Introduction Observers Luenmberger

8/13/2019 Introduction Observers Luenmberger

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596 IEEE TRANSACTIONS O N MJTOMATIC CONTROL, VOL. ac-16 NO. 6, DECEDER 1971

An Introduction to ObserversDAVID G. LUENBERGER, SENOR MEMBER, IEEE

Abstract-Observerswhichapproximately reconstruct m i s s i n gvariable information necessary for control are presen ted n an

special topics of the identity observer, aobserver, linea r functional observers , stability prop-

iscussed.

I. INTRODUCTIONT IS OFTEN convenient when designing feedbackcontroI systems to assume nitially th at t.he entire

ate vector of the system to be controlled isavailableugh measurement. Thus for th e linear time-invariant

i t ) = Ax( t ) + Bu(t) (1.1)X is an n X 1 state vector, u is an I X 1 inputA is an n X n system matrix, and B is an n X Tbution matrix, one might, design a feedback law of

u t ) = t x t ) , t ) which could be implemented ifmere availa.ble. This is, for example, precisely the

rol lam that resu1t.sfrom solution of a quadraticoptimization problem posed for the syst,em ( l) ,

from numerous other echniques ,hat, nsureilit,y and in ome sense improve system performance.If the entirestate vectorcannot be measured, as is

u t ) = d r x t ) , t ) cannot be implemented.us either nen- approachhat, direct.12: accounts

e nonavailabi1it.y of the entire state vector must besed, or a suitable approximation to the sta te vectort be determined t,hat can be substitut,ed into t>hecon-law. I n almost every situation the latt.er approach,

at of developing and using an approximate state vector,vast ly simpler t han a new direct att,ack on the designAdopting this point of view, t,ha t an approximate s tate

orhe unavailable state,in the decomposit,ion of a control design problemo tx o phases. The first phase is design of the control

vector is available. This mayor other design techniques anda control law without dynamics. The

is t.he design of a syst.em th at produces anpproximation to thestate vector. This syst.em, n-hich

deterministic setting is called an observer, or Luen-distinguish i t from t,he Iialman filber,

R. W.Brockett,, Associate Guest Editor . This esearch was supportedManuscript received July 21, 1971. Paper reconunended bypart by theNational Science Foundation under Gran tGK16125.ystems, School of Engineering, Stanford University, Stanford,Theauthor is with he Depart,ment. of Engineering-EconomicCalif. 9430.5, current.ly on leave t Office of Science and Technology,Execut,ive Office of th e President,, Washington, D.C.

ha.s as its inputs the inputs and available outputs of thesystem whose stat e is to be approxima.ted and has a statevector t,hat is linearly related to the desired approxima-tion. The observer is a dynamic syst,em whose charac-teristics are somewhat free t be determined by the de-signer, and it is through its introduction that dyna.micsent.er the overall two-phase design procedure when theentire state is not available.

The observer wa.s first proposed and developed in [ l ]andfurther developed in [2]. Since theseearlypapers,svhich concentrat,ed on observers for purely deterministiccontinuous-time linear ime-invariant sy st em , observertheory has been ext,endedby several researchers to includetime-va.rying systems,discrete ystems, and tochasticsystems [3]-[18]. The effect of an observeron ystemperformance (with respect to a quadratic cost functional)has been examined [ 5 ] , [19]-[22]. New insight,s have beenobtained, and t.he theory has been substantially strea.m-lined [23]-[25]. At. t.he same t.ime, since 1964, observershave formed an integral part of numerous control systemdesigns of which a small percentage have been explicitlyreport,ed [26]-[31]. The simplicity of i ts design and itsresolution of the difficulty imposed by missing measure-mentsmake the observer an attract.ivegeneral designcomponent [ F A ] [32], [33].

In addition to their practical utilit,y, observers offer aunique heoretical ascination. The associated theory sintimatelyelated t.0 the fundamentalinearystemconcepts of cont,rollability, observability, ynamic re-sponse, andst.ability, and provides a simple set.ting inwhich all of these concepts interact. This t,heoreticalrichness has made t.he observer a n at,tractive area of re-search.This paper discusses the basic elements of observerdesign from an elementary ienToint. For simplicityattention isrestricted, as in he earlypapers, to deter-ministic continuous-time linear time-invariant systems.The approa.ch t.aken in this paper, hon-ever, is influencedsubsta,ntially by he simplification and insights derivedfrom the work of several other a.ut.hors during the pastseven years. In order t.o highlight t,he new t.echniques andto provide the opport.unit,y for comparison with theold, many of the example syst,ems presented in t,his paperare the same as in the arlier papers.

11. BASIC HEORSA . Almost any System is an Observer

Initially, consider t,he problem of observing areesystem SI, .e., a syst,em with zero input. If t,he availableoutput,s of this system are used as inputs o drive anot,hersystem Sz, t.he second system will almostalwaysserveas an observer of the first system In that . its state will


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LUENBERGER: AN INTRODUCTION T O OBSERSERS 597and Sz, he observer, is governed by

i t ) = Fz t)+ G g ( t ) 2 .5 )Fig. 1. A simple observer. then H = G C. In designing the observer the m X n

matrix C is fixed and t.he n X m matrix G is arbitrary.tend o t.ra.ck a linear tra.nsformat.ion of thestate of t,he Thus an ident,itv observer is determineduniquelybyfirst, system (see Fig. I ) . This result forms t,he basis of selection of G a,nd takes t,he formobserver theory and explains Ivhy there is a great. deal offreedom inhe design of an observer. i t ) = ( A GC)z( t )+ G g ( t ) .2 . 6 )

z t ) = T x ( t ) + eFf z(O) T x ( O ) ] .2 . 1 ) havearbit.rarydynamics if the original system s com-pletely observable. First, recall t ha t a syst.em ( 2 . 4 ) isProof: We may write immediately . ,completely observable if the matrix

Substituting TA FT = H t,his becomes has rank n. Generally, if an n X n mat.rix A andan m X ni t ) T i t ) = F [ z ( t ) T x ( t ) ]

which has ( 2 . 1 )as a solution.It should be notjed t,hat t.he two systems S and Sp

need not, have the mnle dimension. Also, it can be shown[ l ] hat there is aunique solution T t,o theequationTA FT = H if A and F have no common eigenvalues.Thus a.ny syst>emSzhaving different, eigenvalues from Ais an observer for SI n the ense of Theorem 1 .

Next, we n0t.e that he result of Theorem 1 for freesyst.ems can be easily extended to forced systems byincluding the input. in t,he observer as well as the originalsystem. Thus if S is governed by

X t ) = A x ( t ) + h t ) ( 2 . 2 )a system Sp governed by

i t ) = Fz( t ) + H x ( ~ ) T B u ( t )2 . 3 )\\-ill satisfy ( 2 . 1 ) .Therefore, an observer for a.system canbe designed by first assuming the system is free and thenincorporating the inputs asn ( 2 . 3 ) .B . Identity ObserverAn obviously convenient observer would be one inwhich the tra.nsformation T relating t.he st.ate of theobserver to the st,ateof the original system is t,he identityt.ransformat.ion. This requires t.hat t.he observer S, be ofthe same dynamic order as he original syst.em S1 a,nd t.hat(with T = I ) F = A H. Specification of such an ob-server rests therefore on specificat,ion of t,he matrix H.

The matrix H is determined part.ly by the fixed out,putstructure of t,he original system and part ly by the inputstructure of the observer. If SI, w3h an m-dimensiona,loutput, vectory: is governed by

i t ) = A x ( t )2 .4 a )= C 4 t ) (2.4b)

matrix C satisfy ,his condition we shall say C, ) iscompletely observable.

Lenanza 1: Corresponding to t,he real matrices C and A ,then the set of eigenvalues of A GC can be made tocorrespond to the set of eigenvalues of any n X n realmatrixbysuitable choice of the real mat,rix C if andonly if ( C , A )s completely observable.

This emma, which is now a cornerstone of linearsyst,em theory, was developed in severa.1 st,epsoveraperiod of nearlya decade. For the case 112 = 1 , corre-sponding to single out,put systems, early statements canbe found in Kalman [ 3 4 ] and Luenberger [ l ] , 3 5 ] .Thegeneral result s implicit.ly contained in Luenberger [ 2 ] ,[ 3 6 ] ,and the problem is treabed definitively in Wonham[ 3 7 ] . A nice proof is given byGopinath [ I . (It wasrecent,ly pointed out to me t,hat, Popov [ 3 8 ] published aproof of a result of th is type in 1964.) Calcu1at)ion of theappr0priat.e G ma.t,rix .0 achieve given eigenvalue place-ment for a high-dimensional multivariablesystemcan,however, be a difficult cornputrational chore.

The result. of this basic lemma translat,es directly into aresult on observers.

Th.eorem 2: An identity observer having arbitrarydynamics can be designed for ainear t.ime-invariantsystem if and only if the syet,em s completely observable.

In practice, the eigenvalues of tlhe observer a.re selectedt o be negative, so t.hat t.he state of the observer willconverge to t>he tate of the observed system, and t.hey arechosen t,o be somewhat, more negative t.han the eigen-values of the observed system so t.hat convergence isfaster than other system effects. Theoretically, the eigen-valuescan be moved arbitrarily t,oward minus infinity,yielding ext.remely rapid convergence. This ends, how-ever, t,o make the observer ac t like a differentiator andt,hereby become highly sensitive to noise, and to introduceother difficukies. The general problem of select,ing good


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598 IEEE T R A N S A C T I O N SN -4UTOMATIC C O N T R O L , DECEMBER 1971w+l s+2Fig. 2 . A second-order syst.em.

U

Fig. 3. Struct.ureof reduced-order observer.eigenvalues is still not completely resolved but thepracticeof placing t,hem SO that he observer is slightly faster hanThe reduced-order observer was first ntroduced inthe rest of t,he (closed-loop) sgrstem seems t.0 be a good one. [I]. The simple development present,ed in this section ishasepresentation. We aga.in consider the system

Example: Consider t,he system shown in Fig. 2. This due to Gopinath [251.

(2.7a) X t) = Ax ( t ) + Bu(t )y t ) = CX t) (3.2a)(3.2b)(2.7b) and assume without loss of generality tha t the m outputsof the system are linearly independent-or equiva.lentlg

An identity observer isdeterminedby specifying theobserver input vect,or

that the outputdistribution matrix C has rank-m. In thiscase it can also be assumed, by possibly introducing acha.nge of coordina,tes, that . the matrix C akes the form

The resulting observer system matrix is

which has corresponding characteristic equationX + (3 + g1)X + 2 + g1 + g2 = 0. 2.9)

Suppose we decide to make th e observer have two eigen-valuesequal to -3 . This would give the charact.eristicequation (X + 3)2= X 2 + 6X + 9 = 0. Matching coeffi-cientsfrom 2.9)yields g1 = 3, g2 = 4 he observer is thusgoverned by21 = [ - 54 - 1 1 1 2 1 12 +3 1-

111. REDUCEDIMEKSIOKBSERVERThe ident,ity observer ahhough possessing an ample

measure of simplicity also possesses a certain degree ofredundancy. The redundancy &ems from the fact. thatwhile the observer const,ructs an estimat,e of the entirest.a.te,part of t,he state asgiven by the system outputs arealready available by direct measurement. This redundancycan be eliminat.ed and an observer of lower dimension but.still of arbkrary dynamics can be constructed.

The basic construction of a reduced-order observer sshown in Fig. 3. If y t ) is of dimension m an observer oforder n m is constructed with sta te z t ) t.hat, approxi-mates T x ( t ) for some m x n mat,rix T , as in Theorem 1 .Then an estimate i t ) of x t ) can be determined through

provided tha t he indicat,ed part.itioned mnt,rix is in-ertible. Thus the T associat,ed with the observer must

have n m rows that are linearly independent of t.herows of C.

C = [ZiO] .e., C is partitioned into a.n m X m identitymatrix and anm X n mn) zero matrix. A n appropriatechange of coordinates is obtained by selecting an n m)x n mat.rixD in such a. way that

M = [is nonsingular and using the variables X = M x . ) It is thenconvenient to partit.ion t.he state vector as

x = y ]Wand accordingly writmehe syst.em n the orm

k(t) = Al l y ( t ) + A l d t ) + Blu(t ) (3.3a)G(t ) = A a s t ) + A 2 2 ~ l ( t )+ B 2 ~ ( t ) . (3.3b)

The idea of the construction is thenas ollow.Thevect.or y t) is available for measurement, and if n-e dif-ferentiate it., so is k t ) . Since u t) is also measureable3.3a) provides t.he measurement ~ ( i ) or t,he syst.em

(3.3b) u-hich has state vector w ( t ) and nput. Aely(t) +B2u( t ) .An ident,ity observer of order n m s constructedfor (3.3b) using this measurement. Later t>henecessity todifferent,iat,e is circumvent,ed.

The justifica6ion of t.he const.ruct.ion is based on thefollowing lemma [%I.Lemma. 2: If C, A ) is completely observable, then so is

( A n , Am).The validitmyf t.his stat.ement is, in view of the preceding

discussion, int-uitivelylear. It ran be asily proved direct.lyby applying the definition of comp1et.e observability.

To construct the observer we initially define it in theformh t) = A22 LAn)zir(t) + A P I Y ( ~ ) B 2 ~ ( t )+ L ( m A d t ) ) LBlU(t) . (3.4)In view of Lemmas 1 and 2, L can be selected so that


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LUENBERGER: AN INTRODUCTION TO OBSERVERS 599Y Y

U 82 -LE,

W

Fig. 4. Reduced-order observer using derivative.

Y

Fig. 5 . R.educed-order observer.

Azz EAzl has arbi trary eigenvalues. The configurationof this observer is shown in Fig. 4.

The requireddifferentiation of y can be avoided bymodifying the block diagram of Fig. 4 o that of Fig. 5 ,which is equivalent at th epoint, ~. This yields the desiredfinal form of t,he observer, which can be writteni ( t ) = ( A z z LAlz)z( t )+ (An LAlz)~%(t )+ (Azl LAll)y(t) + (Bz LBl )u( t ) 3.5)with

z t ) = G ( t ) 5 / t ) . 3.6)For this observer T = [-LIZ]. This const,ruction enablesus 60 st,ate the ollowing theorem.

Theorem 3: Corresponding t.0 an nth-order completelycontrollableinearime-invariantystemaving mlinearly ndependent outputs astat.e observer of ordern m ca.nbe constructed having arbit.rary eigenvalues.

It is important to understand that the explicit. form ofthe reduced-order observer given here, obt,ained by par-tiOioning the system, is o d y one way to find the observer.In a.ny specific instance, ot,her techniques such as trans-forming to canonical form or simply hypot,hesizing thegeneral structure and solving for the unknown parametersmay be algebraically simpler. Theorem 3 guarantees that,suchmethods will always yield an appropriate result,.The preceding method used in the derivation is, of course,often a. convenient one.

Example: Consider the syst,em shown in Fig. 2 andt,reated in the exa.mple of Sect,ion 11. This s a second-

Fig. 6. Observer for second-order system.

order system with a. single output so a first-order observerwith an arbitmry eigenvalue can be constructed. The Cmatrix already has he required form, C = 1 0. In thiscase A22 GAB = - 1 G , which gives t.he eigenvalueof the observer. Let. us select G = 2 so tha t the observerwill havets eigenvalue equal t.0 -3 . The resultingobserver a.ttached o the ystem is shown in Fig. 6.

I IT BSERVING A SINGLE LINEARUNCTIONALFor some applications an estimate of a single 1inea.r

functional E = ax of the st,at,e s a.11 t,hat is equired.For example, a linear t,ime-invaria,nt, control law for asingle input, syst?em s by definition determined simply by alinear functional of the system stat.e. The quest,ion arisesthen as 60 m-het.her a less complex observer can be con-structed to yield an estimate of a given linear functionaltha.n an observer tha t estimat,es t.he entire st.ate.Of course,again, it is desired to have freedom in the selection of theeigenvalues of the observer.A major result for bhis problem [ 2 ] is that . any givenlinear functiona,l of the sta te, say, E = Q X , can be esti-mat.ed m&h an observer having v 1 arbitrary eigenvalues.Here v is the obsemability index [ 2 ] defined as the leastpositive integer for which the matrix

[CIAC;(Ar)2Ci . . ( A r ) y - l C r ]has rank n. Since for any completely observa.ble systemv 1 5 n m and for many systems v 1 is far lessthan n m, observing a single linear functional of t,hestat.e may be f a r simpler than observing the entire &atevector.

Th e genera.1 form of the observer is exactly analogousto a reduced-order observer for the entire state vector.The estimate of E = ax is defined by

2 t ) = by t) + cz t ) 4-1)i t ) = Fz t ) + H x ( t ) + T B u ( t ) (4.2)

where F H , T , B a.reas in Sect,ion11-A and where b and care vectors satisfying bC + cT = a.

Again the mportant, result is that the observer needonly haveorder v 1. The precise design technique isdict,ated by considerations of convenience.

We illust.rate the general result 1vit.h a single example.The method used in t.his example ca,n, however, be appliedto any multivariable system.


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600 IEEE TRANS.4CTIONS ON AUTOXL4TIC COSTRO L, DECEM BER 1971

I

Fig. 7. A fourth-order system.

XI 2u

x 3

Fig.8. Funct,ional observer.

Example: Consider t,he fourt,h-order systemshown nFig. 7. This system ni th available measurements x1 a.ndx has observability index 2. Thus any linear functionalcan be observed T1-it.h a first-order observer. Let us decidet o construct an observer with a single eigenvalue equal to-3 to observe the functionalxz+ x. .

Initially neglecting thenput LC we hypothesize anobserver of t he form

i = -32 + 91x1+ 93x3.According to Theorem 1 his has an associated T satisfying

r 2 1 01T I ; ; ; :] + 3T = 9 0 93 p 4.3)

-1 0 0 0If T = tl tz t3 t4 , we would like tz = 1, t 4 = 1. Sub-st,itut.ing hese values n 4.3) we obtain the equation

unknowns tl, g1? t3, g3.tl = , t3 = , g1 = -2, g3 = .

this t,he final observer shonn in Fig. 8 is deduced by

V. CLOSED-LOOPROPERTIESOnce an observerha.sbeen construct.ed for a 1inea.rm which produces an estimate of t,he state vector or

f a linear transformat.ion of the stat.e vector it is impor-t, to consider the effect induced by using this estimate

of t.he true value called for by a cont,rol lau-. Ofimportance in this respect is the effect, of an

on the closed-loop stability properbies of t.heIt would be undesirable, for example, if an other-designbeca.me unsta.ble when it was

ntroduction of an observer. Observers,

introduced. In this section we show that if linearcontrol aw is realized u-ith an observer,of the srst,em are t.hnse of t h e

observer it,self and hose ,hxt mouldbe obtained if thecontrol lax- could be directly implemented. Thus an ob-server does not change the closed-loopeigenvalues of adesign but merely adjoins its own eigenvalues. Similarresults hold for systems with non1inea.r control laws [ a ]

Suppose we have the systemX t) = Ax( t ) + Bu( t ) (5.la)

and the control law

If it were possible t,o realize this control law by use ofavailablemeasurements (whichwoulde possible ifK = RC for some R ) , then the closed-loop system wouldbe governed by

X t) = ( A + B K ) x ( t ) ( 5 . 3 )and hence its eigenvalueswould be the eigenvalues ofA + B K .Now if the control cannot be realized directJy, an ob-server of the form

i t ) = Fz( t ) + G p ( t ) + T B u ( t ) (5.4a)~ t ) Ki t) = Ez t ) + Dg( t ) (5.4b)

whereTA FT = G C (5.5a)

= E T + D C (5.5b)canbeconstructed. From ourprevious theory ( C ,A )completely observable s sufficient. for t.here to be G ,E , D F , T satisfying (5.5) with F having arbit,rary eigen-values. Sett.ingu t ) = K f t ) eads to the composit,esystem;] = [ A + BDCG C + T B D C F + T B E zE ] 1. ( 5 . 6 )This whole structurecan be simplified by ntroducing= z T x and using x and as coordinates. Then (5.6)

becomes, using (5.5)(5.7)

Thus the eigenvalues of the composite system arc thoseofA + BK and of F.me note that in view of Lemma. 1 (applied in its dual

form) if the syst,em (5.1) is completely controllable i t ispossible t,o select. K t.0 place the closed-loopeigenvaluesarbit,rarilg. If this control Ian- is not. realizablebut, thesystem scomplet.dyobservable, an observer (of someorder no great,er than n m ) can be const.ruct,edso thatt,hecontrol law can be estimated. Sincc t,heeigenvaluesof the observer are also arbitrary the eigenvalues of thecompletecomposite system may beseltct,edarbitrarily.We thereforet.ate t.he following important result oflinear syst,em theory [I ], [ 2 ] .

- - ~ ~ ~ ~~~ - _ - ---- Theorem 4: Corresponding tonth order completely


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a.nd hence t,he plant follows the free system. This trackingproperty can be used t.o definea closed-loop system for theplant.

Rather than fix a.t,t,entionon t,he fact that only cert,ainFig. 9. A general servodesign. outputs of thelantre available, we concentratmen thefact. that only certain inputs,as defined by B , are availa.ble.If we ha.d comp1et.e freedom as to where input.s could be

Fig. 10. Compensator for example.

controllable and completely observable system (5.1)having m linearly independent out.puts, a dynamic feed-back syst.em of order n m ca,n be constructed such that,the 2n eigenvalues of the composite syst,em take angpreassigned values.

Alt.hough this eigenvalue result for linear t.ime-invariant,systems is of great theoretical interest, it. should be keptin mind t,hat. ,he more general key result is that. stabilitgis not afect,edby a (stable) observer. Thus even for non-linear or t,ime-varying cont,rol laws an observer can supplya suit,able estimate.Example: Suppose a feedback cont.rol spst,em is t.0 bedesigned for the syst.enl shown in Fig. 2 so t.hat its outputclosely tracks a dist,urbance input d. The general form ofdesign is shown in Fig. 9.

For the particular syst,em shon-n in Fig. 2 let us decideto design a control law that places the eigenvalues at.

1 i. It is easily found t,hat u = x + x2 will ac-complish this. If this ~ R Ws implemented with t,he first.-order observer construct,ed earlier, we obtain the overallsystem shown in Fig. 10, which can be verified t,o haveeigenvalues , + i -i.

VI. DUAL BSERVERSThe funda.menta1 propert): of one syst.em observing

another can be applied in a reverse direction to obtain aspecial kind of controller. Such a c~nt~ro ller ,alled a dualobserver, wa.s introduced by Brasch [33].

Suppose in Fig. 1 he systenl S2 s the given system andSI is a syst,em tha.t we construct t,o control S2. MTe haveshown that the system SZ .ends to follow SI and henceSIca,n be considered as governing the behavior of S,.To make tshisdiscussion specific suppose t,he plant

X t) = Ax ( t ) + Bu( t ) (6.la)y t ) = Cx( t ) (6.lb)

is driven by the free syst,emi t ) = Fz( t ) (6.2a.)~ t ) J z ( t ) (6.2b)

where AP PF = BJ for some P . Then from Theorem 1we see that in thiscase the vector n = x + P z is governedby the equa.tion

n t ) = A n ( t )

supplied, t.he output, imitation would not much matter .Indeed, if t,he out,put y t ) = Cx( t ) could be fed t.0 t.hesystem in the orm

i t ) = A x @ )+ L y ( t ) 03-31t.hen t,he eigenvalues of the system would be the eigen-Iralues of A + LC. By Lemma 1, if the s:stem is observ-able L can be selected to place the eigenvalues arbitrarily.Thedual observer can be thought of assupplying anapproximation to the desired inputs.

To achieve t,he desired result we construct, thedualobserver in the form

z ( t ) = Fz( t ) + M w ( t ) (6.4a.)u t ) = y t ) + C P z ( t ) (6.4b)u t) = J z ( t ) + N w ( t ) (6.4~)

whereAP PF = BJ (6.h)L = P M + B N . (6.5b)

Equations (6.5) are dual o (5.5) and will have solutionJ , MI N , F wit.h F having arbi t,rary eigenvalues if (6.1)is conlpletely controllable.

The composite system isA + BNC BJ + BNCP x . (6.6)41 = [ M C F + M C P ] ]

Introducing n = x + P z and using z and n for coordinatesyields the comp0sit.e syst,em

i [ + LC 01 ]MC F zwhich is the dual of 5.7). The eigenvalues of the com-posite system a,re thus seen to be the eigenvalues of A +LC and the eigenvalues of F . We may therefore st,at.e hedual of Theorem 4.

Th.eorm 5: Corresponding to an nth-order completelycontrollable and complehely observableystem (6.1)having T linearly ndependent. nputs,adynamic feed-back system of order n r can be constructed such t,hatthe 2n eigenvalues of t.he composite system t,alce anypreassigned values.

1711. CONCLUSIOKSIt has been shown t.hat mising &ate-variable infor-

mation, not available for measurement,, ca.n besuit,ablyapproximated by an observer. Generally, as more output,variables aremade available] t.he requiredorder of theobserver is decreased.


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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, DECEMBER 1971

Although the introductory reat,ment given in his [281 P. V. Nadezhdin, The optimal control law in problanswitharbi trary nit ial conditions, Eng. Cybern. USSR),pp. 170174. 1968.a.per is restricted o ime-invariantdeterministic con-syst,ems, much of the t,heory can be [291 E E. APPlicat,ion of observers andopthum filters tot.0 more general sit.uations. The references cited inertial systems, presented at. the I FAC Symp. MultivariableCont,rol SF tems , Dusseldorf, Germany, 1968.this paper should be consulted for these extensions. [30] D.Q : Xayne and,:. Murdock,Modalcontrol of linear timemvarlant systems, Int. J . Contr., vol. 11, no. 2 pp. 223-227,1970.

REFERENCES111 D. G. Luenberger, Observing the st.ate of a inear system,IE EE Trans. X i l , Electron., vol. MIL-8, pp. ?&SO Apr. 1964.[2] ---,Observers formultivariableystems, IEEE Trans.[3] M. Aoki and J. R. Huddle, Est,imation of stat e vector of adubma t. Contr., vol. AC-11, pp. 190-197, Apr. 1966.linear stochastic syst.em wit.h a constrained estimator, IEEETrans. Automat. Contr. (Short. Papers), vol. AC-12, p p. 432-433. A I I P . 1967.

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David G. Luenberger (S57-kI@-S-MJ7l) wasborn in Los Angeles, Cali. , on September 16,1937. He received the B.S. degree from theCalifornia Inst,it.ute of Technology, Pas-adena, in 1959 andhe M.S. and PbD.degrees from Stanfordniversity, PaloAlto, Calif., in 1961 and 1963, respectively,all in electricalengineering.Since 1963 he has been on t.he faculty ofStanford Universit,y, where presently he is aProfessor of Engineering-Economic Systemsand of Electrical Engineering. He is also d i a t e d with the Depart-ment of Operations Research. His activit.ies have been centered pri-marily in t,he graduate program, where he has t,aught ourses in opti-mization, control, mathematical programming, and informationtheory. His research areas have included observability of linear sys-tems, the applicat.ion of functional analysis t.o engineering problems,optimal cont.ro1, mathematical programming, and optimal planning.His experience includes summer employment a t Hughes AircraftCompany and Westinghouse ResearchLaboratories from 1959 to1963, and service as a consultant to West.inghouse, StanfordRe-search Institute, Wolf Management Semices, and Inta sa, Inc. Thisexperience has included work on the control of a large power generat-ing plant, he numerical solution of part id differential equations,optimization of trajectories,optimal planning problems, and nu-merous additionalproblems of optimization and control. He hashelped formdate and solve problems of control, optimization, andgeneral analysis relating t.0 electric power, defense, water resourcemanagement., telecommunications, air traffic control, education, andeconomic planning. He is the author of Optimizationb y Vector SpaceXethods (Wiley, 1969) and has authored or co-authored over 35technical papers. Current lyhe is on eave from Stanford at theoffice of Science and Technology, Executive Office of the President,Washingt.on, D.C.He assist,s th e Science Adviser -with programplanning, review and evaluation, and with formulation of sciencepolicy in areas of civilian technology.Dr. Luenberger s a member of Sigma Xi, TauBetaPi, heSociety for Industrialand Applied Mathematics, he OperationsResearch Society of America, and heManagement Science In-stitute. He served as Chairman (Associate Editor) of the Linearfrom 1969 to 1971.Systems Comnlit.tee of the IEEE Group on Automatic Cont,rol

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