observers for linear multivariable systems with applications

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-16, NO. 6, DECEMBER 1971 603

Observers for Linear Multivariable Systems with Applications

Y. ONDER YUItSEL AND JOSEPH J. BONGIORNO, JR., M E ~ ~ B E R , IEEE

Abstracf-This paper presents an algorithm for the design of asymptotic state estimators (observers) for index-invariant uniformly observable time-varying linear finite-dimensional multivariable systems. The results obtained indicate that asymptotic estimators can be employed in optimally designed regulators provided an increase from the optimal cost is tolerable. I t is also shown that any uniformly observable and uniformly controllable plant with index- invariant observability and controllability matrices can be stabilized with an observer.

W I. INTRODUCTION

HEX modern techniques are employed in the design of a cont,rol system for a dynamic process or plant,

a feedback law often results t,hat requires ava,ilabilit.y of t,he stat,r of the plant. The stat.e of the plant, is not gen- erally accessible in pra.ct,ice, however, and a.n estimator or observer of the plant state variables must be designed. Kalman and Bucy [ 1 ] treated t,he problem of st.ate estima- t.ion for a linear finite-dimensional dynamic plant when all measurement.s are corrupt,ed by white noise. They showed for this kind of plant, that, t,he “best“ estimate of t.he state of the plant is the output from a state estima- t,or t,hat is itself a linear finite-dimensional dynamic system driven by t,he plant input,s and outputs. When the noise is not white or some measurements are “noise-free,” Bryson and Johansen [2] have shown that, the optimal est.imator is a modificat,ion of t.he Kalman-Bucy filter which cont,ains diff erentiators a.nd integrators in general. Since in many control problems a. reasonable assumption is t,hat “noise-free” measurement.s are available and since different.iation of measured data is undesirable for practical rea.sons, an alternative to the Kalman-Bucy filt,er is of interest. Such an alternative is the so-called observer or asymptot.ic st,at.e estimator that has been t.reated in papers [SI-[SI since t.he pioneering work of Lucnberger [3], [4].

1,uenberger showed t.hat, an estimat,e of the n-dimensional state vector of a completely observable linear time- invariant. finite-dimensional dynamic plant with T independent outputs can be generated using an (71-).)-dimensional observer that is it,self a linear t.ime-invaria.nt finite-

Kleinman, Associate Guest Edit,or. Manuscript. received July 2, 19i l . Paper recommended by D. L.

gram under Contract. F44620-69-C-0047. This paper is taken from a This Fork -was supported by the Joint Services Electronics Pro-

dksertation submit.ted by 1’. 0. Piiksel in 1971 t.o the Faculty of the Polytechnic Institute of Brooklyn, Brooklyn, N.Y., in partial f u l - fillment of the requirements for t.he Ph.D. degree in electrical engi- neenng.

Turkey. Y. 0. Yiiksel is with Orta Dogu Teknik Universitesi, Ankara,

Brooklyn, N.Y. 11201. J. J. Bongiorno, Jr., is with t.he Polytechnic Inst ihte of Brooklyn,

dimensional dynamic system. As with t,he 1Za.lma.n-Bucy filter, the Luenberger observer is driven by t,he plant, input.s and 0utput.s. The order of the Luenberger observer is less, however. -4nother essential feature of the Luenberger observer is that t.he norm of the error in the estimate of the plant, state vector is bounded by a deca.ying exponential funct,ion of t.ime. Hence, the error approaches zero as time increases, and for this reason, the observer is also referred to here as an asymptotic state estimator.

Dellon and Sarachik [7] consider the design of compen- sa.t,ors t.o stabilize linear time-varying finite-dimensional dynamic systems. The compensator paramet.ers a.re obtained t,hrough the solut,ion of a “pole-placement” problem. Because Dellon and Sarachik have structured t,heir system so that the only inputs to the compensator are the plant out.puts, there is no guarantee that. the compensat.or will be st,able. In fact, one can cite examples where the contra.ry is true. This difficulty is easily avoided, however, by using an observer in place of the compensator, and considerable insight is obt,ained by exploiting t,he pole-placement character of observers that their work has exposed. This insight. is interpreted from R geometric point of view by Wonham [SI. A complete discussion of this idea is given in Sect.ion I11 from an algebraic point. of view.

Bongiorno and Youla [5] take a different approach in their work. So, a.lso, does Wolovich [6]. In [ 5 ] attention is restrict,ed to observers that have no eigenvalues in common wit,h the plant, and in [6] only observers with dist,inct. eigenvalues are discussed. It. is established in Section VI that. these two approaches are special cases of the general class of ca.nonica1 observers presented in Section V.

The class of canonical observers is a subclass of t.he class of all possible asymptot>ic state estimat.ors. They are ob- t,a.ined by using t,he canonical transformation developed in Sect,ion IV, and they possess the desimble property of having a time-invariant dynamical part. This property makes the canonical observer one which is more easily instru- mented.

When an observer is used in a system t.hat, is opt.ima1 if the state of the plant is observed without. error, one can expect an increase in the cost or performance index. This aspect of observers 1la.s been invest,igat,ed for the optimal regulator with quadrat,ic cost function in [5] and [7]. The contradict,ion between the claims in [SI and [7] regarding the behavior of the cost, increment as t,he real parts of the observer eigenvalues are ma,de highly negative is discussed in [9]. In Section VII, the essentia.1 feat,ures of the development,s in [5], [TI, and [9] are presented and t>he plant st,abilizaton problem is also discussed.

604 IEEE TFL4NSbCTIONS ON AUTOMATIC CONTROL, DECEMBER 1971

The class of linear time-varying finite-dimensional dy- and namical plants treated here is characterized for all t 2 to by the set of equations Qdt> = Qt-l(t)A(t) + Q<-l(t) (8)

i ( t ) = A( t ) x ( t ) + B( t )u ( t ) (1)

= C(t>x(t) (2)

where x is a.n n-vector, u is a.n nz-vector, and y is an r-vector. A ( t ) , B(t) , and C(t ) are real matrices of appropriate size whose elements have con6inuous derivatives with respect. to the time t of orders 1% - 2 , n - 1, and .n - 1, respectively. The measurement mat.rix C ( t ) is assumed to have row rank T without loss of generality. At.tention is restrict.ed to real systenls throughout. t,he paper and all vectors and ma.trices are therefore real unless otherwise stated.

The notation employed is summarized here for easy reference. For a.n a,rbitrary matrix A , the transpose and the inverse a.re denoted by A’ and A-l, respectively. The determinant of a square matrix A is represent,ed by I AI . A diagonal matrix A with diagonal elements X I , X?, . . . , X, is written as

Column vectors are denoted by x , y, et.c., and whenever it, is desirable to exhibit the components of a. vector ex- plicitly, the representat,ion x = [x1x2 . . . x,]‘ is used. The n x 71 identity matrix, the n.-dimensiona,l zero vector, a.nd the n X m zero matrix are denoted, respectively, by l,, on, and On,m. The n.-dimensional column vect$or wit.11 unity in the it,h row and all other element,s equal to zero is denoted by e,(n). Whenever no confusion is likely to result,, one simply writes ei. Finally, the compact notation

e01 { A i ) [Al‘[A2’1 . . . IA,’]‘ (4) i= l , . . . .r

and

row { B , ] [BIIBZl . . . [ B , ] (5 )

is used to represent mat.rices psrtit,ioned according to block rows and block columns, respectivcly.

11. DEFINITIONS

j = 1 . . , ‘,C

Before proceeding with the development of observer theory it is a.dvantageous to define some basic notions. The first concept int.roduced is that of uniform observability and the definition given here is equivalent, to t,he one given in [23].

The definit,ion of uniform observability can be stated in t,erms of the ~ 7 1 X n obseruability matrix

where

for i = 2, 3, . , n.. Dejnition 1: The plant described by (1) and (2) or,

equivalent.ly, the pair f A ( t ) , C ( t ) ] is said to be un.ifomzZy observable if tohe rank of Qo(t) is ‘)z for all t 2 to.

Whenqi,j’(t) represents t.hejth row of Qi(t) and t,he rank of Qo( t ) is n , it is possible at. any time t = tl 2 to to uniquely define an n. X n nonsingular matrix by eliminating from top t.0 bottom t,hose row vect,ors of Qo(t) which are linearly dependent on the preceding rows. In this paper,’ i t is assumed that the rows of t,he said nonsingular matrix are “index invariant,’’ that is t.0 say, if qi,i(tl) is one of the rows selected at t = tl , it remains to be one of those rows for all t 2 to. The elimination procedure thus has t.he property presented in Lemma 1.

Lemma 1: If 4i, i’ is eliminated, then so is qs,j’ for k > i. The proof is given in [19].

Associated with the nonsingular nlat,rix formed from Qo(t) are r scalars defined as follows.

DeJin.ition 2: Each integer qi, i = 1, . . . , T , the observability index of the it.h subsystem, is defined as t,he highest int,eger for which 4pi.i‘ is among the rom of the nonsingular matrix constructed as mentioned above. Clearly pi, the number of qj, i‘ selected are iixed for all t 2 to wit,h the assumpt,ion of index inva.riance. The qi, i = 1, 2, . . , T are thus uniquely specified and sa.tisfy t,he relationship

2 qi = n.. i = l

The nonsingular ma.t.rix Q(t) is const.ructed by ordering t,he selected row vect,ors 4i,i’ according to

where

61(t) = GO1 { 4 9 i , 4 (11) i=l , . . .,r

and &(t) consist,s of q i , j ’ , j = 1, . . -, T , i = 1, -, qi - 1 in any order. This matrix is used in Section IV t,o develop a canonical transformation for the plant, described by (1) and (2).

The notion of an asympt.otic state estimator int.roduced earlier is now given a precise meaning in the following definition.

Dejnitio.n 3: A p-dimensional linear dynamic system

i ( t ) = F ( f ) ~ ( t ) + G(t)y(t) + H ( t ) u ( t ) (12)

(6) is called as asymptotic state estimator for the syst,em of (1) and (‘2) if and only if there exists an n X ( p + T ) matrix

(7) where the case in which t.he system is not, index invariant is treated. * A more general development of observer theory is given in [X],

YUKSEL AND BONGIORNO: OBSERVERS FOR MULTIVARIABLE SYSTEYS

W(t) satisfying

605

and

It is established in t.he sequel t,hat an a.sympt.otic sta.t.e estimat,or exists for any uniformly observable plant, of the kind under st.udy here, and algorithms are given for the design of such est,imators.

111. OBSERVER THEORY The basic concept of observer theory is readily exposed

wit11 the aid of (13). Clearly, the plant and observer st.at.e vectors can a.lways be related by

z ( t ) = T ( t ) x ( t ) - e(t) (14)

through the int,roduction of the error vector e( t ) . Substi- tuting (2) and (14) into (13), one obtains before taking the limit

+ W(t) [ e 3 . (15)

The essence of observer theory is the selection of a matrix T( t ) so that

1, - W(t) - = 0, [:;:;I and

lim e(t) = 0,. t+m

For then it immedia.tely follows from (15) t,hat (13) is mtisfied.

Since the number of r o w of T ( t ) is p , the dimension of the observer state vector, this number should be the sma.11- est possible. Since the rank of the product, of two matrices is less than or equal t,o the rank of either matrix, it immediately follows for p < n - r that the rank of the second t,erm on the left-hand side of (16) is less t,han ? I . Hence, this t.erm can equal 1, only if p 2 n. - r. The smallest, value of p is p = n. - r , therefore, and attention is restricted to this case in the sequel.

The remainder of t.his section is concerned with estab- lishing t.he conditions for the existence of a matrix T( t ) having t.he desired properties. Also, a1gorit)hms are discussed for computing tahe observer matrices F ( t ) , G ( f ) , and

It is not difficult. to verify by direct subst.it.ution of (14) into (12) and utilizat,ion of the relationships (1) and (2) that.

W t ) .

e ( t ) = F(t)e(t) (18)

when

H(t) = T(t )B( t ) . (20)

For any choice of T ( t ) , the observer mat,rix H(t ) is easily comput,ed using (20).

It is now shown that. for any choice of T( t ) satisfying (16), or equivalently,

t1la.t there exist observer mat>rices F ( t ) and G(t) for which (19) is sat.isfied. With W(t) partitioned according to

W ) = [ W ) I N(t) I (22)

where M(t) is an n X T matrix and N(t ) is an n x p matrix, it. follows from (21) t.hat

C(t>W(t ) 7 [ c ( t ) ~ ( t ) I c ( t ) ~ ( t ) I = [1rIo,,,I (23)

T ( t ) WW = [T(t)M(t) I W W ) 1 = [0,,71191 (24)

M(t )C( t ) + N(t)T( t ) = 1,. (26)

and

Multiplying (19) on the right by N(t) and utilizing (23) and (24) yields

F ( t ) = T( t )A( t )N( t ) + F(t)N(t). (26)

Multiplying (19) on the right by M(t ) and utilizing (23) and (24) gives

G(t) = T( t )A( t )M( t ) + P(t)M(t) . (27)

That, (26) and (27) are indeed the solutions of (19) when (23) t,hrough (25) are sat.isfied is easily verified by direct. substitution of (26) and (27) into (19).

Attention is now turned to t.he delineation of the class of T ( t ) mat.rices satisfying (21). Clearly, this cla.ss is determined by the class of all nonsingular W(t) that satisfy (23) . For this class of W(t) , one can compute T ( t ) using (24). The development employs partitioning C(t) and W(t) a.c- cording to

C(t ) = [Cl(t)l W ) l (28)

and

respectively, where Cl(t) is r x Y, C,(t) is r x p , Wll(t) is P X r , W12(t) is r X p , Wzl(t) is p X T, and Wz2(t) is p X p . It, is assumed without loss in generality t,hat C,(t) is nonsingular. This follows from t.he fact t.hat C ( t ) has row rank and the fact. that. t,he state variables can always be renum- bered, if necessary, so t.hat the first r columns of C(t ) are independent for all t 2 to.

Substituting (28) and (29) int.0 (23) yields

T(t )A( t ) - F ( t ) T ( t ) = G(t)C(t) - F( t ) (19) W l d t ) = G - l ( t ) [ L - CZ(t)W21(t)l (30)

606 IEEE TRANSACTIONS ox AUTOMATIC CONTROL, DECEMBER 1971

and where

WlZ(t) = - C,-l(t)Cz(t)Wn(t). (31)

It immediatelv follows that

or, equivalently,

&(t) = - C(t)A(t)&(t). (43)

(33) Adding (39) and (41) gives, finally,

The expression (33) is highly i1luninat)ing. First. of all, it immediately places in evidence that W(t) is nonsingular if, and only if, W,(t) is nonsingular. Second, it. malres the computation of W-I(t) a simple matt.er. The result. is

(34) It is important, to cmphasizc here t,hat WZl(t) and W22(t)

can be chosen arbitrarily except for the requirement t,hat Wn(t) be nonsingular. That is, (21) is satisfied for all W(t) given by (33) with W2,(t) nonsingular.

All t<he information needed to compute T ( t ) , IM(t), and N(t) is now at. ha.nd. One obtains from (24) and (34) that

F ( t ) = W,z-'(t) [Ll(t) - W 2 1 ( ~ ) ~ z ( ~ ) l ~ z z ( ~ ) - w,-yt)Wza(t) (44)

where

U t ) = [C(t)A(O + C ( Q I f i ( 0 . (45)

Inspection of (44) indicates that t,he eigenvalues of F ( t ) can be made to depend solely on t,he choice of W21(t) when W,(t) is chosen to be any constant nonsingular matrix. In t,his case, t,he eigenvalues of F ( i ) are ident.ica1 to t.hose for

P(t ) = [Ll(t) - WZl(t)L(t)I (46)

and w-it.11

G(t) = Wz2e(t) (47)

the error equation (18) becomes T( t ) = W22-yt)[-Wzl(t)llp]

$( t ) = P( t )Z( t ) .

Equa.t,ions (22) and (32) give

and

At this point it should be c1ea.r that once WZI(~) and Wn(t) are selected the comput.at.ion of the observer matrices is stxaightforn-ard using (20), (26), and (27). The only question remaining t,hen is whet.her Wzl(t) and W?Z(t) can be chosen so t.hat t,he solut,ion of t,he error equation (1s) is asgmpt.ot.ica!ly stable.

The error equat.ion matrix F( i ) is given by (26). Using (35) it is not difficult to establish that

f ( t ) = - W22-q) [vlr,,(t)T(f)

+ W21(t)C(t)+ WZl(t.)C(t)I. (33)

When (23) and (24) are recalled, it immediately follows that

f ( t ) N ( t ) = - wn-yt) [W,(t) + WZl(t)C(t)R(t)W?S(t) I (39)

Clearly, e(t) is asymptotically stable if and only if 2(t) is. It is of int.erest t.0 restrict. att.ention to the time-invariant

case for the moment. In this case, the asymptot.ic stability of the solutions of (48) is determined by the eigenvalues of Q or, equivalently,

P I = [L,' - Lz'W211]. (49)

Hence, if { Ll f , Lf ] forms a completely controllable pair or, equivalently, { L1, L] is a completely observable pair one can always find a WZ1 so tha.t any specified set of eigenvalues for P can be realized provided only that complex eigenvalues occur in conjugate pairs [ l l ] . Indeed, one can choose the eigenvalues of P so t.hat they all have negative real parts and t<hus guarantee the asymptotic sta.bility of the solutions of (48).

Algorithms are given in [ lo] and [12] that permit. t.he efficient computation of a Wzl given that { L , , L2] is an observa.ble pair. An efficient computation is one t.hat does not require any transformation t.0 canonical forms. The question of the observability of the pair { L1, Lz} is an- swered by the following lemma.

Lemma 2: The pair { L I , L} is completely observable if and only if the pair {A, C) is. This is al-mys the case when the pla.nt is completely observable. The proof involves several straightforward steps that are omitted for brevit.y.

wI(SEL AND BONGIORNO: OBSERVERS FOR hlULTIVARIM3LE SYSTEMS

The above discussion makes it clear that the design of asymptotic sta.t,e est,imators for'the time-invariant case is not only efficient, but, straightforward as well. In view of the results obtained by U'olovich for uniformly cont.ro1- lable time-varying systems in [lo], it is reasonable t.0 expect that the same is true of observers for uniformly observable time-varying p1ant.s even though observabilit,y and controllability are not duals in t.his case (see the last paragraph on page 53 of [13]). However, an efficient. algorithm which yields a time-varying observer ma,t,rix F ( t ) is not as desira.ble as an algorit,hm which yields a time- invariant ohserver matrix since the latter case is more easily inst,rumented. For t.his reason, the canonical tra.nsformat.ion of the next sect.ion is introduced and utilized to est.ablish a design algorithm for t,he time-varying case.

Iv. A CANONICAL TRANSFORMATION

The linear transformation

x ( t ) = L( t ) i ( t ) (50)

where L(t) is an n x n. mat,rix funct,ion of time, is considered.

Definition 4: L(t) is said to represent a. Lyapunov trans- fomzation. [14] if a) L(t) and t ( t ) a.re cont.inuous and bounded and b) L(t) is nonsingular for all t 2 to.

When the matrix L(t) of (50) represent,s a Lyapunov txansformation, t.he syst,em (1) and (2) is equivalent to the system described by

k( t ) = L-l(AL - L)i( t ) + L-'Bu(t) = A(t ) i ( t ) + i ( t )u( t ) (51)

and

y(t) = CLX = C(t)X(t). (52)

Lemma 3: Any n X n . matrix L(t) const,ructed according

AZgo?-ith.m 1: a) Let p i ( t ) , i = 1, 2, . . ., r , be the ith

b) Define n-dimensional vectors I i , j , i = I , . . , T , j =

to the following algorithm is a Lyapunov transformation.

column of $ - l ( t ) where $ ( t ) is as given by (10).

. . .

607

The proof of Lemma 3 is given in [19]. It is nom possible to stsate the theorem on canonical

transformations. Theorem 1 .- The Lyapunov tra.nsformat,ion constructed

by Algorithm 1 transforms (1) and (2 ) into the equivalent system (51) and (52) where the part.itions of

A( t ) = I : 1 have the following propertiw :

A o i = [e,[ i = 1, . . , 1'; (60)

Ai j = 04i-l,4;-l, i # j and i, j = 1, a , r; (61)

Aio(t) , i = 0, 1, . :, T , contain all the time-varying parame- ters. The ma.t,rix C(t) is given by

C( t> = [ c l ( t ) p T , , l (62)

where C 1 ( f ) is nonsingular for all t 2 to. This ca.nonic form is a generalization of Tuel's [ l s ] , [16]

observation canonic form to t.ime-varying systems. In the case of a plant, wit,h a scalar out.put, the canonical form used reduces to t.he well-known phase-variable form, the t,ime-varying case of which is invest,iga.ted by Silverman [17]. The proof of Theorem 1 is given in [19].

V. CANONICAL OBSERVERS One of the main new results in this paper is the algorit,hm

used below to show the following. Theorem 2: Any index-invariant, uniformly observable

syst,em having the descript,ion given in (1) a,nd (2) has a.n asymptot.ic state estimator of order p = 72. - 'r. Furt.her- more, the edmator can be designed so t.hat. t.he F ( t ) matrix is constant.

The proof is constructive. In fact,, t,he design procedure is given in Algorit,hm 2.

Algorithm 2: a) Eva1uat.e L(t) using Algorit,hm 1. Also, eva1uat.e i ( t ) , &t) , and C(t) in (51) and (52). N0t.e the values of the yi, i = 1, 2, a , r.

b) Let i

I , = ( q j - l), i = 1, . . * , T - 1. (63) j = 1

e) Choose p = n - numbers hi, i = 1, - . . , p , as t.he desired estimator eigenvalues, subject to the two condi- t.ions t.hat they must have negative real parts for error stability and any complex values must be accompanied by their complex conjugat,es for physical realizabilit,y.

d) Determine mi, i = 0, 1, . . . , p - 1 from

fi (X - X,) = cuiXi + XP P - 1

i = l i = O (64)

608 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, DECEMBER 1971

and let

+ = (ap-1, -, a17 (Yo)' . (65)

T o = [-+I row { e k ( p ) } ] . (66)

e) Determine the p X T matrix TO from

i = l , . . . , r - l

f) Let

jq [!&I a.nd

where &(t) is as given by (62). g) Construct the estimator (12) with

F = f i ( t ) f i = [ - + I row { e i ( p ) f ] (70)

G(t) = TA(t)lM(t) (71)

i = l , . . . .p-1

and

H(t) = i;E(t). (72)

The estimate of the state is given by

i ( t ) = W(t) - = L(t)[Ai(t)y(t) + f i z ( t ) ] . (73) [::I For a uniformly observable linear multivariable system,

the Lyapunov transformation L(t), the integers q;, i = 1, . . , r , and the equivalent canonic representation are

uniquely defined by Algorithm 1, and one can apply the results of Sect.ion 111 to the canonical plant (51) and (52). Equations (67) thru (69) follow immediately from (35) through (37) when the choice

[W21(t)IW22(t)l = [ - F o c l - l ( t ) p p l (74)

is made and it is recognized that for the canonical plant, (51) and (52)

[ C l ( O l C 2 ( ~ > I = [ C l ( t ) l O L , l . (75)

Since T is a const.ant matrix, (70) through (72) follow from Theorem 1, ( Z O ) , (26), and (27). It is not difficult t.0 verify that the eigenva.lues of the constant matxix F given by (70) are those chosen in step c of Algorithm 2 which all have negative rea,l parts. It immediately follows from (18) that Algorithm 2 yields a.n asymptotic st.at.e est.imator.

In the case of a plant witlth a single output., the canonic t,ransformation yields the well-known phase-varia.ble form [l'i 1

i ( t ) = [-ci(t)l row { etcn)) 1 (76) i = l ...,n-l

and

t' = [e,'")]' (77)

where

i ' ( t ) = [a",-&) * - &(t)&(t)] (78)

and the &(t) are defined by

1x1, - A(t)l = &(t)Xf + X". (79) n-I

i = l

Since r = 1, the matrix T o becomes

T o = -4 (80 1 with t,he vect,or 4 given by (65). Therefore, in the single- output case Algorithm 2 yields

G(t) = g(t) = TA(t)B(t)

and F is as given by (70) mit.h p = n - 1. N0t.e that no restzictions are imposed on the location of

the estimator eigenvalues except for asymptot,ic stability and realizability considerations. Although the fixed form for F may seem to be restrictive, it is always possible to treat a general form for F via a similarity transformation. This point is worked out in further detail in the following section.

VI. CObfP/iRISON WITH OTHER APPROACHES

The canonic structure of the estimat,or constructed by Algorit,hm 2 can be altered by introducing the transformation

= .(t)t(t>. (82)

In t,his section it, is shown t.hat the estimator described here for t.he single-out,put, plant, can be taramformed t.o the estimators obtained in 1.31 and [6] by a proper choice of

Lemma 4: Let 13 and 6 = i j den0t.e the observer matrices of a Bongiornc-Youla type estimator for a single-out,put plant.2 Let F and G = g denote the matrices of a canonical estimator for the same plant, designed so t h a t F has the sa.me set of eigenvalues as i. Then (82) with

'F 0).

Q = -&-'($) [ p . - Z i j I . . . \ $ i j I i j ] (53)

where

$ J O ( ~ ) =I& - AI (84)

is t.he characteristic polynomial of the plant matrix A , transforms the canonical observer into a Bongiorno- Youla-type observer provided 7 is nonsingular.

For the proof, it. suffices to show that

9 = ~ F 7 - l (85)

2 That the notation here differs from that in 1.51 should be kept in mind.

YUKSEL .4XD BONGIORNO: OBSERVERS FOR NOLTIV-4RIABLE SYSTEMS 609

and A g = 7g. (56)

Using (65), (70), and (53) one obtains

TF = - ~ $ ~ - l ( i ) - aikiij\i+-2i1 . ..liij]. (57) [ n-a i = O

Since F a.nd 13 possess the sa.me characteristic equation, it follows t,hat

n--2 * . = -@-I. (5h)

i = O

Equa.t.ion (57) then yields

C F = $~-~(k)i@C$o(@)c = kc (59)

owing t.0 the fa.ct that j and c#&) commute. Equat,ion (85) immediat.ely follows from (59) when c is nonsingular.

From (Sl) it. follows t,hat n - 3 n - 2

i = O i = O

or

and the proof of Lemma 4 is complete. It is int,erest,ing to not,e that the invertibilit,y condit,ion

on z contains all the extra necessary a.nd sufficient. conditions required by Bongiorno and Youla as opposed t.0 t,he canonica,l estimator. Clearly ~$~(j) is invert,ible if and only if $ and therefore F have no eigenvalues in common with A . On the ot,her hand [ f i - " I . . . liilij] is invertible if and only if t,he pair { j, tj} is cont<rollable. When t,hese conditions arc satisfied, the canonica,l estimatm to be designed is clearly controllable from the y input, since t,he controllability property is preserved under a. nonsingular co- ordinate transformation. It is possible to show by an example, however, that, when A and F ha.ve eigenvalues in common, t.his controllabilit,y property does not have to be sat,isfied.

The tra.nsformation to the R7010vich estimat,or is ob- t,ained when

where X i , i = 1, 3, . . . , n . - 1, are the selected eigenvalues for the estimat,or.

Lemma 5: The t>ransformation given by (52) and (92), when invertible, transforms the canonic estimat,or to a Molovich-type &mator.

For the proof, the approach taken in the proof of Lemma 4 is followed. Using (65), (70), and (92) yields

However, diag { X i ) is nothing else but t.he F matrix

for the Wolovich estimator. i = l , . . . , n - l

Next (81) and (92) are used t.o obtain

= GO1 - E,(t)X,l - (94) i = l ... , ,n -1 j = O

Wolovich expressed the 6 matrix of his estimat.or a.s

T*I = c01 { [l, Xi, . - ., Xi*-l]). e01 {ai - iii) i = l , . . . , n - l i = O . . . . , n - l

(95)

where t,he Pt are defined by n-1 n - 1

n-1

i = O Bjh( = --Xi" (97)

and t.hus

j = T*E = C O ~ - bjX2 - Xi"

This concludes the proof. Note that t,he t,ransformation ma.trix of (92) is invertible

if and only if Xi, i = 1, . . , n - 1, are distinct., since it is basically a Vandermonde mat.rix. This additional condition is a prerequisite to Wolovich's method. However, X i do not have to be real as required by Wolovich.

TrII. FEEDBACK CONSIDERATIONS AND

OPTINAL REGVLATORS The developments in the preceding sections have been

concerned with observers as open-loop estimat.ors of a plant,'s state vector. A main reason in control systems design for obtaining estimates of the plant state vector is t.he fact that unsta.ble plants can often be stabilized with state-

- . . .

610 IEEE TRANSACTIONS ON AUTONATIC CONTROL, DECEMBER 1971

variable feedback. It is now shown that this desirable and feature is ret,ained when instead of state-variable feedback one feeds back the estimate of the state variables generated by an observer. When the rank of Qc( t ) is n and t.he same n columns of

The composite system consisting of the system (1) and Qc(t) are 1inea.rly independent, for all t 2 to, i.e., Qc(t) is (2) followed by an estimator constructed using Algorithm index invariant, then there exists a, matrix k(t) and a

@(t) = A(t)B*-,( t ) - h & l ( t ) , i = 2 , . . ., 12. (111)

2 is characterized by Lyapunov transformation t ( t ) such tha.t

and

where

M(t) = L( t ) f i ( t ) (101)

N(t) = L(t) f i . (102)

T ( t ) = FL-yt) (103)

and

With

the Lyapunov tra,nsformation

reduces this system int.0 the equiva.lent system characterized by

and

Next the feedback law

u(t) = k(t)i(t) (107)

is considered. The closed-loop system is then characterbed by

[%I = [ A( t ) + B ( t ) k ( t ) --B(t)k(t)N(t) O p , n I F 1 [$I.

(10s)

The stabi1it.y of the closed-loop system is determined by the coefficient matrix in (10s). Conditions under x-hich it is possible t.o choose a k(t) so that the system is asympt-oti-

' ca.lly stable ca.n be stated in terms of the coxtrollability ma- t r ix

Q&) = row {s t t ( t ) ] (109) i = l . . . 11

I .

P = t ( t ) [ A ( t ) + B ( t ) k ( t ) ] L - ' ( t ) + e(t)t-'(t) (112)

is a constant matrix which can be made t o have any pre- scribed set of eigenvalues provided complex eigenvalues occur in conjuga.te pairs. This result is the main result established in [lo]. It immediately follom that t-he t,ransformation

a(t> = JWm (113)

reduces the system (108) to [""I = ['I - ~ ( ~ ) B ( t ) ~ ( t ) N ( t ) 1 (114) e (0 0 p . n F 1 [*I

Since the eigenvalues of P and F can all be nmde to have nega.tive real parts, one readily concludes the following from (1 14).

Theorem 3: Any plant of the kind describable by (1) and (2) can always be st.abilized using an observer when the plant is uniformly cont.rollable and uniformly observable with cont,rollabilit.y and observability mat.rices which are index invariant.

Another important reason for obtaining est.imates of the plant state vect,or is t.hat application of the results obtained in opt,imal control theory for linear p1ant.s with a quadratic cost index requires st,ate-variable feedback. The use of observers in systems which would be optimal when the estimated statme variables are free of error is now discussed.

When a perfect estimate of t.he st.ate vector is available for the plant described by (1) and (2), the feedback law (107) can be chosen to minimize tmhe cost, funct.ion

J ( t 0 , t l ) = X'(tlISX(t1)

+ [x'( t )Q(t)x( t ) + u'(t)R(t)u(t)l dt (115)

where S is a real symmetric nonnegative-definite conshnt matrix and, on the interval t o 5 t 5 t l , Q(t) is a real symmetric nonnegative-definite continuous mat.ris and R(t) is a real symmetric positive-definite cont.inuous mat,rix. The optimum feedback mat,rk is given by

to

&(t) = -R-'(t)B'(t)K(t) (116)

where R(t) is the unique solution of t.he Ricatt.i equation

k(t) = -R(t)A(t) - A'(t)K(t) - Q(t ) + R ( t > B ( t ) R - l ( t ) B ' ( t ) R ( t ) (117)

YUKSEL AND BONGIORNO: OBSERVERS FOR MULTIVARIABLE SYSTEXS

It is well known for this case that

J o = min J ( h , tl) = x’(h)k(to)x(fd. (119)

A standard reference for t,he results cited in this paragraph is [25].

This paper, however, considers those cases where neit,her the entire state vect.or nor its initial value is available and an asympt.otic stat.e estimat,or is employed due t.0 t.he absence of a perfect estimate. The closed-loop syst,em described by (108) is considered witrll

u(t)

k(t) = I?&). (120)

The cost function of (1 15) is rearmnged to read

J* = t’(tl)St(tl> + 1 t‘(t)Q(t)t(t) clt (121)

where

T’(t) = [x’(t)[e’(t)l (122)

(123)

and

Q(t) =

Using (1 16) yields

61 1

and

K Z 2 + K 2 2 ~ + F’K= + k l 2 ’ ~ K 0 ~ + N ‘ K ~ ~ B I K ~ = - N ’ K ~ ‘ R K ~ N . (132)

The boundary conditions for the a.bove equations follow from (128) and are

Kn(t1) = S K 1 2 ( t 1 ) = O n , p G ( t 1 ) = O p , p (133)

It. is not, difficult. to verify in vieu- of (1 16), (1 17), and (1 IS) t.hat

Ku(t) = K(t) (134j

is the unique solution of (130) satisfying the appropriate boundary condition in (133). Using this result in (131) immediately yields for the boundary condit,ions (133) that.

K d t ) = o n , p . (135)

K 2 ? + K ~ F + F ’ K ~ ~ = - N ‘ K ~ I R K ~ N . (136)

Hence, (132) reduces to

The solution of (136) satisfying the boundary condit,ion in (133) is

&(t) = eF“N’(7)ko’(~)R(7)ko(7)N(.r)eF‘ d7 (137) I ’

and K2, , ( t ) is, therefore, nonnegat,ive definit,e. Using t,he fa.cts obtained a.bove it follows that

It is not, difficult. t,o shorn that

where K ( t ) is the unique symmet.ric nonnega.tive-definit,e solut.ion of the matrix different,ial equat.ion

k(t) + F’(t)R(t) + K( t )F( t ) = - Q(t) (127)

R(t1) = s. (125)

which satisfies

In (127), F ( t ) is t-he coefficient mat,rix in (10s). With t,he a.id of t,he part.itioning

(129)

J* = x’( to)K(to)x( to) + e’(to)I?22(tO)e(to) (135)

or A J = J ” - J 0 - - e’(to)L(to)e(to). (139)

The quant,it,y AJ represents the cost. increment, incurred as a result of the error in the est,imate of the plant state vec- t,or. One ca.n expect, this cost) increment. to be positive since, in genera.1, &(to) # Op,p and is nonnegative definit.e. The one ca.se of special int,erest, is when

Ko(t)N(t) = 0 r n . P . (140)

From (137), it) t.hen follows that &2( t0 ) = Op,p a,nd the cost increment, AJ is zero for a.11 initial error vectors e(to). However, when (140) is satisfied one has

u(t> = kO(t)W = kOW [ M ( t ) I N ( t ) l [ ~ ] Y ( t )

= j tO( t ) l “Y( t ) (141)

and the control u(t) can be generat,ed directly from t,he plant output. ~ ( t ) with no need for an observer.

Since the rat.e at which the elements of the error vector decay wit,h t,ime is determined by the eigenva.lues of the time-invariant observer matrix F , intuit,ion suggests that

612 IEEE TRANSACTIONS ON AUTObL4TIC CONTROL, DECEMBER 1971

the more negative the real parts of the eigenvalues of F , the smaller will be the cost increment AJ. That, this con- clusion is not. correct., however, is easily demonst>rated Ivith the aid of a simple example. The following case is considered :

(142)

N ’ ( t ) ~ ~ ‘ ( f ) R ( t ) ~ ~ ( t ) N ( t ) = 1 2 (143)

e(to) = e(0) = [l 01’ (144)

and

a1 > 0, a2 > 0, f > 0. (145)

Moreover, attention is restricted to t-he situat.ion in which the eigenvalues of F are negat.ive real and distinct when f = 1. These eigenvalues are denoted by X1 and Xz.

St.art.ing mith (137) and (139) a stra.ight.forward computat.ion leads to

AJ = J C C ( X J j + .Pao2)$i(t)sj(t) & (146) t L 2 2

lo=O i = l j=1

where

and

Since

lim J f ”$ i ( t )$ j ( t ) dt = O D , i, j = 1, 2 (149) f-+m 0

it immediately follows that, choosing t,he eigenvalues of F to be highly negative (i.e., alloningf to become very large) leads to AJ -+ rather t,han AJ -+ 0 as expect,ed. Addi- tional examples and a discussion of this int.eresting phe- nomenon are cont.ained in [5], [9], and [X].

VIII. CONCLUSIONS

This paper presents a.n algorithm for t.he design of asympt.ot,ic stat,e estimators for index-invariant uniformly observable time-varying multivariable systems. The results obt.ained indicate that asgmpt,otic estimators can be employed in opt,imally designed regulat.ors provided an increase from the optimal cost. is tolerable a.nd one recognizes the dangers associated wit.E; estimator eigenvalues wit,h highly negat.ive real part.s. It is also shon-n that any uniformly observable and uniformly cont.rollable plant. wit.11 index-invariant observabi1it.y and cont*rollability mat,rices can be stabilized 1vit.h an observer.

The approach examined in this paper is one in which an estima.t.e of the st.ate is first formed and then operated upon. An alternat>ive approach would be t.0 estimate the desired function of the plant st.ate vector at the outset.. It is sometimes possible in this way t.o obtain observers of lower dimension than t.hose described here. An algorithm

for designing asymptotically stable observers of this type is contained in [19]. The observer dimension obtainable in this xvay compares n-ith t.he dimension of the compensat.ors obtained in [20] and [21]. As in [7], t.he approaches t.alien in [20] and [all, however, do not. guarant.ee the compensator t o be stable.

It should be clear that considerable freedom exists in the design of observers: the specification of t.he observer eigenva.lues does not uniquely define an observer. Of considerable interest,, therefore, is the use of t.he available degrees of freedom to meet certain desired characterist,ics. In [22] a design procedure is described for minimizing the expected value of the cost increment when the means a,nd covari- ances of the elements of the initial pla.nt. stat.e vector are knonn.

REFERENCES [ I ] R. E. Kalman and R. S. Bucy, “New results in linear filtering

and prediction theory,” Trans. ASME, J. Basic Eng., ser. D, pp.

[2] A. E. Bryson, Jr., and D. E. Johansen, ‘Zinear filtering for 9j-108, Mar. 1961.

time-ymying systems using measurements containing coiored

Jan. 1965. noise, ZEEE Trans. Automat. Contr., vol. AC-10, pp. 4-10,

[3] D. G. Luenberger, “Observing the state of a linear system,”

[4] -, “Observers for multivariable systems,” ZEEE Trans. IEEE Trans. Mil. Electron., vol. MIL-8, pp. 74-80, Apr. 1964.

[5] J. J. Bongiorno, Jr., and D. C. Youla, “On observers in multi- Automt. Contr., vol. AC-11, pp. 190-197, Apr. 1966.

variable control systems,” Int. J . Contr., vol. 8, pp. 221-243, Sept. 1968.

[6] W. A. Wolovich, “On state estimation of observable systems,” in 1968 Joint Automatic Control Conf., Preprink, pp. 210-222.

[7] F. Dellon and P. E. Sarachik, “Optimal control of unstable linear plants with inaccessible states,” IEEE Trans. Automat. Contr., vol. AC-13, pp. 491495, Oct. 1968.

[SI W. AI. Wonham, “Dynamic observers-Geometric theory,” ZEEE Trans. Automat. Contr. (Corresp.), vol. AC-15, pp. 258- 259, Apr. 1970.

191 J. J. Bongiorno, Jr., and D. C. Poula, “Discussion of ‘On observers in multi-variable control systems’,” Z n t . J . Conontr., vol.

[lo] W. A. Wolovich, “On the st,abilization of cont,rollable systems,” ZEEE Trans. A7hm.t. Contr. (Short Papers), vol. AC-13, pp.

[ 111 W. h.1. Wonham, “On pole assignment. in multi-input cont.ro1- lable linear systems,” ZEEE Trans. Automat. Contr., vol. AC-12, pp. 660-665, Dec. 1967.

[ la] ill. Heymann, “Comments ‘On pole assignment in mu1t.i-input controllable linear systems’,” ZEEE Trans. Automat. Contr.

12, pp. 183-190, July 1970.

569-572, Oct. 1968.

[13] R. E. Kalman, P. L. Falb, and &I. A. Arbib, Topi’s in Muthe- (Corresp.), pp. 748-749, Dee. 1968.

matical Svstems Theoru. New York: McGram-Hdl. 1969. [14] L. -4. Zideh and C.“A. Desoer, .Linear System Theory-The

State Spa@ Approach.. Kew York: McGraw-Hill, 1963. [15] W . G. Tuel, Jr., “Canonical forms for linear system,” IBM

San Jose Res. Lab., San Jose, Calif., Res. Paper RJ-375, Mar. 1966.

[16] -, “An improved algorithm for the solution of discrete regu- lation problems,” lEEE Tra.ns. Aufmmi. Contr., vol. AC-12,

[17] L. M.,Silvernmn, “Transformation of time-variable systems to canonical (phase-variable) form,” IEEE Tra.ns. Automat.

[18] 31. S$vosudarmo, “Canonlcal observers in linear cont.ro1 sys- Confr. (Short. Papers), vol. AC-11, pp. 300-303, Apr. 1966.

tems, Ph.D. dissertation, Polytechnic Inst.. Brooklyn, Brook- lyn, N.Y., 1969.

[19] Y. 0. Yiiksel, “Canonical estimat.ors for time-varying multivariable systems,” Ph.D. dissertat.ion, Polytechnic Inst. Brook- lyn, Brooklyn, N.Y., 1971.

[20] J. B. Pearson and C. Y. Ding, “Compensator design for multivariable linear systems,” ZEEE Trans. Automat. Contr., vol. AC-14, pp. 130-134, Apr. 1969.

[21] J. D. Ferguson and Z. V. Rekasius, “Opt.imr4j linear control systems w t h incomplete state measurements, ZEEE Trans.

[22] &I. M. N e m a n n , ‘Specific optimal control of the linear regu- Aufum.t. Contr., vol. AC-14, pp. 135-140, Apr. 1969.

pp. 522-528, Oct. 1967.

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. ac-16, NO. 6, D E C E ~ E R 1971 613

lator using a dynamiml controller based on the minimal-order Luenberger observer,” I d . J. Cmtr., vol. 12, pp. 33-48, July

L. 31. Silverman and H. E. Meadows, “%ntrollability and 1970.

observability in t.ime-variable linear systems, J. S I A X Cmtr. , vol. 5, pp. 64-73, 1967. E. T. Tse, “On the optimal control of linear syst.ems with in-

Inst. Technol., Cambridge, Rep. ESL-R-412, Jan. 1970. complete information,” Electron. Syst. Lab., Massachusetts

M. Athans and P. Falb, Optimal Control. New York: McGraw- Hill. 1966.

Joseph J. Bongiorno, Jr. (S’56-IV1’60) was born in Brooklyn, N.Y., on August 3, 1936. He received t.he B.E.E., the M.E.E., and the D.E.E. degrees in 1956, 1958, and 1960, respectively, all from t,he Polytechnic Institute of Brooklyn, Brooklyn, N.Y.

From Sept.ember 1956 to September 1957, he was a Junior Research Fellow at the Micro- wave Research Instit.ute, Polytechnic Insti- tute of Brooklyn. In September 1957 he joined the teaching staff at the Polytechnic 1nst.it.ute.

In March 1959 he began working on his doctoral dkertat.ion at. the Microwave Research Instihte, while supported by the Grumman

Aircraft Engineering Corporat,ion. He is currently an Associate Pro- fessor in the Department of Elect.rica1 Engineering at the Polytechnic 1nst.itute. He was a consultant for t.he Bell Telephone Laboratories, Inc., from 1961-1962 and for the Sperrv G.rmscope Company from 1963 to 1970.

- - - - -

Dr. Bongiorno is a member of Sigma Xi and Eta Kappa Nu.

Y. Onder Yuksel was born in Ankara, Turkey, on May 26, 1942. He received the B.S. and R.I.S. degrees in elect.rical engineering from Orta D o b Teknik Universitesi (the Middle East Technical University), bnkara, Turkey, in 1963 and 1964, respect.ively, and the Ph.D. degree in electrical engineering from the Poly- technic Institut,e of Brooklyn, Brooklyn, N.Y., in 1971.

Employed by the Middle East Technical University since 1964, he received a United

St.ates Government grant and a leave of absence in 1965 to pursue his Ph.D. studies in this country. During part of the time he was an Inst,ructor at the Polytechnic Institute of Brooklyn. He is presently Assistant Professor of electrical engineering at t.he Middle East Technical University. His current interests include muhivariable systems and dynamic compensation techniques.

Optimal Linear Regulators: The Discrete-Time Case

Abstract-In this paper the optimal discrete-time linear-quadratic regulator problem is carefully presented and the basic results are reviewed. Dynamic programming is used to determine the optimization equations. Special attention is given to problems unique to the discrete-time case; this includes, for example, the possibility of a singular system matrix and a singular control-effort weighting matrix. Some problems associated with sampled-data systems are also summarized, e.g., sensitivity to sampling time, and loss of controllability due to sampling.

Computational methods for the solution of the optimization equations are outlined and a simple example is included to illustrate the various computational approaches.

T I. IKTRODUCTIOK

HE pra.ct,ical impetus for the st.udy of discrete-time systems comes from a va.rietg of sources. In some

problems, especially in such fields as biology, economics, etc., discrete-time models are the most natural ones t.0 assume. In other problems, where computat,ion must be done on a digit,al computer, the discret,e-time model often results as suitable approximation t o a continuous-t,ime system. Sampled-dat.a systems form a large class of

Manuscript received July 16, 1971. Paper recommended by D. Kleinman, Associate Guest Editor. This work was supported in Dart bv the National Science Foundat.ion under Grant GK-5.595 and in“part,by the Joint. Services Electronics Program under Con- tract F44620-69-C0047.

Brooklyn, N.Y. 11201. The authors are with t.he Polytechnic Inst ihte of Brooklyn,

practical disc,rete-time systems. Finally, in an increasing number of applications the digital computer funct.ions as a control device. For such syst.ems, time n u s t be quantized to match computer processing time.

Although several studies had been reported on the synthesis of sampled-da.t.a systems [l], [2], the paper by Kalman and Koepcke [3] published in 1958 was one of the first to approach the subject, mat,t,er from what is now called the modern st,at,e-space point of view. The authors formulated the linear quadratic problem and used a dynamic programming approach to obtain a formal solution t o t,he infinite-time problem. The mathematical details of t,his first paper were furt,her developed by Kal- man in a pa.per presented at t,he First Int.ernationa1 Federation on Automatic Control Congress in MOSCOW, 1960 [4]. This latter paper introduced a key concept required for the solution of the opt,imal problem over an in- finit,e time interval, that is the concept, of cont.rollabi1ity. Relatively few nex t.heoret.ica1 results on t.he discrete-time optimal linear regulator problem have been obtained since the publicat,ion of [4] in 1960. The present paper follows t,he theoretical development. of these early papers. For example, dynamic programming is used to derive opt,imiza- tion equations, although other techniques have since been developed, e.g., the discret,e minimum princ.iple, for t,he derivation of these equat,ions. The special intuitive

observers for linear multivariable systems with applications

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