An Introduction To Observers

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CONTROL, VOL.

ac-16,

NO.

6, DECEDER 1971

An Introduction to Observers DAVID G . LUENBERGER,

SENOR MEMBER,

IEEE

Abstract-Observerswhichapproximately reconstruct missing state-variable information necessary for control are presented in an introductory manner. The special topics of the identity observer, a reduced-order observer, linear functional observers, stability properties, and dual observers are discussed.

ha.s as its inputs the inputs and available outputs of the system whose state is to be approxima.ted and has a state vector t,hat is linearly related to the desired approximation. The observer is a dynamic syst,em whose characteristics are somewhat free to be determined by the designer, and it is through its introduction that dyna.mics I. INTRODUCTION ent.er the overall two-phase design procedure when the T IS OFTEN convenient when designing feedback entire state is not available. controI systems to assumeinitially that t.he entire The observer wa.s first proposed and developed in [ l ] state vector of the system to be controlled isavailable andfurther developed in [2]. Since theseearlypapers, through measurement. Thus for the linear time-invariant svhich concentrat,ed on observers for purely deterministic system governed by continuous-time lineartime-invariant system, observer i ( t ) = Ax(t) Bu(t) (1.1) theory has been ext,ended by several researchers to include time-va.rying systems,discretesystems, andstochastic where X is an n X 1 state vector, u is an I’ X 1 input systems [3]-[18]. The effect of an observeronsystem vector, A is an n X n system matrix, and B is an n X T performance (with respect to a quadratic cost functional) distribution matrix, one might, design a feedback law of has been examined [ 5 ] , [19]-[22]. New insight,s have been the form u(t) = t ( x ( t ) , t ) which could be implemented if obtained, and t.he theory has been substantially strea.mx ( t ) mere availa.ble. This is, for example, precisely the lined [23]-[25]. At. t.he same t.ime, since 1964, observers form of control lam that resu1t.sfrom solution of a quadratic have formed an integral part of numerous control system loss optimization problem posed for the syst,em (l), designs of which a small percentage have been explicitly from design techniques that place poles at. prespecified report,ed [26]-[31]. The simplicity of its design and its point,s, and from numerous other techniques t,hat, insure resolution of the difficulty imposed by missing measurestabilit,y and insome sense improve system performance. mentsmake the observer an attract.ivegeneral design If the entirestate vectorcannot be measured, as is component [ F A ] , [32], [33]. typical in most complex systems, the control law deduced In addition to their practical utilit,y, observers offer a in t,he form u(t) = dr(x(t), t) cannot be implemented. uniquetheoreticalfascination. The associated theoryis Thus either a nen- approachthat, direct.12: accounts intimatelyrelated t.0 the fundamentallinearsystem for the nonavailabi1it.y of the entire state vector must be concepts of cont,rollability, observability,dynamic redevised, or a suitable approximation to the state vector sponse, andst.ability, and provides a simple set.ting in must be determined t,hat can be substitut,ed into t>hecon- which all of these concepts interact. This t,heoretical trol law. I n almost every situation the latt.er approach, richness has made t.he observer an at,tractive area of rethat of developing and using an approximate state vector, search. is vastly simpler than a new direct att,ack on the design This paper discusses the basic elements of observer problem. design from an elementaryvienToint. For simplicity Adopting this point of view, t,hat an approximate state attention isrestricted, as in the earlypapers, to detervector will be substitutedforthe unavailable state, ministic continuous-time linear time-invariant systems. result,s in the decomposit,ion of a control design problem The approa.ch t.aken in this paper, hon-ever, is influenced into t x o phases. The first phase is design of the control substa,ntially by the simplification and insights derived law assuming that the statevector is available. This may from the work of several other a.ut.hors during the past be based on opt,imization or other design techniques and seven years. In order t.o highlight t,he new t.echniques and typically resu1t.s in a control law without dynamics. The to provide the opport.unit,y for comparison with the second phase is t.he design of a syst.em that produces an old, many of the example syst,ems presented in t,his paper approximation to the state vector. This syst.em, n-hich are the same as in theearlier papers. in a deterministic setting is called an observer, or Luen11. BASIC THEORS berger observer to distinguish it from t,he Iialman filber,

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Manuscript received July 21, 1971. Paper reconunended by R. W. Brockett,, Associate Guest Editor. Thisresearch was supported in part by theNational Science Foundation under GrantGK 16125. Theauthor is withthe Depart,ment. of Engineering-Economic Systems, School of Engineering, Stanford University, Stanford, Calif. 9430.5, current.ly on leaveat Office of Science and Technology, Execut,ive Office of the President,, Washington, D.C.

A . Almost any System is an Observer Initially, consider t,he problem of observing afree system SI,i.e., a syst,em with zero input. If t,he available output,s of this system are used as inputs to drive anot,her system Sz, t.he second system will almostalwaysserve as an observer of the first system In that. its state will

LUENBERGER: AN INTRODUCTION TO

OBSERS’ERS

597

and Sz,the observer, is governed by

i(t) Fig. 1. A simple observer.

tend to t.ra.ck a linear tra.nsformat.ion of the state of t,he first, system (see Fig. I ) . This result forms t,he basis of observer theory and explains Ivhy there is a great. deal of freedom the in design of an observer.

z(t)

=

Tx(t)

+ eFf[z(O) - T x ( O ) ](.2 . 1 )

Proof: We may write immediately

Substituting TA - FT

=

i(t) - Ti(t)

H t,his becomes =

F[z(t) - T x ( t ) ]

which has (2.1) as a solution.

It should be notjed t,hat t.he two systems SI and Sp need not, have the mnle dimension. Also, it can be shown [ l ] that there is aunique solution T t,o theequation TA - FT = H if A and F have no common eigenvalues. Thus a.ny syst>emSzhaving different, eigenvalues from A is an observer for SIin thesense of Theorem 1 . Next, we n0t.e that the result of Theorem 1 for free syst.ems can be easily extended to forced systems by including the input. in t,he observer as well as the original system. Thus if SI is governed by

=

Fz(t)

+ Gg(t)

(2.5)

then H = GC. In designing the observer the m X n matrix C is fixed and t.he n X m matrix G is arbitrary. Thus an ident,itv observer is determineduniquelyby selection of G a,nd takes t,he form

i(t)

=

( A - GC‘)z(t)

+ Gg(t). (2.6)

havearbit.rarydynamics if the original systemis completely observable. First, recall that a syst.em (2.4) . , is completely observable if the matrix

has rank n. Generally, if an n X n mat.rix A andan m X n matrix C satisfyt,his condition we shall say (C,A ) is completely observable.

Lenanza 1: Corresponding to t,he real matrices C and A , then the set of eigenvalues of A - GC can be made to correspond to the set of eigenvalues of any n X n real matrixbysuitable choice of the real mat,rix C if and only if ( C , A )is completely observable.

Thislemma, which is now a cornerstone of linear syst,em theory, was developed in severa.1 st,epsovera period of nearlya decade. For the case 112 = 1 , corresponding to single out,put systems, early statements can be found in Kalman [34] and Luenberger [ l ] , [35]. The general resultis implicit.ly contained in Luenberger [ 2 ] , X(t) = Ax(t) h ( t ) (2.2) [ 3 6 ] , and the problem is treabed definitively in Wonham [ 3 7 ] . A nice proof is given byGopinath [%I. (It was a system Sp governed by recent,ly pointed out to me t,hat, Popov [38] published a proof of a result of this type in 1964.) Calcu1at)ionof the i ( t ) = Fz(t) H x ( ~ ) TBu(t) (2.3) appr0priat.e G ma.t,rix t.0 achieve given eigenvalue place\\-ill satisfy (2.1).Therefore, an observer for a. system can ment for a high-dimensional multivariablesystemcan, be designed by first assuming the system is free and then however, be a difficult cornputrational chore. incorporating the inputs asin (2.3). The result. of this basic lemma translat,es directly into a result on observers. B . Identity Observer Th.eorem 2: An identity observer having arbitrary An obviously convenient observer would be one in dynamics can be designed for alinear t.ime-invariant which the tra.nsformation T relating t.he st.ate of the system if and only if the syet,emis completely observable. observer to the st,ateof the original system is t,he identity t.ransformat.ion. This requires t.hat t.he observer S, be of I n practice, the eigenvalues of tlhe observer a.re selected the same dynamic order asthe original syst.em S1 a,nd t.hat to be negative, so t.hat t.he state of the observer will (with T = I ) F = A - H . Specification of such an ob- converge to t>hestate of the observed system, and t.hey are server rests therefore on specificat,ionof t,he matrix H. chosen t,o be somewhat, more negative t.han the eigenThe matrix H is determined part.ly by the fixed out,put values of the observed system so t.hat convergence is structure of t,he original system and partly by the input faster than other system effects. Theoretically, the eigenstructure of the observer. If SI, w3h an m-dimensiona,l valuescan be moved arbitrarily t,oward minus infinity, output, vectory: is governed by yielding ext.remely rapid convergence. Thistends, however, t,o make the observer act like a differentiator and i ( t(2.4a) ) = Ax(t) t,hereby become highly sensitive to noise, and to introduce = C4t) (2.4b) other difficukies. The general problem of select,ing good

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-4UTOMATIC CONTROL, DECEMBER

1971

U

s+2

S+l

Fig. 2. A second-order syst.em.

Fig. 3. Struct.ure of reduced-order observer.

eigenvalues is still not completely resolved but thepractice of placing t,hem SO that the observer is slightly faster than The reduced-order observer was firstintroduced in the rest of t,he (closed-loop) sgrstem seems t.0 be a good one. [I]. The simple development present,ed in this section is Example: Consider t,he system shown in Fig. 2. This due to Gopinath [251. state-variable has representation. We aga.in consider the system

X(t) = Ax(t) (2.7a) (2.7b)

An identity observer isdeterminedby

specifying the

observer input vect,or

The resulting observer system matrix is

y ( t ) = CX(t)

X2

+ (3 +

g1)X

+2 + + g1

g2

=

0.

(2.9)

Suppose we decide to make the observer have two eigenvaluesequal to -3. This would give the charact.eristic equation (X 3)2 = X2 6X 9 = 0. Matching coefficientsfrom (2.9) yields g1 = 3, g2 = 4.The observer is thus governed by

21

=

[ --4 5

-1 1 1 2221 1

+

3 .I! -1-

(3.2b)

I;[

is nonsingular and using the variables X = Mx.) It is then convenient to partit.ion t.he state vector as x

+ +

+

(3.2a)

and assume without loss of generality that the m outputs of the system are linearly independent-or equiva.lentlg that the outputdistribution matrix C has rank-m. I n this case it can also be assumed, by possibly introducing a cha.nge of coordina,tes, that. the matrix C takes the form C = [ZiO],i.e., C is partitioned into a.n m X m identity matrix and anm X (n - mn) zero matrix. (An appropriate change of coordinates is obtained by selecting an (n - m) x n mat.rix D in such a. way that

M = which has corresponding characteristic equation

+ Bu(t)

=

y] W

and accordingly writmethe syst.emin theform

+ A l d t ) + Blu(t) Aas(t) + + B2~(t).

k(t) = Ally(t) G(t) =

A22~l(t)

(3.3a) (3.3b)

The idea of the construction is thenasfollow.The vect.or y(t) is available for measurement, and if n-e difThe ident,ity observer ahhough possessing an ample ferentiate it., so is k(t). Since u(t) is also measureable measure of simplicity also possesses a certain degree of (3.3a) provides t.he measurement & ~ ( i )for t,he syst.em redundancy. The redundancy &ems from the fact. that (3.3b) u-hich has state vector w(t) andinput. Aely(t) while the observer const,ructs an estimat,e of the entire B2u(t).An ident,ity observer of order n - m is constructed st.a.te,part of t,he state asgiven by the system outputs are for (3.3b) using this measurement. Later t>henecessity to already available by direct measurement. This redundancydifferent,iat,ey is circumvent,ed. can be eliminat.ed and an observer of lower dimension but. The justifica6ion of t.he const.ruct.ion is based on the still of arbkrary dynamics can be constructed. following lemma [%I. The basic construction of a reduced-order observer is Lemma. 2: If (C, A ) is completely observable, then so is shown in Fig. 3. If y ( t ) is of dimension m , an observer of order n - m is constructed with state z(t) t.hat, approxi- (An, Am). mates Tx(t) for some m x n mat,rix T , as in Theorem 1. The validitmyof t.his stat.ement is, in view of the preceding Then an estimate i ( t ) of x ( t ) can be determined through discussion, int-uitivelyclear. It ran beeasily proved direct.ly by applying the definition of comp1et.e observability. To construct the observer we initially define it in the form provided thatthe indicat,ed part.itioned mnt,rix is in- h(t) = (A22 - LAn)zir(t) A P I Y ( ~ ) B 2 ~ ( t ) vertible. Thus the T associat,ed with the observer must L ( ! m - A d t ) ) - LBlU(t). (3.4) have n - m rows that are linearly independent of t.he rows of C. I n view of Lemmas 1 and 2, L can be selected so that

111. REDUCED DIMEKSIOK OBSERVER

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LUENBERGER: AN INTRODUCTION TO OBSERVERS Y

Y

82 -LE,

U

W

Fig. 6 . Observer for second-order system.

Fig. 4.

order system with a. single output so a first-order observer with an arbitmry eigenvalue can be constructed. The C matrix already has the required form, C = 1 0. In this

Reduced-order observer using derivative.

Y

Y

1

U

1

case A22 - GAB = -1 - G, which gives t.he eigenvalue of the observer. Let. us select G = 2 so that the observer will haveits eigenvalue equal t.0 -3. The resulting observer a.ttachedto thesystem is shown in Fig. 6.

IIT.OBSERVING

Fig. 5. R.educed-order observer.

Azz - EAzl has arbitrary eigenvalues. The configuration of this observer is shown in Fig. 4. The requireddifferentiation of y can be avoided by modifying the block diagram of Fig. 4 to that of Fig. 5 , which is equivalent at thepoint, ~. This yields the desired final form of t,he observer, which can be written

i(t)

+ ( A n - LAlz)~%(t) + (Azl - LAll)y(t) + (Bz - LBl)u(t)

= (Azz -

LAlz)z(t)

(3.5)

A SINGLE

LINEARFUNCTIONAL

For some applications an estimate of a single 1inea.r functional E = a‘x of the st,at,eis a.11 t,hat isrequired. For example, a linear t,ime-invaria,nt, control law for a single input, syst?emis by definition determined simply by a linear functional of the system stat.e. The quest,ion arises then as 60 m-het.her a less complex observer can be constructed to yield an estimate of a given linear functional tha.n an observer that estimat,es t.he entire st.ate.Of course, again, it is desired to have freedom in the selection of the eigenvalues of the observer. A major result for bhis problem [ 2 ] is that. any given linear functiona,l of the state, say, E = Q’X, can be estimat.ed m&h an observer having v - 1 arbitrary eigenvalues. Here v is the obsemability index [ 2 ] defined as the least positive integer for which the matrix

.

[C’IA’C‘;(Ar)2C’i. . \ ( A r ) y - l C r ]

with z(t) = G ( t ) - 5 / ( t ) .

(3.6)

For this observer T = [-LIZ]. This const,ruction enables us 60 st,ate thefollowing theorem. Theorem 3: Corresponding t.0 an nth-order completely controllable linear time-invariant system having m linearlyindependent outputs astat.e observer of order n - m ca.n be constructed having arbit.rary eigenvalues.

has rank n. Since for any completely observa.ble system v - 1 5 n - m and for many systems v - 1 is far less than n - m, observing a single linear functional of t,he stat.e may be f a r simpler than observing the entire &ate vector. The genera.1 form of the observer is exactly analogous to a reduced-order observer for the entire state vector. The estimate of E = a’x is defined by

It is important to understand that the explicit. form of 2 ( t ) = b’y(t) c’z(t) (4-1) the reduced-order observer given here, obt,ained by partiOioning the system, is o d y one way to find the observer. i ( t ) = Fz(t) Hx(t) TBu(t) (4.2) I n a.ny specific instance, ot,her techniques such as transforming to canonical form or simply hypot,hesizing the where F,H , T , B a.re as in Sect,ion 11-A and where b and c general structure and solving for the unknown parameters are vectors satisfying b’C c’T = a’. may be algebraically simpler. Theorem 3 guarantees that, Again the important, result is that the observer need suchmethods will always yield an appropriate result,. only haveorder v - 1. The precise design technique is The preceding method used in the derivation is, of course, dict,ated by considerations of convenience. often a. convenient one. We illust.rate the general result 1vit.h a single example. Example: Consider the syst,em shown in Fig. 2 and The method used in t.his example ca,n, however, be applied t,reated in the exa.mple of Sect,ion 11. This is a second- to any multivariable system.

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observer it,self and those t,hxt mouldbe obtained if the control lax- could be directly implemented. Thus an observer does not change the closed-loopeigenvalues of a design but merely adjoins its own eigenvalues. Similar results hold for systems with non1inea.r control laws [a]. Suppose we have the system

I

Fig. 7. A fourth-order system.

X(t)

=

+ Bu(t)

Ax(t)

(5.la)

-2

XI

u

o

and the control law x3

If it were possible t,o realize this control law by use of

Fig.8. Funct,ional observer.

available measurements (which would be possible if = RC for some R), then the closed-loop system would be governed by

K

Example: Consider t,he fourt,h-order systemshownin Fig. 7. This system nith available measurements x1 a.nd X(t) = ( A B K ) x ( t ) (5.3) x3 has observability index 2. Thus any linear functional can be observed T1-it.h a first-order observer. Let us decide and hence its eigenvalueswould be the eigenvalues of to construct an observer with a single eigenvalue equal to A BK. -3 to observe the functional xz x.%. Now if the control cannot be realized directJy, an obInitially neglecting theinput 'LC we hypothesize an server of the form observer of the form i ( t ) = Fz(t) Gp(t) TBu(t) (5.4a) i = -32 91x1 93x3. ~ ( t= ) Ki(t) = Ez(t) Dg(t) (5.4b) According to Theorem 1this has an associated T satisfying where r-2 1 o TA - FT = GC (5.5a) T I 3T = 0 93 (4.3) K = ET DC (5.5b) -1 0 0 0 canbeconstructed. From ourprevious theory ( C ,A ) If T = tl tz t3 t4, we would like tz = 1, t4 = 1. Sub- completely observableis sufficient. for t.here to be G , E, D! F , T satisfying (5.5) with F having arbit,rary eigenst,itut.ingthese valuesin (4.3) we obtain the equation values. Sett.ing u(t) = K f ( t )leads to the composit,esystem

+

+

+

+

+

+

+ +

01

; -;-; :] +

p

9'

-

+

;] [GCA ++ BDC TBDC =

that ca.n be solved for the fourunknowns tl, g1? t3, g3. This resultsin tl = - 1, t3 = - 3, g1 = -2, g3 = - 5. From this t,he final observer shonn in Fig. 8 is deduced by inspection.

V. CLOSED-LOOP PROPERTIES Once an observerha.sbeen construct.ed for a 1inea.r system which produces an estimate of t,he state vector or of a linear transformat.ion of the stat.e vector it is important, to consider the effect induced by using this estimate in place of t.he true value called for by a cont,rol lau-. Of paramount importance in this respect is the effect, of an observeron the closed-loop stability properbies of t.he system. It would be undesirable, for example, if an otherwise stable cont.ro1designbeca.me unsta.ble when it was realized by introduction of an observer. Observers, fortunat.ely, do not dist.urb stabilit,y properties when they are introduced. I n this section we show that if a linear time-invariantcontrollaw is realized u-ith an observer, the resulting- eigenvalues of the srst,em are t.hnse he ~~~-- - of _ - t---~

~

~

F

"1.

+BETBE] z

(5.6)

This whole structurecan be simplified byintroducing = z - Tx and using x and E as coordinates. Then (5.6) becomes, using (5.5)

(5.7) Thus the eigenvalues of the composite system arc those of

+

BK and of F. m e note that in view of Lemma. 1 (applied in its dual form) if the syst,em (5.1) is completely controllable it is possible t,o select. K t.0 place the closed-loopeigenvalues arbit,rarilg. If this control Ian- is not. realizablebut, the systemiscomplet.dyobservable, an observer (of some order no great,er than n - m ) can be const.ruct,edso that t,hecontrol law can be estimated. Sincc t,heeigenvalues of the observer are also arbitrary the eigenvalues of the completecomposite system may beseltct,edarbitrarily. We thereforest.ate t.he following important result of linear syst,em theory [I], [2]. A

Theorem 4: Correspondingnth an to

order completely

a.nd hence t,he plant follows the free system. This tracking property can be used t.o definea closed-loop system for the plant. Rather than fix a.t,t,entionon t,he fact that only cert,ain outputs of the plant are available, we concentratme on the fact. 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 supplied, t.he output, limitation would not much matter. Indeed, if t,he out,put y(t) = Cx(t) could be fed t.0 t.he system in theform

Fig. 9. A general servodesign.

Fig. 10. Compensator for example.

i(t) =

Ax@)

+ Ly(t)

03-31

t.hen t,he eigenvalues of the system would be the eigenLC. By Lemma 1, if the s:stem is observIralues of A able L can be selected to place the eigenvalues arbitrarily. Thedual observer can be thought of assupplying an approximation to the desired inputs. To achieve t,he desired result we construct, thedual Alt.hough this eigenvalue result for linear t.ime-invariant, observer in the form systems is of great theoretical interest, it. should be kept z ( t ) = Fz(t) M w ( t ) (6.4a.) in mind t,hat.t,he more general key result is that. stabilitg is not af€ect,edby a (stable) observer. Thus even for nonu(t) = y ( t ) CPz(t) (6.4b) linear or t,ime-varying cont,rol laws an observer can supply u(t) = Jz(t) Nw(t) (6.4~) a suit,able estimate. Example: Suppose a feedback cont.rol spst,em is t.0 be where designed for the syst.enl shown in Fig. 2 so t.hat its output AP - PF = BJ (6.h) closely tracks a dist,urbance input d. The general form of design is shown in Fig. 9. L = P M + BN. (6.5b) For the particular syst,em shon-n in Fig. 2 let us decide to design a control law that places the eigenvalues at. Equations (6.5) are dual to (5.5) and will have solution - 1 f i. It is easily found t,hat u = - 2x1 .x2 will acJ, MI N , F wit.h F having arbit,rary eigenvalues if (6.1) complish this. If this ~ R Wis implemented with t,he first.- is conlpletely controllable. order observer construct,ed earlier, we obtain the overall The composite system is system shown in Fig. 10, which can be verified t,o have A BNC BJ BNCP x . (6.6) eigenvalues - 3, - 1 i, - 1 -i. = MC F MCP z] VI. DUALOBSERVERS Introducing n = x P z and using z and n for coordinates The funda.menta1 propert): of one syst.em observing yields the comp0sit.e syst,em another can be applied in a reverse direction to obtain a special kind of controller. Such a c~nt~roller, called a dual i] = LC observer, wa.s introduced by Brasch [33]. F n] z 2 MC Suppose in Fig. 1 the systenl S2is the given system and SI is a syst,em tha.t we construct t,o control S2. MTehave which is the dual of (5.7). The eigenvalues of the comshown that the system SZ t.ends to follow SI and hence posite system a,re thus seen to be the eigenvalues of A LC and the eigenvalues of F. We may therefore st,at.ethe SI ca,n be considered as governing the behavior of S,. dual of Theorem 4. To make tshisdiscussion specific suppose t,he plant Th.eorm 5: Corresponding to an nth-order completely X(t) = Ax(t) Bu(t) (6.la) controllable and complehely observable system (6.1) feedy ( t ) = Cx(t) (6.lb) having T linearlyindependent.inputs,adynamic back system of order n - r can be constructed such t,hat is driven by the free syst,em the 2n - r eigenvalues of t.he composite system t,alce any i ( t ) = Fz(t) (6.2a.) preassigned values. controllable and completely observable system (5.1) having m linearly independent out.puts, a dynamic feedback syst.em of order n - m ca,n be constructed such that, the 2n - 7% eigenvalues of the composite syst,em take ang preassigned values.

+

+ + +

+

41

+

[

+

+ +

]

+

[" +

01

+

+

~ ( t= ) Jz(t)

(6.2b)

1711. CONCLUSIOKS

It has been shown t.hat mising &ate-variable inforwhere AP - PF = BJ for some P . Then from Theorem 1 mation, not available for measurement,, ca.n besuit,ably we see that in thiscase the vector n = x Pz is governed approximated by an observer. Generally, as more output, by the equa.tion variables are made available] t.he requiredorder of the n ( t ) = An(t) observer is decreased.

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Although the introductorytreat,ment given inthis pa.per is restrictedtotime-invariantdeterministic continuous-t,ime 1bea.r syst,ems, much of the t,heory can be ext,ended t.0 more general sit.uations. The references cited for this paper should be consulted for these extensions.

1971

[281 P. V. Nadezhdin,“The optimal control law in problanswith arbitrary initial conditions,’’ Eng. Cybern. (USSR), pp. 170174. 1968. [291 E. E. ‘‘APPlicat,ion of observers a n do p t h u m filters to inertial systems,” presented at. the IFAC Symp. Multivariable Cont,rol SFtems, Dusseldorf, Germany, 1968. [30] D. Q: Xayne and,:. Murdock,“Modalcontrol of linear time mvarlant systems, Int. J . Contr., vol. 11, no. 2, pp. 223-227, 1970. [31] A. Foster and P. A. Orner, “A design procedure for a class of distributedparameter control syst.ems,” ASMEPaper 70WA/Aut-6. [32] D. &I.Wiberg, Schau.m’s Outline S e ~ e s State : Space and Linear Systems. New York: McGraw-Hill, 1971, ch. 8. [33] F. M. Brasch, Jr., “Compensators, observers and controllers,” to be published. [34] R. E. Kalman, “Mat.hematica1 descript.ion of linear dynamical systems,” SIAX(SOC.Ind.Appl.Nath.) J. A p p l . Math., ser. A, vol. 1, pp. 15%192, 1963. [35] D. G. Luenberger, “Invertible solutions to the operator equation T A - BT = C,” in Proc. Am. Math. Soc., vol. 16, no. 6, pp. 1226 - 1229, 1965. [36] --,“Canonical forms for linear multivariable system,” IEEE Trans. Automat. Contr. (Short Papers), vol. AC-12, pp. 290-293, June 196i. [37] R . X. Wonham, “On pole assignment in mult.i-input controllable linearsystem,’’ IEEE Tra.ns.Automat. Contr., vol. AC-12, pp. 660-665, Dec. 1967. of automatic [38] 5’. hI. Popov, Hyperst.abilityandoptimality systems wit,h several control functions,” Rev. Roum. Sci. Tech., Ser. Electrotechn. Energ., vol. 9, pp. 629-690, 1964.

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David G. Luenberger (S’57-kI’@-S-MJ7l) was born in Los Angeles, Cali., on September 16, 1937. H e received the B.S. degree from the California Inst,it.ute of Technology, Pasadena, in 1959 andthe M.S. and P b D . degrees from Stanford University, Palo Alto, Calif., in 1961 and 1963, respectively, all in electricalengineering. Since 1963 he has been on t.he faculty of Stanford Universit,y, where presently he is a Professor of Engineering-Economic Systems and of Electrical Engineering. He is also d i a t e d with the Department of Operations Research. His activit.ies have been centered primarily in t,he graduate program, where he has t,aughtcourses in optimization, control, mathematical programming, and information theory. His research areas have included observability of linear systems, 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 Aircraft Company and Westinghouse ResearchLaboratories from 1959 t o 1963, and service as a consultant to West.inghouse, Stanford Research Institute, Wolf Management Semices, and Intasa, Inc. This experience has included work on the control of a large power generating plant, the numerical solution of partid differential equations, optimization of trajectories,optimal planning problems, and numerous additionalproblems of optimization and control. H e has helped formdate and solve problems of control, optimization, and general analysis relating t.0 electric power, defense, water resource management., telecommunications, air traffic control, education, and economic planning. He is the author of Optimization by Vector Space Xethods (Wiley, 1969) and has authored or co-authored over 35 technical papers. Currentlyhe is onleave from Stanford at the office of Science and Technology, Executive Office of the President, Washingt.on, D.C.He assist,s the Science Adviser -with program planning, review and evaluation, and with formulation of science policy in areas of civilian technology. Dr. Luenbergeris a member of Sigma Xi, TauBetaPi,the Society for Industrialand Applied Mathematics,the Operations Research Society of America, andtheManagement Science Institute. He served as Chairman (Associate Editor) of the Linear Systems Comnlit.tee of the IEEE Group on Automatic Cont,rol from 1969 to 1971.