Lucrare_Luenberger_2

8/3/2019 Lucrare_Luenberger_2

1/8

APRIL, 1966

Observers for Multivariable Systems

D. G. LUENBERGER, MEMBER, IEEE

Abshacf-Often in control design it is necessary o constructestimates of state varia bles which are not available by direct mea-surement. If a system is linear, its state ector can be approximatelyreconstructed by building an observer which is itself a linear systemdrive n by the available outputs and input s of the original system.The state vector of an nth order system with m independent outputscan be reconstructed with an observer of order n - m .

In this paper it is shown that the design of an observer for a sys-tem with M outputs can be reduced to the design of m separate ob-serve rs for single-output subsystems. This result is a consequenceof aspecial canonical form developed in the paper or mult ipleoutput systems.

In the special case of reconstructi on of a single linear functionalof the unknown stat e vector, it is shown that a great reduction inobserver complexity is often possible.

Finally, the application of observers to control design is investi-gated. It is shown that an observers estimate of the system statevector can be used in place of the actual state vector in linear ornonlinear feedback designs without loss of stability.

I. ISTRODUCTION

LM 4K Y COSTROL system designs are based onstate vector feedback, where the input to thesystem is a function only of the cur ren t sta te

vector. Fo r example, i n the case of a con tinuou s, linear,time-invariant system of the form

n-here

x is an n X 1 state vectorZ L is an r X 1 input vector

A is an n x n transition matrixD is an n X r distribution matrix,

such a design xould express u ( t ) as a functio n of x an d t ;u ( f ) = F [ x ( t ) , t ] . Th e function F is determined by theparticular design scheme employed. It m ay b e th e co n-trol function n-hich in some sense optimizes system per-formance or i t may be determined by some other design

techniquyposs ibly tr ial and error. Such state vectorfeedback designs offer many advantage s with respectto both sl-stem performance and analysis. There is, olv-ever, one major disadvantage. In many control situa-t ions the system state vector is not available for directmeasurement. In these situations, i t is not possible oevaluate he ontrol unction F[r ( f ) , ] , an dhence,either he control cheme must be abandoned, or areasonable ubstitute or he tate vector mustbefound.

1966.This

workwas suDDorted in

D a r tbv the Office oi Kava1 Re-

hlanuscript received September 27, 1965; revised Januark- 11,~ ~ . . - ~

search Contract KOI%R-%(83).

Vniversity, Stanford, Calif.The author is with the Dept. of Electrical Engineering, Stanford

Typica lly here will be associated with he system(1) an output vector y of dimension rn < n of the for m

y = H*x (2)

u-here H* is an m X n output matrix. I t will be assumedthat the outputs represented by the components of yare independent-or equivalently, tha t the mat rix H*has rank wz .

T he problem discussed in th is p ape r is t hat of recon-structing the state vector from the available outputs.The system 1%-hich erforms the reconstruction is calledan observer.

One possible method for obtaining he state vectoris t o b uild a model of the given system, drive the modelwith the same inputs as the riginal system, and use thestate vector of the model as an approximatio n to theunknou-n state vector. In this method the dynamic e-havior of the observer is identi cal with that f the sys-tem it ob serves. If initial conditions v-ere not set prop-erly or if ther e n-ere slight d isturb ances, t he mod el gen-erally n-ould not ecover ast enough o provide anestimate suitable for control.

Another approach is to differentiate the available outputs a number of times and then combine these deriva-

tives appropriately to obtain the state vector. In thiscase, the estimate responds instantaneously to distur-bances, bu t it s severely degraded by small quantity ofadditive noise in the measurements.

I t i s im port a nt t o design an observer which is a com-promise bet\\-een these tn-o simple procedures. I t is de-sirable that the dynamic elements of the observer befaster than those of the system it obse rves , but not sofa st a s to possess the undesirable characteristi cs of dif-ferentiat ors (n-hich correspond to poles a t - x ) .

I t has been shou-n [ l ] h a t a n observer for the system(1) can be constru cted which is itself a linear, time-in -

variant system driven by the system i t observes. Theobserver need only contain n - m poles and these maybe chosen arbitrarily.

In Section I1 of t his paper the general theory of ob -servers is reviewed with particular emphasis on single-output spstems.

Section I11 con tain s he main contribution of hispaper. A nen- procedure is developed for designing ob-servers for sys tems which have several outputs. Essen-tially, he problem s reduced o a seri es of obs erve rdesigns for single-output systems-these individual de-signs being relatively straightforu-ard.

the nzXn matrix H* is the transpose of n n X m matrix H.*The star notation represents the transpose of a matrix. Thus

190


2/8

LUENBERGER. OBSERVERS FO R MULTIVARIABLE SYSTEMS 191

The new procedure, based on pecial anonicalorm for multiple-output systems developed in the Ap-endix, not only leads to simpler observer designs butso to stronger theoretical results than obtained previ-

In Section 11, the problem of reconstructing a singlenear unctiona l of the state ather han he entireate vect or is considered. I t is shown th at considerableduction in observer complexity is then possible. This

rocedure is similar to one given by Bass and Gura [2 ]or the design of si ngle-inp ut, linear control systems.

Finally, the stab ility im plications of using t he recon-ructed state vector rather than the actual state vec-

or is discussed. I t is sh oxn tha t observers may be usedo realize both linear and nonlinear control laws withoutss of st abi lit y. Thu s, t s concluded th at observers

an effectively surmount the difficulties associated withontrol design when the state s not measurable.

usly.

I I. OBSERVERSWITH ARBITRARY DYSAMICS

The basic observer configuration is illustrated in Fig.. The system SI is assumed to be a linear time-invari-n t system. I t is assumed initially that the system isee, Le., unforced. The available outputs f this system

rive a second system , which is the observer. Theorem(proved n 111) shows that under hese conditions,

e observers state vector will nearly always tend to-ard a linear transformation of the first systems state

ector.Th.eorenz 1 (Observatiotz of a f r e e s y s t e m ) : Let S1 be a

ee system: 2 = A x which drives S,: i = Bz+ Cx. Sup-ose there s a ransf ormatio n T atisfying T A - BT

C. If z ( 0 ) = T x ( O ) ,then z ( t ) = T?;(t) or all t 2 0 . Orore generally,

z( t ) = T.r( t ) + e B t [ z ( 0 )- Tr(O)]. (3 )Kotice that in Theorem 1 i t is not necessary for A

nd B to be the same size; they only have to be square.A and B have no common eigenvalues, there s alwaysunique T satisfying T A - T = C [ l ] .Assume now that the system SI is governed by

i = A x +Dzt (4)here u s an input ector. An observer driven by bothe plant input and outputs governed by

i = Bz + C X+ TDzt (5)ill behave accordi ng o (3 ) . Therefore, observers de-gned for a free system can be applied o he forcedstem if the i nput is suitably connecte d to the obs erver.The co nstruction of an observer rests on the solutionthe matrix equation T A - B T = C. Th e soIution T

ust have rank great enough to guarantee the recoverythe unmeasurable state variables. A s illustrated be-

w, observer design consists in choosing B and the partC which is unspecified in such a way that T has tha t

operty.

In th is se ction atte ntion is ocused on the problem ofconstructing observers or ingle-output ystems. nSection 111 i t i s shown that he multi ple-o utput casecan be reduced to this form.

Tw o possible constructions for an observer are con-sidered here; both based on a well-known [3 ] canonicalform or an observable single-output system. An ap -proach more suitable for computation, which does not

employ canonical forms, ma y be found n [ l ] .Suppose an nth order system is governed by

and has a single output2 y=kr. I t is assumed that thesystem is completely observable [SI , i .e., the vectors h,A * h , - , (A*) - lh are inearly ndependent. Fo r sucha system there exists a coordinate system in which thesystem is represented in the special canonical form [ 3 ]

The system in this form is represented schematicallyin Fig. 2. Here the vector x is represen ted in the newcoordinate ystem so that he new state variablesXI, x 2 , . . . , x , do not necessarily correspond o th eoriginal state variables-the original variables are sup-

pressed from consideration for the present purpo ses.I t is readily verified that the characte ristic polyno-

mia l of th e matrix n (7) is sn+ansn--l +oln-lsn-2+ - .+Cub

Consider an nth order observer for this system gov-erned by

Fig. 1. A simple observer. 2

Fig. 2. Canonical form of observable system.

resented by h throughout this paper rather than by It*.2 As a matter of convention th e trans pose of a vector h is rep-


3/8

192 IEEE TRANSACTIONS O N AUTOMATICONTROLPRIL

-61 1 o - - . o (P 1 - ad

( P n - a74

= BZ + CS. (8)

I n this case i t is readily verified th at T A - B T = C issatisfied b y T = so that s tate variables f z are in directcorrespondence 11-ith those of x. Since the characteristicpolyno mial of thi s syst em is sn+pnsn-'+ - . . +Dl andthe coefficients i are arbitrary, it is lear that the polesof a n observer of this fo rm are arbit rary.

Consider now the problem of constructing an n - ) thorder observer for (7). If t he observer is taken as

O 1

T A - T = C is satisfied b y

1 .n- 1 1I

sump tion is that of complete observability with respectto the outputs. A system

x = -4 x

y = H* s (1 2)

iscompletely observable [3] if the nx (nnz) matrix[H , A *H, . . . , (A *)n - lH] as rank n. Often a rank ofn will be achieved with a smaller number of powers ofA * times H. The observabili ty index of the system 12)is defined as he east positive nteger or which hematrix [ H , A * H , . . . , ( A*)I-'H] has rank n . The ob-servabil ity index plays a key role in the theo ry of ob-servers for nlultivariable systems.

For some systems the extension to the multiple-out-put case is elementary. For example, consider the fourth-order system shown in Fig. . I t is assumed that the twovariables x1 and x3 are available for direct measurement.

The ourth-order ystem may be egarded as tw ocoupled second-order subsystems as ndicated by hedashed-line boxes in the figure. Th e o ut pu t of the firstbox is the measurable variable x1 and the input is themeasurable variable x3 . Therefore, since it is possible tomeasure the input and output of this second-order sub-system, a first-order observer may be constructed forthis subsystem. Similar considerations apply to the ec-ond box, so i t is seen tha t a n observer for the total sys-tem can be built up from two separate observ ers, eachobserving a single-output subsystem.

Th e seemingly fortuitous situation in the above example is actually commonplace. In fact, all multiple-outputobserving problems can be reduced to the observationof s ingl e-out put subsy stem s. This is a resul t of th e fol-lowing theorem (proved in the Appendix) guaranteeingthe existence of a special canonical form.

Theorem 3 (Canonica l Representat ion of X d t i p l e -Output S y s t e m s ) : Suppose hat he nth order systemi A x n-ith associated out put vect or y = H *x is com-plete1)- observable n-ith observability index v. Supposefurther that y consists of m independent components.Then here is a nonsingular inear coordinate changesuch th at in terms of the neu- coordinates the system hasthe represent ation sho wn in Fig. 4. In this form the sys-tem consists of m component subsystems, each with oneobservable ou tpu t u-hich is a linear combination of thecomponen ts of y. The orders of the subsystem s satisfynl+nl+ . . . +n,=n, and the largest subsystem is oforder v. The subsystems are co upled to each other 0nl>7through their outputs.

From he previous developments t is obvious hatan observer can be construct ed for each of th e single-output subs)-stems, since the inputs to thesubsystemsare available or measurement. Furthermore, he kthobserver can be designed with nk- 1 arbitrar). poles.Thus, by employing Theorems 2 and 3 , i t is easy odeduce

Theorem 4: observer or he ystem 12) an econstructed oforder n--nz. ( . t is he ank of H*. )


4/8

966 LUENBERGER: OBSERVERS FOR MULTIVARI.4BLEYSTEMS 193

Furthermore, the n--m poles of the observe r are arbi -rary.

This result is slightly stronger than the correspondingonclusion of [ l 1. Furthermore, in the form developed

here here s established a simple algorithm for com-uting the observer in terms of the single-ou tput sub-ystems, whereas the method of [ l ] s not so straight-onvard.

Example: Th e sg-stern illu stra ted in Fig. 3 is already inanonical orm pprop riate for esign of a econd-

order observer. The poles of the observer are arbitrarynd will bot h be chosen to be -3 . Th e design is carried

out separately for each subsystem.System S1 is governed by

s2 SIr----------- r - - - - - - - - - - 1

Fig. 3 . Fonrth-order s yt er n of example.

a- RDER

IIhRDER

a

a

a

wRDERFig. 4. Canonical form of multiple-output system.

Fig. 5. Observer for fourt h-ord er s>-stern.

According to t he res ults of Secti on 11, an observer witha pole a t - 3 driven by xl = ,1 0 , x will produce T xwhere

or T = ,1 - , . The obse rver will be governed accord-ing to ( 5 ) which in thi s case is

i = - 3 z + x 1 - sa. (15)The est imate .t 2 of the vari abl e x? is c onstruc ted fromthe measurement x1 an d z according to

A similar procedure applied to the subsystem Sz leadsto the observer

Th e complete observer or he syst em is shown inFig. 5.

IV . OBSERVING SINGLE INEAR FUKCTIONAL

Sometimes it is only necessary o estimate a single(b ut prespecified) inear functional of a sytem's statevector. This is the situ atio n, for example, in the designof line ar, ime-i nvar iant , stat e eedb ack for a ingle-input system. In hese nstances, an obs erv er of con-

siderably reduced complexity can often be constructed\vhich will pro duce this single qua nti ty.

Observa tion of a single linear functional is similar inconcept o a feedback design procedure developed byBass and Gura [2 ] . The method n [ 2 ] is not an observermethod in the sense of this paper and does not enjoy theclosed-loop stability properties developed in Section V .T h e conclusions in th is section concerning the requir eddyn amic orde r of such an observe r, however , coincid e\vith the corresponding conclusions of [2] .

Imagine an observer construct ed for a multiple-out-put system according to to the scheme of Section 111.

T h e o u t p u t of the observer is an esti mate of th e syste mstate vector x. In order to obtain an estimate f a linearfunctio nal of x, sa y u 'x , the same linear functional f theobserver output is taken . The resul t is shon-n in Fig. 6 .

The larg est block in the observer has exactly Y- 1poles which may be chosen arbitrarily . Suppose hesepoles ar e chosen first. Th en th e poles of each of t he ot he rblocks of the observe r can be chosen to be a sub set ofth e poles of this arge st block . Xou;;, corresponding toeach output y k there is a transfer functio n of th e form&(s)/A(s) rom yn through he observer o u'2. T h epolynomial A(s) is the characte ristic poll-n omial of the

largest block in the observer, and &(s) is a polynomialof degree no greater than that of A(s) .


5/8

194 IEEE TRANSACTIOKS ON AUTOX4TICONTROL APRIL

Thus, i t may be concluded that the obs erv er f Fig. 6is equivalent to the form shon-n in Fig. 7 when the in-dividual blocks of the original observer have commonpoles. Xn obs erver of this fo rm can be realized by a sys-tem of order v- 1 the refore the follon-ing theorem isestablished.

T h e o r e m 5: X single linear f unct ional of th e s ta te of alinear system can be observed b>- a system lvith v- 1arbit rarl- poles. (v is the observabi lity index of the s!.s-tern.)

.As pointed out in Bass and Gura [2 ] , v - 1 is oftenconsiderabll; less than n - m , the ord er of a completeobserver. n act, (%,:HZ) -1 < v - 1 I n - m . t\vent!--fifth order s!-stem with five ou tp ut s, for exa mpl e, mayrequire as fen as four arbitrar>- poles to construct anesti mate of a single linear functional of t he sta te vec tor .

E xampl e : Consider again the fourth-order system n Fig.5 . Suppose i t is desired to reconstruct the single linearfunctional x2+x4. &According o T heo rem 5 an observerlvith a single arbitrary pole is sufficient. If the pole ischosen to be a t - 3 , the observer constructed in Section

i - - _ _ - _ _ _ -1Fig. 6. Observing a single linear functional.

1 - 1

Fig. 7. Reduced observer for single linear functio nal.

UA Ax + x q2

1

Fig. 8. First-order observer for example.

I I I for this s ystem can be used as a fir st s tep n the de-sign procedure.

Using t he resul ts ob tained in the previous example

.?? + 2 4 = X I + 3x 3 - z - 6 ~ . (19)By carrying ou t some straightforward manipulations,the observer shovx in Fig. 8 is obtained.

In general, i t is possible to short cut the two-s tep de-sign procedure outlined in this section. One may boldlyhJ-pot hesize the required form of the observe r (accord-ing to Theorem 5 ) and solve for the unknown numeratorpol>-nomials irectll--without irst onstructing hecanonical form. I t turns out that this shortcut proceduresaves only a small percentage of the labor involved anddoes not offer much insight into the observer process.This shortcut procedure follon-s the spirit of Bass an dGura [ 2 ] , however, and is certainll- recommended foractual computations of observers of thi s typ e.

Y. CLOSED-LOOPT-ABILITY ROPERTIES-4s state d in the Introduction , the investigat ion f ob-

servers is aimed a t circumven ting the difficulty of real-izing state vector feedback control when the entire statevector is unavailable for measu rement. Consider a sys-tem governed by

2 = A x + Dld (20)u-ith outputs y = H * x . Suppose hat a (possibly non-linear) control lax of the form 1 = F ( s ) has been derivedfor thi s s!-steIn by s ome design scheme. -An appr opr iat eobserver for (20) is

f = Bz + Cs + T D u (21)\\-here T A - T = C. Cx must be derivable from the out-pu tvector y : hence, C = G H * for ome ppropriatematrix G. The est imated state is a linear combinationof the system outputs and the state vector of th e ob-server

.t = L s + K z (22)\\-here L + K T = I (the dentit>-). he ontrol aw21 = F(.v) can be approximated by a control aw 12 = F ( 2 )based on the estimated state vector. The complete ys-tem is then governed by the equatio ns

f = A x + D F ( . f )i = Bz + CX + T D F ( . t )2 = Lx + kz. (23)

I t is th e purpose of this section to nvestigate hestability properties of the cont rol syst em gover ned by

The system equations (23) can be rearranged so t h a tman >- of the ir stability properties become clearly ap-parent. Defining Z = Z - Tx, = . t - x and then subtract-

ing T times the first equation in (23) from the seco ndleads to the equivalent system

(23).


6/8

966

d = A s + D F ( i ).? = Bz

j = KE. (2.2)

I f , initially, the estimated state vector s equal to thectual state vector, i.e., ~ ( 0 ) = 0 , qual ity n-ill be main-ained for all future time. T his importa nt fact s due to

what migh t be described as the complete u~zcontrollabil-t y of th e observer from zt. I t implies that i f there isnitial equality between the state and its estimate thelosed-loop system using the estimate behaves exactlyike the closed-loop sl-stem using t he actua l s tate to ob-ain the control.

Generally the initial equality between the state andts estim ate will not obtain, and stability properties ofhe complete s>-stem, ncluding he observer, must benvestigated. first result in this regard applies to linearontrol laws. If F ( x )= FN is lin ear, the closed-loop sys-em using th e actua l stat e is i = A + D F ) x . I t has beenhon-n [ l ] , 4] that f an observer with transition matrix

B is used to suppl y a n estimate of the state vector , thelosed-loop poles of the overal l ystem (23) are heigenv-alues of A f D F an d of B . In other words, the ob-erv er does not distur b the poles of the original systemu t merely adds its on-n poles.

In a similar fashion, it is possible to investigate theffect of an observer n realizing a nonlinear control law.

Supp ose t hat th e closed-loop system

i = -4s + D F ( s ) (25)

s asymptoti callp stable in the large [S I . I t is assumedere that the asymptotic stability of (25) is establishedy he onstructi on of a continuously differentiableiapunov function l'(x) for the sys tem n-hich satisfies

he following conditions

1) T'(x) > 0 for N #e, V(8) = 02) T.,'(X)-+= as l l x l : + ~3) ~ ( x ) () =vz~'(x)A x + ~ F ( x ) lO for x# O4) l im, ,z : :+m L ~ ~ ( ~ ) ~ ~ ~ v , T ~ ( . x ) ~ ~+ x .Th e first three assumptions are ufficient to guarantee

s>-mptotic stability f (25), while the fourth is an addi-

ional assumptio n xx-hich is often satisfied n p r a ~ t i c e . ~Theorem 6 sh ow s hat un de r relative117 mild condi-

ions he observer scheme outlined above eads o ansymptoticall>: stable system.

Theorem 4 : Assume that there s available a Liapunovunction for the system 3= . 4x+DF(x ) nhich satisfiesh econd itio ns 1)-4) listed bove. If F ( x ) satisfies aniform Lipschitz condition and the observer is asym p-otically stable in the large , i .e., B has i ts eigenvalues inhe left half plane. The co mplete sys tem (24) is asymp-oticall y stable in the large.

hat - V / Vz.VII>c!ls! l+m.3 For example. if is a pd quadratic form, there is a c>O such

Proof: 4 s a first step in the proof a quadratic Liapunovfunction is constructed for the observer i= BZ by thestandard procedure for stable, inear, ime-invariants1-stems [S I , [ 6 ] .For this purpose define P as the uniquesolution to the matrix equation PB+ B*P= - I . I t iswell known th at he matrix P so defined is positivedefinite and that 8'P8=: : 211 2 p is a Liapunov functionfor he yst em i= B i with derivative (d:: 'dt)II%;p =

-112'12. For he overall ystem (24) define Tl'(.v, 2)= l,'(s) ! I Z ' ~ ~ ~ .V(x, 8) is clearl?; positive definite. ,Mso

W ( x , E ) = G,T'(x)[dx + D F ( P ) ] - ( E j ) 3= L,7(x)+ V,V(x)D[F( . i - ) - F ( s ) ] - l Z I 1 25 c ( x > + c ~ ! ~ G , v ( . x ) ~ ~l l q- i z I (? (26)

where the positive constant c1 is determined by theLipschitz condition.

Using ~ = K Zrom (24) , the above inequality can beconverted to

Ti'(s, z) 5 C ( s ) + G Z l ~ V J ( H ) ( I . 1 1 Z ( 1- ' y . ( 2 7 )L'sing the funct ion T Vit will non- be shon-n that any

traj ecto ry of the system (24) is bounded. Obviouslyisbounded on any rajectory. Condition 4) on theLiapunov function l,r implies t h at for sufficiently largex and bounded 3 the function l P ( x , 2) is negative defi-nite. Therefore, since W(x, E ) + = as I ~ . Y ~ ~ - + =i t is m-possible that l l x l lincrease with out boun d. Thus hereis an K>O such that for all t > O , I l x ( t ) , l O there s afinite time T such t ha t for t > T , L ~ ( . ~ ) + c ~ ~ ~ V ~ l ~ ~ (II&:I< O throughout the annulus E < ' ~ . L ' / ; Tthe unction is negative defin'ite in this annulus an dx mus t tend to\-\-ard the circle 1 . ~ 1 15 E . Since E was arbi-trary, x tends to 8. This establishes asymptotic stabilityin the large.

VI. COXCLVSIOSS

The observer theory developed in this paper can becompared with other meth ods of st at e estimation. Inthe case of noisy measurements an d unknon-n nois>:disturbance inputs, an optimal least-mean-square esti-mat e of the s tate may be g enerated by linear estimator[i]. f the estimator operates on the infinite past and

the noise statistics ar e stationaq-, he estimator is alinear ime-invariant system driven by he measure-ments. Such an estimator will act as an observer in thesense of t h i s paper if the noise disturbances are suddenlydisconnected so that the system becomes free and noise-less. Therefore, optimal estimato rs can be regard ed asobservers with heir pole locations determined by hestatistical pro per ti es of the noise. In many practicalsituations (namely those i n xi-hich the noise level is sig-nificant), statistically optimal estimators offer excellentadvantages over other estimation schemes. As the noiseleveldecreases, however, heoptimal pole location s

move oward - M an d in the imit ing case of per fec t(noise-free) measurements, hetatistically ptimal

d


7/8

196 IEEE TR.4XS.ACTIOXS ON AUTOMATIC CONTROL

estimator consists of a number of differentiators [SI.IYhen noise level is Ion-, noise is not a critical design fac-to randobserv er pole locations hould bebased onother riteria (e.g., parameter variati on ffects, reli-abil ity . ease of s>mthesis, etc.). Genefall>-, here seemsto be little reason to choose observer poles much fasterthan the oth er poles of the closed-loop s>.stem . So far,however, other than in the statisti cal cas e? there s littletheory devoted to the problem f choosing obs erver polelocations.

The procedure outlined in this paper decomposes con-trol design into state reconstruction an d control of acompletely measurable system. Th e effectiveness andpracticali ty of the method lies in the fac t tha t sta ble op-server poles do not affect overall system stability. Theo-rem 2 of [ l and Theorem 6 of this paper are but tworesults concerned with he tability of combined ob-server and control sj-stems. Other questions concerningspeed of re spon se, ime- vary ing systems. an d varioustypes of stability under various assumptions offer fruit -ful areas for research.

One of the most importa nt results describ ed in thispaper is th e canonical form given in the -Appendix. Th iscanonical form can be used to exten d many n-ell- kno \mproperties of single outp ut or single input syst ems omultivariable s>-stems.

A P P E N D I X

Theorem 3 (Canonical Representatiou of Allultiple Out-p u t S y s t e m s ) : Suppose hat he th rder ystem. ? = A x n-ith associated outp ut vect or y = H * s is conl-pletely observable Fvith observabili tJ- ndex u . Supposefurther that y consists of m independent components.Then here is a nonsingular inear coordinate changesuch that in terms of the ne\\- coordinates the systemhas the representation sho\\-n in Fig. 4. I n this form thesystem consists of m. component subs)-stems, each withone observ able outpu t which is a linear combination ofthe comp onen ts of y. Th e orde rs of the ubsyst emssatisfy nl+n%+ . . . +x,=% and the largest subs)-stemis of order v. The sub syst ems are coupled to each otheronly through their outputs.

P r o o f : The firs t step in the proof is the generat ion of a

certain set of n inea r nde pend ent vectors. Th e pro-cedure used here is identical with hat employed byBass and Gura [ 2 ] for anot her purpose.

Since the matrix [ H ,A * H , . . . , A*'-'H] has rank n ,n independent vectors can be taken as a certain n col-umns of th is mat rix. T o define these vectors precisel>-

Start with he columns 121, h2 , . . . , h of thematrix H .hdjo in o these the olumns A * / z l , A * h 2 , . . ,A*h, one b\* one, checking that each nen- columnis linearl y indepe ndent of the previou s ones. (I-sethe Gram-Schmidt orthogonalization procedure.)If an y of th e ne\\- colu mns is found to be depen-dent, omit i t rom the matri x and o o n to the next.

d) -After A*h, has been tested, continue with A * ) 2 h ~ ,(-4 9 2 / 7 2 , . . * , (A*) 'h, , etc., until n linearly inde-pendent column s have been fou nd.

e) If acolumn (d*) 'h, has been skipped because oflinear depen dence, all colum ns of the form (A*)"jwhere k > can be skipped, since they also must bedependent on the previous columns.

-4s a result of thi s proced ure, th ere is defined a n arra5-of 71 independent x-ectors

/ ? I , - 4 * h l , ' . . , ( - 4 * ) " - ' h 1I??, - 4 * h 2 , . . . , ( A * ) i - v / 2

where for each k, v p s u . Furthermore, by constructionther e are coefficients a i j ( k ) such that

m r- 1

(A*)"%: = a z j ( k ) ( A * ) i / z j ( 2 8 )j = 1 i= o

where a i j ( k ) 0 for i > k an d a i j ( k )= 0 for i = p if j 2 k .T h e desired canonical form of the system \\-ill have a

stru ctur e similar to the struc ture of the above arraJ- inthat the kth subs>-stem vi11 be of ord er VI; . Hon-ever, thestate variables of the kth subs>-stem \vi11 be defined interms of vectors from the complete array rather thanjust the kth ron-. Because of t he compl exit y of not atio ndue to th e several indexes required, the explicit trans-formation defining the appropria te nen- stat e vari able sis no t particularly lluminating and herefore n-ill besuppressed. The ne\\- coordin ates are nstead definedimplicitlJ- in terms of a schemat ic d iagram. The kth ub -s>-s tem takes the form shon-n in Fig. 9.

Fig. 9. kth subs!-atetn of canonical form

T h e ou tpu t s of th e nz subs!,stenls are each linea rcom-bination s of the original outp uts and hence are them-selves measurable quantities. Conversely, the new o ut-put s are 1inearlJ- independ ent so the old outputs can berecovered from the ne\v. (The ind epen denc e of th e ne\\-out put s follo\vs from the fact hat he ra nsfo rmati onmatrix relating he old and nen- outputs is triangular\\-ith 1's along th e diagonal .)

In order to establish that the proposed canonical formis i n fact a linear coordinate change of the original sys-tem, i t is only necessary to verify tha t a l l variables ofthe form z = K ' sin the canonical form satisfJ- = ( A * k ) ' x .That thi s requireme nt is satisfied b\- the kth subsystemshonm in Fig. 9 , fo1lon.s directly from ( 2 8 ) .


8/8

ACKNOWLEDGME?JT [4] D. G. Luenberger, Determining the state of a linear system withobservers of low dynamic order, Ph.D. Dissertation, Dept. of

Th e au th or lyould like to ackno\\?ledge the benefit of Elec. Engrg., Stanford University, Stan ford, Calif., 1963.-veral discussions with P. Haley during the course ofis research.

REFERESCES D. G. Luenberger. Observing the state of a inear sys tem, Z E E E Trans . on : l i i l i t a ry E le r t~on ic~ ,ol. MIL-8, pp. 74-80,

] R. 11. Bass and I. Gura, High order system design via stat e--4pril 1964.

space considerations, 1965 Joint &UtOnlQtif ContioI Conf. (Pue-p ~ i n t ~ ,p. 311-318.R . E. K @ l m a n .Mathematical description of linear dpamicalsystems, J . S l d J I , se i . - 4 , on Contiol, vol. 1, no. 2 , pp. 152-192,1963.

[SI J.,LaSalle and S. efshetz, Stability by Liapzm oas Direct :1fethodw t h Applicat ions . Se w York: -Academic, 1961.

[6] R. E. Kalman and J. E. Bertram, Control system analysis anddesign via th e second method of Lyapunov--1. continu ous-tim esystems, Tmns . A S M E , ser. D , J . BasicEzgrg. , rol . 82, pp.371-393, June 1960.

171 R. E. Kalman and R . S . Rucv. N e w results i n linear filtering- an d.prediction theory, TraiZs. A X M E ,s e i . D , J . B a s i c E n g rg . ,C. 95-

108, March 1961.[SI A. E. Bryson. Jr., and I> . E. Johans en. Line ar filtering for ritne-

varying s>-stems using measurements containing colored nois e,Z E E E TTans. on Automati< Control, vol. AC-10, pp. 9-10, Jan-uary 1965.

O n the a priori Information n SequentialEstimation Problems

Abslracf-In this paper, the effect of errors in the a priori infor-tion is studied when the sequential estimations are carried ut on

e state s of linear systems disturbed y white noise. Four theoremse derived o describe he mutual relations among he hree co-riance matrices, namely heoptimum, alculated, nd ctual

variance matrices, where the last tw o are based on the incorrectriori information. By finding the u pper bound for the variance of

e actual estimate, performance of the Kalman filter is prescribedd the knowledge is utilized for design of the combined system ofalog and digital filters. A phase-locked loop receiver is used as anample of analog filter and the considerable mprovement on theimation process is deduced by the theory and it is confirmed by

e experimental simulation on the digital computer.

TI . ISTRODUCTIOS

H E O P T I l I r l L FILTER n-hich has been ntro-duced by Kalman.? into the ield of system theoryJ-ields the minimum variance esti mate of sta tes

line ar yste ms, n.hich are contaminated by whiteaussi an noises, when a set of s eque ntia l obse rva tion se carr ied out . The asic feature of this filter is tha t theti mat e of stat es is up-dated by a se quence of observa -ons so as to minimize its variance, or equivalen tly, i f

is paper represents one phase of research carried out at the Je tirlanuscript eceived Ju ne 17, 1965; evised Januar!. 18, 1966.

opulsion Labora tory, California Ins titu te of Technology, underAS.4 Contract S.AS 7-100.

st itu te of Technology. Pasadena. Calif.The author is with the Jet Propulsion T.aboratory, Califor~lin

n problems. T Y Q T Z S .S M E , ser. D , J . B as ic E n g rg . , vol. 82, pp.K. E. Kalman. X new appr oac h to linear filtering and predic-

-45. March 1960.*R . E. Kalman and R. S. Bucy, Yew results i n linear filtering

d prediction theory, Trans . 4S: l IE , ser. D , J . Basic Engrg. , 1-01,, pp. 95-107, ll ar ch 1961.

noise is Gaussian, to maximize the conditional probabil-it,: density of the current states after having a set ofobservations.

Th e original theor)- assumes an a priori information

on the initial states and its variances. Hon-ever, thereare cases n-hen his apriori nformation may containcertain errors or some information ay not be availablea t t h e beginning of the estimat ing process.

Soong3 examined the effect of errors i n the a priorid a t a on the a posteriori variance of esti mat es for thesingle-stage ase, .e., or the nonsequential ase in

15-hich es ti ma ti on is carried out after a sequence of ob-servation has been completed, an d derived he devia-tions of th e calculated an d actual variances from hetrue minimum variance. Thus o real-time estimation isperformed in this case. Also th e st at e being estimated

in this model is a constant state vector so tha t the s ta tetransition matrix is an identity matrix and no s>-stemnoise is assum ed there.

The same problem is studied n his paper for themultistage case, namely or he case \I-hen sequent ialobservations and estimations are performed n the s ta tein real-time. Also a general transition form of sta tes isassume d subject to excit ation of t he sy stem noise. The nthere is a n investigation of hon- the error i n th e a prioriinformation propagates through the sequential estima-tion process and h o w i t affects the final result of esti ma-

tion problems, Tr a n s .ASME, ser. D , . Bas i cEngrg., vol. 87,pp. 109-112, hlarch 1965.

T. T. Sooyg, O n a priori statistics n minimum variance estima

197

Lucrare_Luenberger_2

Documents

Transcript of Lucrare_Luenberger_2