∈

is the system input and , ,m m mc c cA b f are given in [16].

The engine torque (control input) is restricted by lower and upper bounds and by a torque rate constraint (see [17] for details):

0 ( ) ,

( ) ,

m maxe

m m Me e

u t T

T u t T

≤ ≤

≤ ≤ (3)

where maxeT is the maximum torque that can be generated

by the internal combustion engine and meT , M

eT are torque rate bounds.

The engine and wheel speeds are bounded, i.e.,

1

2

( ) ,

( ) ,

min m maxe emin m maxw w

x t

x t

ω ω

ω ω

≤ ≤

≤ ≤ (4)

where mineω , max

eω are the idle and the maximum speed of

the engine, respectively, and minwω , max

wω are the minimum and the maximum speed of the wheel, respectively.

The control objective is to reach a desired value of the wheel speed, i.e., 2

ssx , as fast as possible and with minimum

overshoot. Note that for a desired wheel speed value 2ssx , it

is possible to obtain the steady-state engine speed 1ssx , axle

wrap 3ssx and engine torque ssu .

To facilitate the implementation of the horizon-1 MPC algorithm of [10], a coordinate transformation is performed on (2), i.e.,

( ) ( ) ( ), ( ) ( ) ( ),m ss m ssx t x t x t u t u t u t= − = − (5)

where ( )1 2 3ss ss ss ssx x x x= , which yields the following

system description

( ) ( ) ( ),c cx t A x t b u t= + (6)

where cA and cb are the transformed system matrices.

B. Networked control system model Consider the continuous –time system (6) with input

delay:

1 1( ) , [ , ],k k k k ku t u t t tτ τ+ += ∀ ∈ + + (7)

where k st kT= , k +∈ , sT +∈ denotes the sampling period and assume that ( ) 0u t = u for all 0[0, ]t τ∈ with

0mu ∈ some predetermined constant vector. ku ∈ is

the control action generated at time kt t= . [0, ]largek ττ ∈

denotes the delay induced by the network at time k +∈ and largeτ +∈ is the maximal possible delay.

Consider the maximum delay as ( )large sTτ υ= Υ + ,

where +Υ ∈ and [0,1)υ ∈ . Assuming that k ku ψ= for

all [ 1, 1]k −Υ− −∈ with [ 1, 1]−Υ− −ψ some predetermined vector, the discretized model is:

1 0 1

1 2 1 1

( )( )( )( ) ( )( ),

k d k d k k k k

k k k k k k

x A x b u u uu u u u

ττ τ

+ −

− − Υ −Υ− −Υ

= + + Δ − +

+ Δ − +…+ Δ − (8)

where c sA TdA e= , ( )

0s c s

T A Td cb e d bθ θ−= ∫ ,

Figure 4.18: Delays management.

sampling periods which are significantly longer than the execution time of Task Ctrl allow the

management of higher communication delays. By means of multiple instances of Task Ctrl,

one can treat successive messages occurred at time intervals less than the execution time of

Task Ctrl (see lower sequence in Fig. 4.18). The limitation is given by the fact that the resulted

command could be available too late, leading to improper behavior of the automatic system.


The performances of the proposed approach are tested according to the hardware-in-the loop

(HIL) technique. In a typical HIL simulator, the plant behavior is deduced from the mathe-

matical models and executed by a dedicated real-time processor. This test-bench mimics the

complexity of real plant, although it allows a simpler, safer and less expensive verification.

Therefore, HIL is strongly recommended for the development of complex critical real-time em-

bedded systems, especially in the automotive industry.

The test-bench setup considered for the experimental trials corresponds to the closed-loop

drivetrain control system and it is consists in a MicroAutoBox system (which simulates in real-

time the physical plant), connected by CAN with the embedded controller. The block diagram

of the HIL based control system is depicted in Fig. 4.17.

MicroAutoBox represents a real-time system based on DS1401 board, able to provide fast

function prototyping - such as chassis control, power train and other rapid control applications.

The model of the plant can be ported from MATLAB, using certain dSPACE libraries. Fur-

thermore, specific dSPACE tools could be employed in order to integrate all necessary CAN

messages which must be communicated with the ECU. It should be noticed that this available

setup is designed to behave very closely to the real drivetrain system.

The embedded controller is implemented on a Freescale 16-bit MC9S12XDP512 unit, mak-

ing use of OSEK/VDX facilities. This microcontroller has a large palette of on-chip peripherals

117


0 2 4 6 8 100

10

20

30

40

Time [s]

Veh

icle

spe

ed [k

m/h

]

Figure 4.19: HIL: Vehicle velocity.

(5 CAN modules, two 10-bit analog-to-digital converters, an 8-channel pulse-width modula-

tor, three serial peripheral interfaces, two 8-bit micro-timers and four 16-bit periodic interrupt

timers) and satisfies all the hardware requirements imposed by the approach.

Having in mind the dynamic behavior of the plant, all experiments have been carried out

for the sampling period Ts = 0.01s , assuming that certain delays can occur during data trans-

mission. The upper bound of the delays that are induced by CAN are calculated using the

methodology described in 2.5.3.3, which results in τmax = 1.7Ts = 0.017s . The delays are

time-varying and uniformly distributed within [0, τmax].

The one step ahead predictive controller uses the weight matrices Q = 11I3 , R = 0.5

and G = 1 . The technique presented in [Gielen and M.Lazar, 2009b] is used for the off-line

computation of the infinity norm based local CLRF V (x) = ‖Px‖∞ for ρ = 0.99 and the affine

model of the drivetrain in closed-loop.

The experimental trials are analyzed having in mind that the control objective is to reach

a desired speed reference in a short amount of time, although, at the same time, to increase

the passenger comfort by reducing the oscillations occurred when the vehicle is subjected to

accelerations.

The vehicle velocity illustrates that the designed one step ahead predictive controller is able

to ensure a fast response (short rising time), reduced overshoot and a tracking error close to

0 during the stationary regime (Fig. 4.19). The control input, namely the engine torque, is

represented in Fig. 4.20. Here one can see that the constraint on the upper bound of the control

signal is satisfied by the tested control strategy.

For this experiment the evolution of the CLF relaxation variable λ∗k and the corresponding

upper bound defined by (4.14) are shown in Fig. 4.21.

Moreover, the controller ensures acceptable torsions of the drive shaft, as indicated in Fig.

4.22 by plotting the difference between the wheel speed and the engine speed divided by the

118

Conclusions

0 2 4 6 8 100

50

100

Time [s]

Engin

e torq

ue [N

m]

Figure 4.20: HIL: Engine torque (control signal).

0 2 4 6 8 100

100

200

300

400

Time [s]

Lam

bda


Figure 4.21: HIL: History of λ∗k throughout the simulation.

total gear ratio itot .

4.5 Conclusions

Different strategies for damping the drivetrain oscillations and different methods to compensate

the delays that appear in a communication network were reported in the literature, but none in

which the drivetrain is a component of a networked control system, being controlled through a

0 2 4 6 8 10−10

−5

0

5

10

Time [s]

Spe

ed d

iffer

ence

[rpm

]

Figure 4.22: HIL: Speed difference.

119


communication network.

This chapter proposes a novel method of modeling a CAN-based network control system

that accounts for time-varying delays within a provided bound based on polypotic approxima-

tions for controller design purposes. Then, a one step ahead predictive controller based on

flexible control Lyapunov functions [M.Lazar 2009] is designed for the resulting model with

polytopic uncertainty and hard constraints. The developed algorithms have the potential to sat-

isfy the timing requirements, due to the short horizon, while it can still offer a non-conservative

solution to stabilization due to the flexibility of the Lyapunov function. Moreover, it is shown

that for an appropriately chosen Lyapunov function, the MPC algorithms amount to solving a

single, low-complexity linear program each sampling instant. This means that the proposed

LMPC algorithm can be applied for the class of fast processes, which require short sampling

periods. Furthermore, a new limiting condition on the auxiliary optimization variable that adds

flexibility to the CLF is proposed.

A simulation experiment designed in collaboration with Ford Research and Advanced Engi-

neering, US, validated the proposed approach and indicated that the developed scheme has the

potential to meet the required real-time control specifications. Using TrueTime Toolbox as the

simulation environment, the results obtained with this method were compared with the results

obtained with other strategies and the comparison illustrates that the proposed controller has an

overall superior performance and it meets the required timing constraints.

The last contribution is represented by a new OSEK based networked controller designed

according to the one step ahead predictive control algorithm with flexible control Lyapunov

function. The proposed embedded controller can manage certain communication delays, at

convenient computational costs, as the command results by solving a linear program.

The experiments were carried out according to the HIL technique, the industrial HIL test-

bench setup consisting of a Freescale based electronic control unit (which implements the con-

troller) linked via a CAN bus with a dSPACE MicroAutoBox plant simulator. The results ob-

tained for vehicle driveline oscillations damping illustrate the potential of the proposed con-

troller for real-time applications, indicating the efficiency of the approach in dealing with time

varying delays, whilst preserving a fast response and reduced overshoot on the vehicle speed,

as well as reduced torsions of the driveshaft.

The results presented in this chapter were published in:

• [Caruntu et al., 2011f]: C. F. Caruntu, M. Lazar, S. Di Cairano, R. H. Gielen and P. P. J. van den

Bosch. Horizon-1 predictive control of networked controlled vehicle drivetrains. In 18th IFAC

World Congress, Milano, Italy, 2011.

120

Conclusions

• [Braescu, Caruntu et al., 2011]: F. C. Braescu, L. Ferariu, C. F. Caruntu and C. Lazar. OSEK

Based Embedded Networked Controller Designed to Handle Communication Delays. In 2nd East-

ern European Regional Conference on the Engineering of Computer Based Systems, Bratislava,

Slovakia, 2011.

121


122

Chapter 5

Robust nonlinear networked predictivecontrol

In this chapter, a new bounding technique for the perturbations introduced by the network-

induced time-varying delays and a robust one step ahead predictive controller, which takes into

account the bounds of the disturbances, are proposed.

5.1 Introduction

In networked control systems there are many network-induced imperfections and constraints:

quantization errors, packet dropouts, variable sampling periods, time-varying delays and com-

munication constraints. The goal of this chapter is to provide a control design methodology

that can assure the closed-loop performances of a physical plant, while compensating the time-

varying delays introduced by the communication network that links the controller with the re-

mote process. Firstly, starting from the disturbances model described in Subsection 2.4.1.2, the

error caused by the time-varying delays is modeled as a disturbance and two novel methods

of bounding the disturbances are proposed. Then, based on the robust controller presented in

Subsection 2.4.2.4, a robust one step ahead predictive controller based on flexible control Lya-

punov functions is designed, which explicitly takes into account the bounds of the disturbances

caused by time-varying delays and guarantees also the input-to-state stability of the system in

a non-conservative way. In addition, it is shown that by choosing an appropriately Lyapunov

function, the MPC algorithm amounts solving a single, low-complexity linear program each

sampling instant.

To illustrate the potential for application to fast processes, the proposed robust LMPC strat-

123

Robust nonlinear networked predictive control

caτ

scτ

r

kyku kk ay

−

k kk a by− −


Controller

ca scτ τ+


−

++


Network

Network

r

kyku k kk a by

− −

k kk a by− −


Controller

Remote process

r

kyku k kk a by

− −

k kk a by− −


Controller

Remote process

Continuous-

time Plant ZOH

A Ts

S

Continuous-

time Plant ZOH

A Ts

S

Continuous-

time Plant ZOH

A Ts

S

Figure 5.1: Control system with network disturbance.

egy is applied to control an Ethernet-controlled DC motor and a vehicle drivetrain through

CAN. The methodology is firstly applied for an Ethernet-controlled DC motor and the Mat-

lab/Simulink and TrueTime simulations illustrate the effectiveness and robustness of the pro-

posed delay modeling and control strategy. Secondly, the modeling method and the control

strategy were tested on a vehicle drivetrain controlled through CAN, with the aim of damping

driveline oscillations, which is crucial in improving driveability ans passenger comfort. Several

Matlab/Simulink and TrueTime simulations based on realistic scenarios show that the proposed

control scheme can handle the performance/physical constraints. Moreover, the drivetrain con-

trol strategy was tested in a real-time simulation setup, the experimental results showing that

the control strategy can handle the strict limitations on the computational complexity.

5.2 Modeling input delays as disturbances

Considering the structure represented in Fig. 2.2, it is easy to observe that it is equivalent to the

structure represented in Fig. 5.1, in which the difference between the actual control signal and

the delayed one is regarded as a hypothetical disturbance [Natori et al., 2008, Gadamsetty et

al., 2009]. In other words, the hypothetical disturbance exerted on the physical plant delays the

feedback signal, the discrete-time disturbance representation being given by

udk = uk−ak−bk − uk, (5.1)

with ak and bk from (2.1).

In what follows, the goal is to find a bounded set W, in which to include all possible distur-

bances that can appear due to the time-varying delay introduced by the communication network,

knowing that the input of the remote process is bounded. The proposed method is different from

the one presented in [Natori et al., 2008], in which, a communication delay observer is used to

determine the difference between the computed control signal and the control signal that pro-

124

Modeling input and output delays as disturbances

duced the measured output.

Now consider that the control signal is bounded by lower and upper bounds

umin ≤ uk ≤ umax, (5.2)

where umin and umax are the minimum and the maximum control signal values that can be given

as input to the remote process. Then, the disturbance can be bounded as [Caruntu and C.Lazar, 2011e,

Caruntu and C.Lazar, 2011d]

umin − umax ≤ udk ≤ umax − umin. (5.3)

Furthermore, if the discrete control signal is also restricted by an incremental bound

−u∆ ≤ ∆uk ≤ u∆, (5.4)

where ∆uk := uk − uk−1, for all k ∈ Z≥1, with u0 some predetermined value and u∆ is the

maximum increase/decrease of the control signal at each sampling time instant k ∈ Z≥1, the

disturbance can be rebounded as [Caruntu and C.Lazar, 2011e, Caruntu and C.Lazar, 2011d]

−(a+ b)u∆ ≤ udk ≤ (a+ b)u∆, (5.5)

where a and b are defined as in (2.2).

Even though the time-varying delay gives a time-varying disturbance, the set W, which is

defined by Bdudk, with udk from (5.3) or from (5.5), remains fixed, so this modeling technique is

suitable for the use of the results presented in [M.Lazar and Heemels, 2008], in which the dis-

turbances are explicitly taken into account during the design phase of the predictive controller.

5.3 Modeling input and output delays as disturbances

Consider a physical plant given by a linear state-space model

x(t) = Acx(t) +Bcu(t− τ ca) (5.6)

where x(t) is the system state, u(t) is the control signal and Ac and Bc are the continuous-time

system matrices.

It is easy to observe that Fig. 2.1 is equivalent to Fig. 5.2, in which the difference between

125


caτ

scτ

r

kx

ku

kk ax

−

k kk a bx

− −

kk au

−Discrete-time

Controller


Continuous-

time Plant ZOH

A Ts

S

kk a ku u

−−

++

Network

r

kx

ku

kk ax

−

k kk a bx

− −

kk au

−Discrete-time

Controller

Remote process

Continuous-

time Plant ZOH

A Ts

S

++

k k kk a b k ax x

− − −−

ca scτ τ+

Network r

kx

ku

k kk a bx

− −

k kk a bx

− −

k kk a bu

− −Discrete-time

Controller

Remote process

Continuous-

time Plant ZOH

A Ts

S

Figure 5.2: Control system with network-induced time delay as disturbance.

the actual control signal and the delayed one is regarded as a hypothetical disturbance [Na-

tori et al., 2008]. Furthermore, the difference between the actual output and the delayed one is

regarded as another hypothetical disturbance.

5.3.1 Forward channel delays

The discrete-time forward channel disturbance representation is given by

udk = uk−ak − uk. (5.7)

The goal is to find a bounded set Wu, in which to include all possible disturbances that

can appear due to the time-varying delays introduced in the forward channel, knowing that

the input of the remote process is bounded. The proposed method is different from the one

presented in [Natori et al., 2008], in which a communication delay observer is used to determine

the difference between the computed control signal and the control signal that produced the

measured output.

Now consider that the control signal is bounded by lower and upper bounds

umin ≤ uk ≤ umax, (5.8)

where umin and umax are the minimum and the maximum control signal values that can be given

as input to the remote process. Then, the disturbance can be bounded as [Caruntu and C.Lazar, 2011e,

Caruntu and C.Lazar, 2011d]

umin − umax ≤ udk ≤ umax − umin. (5.9)

126

Modeling input and output delays as disturbances

Furthermore, if the discrete control signal is also restricted by an incremental bound

−u∆ ≤ ∆uk ≤ u∆, (5.10)

where ∆uk := uk − uk−1, for all k ∈ Z≥1, with u0 some predetermined value and u∆ is the

maximum increase/decrease of the control signal at each sampling time instant k ∈ Z≥1, the

disturbance can be rebounded as [Caruntu and C.Lazar, 2011e, Caruntu and C.Lazar, 2011d]

−au∆ ≤ udk ≤ au∆. (5.11)

5.3.2 Feedback channel delays

The hypothetical disturbance exerted on the physical plant delays the feedback signal, the

discrete-time feedback channel disturbance representation being given by

xdk = xk−ak−bk − xk−ak . (5.12)

The goal is now to find a bounded set Wx, in which to include all possible disturbances that

can appear due to the time-varying delays introduced in the feedback channel, knowing that the

input of the remote process is bounded.

Consider the physical plant (5.6) with input disturbance

x(t) = Acx(t) +Bc(u(t) + ud(t)) (5.13)

Discretizing (5.13) yields

xk+1 = Adxk +Bd(uk + udk) =

= A2dxk−1 + AdBd(uk−1 + udk−1) +Bd(uk + udk) =

= Ajdxk−j+1 + Aj−1d Bd(uk−j+1 + udk−j+1) + . . .

· · ·+ AdBd(uk−1 + udk−1) +Bd(uk + udk) =

= Ajdxk−j+1 +

j−1∑

i=0

AidBd(uk−i + udk−i)

(5.14)

127


Calculating

xk−ak = Adxk−ak−1 +Bd(uk−ak−1 + udk−ak−1) =

= Abkd xk−ak−bk + Abk−1d Bd(uk−ak−bk + udk−ak−bk) + . . .

· · ·+ AdBd(uk−ak−2 + udk−ak−2)+

+Bd(uk−ak−1 + udk−ak−1) =

= Abkd xk−ak−bk +

bk−1∑

i=0

AidBd(uk−ak−i−1 + udk−ak−i−1)

(5.15)

(5.12) becomes [Caruntu and C.Lazar, 2011c, Caruntu and C.Lazar, 2011j]

xdk = xk−ak−bk − Abkd xk−ak−bk −bk−1∑

i=0

AidBd(uk−ak−i−1 + udk−ak−i−1) =

= (In − Abkd )xk−ak−bk −bk−1∑

i=0

AidBd(uk−ak−i−1 + udk−ak−i−1).

(5.16)

xk−ak−bk is known at time instant k, all uk−i, i ∈ Z≥1 are known, all udk−i, i ∈ Z≥1 are

bounded, ak and bk are bounded by a and b, respectively, so xdk can be dynamically bounded at

each sampling instant k ∈ Z+.

Now, (5.14) becomes

xk+1 = Adxk +Bduk +Bdudk + xdk. (5.17)

Even though the time-varying delay gives a time-varying disturbance, the sets Wu, which

is defined by Bdudk, with udk from (5.9) or from (5.11), and Wx, which is defined by xdk from

(5.16), remain fixed, so this modeling technique is suitable for the use of the results presented

in [M.Lazar and Heemels, 2008], in which the disturbances are explicitly taken into account

during the design phase of the predictive controller, which will be accomplished in the next

chapter.

5.4 Robust Lyapunov-based Model Predictive Control

Recently, an extension and relaxation of the conventional Lyapunov condition was proposed in

[M.Lazar 2009], yielding so-called flexible Lyapunov functions which offer a stability guarantee

(under a recursive feasibility assumption) even for a short prediction horizon. In this section, an

128

Robust Lyapunov-based Model Predictive Control

one step ahead MPC scheme that employs a flexible Lyapunov function [M.Lazar 2009] to attain

stability and performance, in which the disturbances are explicitly taken into account during the

design phase by using input-to-state stability (ISS) concepts (see, e.g. [Jiang and Wang, 2001]),

is proposed. Furthermore, it is shown that for a particular choice of the Lyapunov function

candidate the proposed design leads to a low-complexity linear program (LP) that can be solved

efficiently within the required time range, while strictly enforcing the constraints and consider-

ing the delays induced by the communication network, via the usage of the disturbance models

described in Subsections 5.2 and 5.3.

5.4.1 Robust Lyapunov-based predictive controller

Consider the perturbed discrete-time constrained nonlinear system of the form

xk+1 = φ(xk, uk, wk) := φ(xk, uk) + wk

:= Adxk +Bduk + wk, k ∈ Z+,(5.18)

where xk ∈ X ⊆ Rn is the state, uk ∈ U ⊆ Rm is the control input and wk ∈ W ⊆ Rn is the

unknown disturbance introduced by the network-induced time-varying delays at the discrete-

time instant k. φ : Rn×Rm×Rn → Rn and φ : Rn×Rm → Rn are arbitrary nonlinear, possibly

discontinuous, functions with φ(0, 0, 0) = 0 and φ(0, 0) = 0. Ad ∈ Rn×n and Bd ∈ Rn.

Naturally, it is assumed that the set of feasible states X, the set of feasible inputs U and the

disturbance set W are bounded polyhedra with non-empty interiors containing the origin. Next,

let α1, α2, α3 ∈ K∞ and let σ ∈ K.


α1(‖x‖) ≤ V (x) ≤ α2(‖x‖), ∀x ∈ Rn (5.19)

and for which there exists a possibly set-valued control law π : Rn ⇒ U such that

V (Adxk +Bduk + wk)− V (x) ≤ −α3(‖x‖) + σ(‖w‖),∀x ∈ X,∀u ∈ π(x),∀w ∈W,

(5.20)

is called an input-to-state stability control Lyapunov function (ISS-CLF) in X for system (5.18)

and disturbances in W.

ISS theory (see [Jiang and Wang, 2001]) can be used to derive an input-to-state stabilizing

predictive control scheme with improved disturbance rejection, as done in [M.Lazar and Heemels, 2008],

129


where this property is refereed to as optimized ISS.

As such, let W be a convex hull of the vertices we, e = 1, . . . , E, and let λek, k ∈ Z+, be

optimization variables associated with each vertex we. Let J(λ1, . . . , λE, λ) : RE+ × R+ → R+

be a strictly convex, radially unbounded function (i.e. J(·) tends to infinity when its arguments

tend to infinity) and let J(λ1, . . . , λE, λ) → 0 ⇒ λe → 0 for all e = 1, . . . , E and λ → 0, and

J(0, . . . , 0, 0) = 0.

Choose off-line a CLF V (·) for system (5.18) without disturbances and let α3 ∈ K∞ and

x ∈ X be given. At each control sampling instant k ∈ Z+ the one step ahead ISS MPC

controller solves the following problem.

Problem 5.4.2 At time k ∈ Z+ measure the state xk and minimize the cost J(λ1k, . . . , λ

Ek , λk)

over uk, λ1k, . . . , λ

Ek and λk, subject to the constraints

uk ∈ U, (Adxk +Bduk) ∈ X, λek ≥ 0, λk ≥ 0, (5.21a)

V (Adxk +Bduk)− V (xk) + α3(‖xk‖) ≤ λk, (5.21b)

V (Adxk +Bduk + we)− V (xk) + α3(‖xk‖)− λek ≤ 0, (5.21c)

for all e = 1, . . . , E. 2

Let π(xk) := uk ∈ Rm | ∃λk, λek, e ∈ Z[1,E] s.t. (5.21) holds and let φcl(xk, π(xk), wk) :=

φ(xk, uk, wk) | uk ∈ π(xk) denote the difference inclusion corresponding to system (5.18)

in closed-loop with the set of feasible solutions obtained by solving Problem 5.4.2 at each

sampling instant k ∈ Z+.

Next, the main robust stability result in terms of ISS is stated. This result is an adaptation

of the main result in [M.Lazar and Heemels, 2008], to fit the relaxation (5.21b) of Problem 5.4.2,

i.e., λk = 0 for all k ∈ Z+ corresponds to the problem considered in [M.Lazar and Heemels, 2008].

Theorem 5.4.3 Let α1, α2, α3 ∈ K∞, a continuous and convex CLF V (·) and a cost J(·) be

given. Suppose that Problem 5.4.2 is feasible for all states x in X and assume that limk→∞ λ∗k =

0. Then, the trajectories generated by the difference inclusion

xk+1 ∈ φcl(xk, π(xk), wk), k ∈ Z+, (5.22)

with initial state x0 ∈ X converge in finite time to a RPI subset of X, in which the difference

inclusion is ISS for disturbances in W.

130



to-state stability and Lyapunov stability and is therefore omitted here. The interested reader is

referred to [M.Lazar and Heemels, 2008] and [M.Lazar 2009] for more details. Advantageous

properties of the proposed robust controller are that ISS is guaranteed for any (feasible) solution

of the optimization problem, state and input constraints can be explicitly accounted for, and

feedback to disturbances is provided actively, on-line. The key of the stability proof is the

limiting condition limk→∞ λ∗k = 0.

In what follows a new non-conservative solution for guaranteeing this condition is provided,

such that λk can be small or even 0 for some time after which it is allowed to increase again, as

long as this does not violate the upper bound.

Lemma 5.4.4 Let Nλ ∈ Z≥1 be a fixed constant to be chosen a priori and let ρ ∈ R[0,1). If

0 ≤ λk ≤ maxi∈[1,Nλ]

ρiλ∗k−i, ∀k ∈ Z≥Nλ , (5.23)


Lemma 5.4.4 is proven in [Gielen and M.Lazar, 2009a, Gielen and M.Lazar, 2009b] and is

omitted here.

Constraint (5.23) is not active for the first Nλ discrete-time instants, and then λk can take

any value in R[0,M ]. Then, starting with k = Nλ it provides a monotonically decreasing upper

bound Mk which allows λk to take any value in R[0,Mk] for all k ∈ Z≥Nλ . Obviously, if Mk = 0

for some k, which happens if λ∗j = 0 for all j ∈ Z[k−Nλ+1,k], constraint (5.23) is equivalent to

λk = 0, which is always feasible at the equilibrium. This means that constraint (5.23) allows

a non-monotone evolution of λk for the first Nλ sampling instants and furthermore, as long as

λ∗k−i > 0 for at least one i ∈ Z[1,Nλ].

Lemma 5.4.5 Let Ω ∈ R+ be a fixed constant to be chosen a priori and let ρ ∈ R[0,1) and

M ∈ Z>0. If

0 ≤ λk ≤ ρ1M (λ∗k−1 + ρ

k−1M Ω), ∀k ∈ Z+, (5.24)


Proof: Since λ∗k is bounded for all k ∈ Z+ [M.Lazar 2009], there exists an Nλ ∈ R+ such that

λ∗0 ≤ Nλ. Applying (5.24) recursively, the following inequality is obtained:

λk ≤ ρkM (Nλ + kΩ), ∀k ∈ Z+. (5.25)

131


Since ρ ∈ R[0,1) it holds that limk→∞ ρkM = 0. Moreover,

limk→∞

kρkM = lim

k→∞k(ρ−1)−

kM = lim

k→∞

k

(ρ−1)kM

(5.26)

and applying l’Hospital’s rule [Stewart, 2003] it follows that

limk→∞

k

(ρ−1)kM

= limk→∞

ddkk

ddk

(ρ−1)kM

=

= limk→∞

1

(ρkM )k ln(ρ−1)

M

= 0,

(5.27)

which completes the proof.

By augmenting Problem 5.4.2 with constraint (5.24) the property limk→∞ λ∗k = 0 is thus

guaranteed, which is sufficient for asymptotic stability. Note that, in constraint (5.24), ρk−1Ω

gives a decreasing ramp and λ∗k−1 dictates with what value the ramp is allowed to be violated in

order to maintain the feasibility of the closed-loop control system. It is worth to mention that the

proposed limiting condition is less conservative than the solution presented in [M.Lazar 2009],

which corresponds to setting Ω = 0.

In the next subsection it is described how the developed one step ahead MPC scheme for

the constrained system (5.18) can be implemented by solving a single LP during each control

cycle.

5.4.2 Implementation issues

Next, consider the following cost function to be minimized

J1(xk, uk, λk,λk) := JMPC(xk, uk) + J(λk, λk)

:=‖Px(Adxk +Bduk)‖∞+‖Qxxk‖∞ + ‖Ruk‖∞ + J(λk, λk),

(5.28)

where the cost on the optimization variables λk := [λ1k, . . . , λ

Ek ]T and λk is defined as J(λk, λk) :=

‖Λλk‖∞+ |Λλ|, where Λ is a full-column rank matrix of appropriate dimensions and Λ ∈ R>0.

The cost J(·) is chosen as required in Problem 5.4.2 and the matrices Px, Qx and R are known

full-column rank matrices of appropriate dimensions.

132


The cost function (5.28) is subjected to the following constraints:

umin ≤ uk ≤ umax,

−u∆ ≤ ∆uk ≤ u∆,

xmin ≤ Adxk+Bduk ≤ xmax,

(5.29)

where umin and umax are the control command lower and upper bounds, u∆ is the incremental

constraint on the control command and xmin and xmax are appropriate vectors that bound the

states of the system.

Now consider the following infinity-norm based candidate CLF, i.e.,

V (x) = ‖Px‖∞, (5.30)

where P ∈ Rp×n is a full-column rank matrix to be determined. This function satisfies (5.19)

with α1(s) = σ√ps, where σ is the smallest singular value of P , and with α2(s) = ‖P‖∞s. For

xk ∈ Ωi, substituting (5.56) and (5.30) in (5.21b) and (5.21c) yields

‖P (Adxk +Bduk)‖∞−− ‖Pxk‖∞ + ‖Qxk‖∞ ≤ λk,

(5.31)

‖P (Adxk +Bduk + we)‖∞−− ‖Pxk‖∞ + ‖Qxk‖∞ − λek ≤ 0,

(5.32)

where P , xk and xssk are known, e = 1, . . . , E and ‖Qxk‖ ≤ ξ‖xk‖, for α3(s) := ξx, ∀x ∈ X.

Although (5.28), (5.31) and (5.32) appear to be nonlinear in the optimization variables, due

to using only horizon 1, the optimization problem can be recast as a linear program (LP) via a

particular set of equivalent linear inequalities. By definition of the infinity norm, for ‖x‖∞ ≤ c

to be satisfied, it is necessary and sufficient to require that ±[x]j ≤ c for all j ∈ 1, 2, ..., n.So, for (5.31) and (5.32) to be satisfied, it is necessary and sufficient to require that

± [P (Adxk +Bduk)]j−− ‖Pxk‖∞ + ‖Qxk‖∞ ≤ λk,

(5.33)

133


± [P (Adxk +Bduk + we)]j−− ‖Pxk‖∞ + ‖Qxk‖∞ − λek ≤ 0,

(5.34)

for j ∈ 1, 2, . . . , n and e = 1, . . . , E. This yields a total of 2p+ 2p(E+ 1) linear inequalities

in the optimization variables uk, λ1k, . . . , λ

Ek and λk. Solving Problem 5.4.2, which includes

minimizing the cost function (5.28), can be reformulated as the following problem.

Problem 5.4.6min

uk,λk,λk

(ε1k + ε2k + ε3k + ε4k) (5.35)

subject to (5.24), (5.29), (5.31), (5.32), and

±[Px(Adxk +Bduk)]j+

+‖Qxxk‖∞ ≤ ε1k, ∀j ∈ 1, 2, . . . , n (5.36a)

±Ruk ≤ ε2k, (5.36b)

±Λλk ≤ ε3k, (5.36c)

Λλk ≤ ε4k. (5.36d)

Notice that Problem 5.4.6 is a linear program, since xk and all λ?k−i, i ∈ Z[1,Nλ], for Lemma

5.4.4, are known at time k ∈ Z>Nλ and thus, all constraints are linear in the unknowns uk, λk,

λk and εlk, l ∈ Z[1,4].

Notice that Problem 5.4.6 is a linear program, since xk and λ?k−1, for Lemmas 4.3.4 and

5.4.5, are known at time k ∈ Z+ and thus, all constraints are linear in the unknowns uk, λk, λkand εlk, l ∈ Z[1,4].

This implies the control algorithm can be executed in a shorter time than standard MPC

schemes, which typically require an optimal solution to be found. This means that the proposed

controller can be applied for the class of fast processes, with short sampling periods, implying

that the controller can be used in real-time applications.

The one step ahead MPC algorithm is stated next.

Algorithm 5.4.7 At each sampling instant k ∈ Z+:

Step 1: Measure the current state xk;

Step 2: Solve the LP Problem 5.4.6 and pick any feasible control action u∗(xk);

Step 3: Implement uk := u∗ as control action.

134

Robust LMPC applied to control a DC motor through Ethernet

Using the technique of rewriting inequalities that include∞-norms as linear inequalities in

combination with the above optimization problem, one can formulate a linear program whose

solution minimizes the cost J(λ1k, . . . , λ

Ek , λk) for a horizon equal to one and it satisfies all

constraints in (5.21).

An advantageous feature of the proposed scheme is that only a feasible solution of (5.35)

is required in order to assure stability. This implies that the control algorithm can be executed

in a shorter time than standard MPC schemes, which typically require optimality to guarantee

stability.

5.5 Robust LMPC applied to control a DC motor through

Ethernet

In the last decades, the computer networks and their applications passed through major changes,

reaching nowadays to be used in the area of high performance distributed/networked control and

automation [Neumann, 2007]. This area is still developing with attractive and widespread appli-

cations ranging from military unmanned vehicles and large manufacturing plants to telerobotics

and telemedicine [Tang and Yu, 2007].

Nowadays, the networked control systems are used even for time-sensitive applications,

such as remote DC motor actuation control, being well known that DC motors are used exten-

sively in industry for applications such as robot arm drives, machine tools, steel rolling mills,

electric trains and even aircraft control [Chow and Tipsuwan, 2003], requiring speed controllers

to perform the desired tasks.

In this Section, the remote DC motor speed control application is used to illustrate the effec-

tiveness of the proposed disturbance bounding methodology (Section 5.2) and the performance

of the previously proposed one step ahead predictive control scheme (Section 5.4). The goal of

the control algorithm is to achieve zero steady-state error and smooth, fast response of the DC

motor to step inputs, which are popular desired DC motor performance characteristics for many

industrial applications.

Firstly, the networked DC motor control architecture is presented and then the dynamics of

the DC motor, which is the remote process, are briefly described.

135


Remote process

LOAD

Actuator node

E T H E R N E T

y

ucy

au

Remote process

Remote process

ry Controller node

(ECU)

Sensor node

Remote process

DC motor

M

Figure 5.3: DC motor networked control system.

5.5.1 Networked control architecture

The complete networked DC motor control architecture, which is graphically depicted in Fig.

5.3, consist of the following actions:

• The sensors measure the outputs of the physical plant, convert the signal with the A/D

converter, and send the samples y to the controller through Ethernet;

• The DC motor controller receives the delayed sampled measurements from the sensors yc

and the desired reference yr and computes the required control signal, while handling the

physical/control constraints and the delays induced by Ethernet;

• The control signal computed by the controller, i.e., u, is sent to the remote process through

Ethernet;

• The actuator, which is a power conversion unit with a D/A converter, receives the delayed

control signal ua and acts the DC motor.

In Fig. 5.3 the dashed lines represent the direction of the messages sent to and from the

controller.

Note that the sensors send the monitored signals y to the controller and the value that reaches

the controller, yc, can be related to y as

yck = y(kTs − τ sc), (5.37)

136


where τ sc is the delay time to transmit the measured signal y from the sensor to the controller.

Also, the control value that reaches the DC motor ua can mathematically be expressed as

uak = u(kTs − τ ca), (5.38)

where τ ca is the delay time to transmit the control signal u from the controller to the actuator.

5.5.2 DC motor modeling

The loop equation for the electrical circuit of the DC motor is [Chow and Tipsuwan, 2003]

ea = Ldiadt

+Ria +Kbω (5.39)

and the mechanical torque balance based on Newton’s law is

Jdω

dt+Bω + Tl = Kia, (5.40)

where ea is the armature input voltage, L is the armature inductance, ia is the armature current,

R is the armature resistance, J is the system moment of inertia, B is the system damping

coefficient, K and Kb are the torque constant and the back emf constant, respectively, Tl is the

load torque and ω is the angular velocity of the rotor. Note that the DC motor has a driven load,

that can be a robot arm or an unmanned electric vehicle.

Using u = ea as the control signal for the DC motor and introducing two state variables, the

armature current and the angular velocity of the rotor, i.e.,

x1 = ia, (5.41a)

x2 = ω, (5.41b)

the electro-mechanical dynamics of the DC motor can be described by the following continuous-

time affine state-space description

x(t) = Acx(t) + bcu(t) + fc,

y(t) = Ccx(t),(5.42)

where x(t) = ( x1(t) x2(t) )> is the system state, u(t) ∈ R is the system input, y(t) ∈ R2 is the

system output, Ac =

(−RL−Kb

LKJ−BJ

), bc =

(1L0

)and Cc = ( 1 0

0 1 ) are the system matrices and

137


fc =(

0TlJ

)is the affine term.

The armature voltage (control signal) for driving the motor is restricted by lower and upper

bounds as

emina ≤ u(t) ≤ emaxa , (5.43)

where emina and emaxa are the minimum and the maximum voltages that can be given as input

to system (5.42), respectively, in order to maintain the motor’s linear characteristics. Also, the

armature current and the rotor angular velocity are bounded, i.e.,

imina ≤ x1(t) ≤ imaxa , (5.44a)

ωmin ≤ x2(t) ≤ ωmax, (5.44b)

where imina and imaxa are the minimum and the maximum currents and ωmin and ωmax are the

lowest and the highest angular velocities of the rotor, respectively.

Discretizing model (5.42) yields

xmk+1 = Amd xmk + bmd u

mk + fmd , (5.45)

for all k ∈ Z+, where Amd and bmd are the discretized system matrices, fmd is the discretized

affine term and xmk and umk are the state and the input of the system at time instant k ∈ Z+.

The constraints (5.43) and (5.44) now become:

emina ≤ umk ≤ emaxa , (5.46)

imina ≤ xm1,k ≤ imaxa ,

ωmin ≤ xm2,k ≤ ωmax.(5.47)

The control objective is to reach a desired value of the rotor angular velocity, i.e., xss2 , as

fast as possible and with minimum overshoot. Note that for a desired rotor velocity value xss2 ,

one can obtain the steady-state armature current as

xss1 =B

Kxss2 −

1

KTl, (5.48)

138


and the steady-state armature voltage as

uss =

(RB

K+Kb

)xss2 −

R

KTl. (5.49)

To implement the one step ahead MPC algorithm, firstly, a coordinate transformation must

be performed on (5.45)

xk = xmk − xss, uk = umk − uss, (5.50)

where xss = ( xss1 xss2 )>, obtaining the following system description

xk+1 = Adxk + bduk, (5.51)

where Ad and bd are the discretized and transformed system matrices.

Using (5.50) and (5.51), the constraints given in (5.46) and (5.47) can be converted to:

x1,k ∈ [bx1 , bx1 ],

x2,k ∈ [bx2 , bx2 ],

uk ∈ [bu, bu],

(5.52)

for suitably chosen constants.

The control objective can now be formulated as to stabilize (5.51) around the equilibrium [0

0]> while fulfilling the constraints given in (5.52).

Now, considering that the DC motor, modeled as in (5.42), is controlled through a commu-

nication network, e.g., Ethernet, for which the structure is represented in Fig. 2.1, and applying

the procedure proposed and described in Section 5.2, the disturbed state-space representation is

x(t) = Acx(t) + bcu(t) + fc + w(t),

y(t) = Ccx(t),(5.53)

where the error caused by the time-varying delay introduced by the communication network is

modeled as a disturbance (Fig. 5.1) and computed as in (5.1), w(t) = bcud(t).


In this subsection the predictive controller presented in Section 5.4 is tested on the DC motor

model, subject to uncertain time-varying input and output delay. The network-based DC motor

139


0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

Time [s]

Com

mun

icat

ion

dela

ys [s

]

Figure 5.4: Delays introduced by the communication network.

control system was simulated using Matlab, where the sampling period of the system was cho-

sen as Ts = 0.02s and the values of the parameters used in simulations are given in Table 3 in

Appendix A.

The upper bound of the delays that are induced by Ethernet was chosen as τmax = 3Ts =

0.06s and it was considered that the delays are time-varying and uniformly distributed in the

interval [0, τmax], as shown in Fig. 5.4.

The one step ahead predictive controller was designed using the following weight matrices

of the cost (5.28): Px = 0.5 · I2, Qx = 0.1 · I2, R = 0.2, Λ = 0.1 · I2 and Λ = 0.1. The

technique presented in [M.Lazar 2006] was used for the off-line computation of the infinity

norm based local CLF V (x) = ‖Px‖∞ and the affine model of the DC motor in closed-loop

with uk := Kxk for Q = ( 1 00 0.01 ). The following matrices were obtained:

P =

(44.9069 1.8479

2.1884 3.1199

),

K =(−1.9784 −0.3707

).

(5.54)

The bounds for the constraints in (5.29) are defined as umin = emina −uss, umax = emax

a −uss,u∆ =∞, xmin =

(imina −xss1ωmin−xss2

)and xmax =

(imaxa −xss1ωmax−xss2

).


matrix P of the local CLF V (·), and it was never used for controlling the system.

For the proposed one step ahead MPC scheme, recursive feasibility implies asymptotic sta-

bility. However, recursive feasibility is not a priori guaranteed and hinges mainly on the con-

straint (5.23) on the future evolution of λ∗k. For the considered case study, the employedNλ = 4

proved to be large enough to guarantee recursive feasibility for a wide range of operating sce-

narios. The evolution of the CLF relaxation variable λ∗k and the corresponding upper bound

140


0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

Time [s]

Lam

bda


Figure 5.5: History of λ∗k.

0 0.2 0.4 0.6 0.8 10

50

100

150

200

250

Time [s]

Ang

ular

vel

ocity

[rad

/s]

referenceGielen2009aFCLF MPC

Figure 5.6: Rotor angular velocity.

defined by (5.23) for ρ = 0.99 and Nλ = 4 is shown in Fig. 5.5, for this particular simulation.

Therein, it can also be observed that λ∗k may increase or decrease as it is needed for the first

Nλ sampling periods after which it is allowed to increase only as long as this does not violate

the upper bound. However as k → ∞, λ∗k is forced to converge to 0, which in turn implies

input-to-state stability as guaranteed by Theorem 5.4.3.

In what follows the performance of the resulting closed-loop system is analyzed using the

trajectories plotted in Fig. 5.6, Fig. 5.7 and Fig. 5.8. Also, the time needed for computation

of the control input was calculated and resulted that the maximum search time is less than

0.5ms. Note that, although the proposed one step ahead controller takes into account in the

designing phase the time-varying delays that appear on Ethernet, the worst case time needed for

computation of a control input is much less than a sampling period. Also, a comparison is made

with the results obtained by applying the method proposed in [Gielen and M.Lazar, 2009a].

Fig. 5.6 shows the closed-loop response of the networked control system subject to track-

ing a 200rad/s reference signal, where one can see that the rotor angular velocity reference is

reached in a short time and with no overshoot.

The armature voltage is represented in Fig. 5.7, where it can be seen that the constraint on

141


0 0.2 0.4 0.6 0.8 1

−10

0

10

Time [s]

Arm

atur

e vo

ltage

[V]

Gielen2009aFCLF MPC

Figure 5.7: Armature voltage (control signal).

0 0.2 0.4 0.6 0.8 1−1

0

1

2

3

Time [s]

Arm

atur

e cu

rren

t [A

]

Gielen2009aFCLF MPC

Figure 5.8: Armature current.

the upper and lower bounds of the control signal are satisfied.

Note that, although the results are similar to the ones obtained in [Gielen and M.Lazar, 2009a],

in the case considered in this section the maximum delay is three times higher than the upper

bound of the delays considered in [Gielen and M.Lazar, 2009a].


In this subsection the predictive controller presented in Section 5.4 is tested on the DC motor

model, subject to uncertain time-varying input and output delay. Also, a PID controller was

designed and implemented to make a comparison between a conventional controller and an

advanced controller. The network-based DC motor control system was simulated using a True-

Time simulator developed in Matlab, where the sampling period of the system was chosen as

Ts = 0.02s and the values of the parameters used in simulations are given in Appendix A, Table

3.

142


5.5.4.1 Simulation environment

The PID and the MPC control strategies were simulated using TrueTime simulator [Henriks-

son et al., 2003], which is a very powerful MATLAB-based network simulation toolbox that

can effectively simulate real-time networked control systems. There are two primary Simulink

blocks in the TrueTime package: the computer block and the network block, both being easy

to customize in order to obtain a practical NCS. The computer block simulates an ECU with a

flexible real-time kernel, A/D and D/A converters, a network interface, and external interrupt

channels. The user has to write code functions, which define the evolution of the control tasks

and the interrupt handlers. The network block, which allows nodes to communicate over the

simulated network, is event-driven, so it executes every time a message enters or leaves the net-

work. To simulate a required network condition, different parameters such as: medium access

control (MAC) protocol (Ethernet, CAN, etc.), transmission rate, minimum frame size, etc., can

be customized [Henriksson et al., 2003].

In the designed NCS simulation platform (see Fig. 5.9), the sensors (Node 4, 5), controller

(Node 3) and actuator (Node 2) are implemented using computer blocks and the Ethernet com-

munication network is realized using a network block in which the MAC protocol is specified

as CSMA/CD. The sensors and the actuator are connected to the plant through their A/D con-

verters and D/A converter, respectively. The platform also includes an interfering node (Node

1) sending disturbing traffic over the network.


A discrete PID controller was designed and tuned to have a fast response, which yielded the

proportional, integral and derivative terms KR = 0.1, Ki = 0.25 and Kd = 0, respectively.

The one step ahead predictive controller was designed using the following weight matrices of

the cost (5.28): Px = 0.5 · I2, Qx = 0.1 · I2, R = 0.2, Λ = 0.1 · I2 and Λ = 0.1. The


norm based local CLF V (x) = ‖Px‖∞ and the affine model of the DC motor in closed-loop

with uk := Kxk for Q = ( 1 00 0.01 ). The following matrices were obtained:

P =

(44.9069 1.8479

2.1884 3.1199

),

K =(−1.9784 −0.3707

).

(5.55)


143


Network

(Ethernet )

Speed

reference

Node 5

(Sensor)

A/D

Snd

Node 4

(Sensor)

A/D

Snd

Node 3

(Controller )

Rcv

rSnd

Node 2

(Actuator )

Rcv

D/A

Node 1

(Interference )

Rcv

Snd

snd1

snd3

snd4

snd5

rcv1

rcv2

rcv3

DC motor model

u x

Figure 5.9: TrueTime Simulink diagram - DC motor.

matrix P of the local CLF V (·), and it was never used for controlling the system.

Clearly, no stability guarantee can be obtained for the DC motor system in closed-loop

with the PID controller. For the proposed one step ahead MPC scheme, recursive feasibility

implies asymptotic stability. However, recursive feasibility is not a priori guaranteed and hinges

mainly on the constraint (4.14) on the future evolution of λ∗k. Different experiments have been

conducted, simulating the DC motors behavior in response to accelerations and decelerations,

while the percentage of the network bandwidth occupied by the interfering node varies, which

are presented in the following. For the considered case studies, through extensive simulations,

the employed Ω = 15 proved to be large enough to guarantee recursive feasibility for a wide

range of operating scenarios.

In all figures (Fig. 5.10 - Fig. 5.12) in the top left sub-figures, the green (dashed) line repre-

sents the upper bound for variable λk and the black (continuous) line represents the actual value

of λ. In the top right sub-figures, the rotor angular velocity reference (black (dark) continuous

line), the DC motor output velocity for the PID controller (dashed line) and the DC motor output

velocity for the one step ahead predictive controller (red (light) continuous line) are represented

and in the middle left sub-figures, the armature voltage (control signal) for the two controllers

is illustrated. The middle right sub-figures illustrate the armature current for both controllers

and in the bottom sub-figures, the variable λ1 and λ2 are represented.

For the first simulation scenario (Fig. 5.10), the percentage of the network bandwidth occu-

144


0 1 2 30

20

Time [s]

λ

0 1 2 3−400

−200

0

200

400

Time [s]

ω[r

ad/s

]

0 1 2 3

−10

0

10

Time [s]

e a[V

]

0 1 2 3−5

0

5

Time [s]

i a[A

]

0 1 2 30

50

100

150

Time [s]

λ1

0 1 2 30

50

100

150

Time [s]

λ2

Figure 5.10: Simulation results: 10% network bandwidth occupied by the interfering node.

pied by the interfering node was set to 10%, for the second scenario (Fig. 5.11) to 40% and for

the third scenario (Fig. 5.12) to 70%.

A sequence of stairs was applied as reference for the rotor angular velocity and the one step

ahead predictive controller performances can be observed in the top right sub-figures: the ref-

erence is reached in a short time and with an overshoot less than 2% for the first two simulation

scenarios. For the third simulation scenario the overshoot increases and oscillations appear in

reaching the reference. In all simulation scenarios, the PID controller barely manages to cope

with the network-induced time-varying delays.

Note that, although the results obtained with the one step ahead predictive controller are

similar to the ones obtained in [Gielen and M.Lazar, 2009a], in this section the maximum

delay is three times higher (0.06s) in the second simulation scenario and seven times higher

(0.14s) in the third simulation scenario than the upper bound of the delays considered in [Gie-

len and M.Lazar, 2009a].

The evolution of the CLF relaxation variable λ∗k and the corresponding upper bound defined

by (4.14) for ρ = 0.99 and Ω = 15 are shown in the top left sub-figures. Therein, it can

be observed that λ∗k may increase or decrease as it is needed as long as this does not violate

the upper bound. Due to changing the reference rotor angular velocity, the upper bound of

the CLF relaxation variable defined by (4.14) may become unfeasible, so whenever a change

145


0 1 2 30

20

Time [s]

λ

0 1 2 3−400

−200

0

200

400

Time [s]

ω[r

ad/s

]

0 1 2 3

−10

0

10

Time [s]

e a[V

]

0 1 2 3−5

0

5

Time [s]

i a[A

]

0 1 2 30

50

100

150

Time [s]

λ1

0 1 2 30

50

100

150

Time [s]

λ2


0 1 2 30

20

Time [s]

λ

0 1 2 3−400

−200

0

200

400

Time [s]

ω[r

ad/s

]

0 1 2 3

−10

0

10

Time [s]

e a[V

]

0 1 2 3−5

0

5

Time [s]

i a[A

]

0 1 2 30

50

100

150

Time [s]

λ1

0 1 2 30

50

100

150

Time [s]

λ2


146

Robust LMPC for a CAN-controlled drivetrain including the clutch

in the reference rotor angular velocity occurs, the value of the upper bound is re-initialized.

However as k →∞, λ∗k is forced to converge to 0, which in turn implies input-to-state stability

as guaranteed by Theorem 5.4.3.

In all figures (Fig. 5.10 - Fig. 5.12) it can be seen that the constraints imposed on the upper

and lower bounds of the control signal (armature voltage) and the outputs (angular velocity and

armature current) are satisfied. Note that, although the PID controller does not enforce con-

straints on control command, its output was saturated in order to enforce the armature voltage

limits emina and emaxa .

In the bottom sub-figures it can be seen that, due to the continuously time-varying delays,

the optimization variables λ1 and λ2 reach values greater than 0 in steady-state.

Also, the time needed for computation of the control input for the one step ahead predictive

controller was calculated and resulted that the maximum search time is less than 0.5ms. Note

that, although the proposed one step ahead controller takes into account in the designing phase

the time-varying delays that appear on Ethernet, the worst case time needed for computation

of a control input is much less than a sampling period. Simulations were performed using the

CDD Dual Simplex LP solver in Matlab R2008a on Windows 7 running on a Pentium Dual

Core 2.00 GHz CPU and 3GB RAM mobile PC.

5.6 Robust LMPC for a CAN-controlled drivetrain including

the clutch

In this section, firstly, an accurate piecewise affine drivetrain model with three inertias is de-

rived and, secondly, a one step ahead robust predictive controller based on flexible Lyapunov

functions is designed. Several simulations based on realistic scenarios show that the proposed

control scheme can handle both the performance and physical constraints, and the strict limita-

tions on the computational complexity.

5.6.1 Discrete-time model

To obtain a discrete-time PWA model, each affine subsystem in (3.53) is discretized with sam-

pling period Ts using the Tustin transform, which yields

xmk+1 = Amdixmk + bmd u

mk + fmd if xk ∈ Ωi, (5.56)

147


0 2 4 6 8 101

2

3

4

Time [s]

Clu

tch

mod

e

discretecontinuous

0 2 4 6 8 100

10

20

30

40

50

Time [s]

Veh

icle

vel

ocity

[km

/h]

discretecontinuous

Figure 5.13: Clutch active mode and velocity histories.

for all k ∈ Z+, where Amdi and bmd are the corresponding discretized system matrices, fmd is the

discretized affine term and xmk , umk are the state and input of the system at time instant k ∈ Z+.

The active mode i is selected for the discrete-time PWA system using (3.54), the same as done

for the continuous-time PWA system. Letting I15 denote I5 with the first element on the diagonal

equal to zero, the reset condition (3.55) now becomes

∀k ∈ Z≥1, if (xmk−1, xmk ) ∈ Ω1 × Ω2, set xmk := I1

5xmk . (5.57)

Note that the utilized discretization method is more or less standard for PWA systems; for ex-

ample, it is also employed in the previous works on predictive control for driveline oscillations

damping [Saerens et al., 2008, Rostalski et al., 2007, Van Der Heijden et al., 2007]. A simu-

lation comparison, see Fig. 5.13, indicated that the so-obtained discretized model produces a

similar response as the original continuous-time model, for a sampling period of 5ms. This

value of the sampling period corresponds to an acceptable compromise between the industrial

standard sampling period for such an application (around 10ms) and the corresponding error in

terms of velocity and clutch active mode. The more frequent transitions between the closing

and the closed mode, and between the closed and the locked mode, for the discrete-time model,

which can be observed in Fig. 5.13, are due to the fact that within one sampling period the state

exceeds the corresponding mode thresholds.

148


The engine torque rate constraint now becomes

−T∆e ≤ ∆umk ≤ T∆

e , ∀k ∈ Z≥1, (5.58)

where ∆umk := umk − umk−1 and T∆e is the maximum allowed increase (decrease) in torque at

each sampling instant. Torque rate constraints are important to allow full usage of the airflow

to maintain the torque reserve, so that torque variations can be actuated instantaneously.

The desired objective is to reach a desired value of the wheel speed, i.e., xss5 , as fast as

possible and with minimum overshoot, while damping driveline oscillations. Note that for a

desired wheel speed value xss5 , one can obtain the corresponding steady-state values of the

system states and input, i.e., xss1 , xss2 , xss3 , xss4 and uss. As such, the above objective can be

formulated as asymptotic stabilization of the desired steady-state point while satisfying the

required constraints.

In what follows, for simplicity of exposition, a coordinate transformation is performed in

(5.56) to translate the problem into stabilization of the origin, i.e.,

xk = xmk − xss, uk = umk − uss, (5.59)

where xss =(xss1 xss2 xss3 xss4 xss5

)>, which yields the following system description:

xk+1 = Adixk + bdiuk + fdi if xk ∈ Ωi, (5.60)

along with the corresponding reset condition (5.57). Here Adi and bdi are the discretized and

transformed system matrices and fdi are the discretized and transformed affine terms. Notice

that the transformed PWA model (5.59) has zero as an equilibrium within region Ω2, i.e., fd2 =

0. Also, observe that uss can be interpreted as the feedforward component of the control action.

The control objective can now be formulated as to stabilize the discretized system (5.60)

around the zero equilibrium while fulfilling the system constraints.


In this subsection, the results obtained when applying the robustly stabilizing Lyapunov-based

predictive controller proposed in Section 5.4 on the drivetrain model described in Subsection

3.5.2 are presented. The proposed control structure was implemented in Matlab/Simulink and

the delays introduced by the communication network were simulated using Variable Time Delay

blocks.

149


The continuous-time PWA model (3.53)-(3.55) with

Ac1 = 104 ·

0 0 0.0001 −0.0004 0

0 0 0 0.0000 −0.0001

0 0 −0.0001 0 0

0 −8.3086 0 −0.0299 0.1080

0 0.0035 0 0.0000 −0.0000

,

Ac2 = 105 ·

0 0 0.0000 −0.0000 0

0 0 0 0.0000 −0.0000

−0.0471 0 −0.0002 0.0006 0

1.7215 −0.8309 0.0065 −0.0256 0.0108

0 0.0003 0 0.0000 −0.0000

,

Ac3 = 105 ·

0 0 0.0000 −0.0000 0

0 0 0 0.0000 −0.0000

−0.0941 0 −0.0004 0.0012 0

3.4431 −0.8309 0.0129 −0.0482 0.0108

0 0.0003 0 0.0000 −0.0000

,

Ac4 = 105 ·

0 0 0.0000 −0.0000 0

0 0 0 0.0000 −0.0000

−0.1882 0 −0.0006 0.0021 0

6.8862 −0.8309 0.0215 −0.0783 0.0108

0 0.0003 0 0.0000 −0.0000

,

bc =

0

0

5.8824

0

0

and fc =

0

0

0

0

−0.3103

(5.61)

was implemented in Matlab/Simulink and two different control strategies were applied to damp

driveline oscillations, i.e., the one step ahead robust predictive controller proposed in this thesis

and a PID controller. The sampling period of the system was chosen to be Ts = 5ms. The

values of the parameters used in simulations, which relate to a medium size passenger car, were

obtained with the help of Ford Research and Advanced Engineering, US, and they are given in

Table 2 in Appendix A.

150



ology described in Subsection 2.5.3.3, resulting that τmax = 3.4Ts = 0.017s < 4Ts, which is

used to apply the methodology described in Section 5.3 to model the effects of the variable time

delays as disturbances. Then, the bounds of the disturbances are explicitly taken into account

by the robust one step ahead predictive control strategy described in Section 5.4. The delays are

time-varying and uniformly distributed in the interval [0, τmax].


time, to increase the passenger comfort by reducing the oscillations that appear in the driveline.

The axle wrap is calculated as the difference between the engine speed (divided by the total

transmission ratio) and the wheel speed, and it is used as a measure of the driveline oscillations.

A PID controller was designed based on [O’Dwyer, 2006] and it was tuned to have a fast

response, which yielded the proportional, integral and derivative terms KR = 30, Ti = 10−3

and Td = 9 · 10−5, respectively.

Remark 5.6.1 The more common approach to the design of an explicit MPC controller was

also applied. For the considered discrete-time PWA model and operating constraints, using

the Multi Parametric Toolbox for Matlab, a feasible solution to the corresponding mpMILP

problem was only obtained for the prediction horizon equal to 1, but the resulting performance

was substandard. For a prediction horizon larger than 1, despite using a powerful working

station and several robust mpMILP solvers, a solution could not be obtained. This indicates the

non-trivial nature of the considered case study. 2

The one step ahead robust predictive controller proposed in this thesis uses the following weight

matrices of the cost (5.28): Px = 0.01I5, Qx = 0.1I5, R = 0.02, Λ = 1 and Λ = 1. The


norm based local CLF V (x) = ‖Px‖∞ and the PWA model of the drivetrain in closed-loop

151


with uk := Kixk if xk ∈ Ωi, i ∈ Z[1,4]. The following matrices were obtained

P =

38.5 −54.85 −1.44 1 −4.89

−26.9 18.13 0.91 −4.41 5.18

22.31 287.69 0.72 −0.53 −47.14

−62.38 −17.33 9.06 −9.02 26.48

25.92 263.26 −0.11 0.37 55.71

,

K1 =(

45.54 −17.58 −1.69 −8.42 46.33),

K2 =(

13.27 −30.15 −5.35 −6.83 94.15),

K3 =(

17.84 −26.83 −7.02 −6.78 88.49),

K4 =(

23.40 −30.89 −6.52 −7.21 31.03).

(5.62)

The above control law was only employed off-line, to calculate the weight matrix P of the

local CLF V (·), and it was never used for controlling the system.

The bounds for the constraints in (5.29) are defined as umin = 0− uss, umax = Tmaxe − uss,

u∆ = T∆e , xmin =

−∞−∞

ωmine −xss3−∞

ωminw −xss5

and xmax =

( ∞∞

ωmaxe −xss3∞ωmaxw −xss5

).

The following paragraph is dedicated to the stability of the resulting closed-loop system for

each technique. Clearly, no stability guarantee can be obtained for the PWA system in closed-

loop with the PID controller. For the one step ahead MPC scheme developed in this chapter,

recursive feasibility implies asymptotic stability. However, recursive feasibility is not a priori

guaranteed and hinges mainly on the constraint (5.24) on the future evolution of λ∗k. For all

simulation scenarios case studies, the values Ω = 15000 and M = 5 proved to be large enough

to guarantee recursive feasibility for the desired operating scenarios.

The worst case time (over numerous simulation scenarios) needed for computation of the

control input for the proposed one step ahead predictive controller was less than 2ms, which

meets the timing constraints.

Different simulations were conducted, to evaluate the vehicle behavior in response to accel-

eration, deceleration, tip-in and tip-out maneuvers and a stress test, which are presented in the

following subsections. Note that, although the PID controller does not enforce constraints on

control command, its output was saturated in order to enforce the engine limitations, i.e., the

torque limit Tmaxe .

In all figures (Fig. 5.14 - Fig. 5.18), the thick black dashed-line represents the vehicle

velocity reference, the light blue line represents the PID results and the dark red line represents

152


0 5 10 15 200

2000

4000

6000

8000

10000

Time [s]

Lam

bda

0 5 10 15 200

10

20

30

40

Time [s]

Veh

icle

vel

ocity

[km

/h]

0 5 10 15 20−100

−50

0

50

100

Time [s]

Axl

e w

rap

angu

lar

spee

d [r

pm]

0 5 10 15 200

50

100

150

200

Time [s]

Eng

ine

torq

ue [N

m]

Figure 5.14: Scenario 1: Acceleration test.

0 5 10 15 201

2

3

4

Time [s]

Clut

ch m

ode

Figure 5.15: Scenario 1: Clutch mode of operation.

the one step ahead predictive controller results. Also, in the top left sub-figure of each figure,

the green dashed-line represents the upper bound for the variable λk given by (5.24) and the

black line represents the optimal value λ∗k.

5.6.2.1 Scenario 1: Acceleration test

A first simulation test is performed on an acceleration scenario where the vehicle has to accel-

erate from 0 km/h to 30 km/h.

In what follows the performance of the resulting closed-loop systems for the acceleration

scenario is analyzed using the trajectories plotted in Fig. 5.14. The amplitude of the axle wrap

is represented in Fig. 5.14, bottom left. The evolution of the CLF relaxation variable λ∗k and the

corresponding upper bound defined by (5.24) for ρ = 0.99, Ω = 15000 and M = 5 is shown

in Fig. 5.14, top left. It can be observed that λ∗k may decrease or even go to 0, after which

153


0 5 10 15 200

10

20

30

40

50

Time [s]

Lam

bda

0 5 10 15 200

10

20

30

40

Time [s]

Veh

icle

vel

ocity

[km

/h]

0 5 10 15 20−20

−10

0

10

20

Time [s]

Axl

e w

rap

angu

lar s

peed

[rpm

]

0 5 10 15 200

20

40

60

Time [s]

Eng

ine

torq

ue [N

m]

Figure 5.16: Scenario 2: Deceleration test.

it is allowed to increase again, as long as this does not violate the upper bound. However as

k →∞, λ∗k is forced to converge to 0. In Fig. 5.15 the clutch mode history was represented for

the PID controller and for the one step ahead predictive controller, to show that in the transient

the closed-loop system frequently switches between the operating modes.

5.6.2.2 Scenario 2: Deceleration test

The second simulation scenario consist of decelerating the vehicle from 30 km/h to 10 km/h and

the results obtained are illustrated in Fig. 5.16. Although both controllers obtained almost the

same settling time, the PID controller produces some undesired axle wrap oscillations, indicat-

ing that the one step ahead predictive controller has a superior behavior in terms of damping the

drivetrain oscillations. The evolution of the CLF relaxation variable λ∗k and the corresponding

upper bound is illustrated in Fig. 5.16, top left. Although the upper bound starts from 15000,

due to Ω, only the values below 50 were plotted. This makes it possible to observe the evolution

of λ∗k.

5.6.2.3 Scenario 3: Tip-in tip-out maneuvers

The results of a tip-in, tip-out maneuver simulation, in which the reference vehicle velocity goes

from 30 km/h to 10 km/h and back to 30 km/h, are presented in Fig. 5.17. It can be seen that

the one step ahead predictive controller has a slightly faster response with no overshoot when it

approaches the reference velocity. Moreover, the oscillations of the axle wrap are damped much

faster in the acceleration phase. In the deceleration phase, again, the PID controller produces

154


0 10 20 30 40 50 600

50

100

150

200

Time [s]

Lam

bda

0 10 20 30 40 50 600

10

20

30

40

Time [s]

Veh

icle

vel

ocity

[km

/h]

0 10 20 30 40 50 60−50

0

50

Time [s]

Axl

e w

rap

angu

lar s

peed

[rpm

]

0 10 20 30 40 50 600

50

100

150

200

Time [s]

Eng

ine

torq

ue [N

m]

Figure 5.17: Scenario 3: Tip-in, tip-out maneuver simulation.

undesired oscillations of the axle wrap. Note that the controller performance during decelera-

tion is limited by the actuator authority. For this experiment the evolution of the CLF relaxation

variable λ∗k and the corresponding upper bound defined by (5.24) are shown in Fig. 5.17, top

left. Due to changing the reference vehicle velocity, the upper bound of the CLF relaxation

variable defined by (5.24) may become unfeasible, so whenever a change in the reference vehi-

cle velocity occurs, the value of the upper bound was re-initialized. However as k → ∞, λ∗k is

forced to converge to 0.

5.6.2.4 Scenario 4: Stress test

The results of a stress test, in which the reference velocity is a square wave that changes rapidly

between 30 km/h and 20 km/h, are presented in Fig. 5.18. The purpose is to check what happens

to the axle wrap speed if it does not have enough time to settle between two set-point changes,

which means that continuous perturbations may occur. The results illustrate how the one step

ahead predictive controller has again a smaller amplitude for the axle wrap angular speed, while

the PID barely manages to cope with this kind of maneuver. Whenever a change in the reference

vehicle velocity occurs, the value of the upper bound was re-initialized as done in the previous

scenario. This is not visible in Fig. 5.18, top left, because the upper bound starts again from 500

and it does not reach values below 50 in such a short amount of time.

155


0 1 2 3 4 5 60

10

20

30

40

50

Time [s]

Lam

bda

0 1 2 3 4 5 620

25

30

35

40

Time [s]

Veh

icle

vel

ocity

[km

/h]

0 1 2 3 4 5 6−50

0

50

Time [s]

Axl

e w

rap

angu

lar s

peed

[rpm

]

0 1 2 3 4 5 60

50

100

150

200

Time [s]

Eng

ine

torq

ue [N

m]

Figure 5.18: Scenario 4: Stress test.

Node 8

(Interference )

Rcv

Snd

Node 7

(Actuator )

Rcv

D/A

Node 6

(Controller )

Rcv

r

Snd

Node 5

(Sensor )

A/D

Snd

Node 4

(Sensor )

A/D

Snd

Node 3

(Sensor )

A/D

Snd

Node 2

(Sensor )

A/D

Snd

Node 1

(Sensor )

A/D

Snd

Networksnd1

snd2

snd3

snd4

snd5

snd6

snd8

rcv6

rcv7

x_5x_4x_3x_1 x_2

u

Drivetrain model

ux

Figure 5.19: TrueTime Simulink diagram - three inertias drivetrain model.


The one step ahead predictive controller was also tested with TrueTime simulation Toolbox.

In the designed NCS simulation platform (see Fig. 5.19), the sensors (Node 1, 2, 3, 4, 5),

controller (Node 6) and actuator (Node 7) are implemented using computer blocks and the

CAN communication network is realized using a network block in which the MAC protocol is

specified as CSMA/AMP. The sensors and the actuator are connected to the plant through their

A/D converters and D/A converter, respectively. The platform also includes an interfering node

(Node 8) sending disturbing traffic over the network.

The one step ahead predictive controller proposed in this thesis uses the following weight

matrices of the cost (5.28): Px = 0.0001I5, Qx = 0.002I5, R = 0.25, Λ = 0.01 and Λ = 1.

156


0 2 4 6 8 100

1

2

3x 10

4

Time [s]

Lam

bda

0 2 4 6 8 100

10

20

30

Time [s]

Veh

icle

vel

ocity

[km

/h]

0 2 4 6 8 10−100

−50

0

50

100

Time [s]

Axl

e w

rap

angu

lar s

peed

[rpm

]

0 2 4 6 8 100

50

100

150

200

Time [s]

Eng

ine

torq

ue [N

m]

Figure 5.20: TrueTime Scenario 1: Acceleration test.

0 2 4 6 8 101

2

3

4

Time [s]

Clut

ch m

ode

Figure 5.21: TrueTime Scenario 1: Clutch mode of operation.

For all simulation scenarios case studies, the values Ω = 15000 and M = 5 proved to be

large enough to guarantee recursive feasibility for the desired operating scenarios.

It can be seen in Figs. 5.20 - 5.24 that the obtained results are similar to the ones obtained

with Matlab/Simulink.

5.6.4 Real-time results

The robust one step ahead predictive controller was also tested using a real-time simulation

test-bench similar to the one described in Chapter 3, Section 3.5.

The predictive controller proposed in this paper uses the following weight matrices of the

cost (5.28): Px = 0.01I5, Qx = 0.1I5, R = 0.2, Λ = 0.01 and Λ = 1.

For the robust one step ahead MPC scheme developed in this chapter, recursive feasibility

implies asymptotic stability. However, recursive feasibility is not a priori guaranteed and hinges

157


0 2 4 6 8 100

100

200

300

400

500

Time [s]

Lam

bda

0 2 4 6 8 100

10

20

30

40

Time [s]

Veh

icle

vel

ocity

[km

/h]

0 2 4 6 8 10−20

−10

0

10

20

Time [s]

Axl

e w

rap

angu

lar s

peed

[rpm

]

0 2 4 6 8 100

20

40

60

Time [s]

Eng

ine

torq

ue [N

m]

Figure 5.22: TrueTime Scenario 2: Deceleration test.

0 5 100

1000

2000

3000

4000

5000

Time [s]

Lam

bda

0 5 100

10

20

30

40

Time [s]

Veh

icle

vel

ocity

[km

/h]

0 5 10−50

0

50

Time [s]

Axl

e w

rap

angu

lar s

peed

[rpm

]

0 5 100

50

100

150

200

Time [s]

Eng

ine

torq

ue [N

m]

Figure 5.23: TrueTime Scenario 3: Tip-in, tip-out maneuver simulation.

158

Conclusions

0 1 2 3 4 5 60

100

200

300

400

500

Time [s]

Lam

bda

0 1 2 3 4 5 620

25

30

35

Time [s]

Veh

icle

vel

ocity

[km

/h]

0 1 2 3 4 5 6−50

0

50

Time [s]

Axl

e w

rap

angu

lar s

peed

[rpm

]

0 1 2 3 4 5 60

50

100

150

200

Time [s]

Eng

ine

torq

ue [N

m]

Figure 5.24: TrueTime Scenario 4: Stress test.

mainly on the constraint (5.24) on the future evolution of λ∗k. For all simulation scenarios case

studies, the values Ω = 15000 and M = 5 proved to be large enough to guarantee recursive

feasibility for the desired operating scenarios.

The results of a tip-in, tip-out maneuver simulation, in which the reference vehicle velocity

goes from 0 km/h to 30 km/h to 10 km/h and back to 30 km/h, are presented in Fig. 5.25. The

vehicle velocity reference and the response of the system are illustrated in Fig. 5.25, top right,

where it can be seen that the system reaches the reference in a short time, having no overshoot

when it approaches the reference velocity. The amplitude of the axle wrap is represented in

Fig. 5.25, top right. In Fig. 5.25, bottom left, the engine torque (control command) is illus-

trated and it can be seen that the lower and upper constraints are satisfied. The clutch mode is

represented in Fig. 5.25, bottom right.

5.7 Conclusions

In this chapter, a robustly stabilizing MPC algorithm for delay-perturbed systems is proposed.

Firstly, the error caused by the time-varying network-induced delays is considered as a dis-

turbance and two novel methods of finding the bounds of the disturbance are proposed for

controller design purposes. Secondly, an one step ahead MPC scheme is designed using the

concept of flexible control Lyapunov functions [M.Lazar 2009] that can handle the perfor-

mance/physical constraints and that explicitly accounts for rejection of disturbances introduced

by the time-varying delay in the communication network and do not require a precise model

159


10 20 30 40 500

10

20

30

Time [s]

Veh

icle

vel

ocity

[km

/h]

10 20 30 40 50−100

−50

0

50

100

Time [s]

Axl

e w

rap

angu

lar s

peed

[rpm

]

10 20 30 40 500

50

100

150

200

Time [s]

Eng

ine

torq

ue [N

m]

10 20 30 40 501

2

3

4

Time [s]

Clu

tch

mod

eFigure 5.25: Real-time results: Tip-in, tip-out maneuver simulation.

of time delay. Also, a flexible control Lyapunov function was employed to obtain a non-

conservative ISS stability guarantee for the developed one step ahead MPC scheme. Moreover,

it is shown that for an appropriately chosen Lyapunov function, the MPC algorithms amount

to solving a single, low-complexity linear program each sampling instant. This means that the

proposed LMPC algorithm can be applied for the class of fast processes, which require short

sampling periods. Furthermore, a new limiting condition on the auxiliary optimization variable

that adds flexibility to the CLF is proposed.

The proposed modeling method and control strategy were applied to control through Ether-

net the angular velocity of a DC motor, which is one of the most popular devices for actuation

and propulsion systems. The results obtained using the Matlab/Simulink and TrueTime simula-

tors for different simulation scenarios illustrate the effectiveness and robustness of the proposed

delay modeling and control strategy and that the proposed controller has good performances

and it meets the required timing constraints.

The proposed control solution was tested on the validated vehicle drivetrain model inte-

grated with CAN and the simulation scenarios indicate that the proposed scheme, besides yield-

ing a feasible algorithm, outperforms controllers obtained using classical approaches, such as

explicit MPC and PID control. Moreover, the performed tests indicate that the proposed design

methodology can handle both the performance/physical constraints and the strict limitations on

the computational complexity corresponding to vehicle drivetrain oscillations damping. Fur-

thermore, the proposed robust one step ahead predictive control strategy was tested on a real-

time simulation test-bench and the obtained results illustrate the potential of the proposed con-

troller for real-time applications, indicating that the strategy can be implemented to control the

160

Conclusions

actual physical plant.The results presented in this chapter were published in:

• [Caruntu et al., 2010a]: C. F. Caruntu, A. E. Balau and C. Lazar. Cascade based Control of

a Drivetrain with Backlash. In 12th International Conference on Optimization of Electrical and

Electronic Equipment, Brasov, Romania, 2010.

• [Caruntu et al., 2011a]: C. F. Caruntu, A. E. Balau, M. Lazar, P. P. J. van den Bosch and S. Di

Cairano. A Predictive Control Solution for Driveline Oscillations Damping. In Hybrid Systems:

Computation and Control, pages 181–190, Chicago, USA, 2011.

• [Caruntu and C.Lazar, 2011e]: C. F. Caruntu and C. Lazar. Stabilizing MPC for network-

controlled systems with an application to DC motors. In IEEE International Conference on

Mechatronics, Istanbul, Turkey, 2011.

• [Caruntu and C.Lazar, 2011d]: C. F. Caruntu and C. Lazar. Robustly stabilizing MPC design for

networked control systems with an application to DC motors. IET Control Theory and Applica-

tions, 2011.

• [Balau, Caruntu and C.Lazar, 2011a]: A. E. Balau, C. F. Caruntu and C. Lazar. Driveline oscilla-

tions modeling and control. In 18th International Conference on Control Systems and Computer

Science, pages 332–338, Bucuresti, Romania, 2011.

• [Caruntu and C.Lazar, 2011c]: C. F. Caruntu and C. Lazar. Robust MPC for TrueTime Simula-

tion of a Vehicle Drivetrain Controlled through CAN. In 16th IEEE International Conference on

Emerging Technologies and Factory Automation, Toulouse, France, 2011.

• [Caruntu et al., 2011i]: C. F. Caruntu, D. Onu and C. Lazar. Real-time Simulation of a Vehicle

Drivetrain Controlled through CAN using a Robust MPC Strategy. In 15th International Confer-

ence on System Theory, Control and Computing, Sinaia, Romania, 2011.

161


162

Chapter 6

Conclusions

From the wide variety of fast processes, in this thesis, the focus is on the subclass of automotive

systems, which have fast dynamics, requiring short sampling periods.

Networks and their applications are promising for wide deployment of future industrial ap-

plications. There are still many important issues that need to be addressed so that economical

and reliable network-based control can be used much more extensively. The adverse effect of

the network on the system performance lead researchers seek for new techniques to compen-

sate for this effect. It is noticeable that these techniques can have their own advantages and

disadvantages. In order to apply one of these techniques, designers have to consider both the

network and the control algorithms. Nevertheless, none of these techniques is perfect because

many assumptions are extensively used.

In this chapter, a summary of the main contributions proposed in this thesis and several

possible directions for future research are presented.

6.1 Contributions

The major contributions are in the domains of

• Delay modeling;

• Linear networked predictive control;

• Nonlinear and Robust nonlinear predictive control;

• Automotive networked applications.

6.1.1 Delay modeling

In this thesis, different delay modeling methods for controller design purposes were proposed.

163

Conclusions

Firstly, it is assumed that the delays in the communication network are bounded and three

methods of considering the delays by the predictive control algorithm are proposed: average

method, identification method and adaptation method. The average method considers the mean

value of the delays that can appear in the communication network. For the identification method,

the delay is considered equal to the minimum delay that can appear in the communication

network and a model that includes the time-varying delays in the plant model is proposed.

The adaptation method is designed to adapt the control algorithm to the network-induced time-

varying delays at each sampling instant.

The second modeling method considers a polytopic approximation technique to obtain a

discrete-time model of the closed-loop CAN system. This technique requires an estimation of

the maximum delay that can be induced by a CAN, which enables the usage of the modeling

technique developed in [Gielen et al., 2010] for networked control systems (NCS) via polytopic

approximations.

The third method considers the error caused by the time-varying delays as a disturbance

and two novel methods of bounding the disturbances are proposed. The bounds are used in the

design phase of a robustly stabilizing predictive controller.

6.1.2 Linear networked predictive control

In this thesis, a new networked predictive control strategy is proposed with the aim of control-

ling the output of a physical plant, while compensating the effects of the time-varying delay.

The strategy is based on the delay modeling methods proposed in Section 3.2: average, identifi-

cation and adaptation method. The average delay modeling method considers the average value

of the minimum and the maximum delays that can appear in the communication network. The

identification modeling method considers that the delay is equal to the minimum delay that can

appear in the communication network and uses a different model of the physical plant that is

identified in order to capture the dynamics of the system including the delays between the min-

imum and the maximum delays that can appear in the communication network. The adaptation

method is designed in order to adapt the control algorithm to the network-induced time-varying

delays. The adaptation algorithm is derived with assumption that the average communication

delays from sensor-to-controller and from controller-to-actuator are equal, both being variables.

Unlike for the first two modeling methods, in the case of the adaptation method there is no need

to know the upper bound of the delays. Moreover, a new method by which the controller is

adapted to the difference between the desired reference and the plant output is proposed.

The LNPC can be analytically determined off-line, which means that the actual control com-

164

Contributions

mand can be computed very fast, so this method is capable of satisfying the timing constraints

associated to the fast processes that require short sampling periods, which are used as case stud-

ies to prove the performances of the proposed delay modeling methods and predictive control

strategy.

The strategy was then applied to control several physical plants from automotive applica-

tions (DC motor, ICE, valve-clutch system, drivetrain) with the aim of decreasing the influence

of the communication time-varying delays on the closed-loop control performances over CAN.

Comparisons were made with a PI controller and a Smith-like predictive controller with adapta-

tion to communication delay developed in [Velagic, 2008], in order to illustrate the performance

of the proposed approaches. The experiments designed to test the strategies developed in this

thesis verify the better performances of the proposed methods. The proposed linear networked

predictive control strategy was also tested on a real-time test-bench and the obtained results

prove that it can be implemented to control the actual physical plant (drivetrain).

6.1.3 Nonlinear and Robust nonlinear networked predictive control

Current available control techniques generally focus on conventional control techniques. In or-

der to consider the uncertainty of communication delays, computational intelligent approaches

as Lyapunov-based model predictive control can be used. These are supposed to work in sit-

uations where there is a large uncertainty and/or unknown variation in plant parameters and

structure.

The fourth chapter proposed a novel method of modeling the CAN-induced delays for con-

trol purposes and a one step ahead MPC strategy that can handle both the performance/physical

constraints and the time-varying delays. A flexible Lyapunov function was employed to obtain

a non-conservative stability guarantee for the developed one step ahead MPC scheme. The re-

sults obtained with this method applied to minimize the drivetrain oscillations were compared

with the results obtained with other strategies (PID, explicit MPC) and the comparison illus-

trates that the proposed controller has an overall superior performance and it meets the required

timing constraints.

In the fifth chapter, a novel method of bounding the disturbances caused by the network-

induced time-varying delays for controller design purposes and a robust one step ahead MPC

strategy, that can handle the performance/physical constraints and that can explicitly take into

account the bounds of the disturbances caused by the time-varying delays, are proposed. Also, a

flexible Lyapunov function was employed to obtain a non-conservative ISS stability guarantee

for the developed one step ahead MPC scheme. Moreover, it is shown that by choosing an

165

Conclusions

appropriately Lyapunov function, the MPC algorithm amounts solving a single, low-complexity

linear program each sampling instant.

Two new limiting conditions on the auxiliary optimization variable that adds flexibility to

the CLF were proposed.

The possibility of implementing the LMPC and Robust LMPC algorithms for fast processes

is achieved via simpler stabilizing and performance constraints, that can be implemented as a

finite number of linear inequalities. The LP problems equivalent to the LMPC optimization

problem in Step 1 of Algorithm 4.3.6 and Robust LMPC optimization problem in Step 1 of Al-

gorithm 5.4.7 were always solved, for the considered case studies, within the allowed sampling

interval.

The proposed modeling method and control strategy were applied to control through Eth-

ernet the angular velocity of a DC motor and the results obtained using Matlab/Simulink and

the TrueTime simulator for different simulation scenarios illustrate that the proposed controller

has good performances and it meets the required timing constraints. Furthermore, the controller

needs only an approximation of the maximum delay time that can be introduced by the com-

munication network and does not require a precise model of it, therefore it can be applied to

control systems with time-varying and unpredictable time delay, the controller being robust to

these types of time delays.

6.1.4 Automotive networked applications

Until now, different strategies for controlling the automotive systems and different methods to

compensate the delays that appear in communication networks were reported in the literature,

but none in which these systems are considered components of a networked control system,

being controlled through a communication network.

The proposed LNPC strategy was applied to control several physical plants from automotive

applications (DC motor, ICE, valve-clutch system, drivetrain) and to decrease the influence of

the variable-time delay induced in the NCS on the control performance. The performance of the

proposed strategy is demonstrated by simulation results and corresponding comparisons prove

the significance of this method.

The one step ahead predictive control strategy proposed in Chapter 4 was applied to mini-

mize the drivetrain oscillations, while compensating the CAN-induced time-varying delays via

the modeling technique proposed in Section 4.2.

Several TrueTime simulation experiments defined in collaboration with Ford Research and

Advanced Engineering, US validate the proposed nonlinear networked predictive control ap-

166

Future research

proach, indicating that the developed scheme has the potential to meet the required real-time

control specifications. The simulation results also indicate that the proposed scheme can out-

perform other types of controllers, such as PID or explicit MPC.

Moreover, the predictive control strategy was experimentally verified using a HIL test-bench

consisting of a Freescale based electronic control unit (which implements the controller) linked

via a CAN bus with a dSPACE MicroAutoBox plant simulator. The experimental results illus-

trate that the proposed scheme satisfies the required timing constraints.

The robust nonlinear predictive control methodology was applied for a network-controlled

DC motor and the Matlab/Simulink and TrueTime simulations illustrate the effectiveness and

robustness of the proposed delay modeling and control strategy. Moreover, the modeling method

and the control strategy were tested on a vehicle drivetrain controlled through CAN, with the

aim of damping driveline oscillations, which is crucial in improving driveability ans passen-

ger comfort. Several Matlab/Simulink and TrueTime simulations based on realistic scenarios

show that the proposed control scheme can handle the performance/physical constraints. Fur-

thermore, the drivetrain control strategy was tested on a real-time test-bench, the experimental

results showing that the control strategy can handle the strict limitations on the computational

complexity.

The last contribution is concerned with the development of a novel, more accurate state-

space piecewise affine drivetrain model with three inertias, in which both driveshafts and clutch

flexibilities were considered.

6.2 Future research

The research presented in this thesis provides several interesting research directions, which are

briefly described in the following:

• In Chapter 3, Section 3.2, Chapter 4, Section 4.2 and Chapter 5, Sections 5.2 and 5.3,

several modeling techniques for the network-induced time-varying delays are proposed,

but it is assumed that the delays introduced by the communication network are bounded

and the bounds are used in the controller design. Still, for the adaptation method (Sub-

section 3.2.3), designed in order to adapt the control algorithm to the network-induced

time-varying delays, there is no need to know the exact upper bound of the delays, but

just only an approximation of it. The remaining questions are how to predict the de-

lay without a model and how to design a controller without knowing the bounds of the

time-varying delays ?

167

Conclusions

• In Chapter 4, Section 4.2, only input delays are considered, but, it is well known, that

in NCSs also output delays are present. So, the task will be to consider output delays in

using the polytopic over-approximation technique [Gielen et al., 2010].

• The identification method proposed in Chapter 3, Subsection 3.2.2, which uses a different

model of the physical plant that is identified in order to capture the dynamics of the system

including the delays between the minimum and the maximum delays that can appear in

the communication network could be extended to the state-space models, which can be

further used by the LMPC strategy.

• There are many network-induced imperfections and constraints which are categorized in

five types in [Heemels et al., 2010], but, usually, the available literature on NCS considers

only some of them in the analysis of NCS. Throughout this thesis, only network-induced

time-varying delays are considered in the controller design. It would be also useful to

consider data-packet dropouts, quantization errors, variable sampling/transmission inter-

vals and the network load (limited communication) in the controller design.

• Although the case studies considered throughout this thesis are mainly fast processes

from automotive applications, the theoretical results presented could be applied for other

network-controlled applications from different industry fields: aeronautics, automated

manufacturing facilities, military unmanned vehicles, telerobotics, telemedicine.

• The communication protocols used for the simulations and experiments illustrated in this

thesis are CAN and Ethernet. Also, the theoretical results presented could be applied for

other communication protocols and even for wireless networks.

168

Appendix A

Table 1: Valve-clutch parameter values

Symbol Value UnitKe 1000 [N/m]K 900 [N/m]Mv 25e-3 [kg]βe 1.6e+9 [N/m2]

KC = KD 7.58e-11 [(m3/s)/(N/m2)]K1 5.50e-10 [(m3/s)/(N/m2)]K2 3.52e-9 [(m3/s)/(N/m2)]K3 1.26e-8 [(m3/s)/(N/m2)]Kq 5.3418 [(m3/s)/(N/m2)]w 3e-3 [m]PS 1e+6 [N/m2]PT 0 [N/m2]kl 2e-9 [(m3/s)/(N/m2)]VC 7.53e-8 [m3]VD 1.04e-7 [m3]Vt 3.2e-4 [m3]VL 2.51e-5 [m3]AC 3.66e-5 [m2]AD 2.94e-5 [m2]AL 7.75e-4 [m2]Mp 0.5 [kg]ka 0.005 [Nm2/A2]kb 0.01 [m]Ls 0.01 [H]Rs 0.5 [Ω]

169

Appendix A

Table 2: Simulation vehicle parameter values

Symbol Value Measure Unit DescriptionJe 0.17 [kg m2] Engine inertiaJt 0.014 [kg m2] Transmission inertiaJf 0.031 [kg m2] Final drive inertiaJw 1 [kg m2] Wheel inertiadd 65 [Nms/rad] Flexible driveshaft dampingkd 5000 [Nm/rad] Flexible driveshaft stiffnessde 0.159 [Nms/rad] Engine dampingdt 0.1 [Nms/rad] Transmission dampingdf 0.1 [Nms/rad] Final drive dampingdw 0.1 [Nms/rad] Wheel dampinggt 3.5 Gearbox ratio (1st gear)gf 3.7 Final drive ratio

mCOG 1400 [kg] Vehicle massrw 0.32 [m] Wheel radiuscr1 0.01 [Nm/kg] Rolling coefficientcr2 0.36 [Nms/rad] Approximation coefficientcd 0.3 [rad−2] Airdrag coefficientρair 1.2 [kg/m3] Air densityAf 2.7 [m2] Frontal area of the vehicleg 9.8 [m/s2] Gravitational acceleration

χroad 0 [rad] Road slopeθ1 0.1745 [rad] Clutch switching boundaryθ2 0.2094 [rad] Clutch switching boundarydc1 0 [Nms/rad] Clutch damping (open)dc2 3 [Nms/rad] Clutch damping (closing)dc3 6 [Nms/rad] Clutch damping (closed)dc4 10 [Nms/rad] Clutch damping (locked)kc1 0 [Nm/rad] Clutch stiffness (open)kc2 800 [Nm/rad] Clutch stiffness (closing)kc3 1600 [Nm/rad] Clutch stiffness (closed)kc4 3200 [Nm/rad] Clutch stiffness (locked)

Tmaxe 200 [Nm] Maximum engine torqueT∆e 6 [Nm] Maximum engine torque increase/decrease

ωmine 62.83 [rad/s] Engine idle speed

ωclosinge 83.76 [rad/s] Engine closing speedωmaxe 628.3 [rad/s] Maximum engine speedωminw 0 [rad/s] Minimum wheel speed

ωmaxw 250 [rad/s] Maximum wheel speed

170

Table 3: Simulation DC motor parameter values

Symbol Value DescriptionJ 42.6 · 10−6 kg-m2 InertiaL 170 · 10−3 H InductanceR 4.67 Ω Terminal resistanceB 47.3·10−6 N-m-s/rad Damping coefficientK 14.7 · 10−3 N-m/A Torque constantKb 14.7 · 10−3 V-s/rad Back-EMF constantTl 0 N-m Load torque

emina -15 V Minimum armature voltageemaxa 15 V Maximum armature voltageimina -5 A Minimum armature currentimaxa 5 A Maximum armature currentωmin -500 rad/s Minimum angular velocityωmax 500 rad/s Maximum angular velocity

171

Appendix A

Table 4: Simulation vehicle parameter values

Symbol Value Measure Unit DescriptionJe 0.184 [kg m2] Engine inertiaJg 1.1828 [kg m2] Transmission parts inertiaJw 5.38 [kg m2] Wheel inertiadd 42 [Nms/rad] Flexible driveshaft dampingkd 6000 [[Nm/rad] Flexible driveshaft stiffnessde 0.15 [Nms/rad] Engine damping coefficientda 0.26 [Nms/rad] Approximation factorig 3.778 Gearbox ratio (1st gear)if 3.667 Final drive ratiomv 1094 [kg] Vehicle massrw 0.281 [m] Wheel radiuscr 0.01 Rolling constantcd 0.3 [rad−2] Airdrag coefficientρair 1.2 [kg/m3] Air densityAf 2.7 [m2] Frontal area of the vehicleg 9.8 [m/s2] Gravitational acceleration

χroad 0 Hill gradientTmaxe 120 [Nm] Maximum engine torqueT∆e 2.5 [Nm] Maximum engine torque increase/decrease

ωmine 62.83 (600) [rad/s] ([rpm]) Engine idle speed

ωmaxe 523.6 (5000) [rad/s] ([rpm]) Maximum engine speedωminw 0 (0) [rad/s] ([km/h]) Minimum wheel speed

ωmaxw 247.1 (250) [rad/s] (km/h]) Maximum wheel speed

172

References

[Angleviel et al., 2006] D. Angleviel, J. D. Alzingre, L. Billet and D. Franchon. Development

of a contactless brushless DC actuator for engine management. In SAE 2006 World

Congress & Exhibition, Michigan, USA, 2006. 59

[Anwar, 2007] S. Anwar. Predictive Yaw Stability Control of a Brake-by-Wire Equipped Vehicle

via Eddy Current Braking. In American Control Conference, New York, USA, 2007.

40

[Balau, Caruntu et al., 2009a] A. E. Balau, C. F. Caruntu, C. Lazar and D. I. Patrascu. New

Model for Predictive Control of an Electro-Hydraulic Actuated Clutch. In 8th Interna-

tional Conference Fuel Economy, Safety and Reliability of Motor Vehicles, volume 1,

pages 463–472, Bucuresti, Romania, 2009. 9, 67, 70, 87

[Balau, Caruntu et al., 2009b] A. E. Balau, C. F. Caruntu, D. I. Patrascu, C. Lazar, M. H. Mat-

covschi and O. Pastravanu. Modeling of a Pressure Reducing Valve Actuator for Auto-

motive Applications. In 18th IEEE International Conference on Control Applications,

Part of 2009 IEEE Multi-conference on Systems and Control, Saint Petersburg, Russia,

2009. 9, 67, 68, 87

[Balau, Caruntu and C.Lazar, 2011a] A. E. Balau, C. F. Caruntu and C. Lazar. Driveline oscil-

lations modeling and control. In 18th International Conference on Control Systems and

Computer Science, pages 332–338, Bucuresti, Romania, 2011. 11, 161

[Balau, Caruntu and C.Lazar, 2011b] A. E. Balau, C. F. Caruntu and C. Lazar. Simulation and

Control of an Electro-Hydraulic Actuated Clutch. Mechanical Systems and Signal Pro-

cessing, vol. 25, pages 1911–1922, 2011. 10, 53, 68, 69, 70, 88

[Baumann et al., 2006] J. Baumann, D.D. Torkzadeh, A. Ramstein, U. Kiencke and T. Schlegl.

Model-based predictive anti-jerk control. Control Engineering Practice, vol. 14, pages

259–266, 2006. 42, 43

173

REFERENCES

[Bemporad et al., 2001] L. Bemporad, F. Borrelli, L. Glielmo and F. Vasca. Optimal piecewise-

linear control of dry clutch engagement. In 3rd IFAC Workshop Advances in Automo-

tive Control, Karlsruhe, Germany, 2001. 42

[Bernardini et al., 2010] D. Bernardini, M. C. F. Donkers, A. Bemporad and W. P. M. H.

Heemels. A Model Predictive Control Approach for Stochastic Networked Control

Systems. In 2nd IFAC Workshop on Estimation and Control of Networked Systems,

Annecy, France, 2010. 3, 5, 14

[Berriri et al., 2007] M. Berriri, P. Chevrel, D. Lefebvre and M. Yagoubi. Active damping

of automotive powertrain oscillations by a partial torque compensator. In American

Control Conference, pages 5718–5723, New York City, USA, 2007. 43

[Berriri et al., 2008] M. Berriri, P. Chevrel and D. Lefebvre. Active damping of automotive

powertrain oscillation by a partial torque compensator. Control Engineering Practice,

vol. 16, pages 874–883, 2008. 40, 43

[Braescu, Caruntu et al., 2011] F. C. Braescu, C. F. Caruntu, L. Ferariu and C. Lazar. OSEK

Based Embedded Networked Controller Designed to Handle Communication Delays.

In 2nd Eastern European Regional Conference on the Engineering of Computer Based

Systems, pages 71–77, Bratislava, Slovakia, 2011. 10, 121

[Bruce et al., 2005] L. Bruce, G. De Nicolao and R. Scattolini. On powertrain oscillation

damping using feedforward and LQ feedback control. In IEEE Conference on Con-

trol Applications, pages 1415–1420, Toronto, Canada, 2005. 42, 43

[Buttazzo, 2006] G. Buttazzo. Research Trends in Real-Time Computing for Embedded Sys-

tems. ACM SIGBED Review, vol. 3, 2006. 113

[Camacho and Bordons, 2004] E. F. Camacho and C. Bordons. Model Predictive Control.

Springer Verlag, 2004. 4, 52

[Canale et al., 2006] M. Canale, L. Fagiano, M. Milanese and P. Borodani. Robust Vehicle Yaw

Control using Active Differential and Internal Model Control Techniques. In American

Control Conference, Minneapolis, USA, 2006. 40

[Canale et al., 2008a] M. Canale, L. Fagiano, A. Ferrara and C. Vecchio. A Comparison Be-

tween IMC and Sliding Mode Approaches to Vehicle Yaw Control. In American Control

Conference, Seattle, USA, 2008. 40

174

REFERENCES

[Canale et al., 2008b] M. Canale, L. Fagiano, A. Ferrara and C. Vecchio. Vehicle Yaw Control

via Second-Order Sliding-Mode Technique. IEEE Transactions on Industrial Electron-

ics, vol. 55, 2008. 40

[Cao and Zhang, 2006] Y.-can Cao and W.-dong Zhang. Modified fuzzy PID control for net-

worked control systems with random delays. In World Academy of Science, Engineer-

ing and Technology, volume 12, pages 312–315, Bath, UK, 2006. 31

[Caruntu and Lazar, 2009a] C. F. Caruntu and C. Lazar. Network-Induced Variable Time Delay

Compensation Technique Based on Predictive Control. In 17th International Conference

on Control Systems and Computer Science, pages 65–71, 2009. 9, 87

[Caruntu and Lazar, 2009b] C. F. Caruntu and C. Lazar. Predictive Control for Time-Varying

Delay in Networked Control Systems. In 8th IFAC Workshop on Time Delay Systems,

pages 49–54, Sinaia, Romania, 2009. 9, 53, 87

[Caruntu et al., 2009c] C. F. Caruntu, M. H. Matcovschi, A. E. Balau, D. I. Patrascu, C. Lazar

and O. Pastravanu. Modeling of an Electromagnetic Valve Actuator. Buletinul Institu-

tului Politehnic Iasi, vol. LV (LIX), pages 9–28, 2009. 9, 67, 87

[Caruntu et al., 2010a] C. F. Caruntu, A. E. Balau and C. Lazar. Cascade based Control of a

Drivetrain with Backlash. In 12th International Conference on Optimization of Electri-

cal and Electronic Equipment, Brasov, Romania, 2010. 10, 161

[Caruntu et al., 2010b] C. F. Caruntu, A. E. Balau and C. Lazar. Networked Predictive Control

Strategy for an Electro-Hydraulic Actuated Wet Clutch. In 6th IFAC Symposium Ad-

vances in Automotive Control, pages 419–424, Munchen, Germany, 2010. 10, 53, 69,

70, 88

[Caruntu et al., 2011a] C. F. Caruntu, A. E. Balau, M. Lazar, P. P. J. van den Bosch and S. Di

Cairano. A Predictive Control Solution for Driveline Oscillations Damping. In Hybrid

Systems: Computation and Control, pages 181–190, Chicago, USA, 2011. 10, 161

[Caruntu and C.Lazar, 2011b] C. F. Caruntu and C. Lazar. Networked Predictive Control for

Time-varying Delay Compensation with an Application to Automotive Mechatronic Sys-

tems. accepted for Journal of Control Engineering and Applied Informatics, 2011. 10,

53, 58, 69, 88

175

REFERENCES

[Caruntu and C.Lazar, 2011c] C. F. Caruntu and C. Lazar. Robust MPC for TrueTime Simula-

tion of a Vehicle Drivetrain Controlled through CAN. In 16th IEEE International Con-

ference on Emerging Technologies and Factory Automation, Toulouse, France, 2011.

11, 128, 161

[Caruntu and C.Lazar, 2011d] C. F. Caruntu and C. Lazar. Robustly stabilizing MPC design for

networked control systems with an application to DC motors. accepted for IET Control

Theory and Applications, 2011. 11, 125, 126, 127, 161

[Caruntu and C.Lazar, 2011e] C. F. Caruntu and C. Lazar. Stabilizing MPC for network-

controlled systems with an application to DC motors. In IEEE International Conference

on Mechatronics, pages 973–978, Istanbul, Turkey, 2011. 10, 125, 126, 127, 161

[Caruntu et al., 2011f] C. F. Caruntu, M. Lazar, S. Di Cairano, R. H. Gielen and P. P. J. van den

Bosch. Horizon-1 predictive control of networked controlled vehicle drivetrains. 18th

IFAC World Congress, pages 3824–3830, Milano, Italy, 2011. 10, 90, 120

[Caruntu et al., 2011g] C. F. Caruntu, M. Lazar, S. Di Cairano, R. H. Gielen and P. P. J. van

den Bosch. Lyapunov based Predictive Control of Vehicle Drivetrains over CAN. to be

submitted to a journal, 2011.

[Caruntu et al., 2011h] C. F. Caruntu, D. Onu, F. C. Braescu and C. Lazar. Model Predictive

Control for Real-time Simulation of a Network-controlled Vehicle Drivetrain. In 2nd

Eastern European Regional Conference on the Engineering of Computer Based Sys-

tems, pages 115–123, Bratislava, Slovakia, 2011. 10, 88

[Caruntu et al., 2011i] C. F. Caruntu, D. Onu and C. Lazar. Real-time Simulation of a Vehicle

Drivetrain Controlled through CAN using a Robust MPC Strategy. In 15th International

Conference on System Theory, Control and Computing, Sinaia, Romania, 2011. 11, 161

[Caruntu and C.Lazar, 2011j] C. F. Caruntu and C. Lazar. Network-induced time-varying delay

modeling and predictive compensation with stability guarantee. submitted to IEEE

Transactions on Industrial Informatics, 2011. 11, 128

[Cerone et al., 2007] V. Cerone, M. Milanese and D. Regruto. Hardware-in-the-loop (HIL)

results on yaw stability control. In IEEE Multi-conference on Systems and Control,

Singapore, 2007. 40

176

REFERENCES

[Chan and Ozguner, 1995] H. Chan and U. Ozguner. Closed-loop control of systems over a

communications network with queues. International Journal of Control, vol. 62, pages

493–510, 1995. ix, 18, 29, 49

[Choi, 2008] S. Choi. Antilock Brake System with a Continuous Wheel Slip Control to Max-

imize the Braking Performance and the Ride Quality. IEEE Transactions on Control

Systems Technology, vol. 16, 2008. 40

[Chow and Tipsuwan, 2001] M. Y. Chow and Y. Tipsuwan. Network-based control systems:

a tutorial. In 27th Annual Conference of the IEEE Industrial Electronics Society

(IECON‘01), volume 3, pages 1593–1602, Denver, USA, 2001. 2, 3, 14

[Chow and Tipsuwan, 2003] M. Y. Chow and Y. Tipsuwan. Gain Adaptation of Networked DC

motor controllers based on QoS variations. IEEE Transactions on Industrial Electron-

ics, vol. 50, pages 936–943, 2003. 135, 137

[Cloosterman et al., 2006] M. B. G. Cloosterman, N. van de Wouw, W. P. M. H. Heemels and

H. Nijmeijer. Robust stability of networked control systems with time-varying network-

induced delays. In 45th Conference on Decision and Control, pages 4980–4985, San

Diego, USA, 2006. 38

[Cloosterman et al., 2009] M. B. G. Cloosterman, N. van de Wouw, W. P. M. H. Heemels and

H. Nijmeijer. Stability of networked control systems with uncertain time-varying delays.

IEEE Transactions on Automatic Control, vol. 54, pages 1575–1580, 2009. 3, 5, 14, 38

[Cloosterman et al., 2010] M. B. G. Cloosterman, L. Hetel, N. van de Wouw, W. P. M. H.

Heemels, J. Daafouz and H. Nijmeijer. Controller synthesis for networked control sys-

tems. Automatica, vol. 46, pages 1584–1594, 2010. 3, 5, 14, 38

[de la Pena and Cristofides, 2008] D. M. de la Pena and P. D. Christofides. Lyapunov-based

model predictive control of nonlinear systems subject to data losses. IEEE Transactions

on Automatic Control, vol. 53, pages 2076–2089, 2008. 3, 5, 14

[Di Cairano et al., 2007] S. Di Cairano, A. Bemporad, I. V. Kolmanovsky and D. Hrovat.

Model Predictive Control of Magnetically Actuated Mass Spring Dampers for Auto-

motive Applications. International Journal of Control, vol. 80, pages 1701–1716, 2007.

40

177

REFERENCES

[Di Cairano et al., 2010] S. Di Cairano, D. Yanakiev, A. Bemporad, I.V. Kolmanovsky and

D. Hrovat. Model Predictive Powertrain Control: an Application to Idle Speed Reg-

ulation. Lecture Notes in Control and Information Sciences. Springer, 2010. 40, 80,

99

[Drakunov et al., 2000] S. V. Drakunov, B. Ashrafi and A. Rosiglioni. Yaw Control Algorithm

via Sliding Mode Control. In American Control Conference, Chicago, USA, 2000. 40

[Du and Qian, 2008] F. Du and Q. Q. Qian. Fuzzy immune self-regulating PID control based

on modified Smith Predictor for networked control systems. In IEEE International Con-

ference on Networking, Sensing and Control, Sanya, China, 2008. 6

[Fredriksson et al., 2002] J. Fredriksson, H. Weiefors and B. Egardt. Powertrain control for

active damping of driveline oscillations. Vehicle System Dynamics, vol. 37, pages

359–376, 2002. 40, 43

[Gadamsetty et al., 2009] B. Gadamsetty, S. Bogosyan, M. Gokasan and A. Sabanovic. Novel

observers for compensation of communication delay in bilateral control systems. In

35th Annual Conference of the IEEE Industrial Electronics Society, Porto, Portugal,

2009. 6, 124

[Gao et al., 2008] T. G. Gao, S. K. Saha and I. N. Kar. Sensor-actuator based smart yoke for

a rack and pinion steering system. In 5th Int. Mobility Conf. on Emerging Automotive

Technologies: Global & Indian Perspective, pages 358–364, 2008. 59

[Ge et al., 2007] Y. Ge, L. Tian and Z. Liu. Survey on the stability of networked control systems.

Journal of Control Theory and Applications, vol. 5, 2007. 22

[Gerling, 2001] D. Gerling. Challenges and perspectives of electrical drives in the automotive

industry. In 9th European Conference on Power Electronics and Applications (EPE),

Graz, Austria, 2001. 59

[Gielen and M.Lazar, 2009a] R.-H. Gielen and M. Lazar. Further results on stabilization of

linear systems with time-varying input delay. In 8th IFAC Workshop on Time Delay

Systems, Sinaia, Romania, 2009. 3, 5, 14, 16, 131, 141, 142, 145

[Gielen and M.Lazar, 2009b] R.-H. Gielen and M. Lazar. Stabilization of networked control

systems via non-monotone control Lyapunov functions. In 48th IEEE Conference on

Decision and Control, pages 7942–7948, Shanghai, China, 2009. 6, 25, 35, 36, 91, 94,

95, 104, 118, 131

178

REFERENCES

[Gielen et al., 2010] R.-H. Gielen, S. Olaru, M. Lazar, W.-P.-M.-H. Heemels, N. van de Wouw

and S.-I. Niculescu. On polytopic inclusions as a modeling framework for systems with

time-varying delays. Automatica, vol. 46, pages 615–619, 2010. 5, 7, 26, 35, 90, 91,

92, 164, 168

[Grotjahn et al., 2006] M. Grotjahn, L. Quernheim and S. Zemke. Modelling and identifica-

tion of car driveline dynamics for anti-jerk controller design. In IEEE International

Conference on Mechatronics, pages 131–136, Budapest, Hungary, 2006. 42, 77

[Gu et al., 2009] R. Gu, S. S. Bhattacharyya and W. S. Levine. Dataflow-based implementation

of model predictive control. In 28th American Control Conference, St. Louis, USA,

2009. 4

[Gupta and Chow, 2008] R. A. Gupta and M. Y. Chow. Overview of networked control sys-

tems. In Networked Control Systems - Theory and Applications, pages 1–23. Springer,

London, 2008. 2, 3, 14

[Halevi and Ray, 1990] Y. Halevi and A.Ray. Performance analysis of integrated communica-

tion and control systems networks. ASME, Journal of Dynamic Systems, Measurement

and Control, vol. 112, pages 365–632, 1990. 3, 14

[Heemels et al., 2010] W. P. M. H. Heemels, A. R. Teel, van de Wouw N and D. Nesic. Net-

worked Control Systems With Communication Constraints: Tradeoffs Between Trans-

mission Intervals, Delays and Performance. IEEE Transactions on Automatic Control,

vol. 55, pages 1781–1796, 2010. 3, 14, 168

[Henriksson et al., 2003] D. Henriksson, A. Cervin and K.E. Arzn Pierrot. TrueTime: Real-

time control system simulation with MATLAB/Simulink. In Nordic MATLAB Confer-

ence, Copenhagen, Denmark, 2003. 108, 143

[Herpel et al., 2009] T. Herpel, K.-S. Hielscher, U. Klehmet and R. German. Stochastic and

deterministic performance evaluation of automotive CAN communication. Computer

Networks, vol. 53, pages 1171–1185, 2009. 47

[Hespanha et al., 2007] J. Hespanha, P. Naghshtabrizi and Y. Xu. A survey of recent results

in networked control systems. In Proc. of IEEE - Special Issue on Technology of Net-

worked Control Systems, volume 95, pages 138–162, 2007. 4, 14

179

REFERENCES

[Hong et al., 2006] D. Hong, P. Yoon, H.-J. Kang and K. Hwang I.and Huh. Wheel Slip Con-

trol Systems Utilizing the Estimated Tire Force. In American Control Conference, Min-

neapolis, USA, 2006. 40

[Hrovat et al., 2000] D. Hrovat, J. Asgari and M. Fodor. Automotive mechatronic systems. In

Mechatronic systems techniques and applications (vol. 2), pages 1–98. Gordon and

Breach Science Publishers, Inc., 2000. 42, 80

[Huang et al., 2010] C. Huang, Y. Bai and X. Liu. H-Infinity State Feedback Control for a Class

of Networked Cascade Control Systems with Uncertain Delay. IEEE Transactions on

Industrial Informatics, vol. 6, pages 62–72, 2010. 3, 14, 16, 38

[Ibeas et al., 2007] A. Ibeas, R. Vilanova and P. Balaguer. Multiple-delay Smith Predictor

based control of LTI systems with bounded uncertain delay. In 22nd IEEE International

Symposium on Intelligent Control, Singapore, 2007. 6

[Jiang and Wang, 2001] Z.-P. Jiang and Y. Wang. Input-to-state stability for discrete-time non-

linear systems. Automatica, vol. 37, pages 857–869, 2001. 37, 129

[Jiangang et al., 2007] L. Jiangang, L. Biyu, Z. Ruifang and L. Meilan. The New Variable-

period Sampling Scheme for Networked Control Systems with Random Delay Based

on BP Neural Network Prediction. In 26th Chinese Control Conference, Zhangjiajie,

China, 2007. 1, 3, 5, 6, 14

[Johansen et al., 2001] T. A. Johansen, J. Kalkkuhl J.and Ludemann and I. Petersen. Hybrid

Control Strategies in ABS. In American Control Conference, Arlington, USA, 2001. 40

[Johansson et al., 2005] K. H. Johansson, M. Torngren and L. Nielsen. Vehicle Applications

of Controller Area Network. Handbook of Networked and Embedded Control Systems.

Springer, 2005. 1, 13

[Johnson and Fontaine, 2001] E. N. Johnson and S. Fontaine. Use of Flight Simulation to Com-

plement Flight Testing of Low-cost UAVs. In AIAA Modeling and Simulation Technolo-

gies Conference and Exhibit, Montreal, Canada, 2001. 114

[Juanole and Mouney, 2005] G. Juanole and G. Mouney. QoS in real time distributed sys-

tems and process control applications. In Workshop on Networked Control Systems

(NeCST), Corsica, France, 2005. 3, 14

180

REFERENCES

[Keshmiri and Shahri, 2007] R. Keshmiri and A. M. Shahri. Intelligent ABS Fuzzy Controller

for Diverse Road Surfaces. International Journal of Mechanical Systems, Science and

Engineering, vol. 1, 2007. 40

[Kiencke and Nielsen, 2005] U. Kiencke and L. Nielsen. Automotive control systems: for en-

gine, driveline, and vehicle, volume 290. Springer Verlag, 2005. 43, 63, 77, 100

[Klehmet et al., 2008] U. Klehmet, T. Herpel, K.-S. Hielscher and R. German. Delay Bounds

for CAN Communication in Automotive Applications. In 14th GI/ITG Conference Mea-

surement, Modelling and Evaluation of Computer and Communication Systems, Dort-

mund, Germany, 2008. 47, 63, 71

[Krtolica et al., 1994] R. Krtolica, U. Ozguner, H. Chan, H. Goktas, J. Winkelman and M. Li-

ubakka. Stability of linear feedback systems with random communication delays. Inter-

national Journal of Control, vol. 59, pages 925–953, 1994. ix, 17, 18, 24, 49

[Kvasnica et al., 2006] M. Kvasnica, P. Grieder, M. Baotic and F.J. Christophersen. Multi-

Parametric Toolbox (MPT), 2006. 34

[Lagerberg and Egardt, 2005] A. Lagerberg and B. Egardt. Model predictive control of au-

tomotive powertrains with backlash. In 16th IFAC World Congress, Prague, Czech

Republic, 2005. 43

[Langjord et al., 2008] H. Langjord, T. A. Johansen and J. P. Hespanha. Switched control of

an electropneumatic clutch actuator using on/off valves. In 27th American Control

Conference, Seattle, USA, 2008. 67

[Lauer et al., 2006] V. Lauer, M. Hiller, M. Osella, M. Auerswald and J. Lucas. EASIS - Elec-

tronic Architecture and Systems Engineering for integrated Safety-Systems. In Diskus-

sionskreis Fehlertoleranz, Munich, Germany, 2006. 43

[M.Lazar 2006] M. Lazar. Model predictive control of hybrid systems: Stability and robustness.

PhD thesis, Eindhoven University of Technology, The Netherlands, 2006. 140, 143, 151

[M.Lazar and Heemels, 2008] M. Lazar and W. P. M. H. Heemels. Optimized Input-to-State

Stabilization of Discrete-time Nonlinear Systems with Bounded Inputs. In 27th Ameri-

can Control Conference, Seattle, USA, 2008. 37, 38, 125, 128, 129, 130, 131

181

REFERENCES

[M.Lazar 2009] M. Lazar. Flexible control Lyapunov functions. In 28th American Control

Conference, pages 102–107, St. Louis, MO, USA, 2009. 7, 8, 36, 38, 90, 92, 94, 95,

102, 120, 128, 129, 131, 132, 159

[C.Lazar, Caruntu and Balau, 2010] C. Lazar, C. F. Caruntu and A. E. Balau. Modelling and

Predictive Control of an Electro-Hydraulic Actuated Wet Clutch for Automatic Trans-

mission. In IEEE Symposium on Industrial Electronics, Bari, Italy, 2010. 9, 67, 68, 69,

70, 87

[Le Boudec and Thiran, 2001] J.-Y. Le Boudec and P. Thiran. Network Calculus. Lecture

Notes in Computer Science. Springer Verlag, 2001. 48

[Lee and Zak, 2001] Y. Lee and S. H. Zak. Genetic Neural Fuzzy Control of Anti-Lock Brake

Systems. In American Control Conference, Arlington, USA, 2001. 40

[Leen et al., 1999] G. Leen, D. Heffernan and A. Dunne. Digital Networks in the Automotive

Vehicle. Computing & Control Engineering Journal, 1999. 43, 45

[Lefebvre et al., 2003] D. Lefebvre, P. Chevrel and S. Richard. An Hinfinity-based control

design methodology dedicated to the active control of vehicle longitudinal oscillations.

IEEE Transactions on Control System Technology, vol. 11, 2003. 40, 43

[Li and Mills, 2001] Q. Li and D. L. Mills. Jitter based delay boundary prediction of wide-area

networks. IEEE/ACM Transactions on Networking, vol. 9, 2001. 24

[Li and Cao, 2003] Q. Li and C. Yao. Real-time concepts for embedded systems. CMP Books,

USA, 2003. 113

[Li et al., 2008a] H. Li, Z. Sun, H. Liu and M. Y. Chow. Predictive observer-based control

for networked control systems with network-induced delay and packet dropout. Asian

Journal of Control, vol. 10, pages 638–650, 2008. 3, 5, 14, 38, 39

[Li and Zhao, 2008b] W. Li and W. Zhao. Challenges of Automotive Network Systems. In

National Workshop on High-Confidence Automotive Cyber-Phisical Systems, Troy, MI,

USA, 2008. 44

[Lin and Hsu, 2003] C. M. Lin and A. F. Hsu. Self-Learning Sliding-Mode Control for Antilock

Braking Systems. IEEE Transactions on Control Systems Technology, vol. 11, 2003. 40

182

REFERENCES

[Lin and Antsaklis, 2004] H. Lin and P. J. Antsaklis. Persistent disturbance attenuation prop-

erties for networked control systems. In 43rd Conference on Decision and Control,

Atlantis, Bahamas, 2004. 16

[Lin and Zhang 2008] C. Lin and L. Zhang. Hardware-in-the-loop Simulation and Its Applica-

tion in Electric Vehicle Development. In IEEE Vehicle Power and Propulsion Confer-

ence (VPPC), Harbin, China, 2008. 114

[Liu et al., 2004] G. P. Liu, J. X. Mu and D. Rees. Networked predictive control of systems

with random communication delay. In UKACC International Conference on Control,

Bath, UK, 2004. 30

[Liu et al., 2010] X. Liu, Y. Xia, M. S. Mahmoud and Z. Deng. Modeling and stabilization

of MIMO networked control systems with network constraints. International Journal of

Innovative Computing, Information and Control, vol. 6, pages 4409–4419, 2010. 3, 14,

38

[Lopez et al., 2006] I. Lopez, J. L. Piovesan, C. T. Abdallah, D. Lee, O. Martinez, M. Spong

and R. Sandoval. Practical issues in networked control systems. In American Control

Conference, Minneapolis, USA, 2006. 16

[Luck and Ray, 1990] R. Luck and A. Ray. An observer-based compensator for distributed

delays. Automatica, vol. 26, pages 903–908, 1990. ix, 16, 17, 24, 28, 49

[Luck and Ray, 1994] R. Luck and A. Ray. Experimental verification of a delay compensation

algorithm for integrated communication and control systems. International Journal of

Control, vol. 59, 1994. 17, 28

[Lv et al., 2007] H. Lv, Y. Jia, J. Du and Q. Du. ABS Composite Control Based on Optimal

Slip Ratio. In American Control Conference, New York, USA, 2007. 40

[Maciejowski, 2002] J. M. Maciejowski. Predictive control with constraints. Prentice Hall,

Harlow, 2002. 4, 5, 89

[Matsura et al., 2004] T. Matsura, S. Ichikawa, F. Katsu and M. Sato. The development of

a controller confirmation system for automatic transmissions and its applications. In

IEEE International Conference on Control Applications, pages 1415–1419, Taipei, Tai-

wan, 2004. 114

183

REFERENCES

[McNeal and Belkhayat, 2007] R. McNeal and M. Belkhayat. Standard Tools for Hardware-

in-the-Loop (HIL) Modeling and Simulation. In IEEE Electric Ship Technologies Sym-

posium, pages 130–137, Arlington, USA, 2007. 114

[Merritt, 1967] H. E. Merritt. Hydraulic control systems. John Wiley & Sons, USA, 1967. 67

[Morselli et al., 2003] R. Morselli, R. Zanasi, R Cirsone, R. Sereni, R. Bedogni and E. Sedoni.

Dynamic modeling and control of electro-hydraulic wet clutches. In IEEE Conference

on Intelligent Transportation Systems, Shanghai, China, 2003. 67

[Mu et al., 2005] J. Mu, G. P. Liu and D. Rees. Design of robust networked predictive control

systems. In American Control Conference, Portland, USA, 2005. 16

[Natori et al., 2008] K. Natori, R. Oboe and K. Ohnishi. Stability Analysis and Practical De-

sign Procedure of Time Delayed Control Systems With Communication Disturbance

Observer. IEEE Transactions on Industrial Informatics, vol. 4, pages 185–197, 2008.

6, 27, 124, 126

[Natori et al., 2010] K. Natori, T. Tsuji, K. Ohnishi, A. Hace and K. Jezernik. Time-Delay

Compensation by Communication Disturbance Observer for Bilateral Teleoperation

Under Time-Varying Delay. IEEE Transactions on Industrial Electronics, vol. 57, pages

1050–1061, 2010. 27, 31

[Neelekantan, 2008] V. A. Neelekantan. Model predictive control of a two stage actuation

system using piezoelectric actuators for controllable industrial and automotive brakes

and clutches. Journal of Intelligent Material Systems and Structures, vol. 19, pages

845–857, 2008. 67

[Neumann, 2007] P. Neumann. Communication in industrial automation - what is going on ?

Control Engineering Practice, vol. 15, pages 1332–1347, 2007. 135

[Nilsson et al., 1998] J. Nilsson, B. Bernhardson and B. Wittenmark. Stochastic analysis and

control of real-time systems with random time delays. Automatica, vol. 34, pages 57–64,

1998. ix, 18, 19, 24, 25, 29

[O’Dwyer, 2006] A. O’Dwyer. Handbook of PI and PID controller tuning rules, volume 2.

Imperial College Press, 2006. 83, 103, 109, 151

184

REFERENCES

[Onat and Parlakay, 2007] A. Onat and M. E. Parlakay. Implementation of model based net-

worked Predictive control system. In 9th Real-Time Workshop (RTLW09), pages 85–

93, Linz, Austria, 2007. 2, 13

[Onat et al., 2008] A. Onat, A. T. Naskali and E. Parlakay. Model based predictive networked

control systems. In 17th IFAC World Congress, pages 13000–13005, Seoul, Korea,

2008. 3, 5, 14

[Onat et al., 2010] A. Onat, T. Naskali, E. Parlakay and O. Mutluer. Control over imperfect

networks: model based predictive networked control systems. IEEE Transactions on

Industrial Electronics, 2010. 3, 14

[Oudghiri et al., 2007] M. Oudghiri, M. Chadli and A. El Hajjaji. Vehicle yaw control using a

robust H observer-based fuzzy controller design. In 46th IEEE Conference on Decision

and Control, New Orleans, USA, 2007. 40

[Patrascu et al., 2009] D. I. Patrascu, A. E. Balau, C. F. Caruntu, C. Lazar, M. H. Matcovschi

and O. Pastravanu. Modelling of a Solenoid Valve Actuator for Automotive Control

Systems. In 17th International Conference on Control Systems and Computer Science,

pages 541–546, 2009. 9, 67, 87

[Pettersson, 1997] M. Pettersson. Driveline modeling and control. PhD thesis, Linkoping

University, 1997. 40, 42, 43

[Ping et al., 2010] E. P. Ping, K. Hudha, M. H. Bin and H. Jamaluddin. Hardware-in-the-loop

Simulation of Automatic Steering Control. In 11th International Conference Control,

Automation, Robotics and Vision, pages 964–969, Singapore, 2010. 115

[Rabbath et al., 2001] C. A. Rabbath, H. Desira and K. Butts. Effective Modeling and Simula-

tion of the Internal Combustion Engine Control Systems. In American Control Confer-

ence, pages 1321–1326, Arlington, USA, 2001. 114

[Rawlings and Mayne, 2009] J. B. Rawlings and D. Q. Mayne. Model predictive control: The-

ory and design. Nob Hill Publishing, 2009. 4

[Richard et al., 1999] S. Richard, P. Chevrel and B. Maillard. Active control of future vehi-

cle drivelines. In 38th IEEE Conference on Decision and Control, pages 3752–3757,

Phoenix, Arizona, USA, 1999. 40, 43

185

REFERENCES

[Richard, 2003] J. P. Richard. Time delay systems: an overview of some recent advances and

open problems. Automatica, vol. 99, pages 1667–1694, 2003. 16

[Rodriguez and Menendez, 2007] R. V. Rodriguez and R. M. Menendez. Network-induced de-

lay models for CAN-based networked control systems. In 7th IFAC Conference on

Fieldbuses and Networks in Industrial and Embedded Systems, 2007. 22

[Rostalski et al., 2007] P. Rostalski, T. Besselmann, M. Maric, F. Van Belsen and M Morari. A

hybrid approach to modelling, control and state estimation of mechanical systems with

backlash. International Journal of Control, vol. 80, pages 1729–1740, 2007. 40, 42, 43,

148

[Saerens et al., 2008] B. Saerens, M. Diehl, J. Swevers and E. Van den Bulck. Model predictive

control of automotive powertrains - first experimental results. In 47th IEEE conference

on Decision and Control, pages 5692–5697, Cancun, Mexico, 2008. 40, 42, 148

[Salehi et al., 2008] S. Salehi, L. Xie and W. Cai. Robust controller design for networked

control systems with uncertain time delays. In 10th International Conference on Control,

Automation, Robotics and Vision, pages 1407–1412, Hanoi, Vietnam, 2008. 3, 5, 14,

16, 38

[Schinkel and Hunt, 2002] M. Schinkel and K. Hunt. Anti-lock Braking Control using a Sliding

Mode like Approach. In American Control Conference, Anchorage, USA, 2002. 40

[Shen et al., 2010] Q. Shen, W. Gui, Y. Xiong and C. Yang. Predictive control and scheduling

codesign in network control systems. Journal of Control Theory Applications, vol. 8,

pages 239–244, 2010. 3, 14

[Sheng and Pan, 2010] L. Sheng and Y.-J. Pan. Stochastic stabilization of sampled-data net-

worked control systems. International Journal of Innovative Computing, Information

and Control, vol. 6, pages 1949–1972, 2010. 3, 14, 38

[Stewart, 2003] J. Stewart. Calculus, early transcendentals. Thomson Brooks/Cole, 2003. 94,

132

[Stewart et al., 2005] P. Stewart, J.C. Zavala and P.J. Flemming. Automotive drive by wire

controller design by multi-objectives techniques. Control Engineering Practice, vol. 13,

pages 257–264, 2005. 40, 43

186

REFERENCES

[Stroop and Stolpe, 2006] J. Stroop and R. Stolpe. Prototyping of Automotive Control Systems

in a Time-Triggered Environment Using FlexRay. In IEEE Conference on Computer

Aided Control Systems Design, Munich, Germany, 2006. 43

[Tang and Yu, 2007] X. Tang and J. Yu. Networked control system: survey and directions.

In Bio-Inspired Computational Intelligence and Applications, volume 4688 of Lecture

Notes in Computer Science, pages 473–481. Springer Verlag, 2007. 135

[Templin, 2008] P. Templin. Simultaneous estimation of driveline dynamics and backlash size

for control design. In IEEE International Conference on Control Applications, pages

13–18, San Antonio, Texas, USA, 2008. 42

[Templin and Egardt, 2009] P. Templin and B. Egardt. An LQR torque compensator for driv-

eline oscillation damping. In IEEE International Conference on Control Applications,

pages 352–356, Saint Petersburg, Rusia, 2009. 42, 43

[Tipsuwan and Chow, 2003] Y. Tipsuwan and M. Y. Chow. Control methodologies in net-

worked control systems. Control Engineering Practice, vol. 11, pages 1099–1111, 2003.

1, 2, 14, 22, 23, 27

[Tondel and Johansen, 2005] P. Tondel and T. A. Johansen. Control Allocation for Yaw Sta-

bilization in Automotive Vehicles using Multiparametric Nonlinear Programming. In

American Control Conference, Portland, USA, 2005. 40

[Tran and Vlacic, 2006] T. Tran and L. Vlacic. Practical process control techniques training.

In 7th IFAC Symposium on Advances in Control Education, Madrid, Spain, 2006. 4

[Van Der Heijden et al., 2007] A.C. Van Der Heijden, F.A. Serrarens, M.K. Camlibel and

H. Nijmeijer. Hybrid optimal control of dry clutch engagement. International Jour-

nal of Control, vol. 80, pages 1717–1728, 2007. 43, 67, 77, 148

[Vatanski et al., 2007] N. Vatanski, J. P. Georges, C. Aubrun, E. Rondeau and S. L. Jamsa-

Jounela. Compensating the transmission delay in networked control systems. In 14th

Nordic Process Control Workshop (NPCW07), Espoo, Finland, 2007. 3, 14

[Velagic, 2008] J. Velagic. Design of Smith-like Predictive Controller with Communication

Delay Adaptation. In World Congress on Science, Engineering and Technology, 5th

International Conference on Control and Automation, Paris, France, 2008. 6, 7, 24, 32,

59, 60, 65, 66, 72, 74, 87, 165

187

REFERENCES

[Wang and Longoria, 2006] J. Wang and R. G. Longoria. Coordinated Vehicle Dynamics Con-

trol with Control Distribution. In American Control Conference, Minneapolis, USA,

2006. 40

[Wang et al., 2010] X.-L. Wang, C.-G. Fei and Z.-Z. Han. Adaptive predictive functional con-

trol for networked control systems with random delays. International Journal of Au-

tomation and Computing, 2010. 3, 5, 14, 16

[Wei et al., 2002] W. Wei, B. Wang and D. Towsley. Continuous-time hidden Markov models

for network performance evaluation. Performance Evaluation, vol. 49, 2002. 25

[Witrant et al., 2007] E. Witrant, D. Georges, C. Canudas-de-Wit and M. Alamir. On the use

of state predictors in networked control systems. In Applications of Time Delay Sys-

tems, volume 352 of Lecture Notes in Control and Information Sciences, pages 17–35.

Springer Verlag, 2007. 16

[Wolf et al., 2004] Marko Wolf, Andre Weimerskirch and Christof Paar. Security in automotive

bus systems. In Workshop on Embedded Security in Cars, 2004. 44

[Wu et al., 2008] D. Wu, H. Ogai, M. Ogawa, M. Maruto, J. Kusaka and P. Jiao. Engine Control

Education System. In SICE Annual Conference, pages 2901–2904, Nagoya , Japan,

2008. 114

[Wu et al., 2009] J. Wu, L. Zhang and T. Chen. Model predictive control for networked control

systems. International Journal of Robust and Nonlinear Control, vol. 19, pages 1016–

1035, 2009. 5

[Wu et al., 2010] D. Wu, J. Wu and S. Chen. Robust stabilisation control for discrete-time

networked control systems. International Journal of Control, vol. 83, pages 1885–1894,

2010. 38

[Yang et al., 2006] C. Yang, S. Zhu, W. Kong and L. Lu. Application of generalized predictive

control in networked control systems. Journal of Zhejiang University, vol. 7, pages

225–233, 2006. 2

[Yoo and Wang, 2007] D. K. Yoo and L. Wang. Model based Wheel Slip Control via Con-

strained Optimal Algorithm. In IEEE Multi-conference on Systems and Control, Sin-

gapore, 2007. 40

188

REFERENCES

[Yu et al., 2002] F. Yu, J. Z. Feng and J. Li. A Fuzzy Logic Controller Design for Vehicle ABS

with a on-line Optimized Target Wheel Slip Ratio. International Journal of Automotive

Technology, vol. 3, pages 165–170, 2002. 40

[Yue et al., 2009] D. Yue, E. Tian, Y. Zhang and C. Peng. Delay-distribution-dependent robust

stability of uncertain systems with time-varying delay. International Journal of Robust

and Nonlinear Control, vol. 19, pages 377–393, 2009. 5, 38

[Zhang et al., 2001] W. Zhang, M. S. Branicky and S. M. Philips. Stability of networked sontrol

systems. IEEE Control Systems Magazine, vol. 21, 2001. 3, 14, 20, 21, 29, 49

[Zhang et al., 2005] Q. Zhang, C. Lin and P. Chen. Time-stamped dynamic matrix control for

networked control systems. In IEEE International Conference on Industrial Technology,

pages 1159–1163, Hong Kong, 2005. 3, 4, 14, 16

[Zhang et al., 2006] Q. Z. Zhang, Y. C. Tang and W. D. Zhang. Adaptive dynamic matrix con-

trol for network-based control systems with random delays. Asian Journal of Control,

vol. 8, 2006. 24

[Zhang et al., 2008] Y. Zhang, F. Yu, K. Huang and Y. Gu. Permanent-magnet DC motor ac-

tuators application in automotive energy-regenerative active suspensions. In FISITA

2008 World Automotive Congress, Munich, Germany, 2008. 59

[Zhao et al., 2010] Y.-B. Zhao, G.-P. Liu and D. Rees. Packet-based deadband control for

Internet-based networked control systems. IEEE Transactions on Control Systems Tech-

nology, vol. 18, pages 1057–1067, 2010. 39

[Zurawski, 2006] R. Zurawski. Embedded systems handbook. CRC Press, 2006. 113

189

Tipărit la Tipografia Rotaprint a Universităţii Tehnice “Gheorghe Asachi” din Iaşi

Proiectul „Burse Doctorale - O Investiţie în Inteligenţă

(BRAIN)”, POSDRU/6/1.5/S/9, ID 6681, este un proiect

strategic care are ca obiectiv general „Îmbunătățirea

formării viitorilor cercetători în cadrul ciclului 3 al

învățământului superior - studiile universitare de doctorat -

cu impact asupra creșterii atractivității şi motivației pentru

cariera în cercetare”.

Proiect finanţat în perioada 2008 - 2011.

Finanţare proiect: 14.424.856,15 RON

Beneficiar: Universitatea Tehnică “Gheorghe Asachi” din

Iaşi

Partener: Universitatea “Vasile Alecsandri” din Bacău

Director proiect: Prof. univ. dr. ing. Carmen TEODOSIU

Responsabil proiect partener: Prof. univ. dr. ing. Gabriel LAZĂR

2011 Caruntu Constantin

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