Feedback Realization of Open Loop Diagonalizers

FEEDBACK REALIZATION OF OPEN LOOP DlAGONALIZERS VASFI ELDEM Tubitak, Marmara Research Centre Division of Mathematics Gebze, Kocaeli 41470, Turkey

Abstract. In this work the feedback realization of open loop diagonalizers (old) of a linear, time-invariant multivariable system is considered. In the first part of the paper, the properties of old's which admit i) dynamic state feedback, ii) constant state feedback, Hi) dynamic output feedback and iv) constant output feedback are investigated. Then, in the second part, dynamic (constant) output feedback decoupling problems are formulated as determining an open loop diagonalizer which admit the desired feedback realization. Key Words. Linear systems, diagonalizers, feedback realization, interactor. field of rational . polynomials with real numbers. The matrix in RPxm(s)

1. INTRODUCTION

During the last decade a renewed interest has been witnessed in the decoupling problems. This interest is mainly focused on state feedback decoupling as in, for instance, Descusse, Lafay and Malabre (1988), Dion a nd Commault (1988) or block decoupling as in Descusse, Lafay and Malabre (1983), Dion, Torres and Commault (1990) . Dynamic output feedback decoupling, on the other hand, is considered by Hammer and Kh argonekar (1984) and by Eldem and Ozguler (1989). Also, Kucera (1983) considered block decoupling by dynamic compensation with internal stability. Open loop block and scaler diagonalization are taken up in the works of Ozguler and Eldem (1989). This paper is basically a continuation of the works of Ozguler and Eldem (1989). It starts out with the investigation of the properties of old's which admit a feedback realization. The Objective is to formulate the feedback decoupling problems as determining a specific subset of the class of old's which admit a particular feedback realization. Throughout the paper linear, time-invariant, multivariable systems described by the following state space equations and input-output relations x(t) =A x(t) + B u(t) yet) =Cx(t) yes) =Z(s)u(s)

functions · and the ring of coefficients from the field of Laurent series expansion of a is given by

B(s):=

L

(2)

Bjs-;

;= -I:

where B:s are constant matrices. If B k= .. ·=B 1 1 - =0, then B(s) is called proper. If Bo is also zero then B(s) is called strictly proper. B(s) is called right (left) biproper if it is proper and Bo has full column (row) rank. B(s) is called simply biproper if Bo is square and nonsingular. Note that a right (left) biproper B(s) has a left (right) biproper left (right) inverse which will be denoted as ordinary inverse B(sr1. Using the Laurent series expansion above, strictly proper (B-(s» and strictly polynomial (B + (5» parts of B(s) are defined as B-(s): =

L ;sl

-1

Bj

s-;

B+(s): =

L

B;

s-;

;s-I:

Static left (right) kernel of a rational matrix B(s) is a linearly independent set of row (column) vectors {Xi} such that xjB(s)=O (8(S)xi=0). (In the rest of the paper kernel will be used instead of right kernel). A basis for a static left (right) kernel can be obtained by picking out the zero order rows (columns) of a minimal pol.}nomjal basis (Forney (1975» for the kernel. It is well known in the literature that the interactor, first defined by Wolovich and Falb (1976), plays a crucial role in decoupling problems. For a given strictly proper, pxm full row rank transfer matrix Z(s), the interactor is defined as a lower left triangular polynomial matrix X(s) such that

(1)

, Z(s):=C(sl-ArIB

are considered. In the above equations x(.), u(.) and y(.) take values from n, m and p dimensional linear spaces over the field of real numbers. A, B and C are constant matrices of appropriate dimensions. As usual R(s) and R[s] denote the

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2nd If'AC W Olok .. h o p o n SYSTEM S TRU C TURE AND C ONTROl. 3 - 5 S " ptenoh " ,- 100 2 . PRAGUE. CZE C HOSLOVAKIA

X(s)Z(s) is left biproper. The interactor call be .. expressed uniguely as X(s):=H(s)D(s) where D(s):=diag{snI} and H(s) is a lower left triangular matrix with ones on the diagonal. The set of integers {ni, i=I,2, ... ,p} are called the infinite zero orders of Z(s) and the column degrees of X(s) (a(X(s»ci) are called essential orders and denoted as ne i in Commault etal (1986). In view of the fact' that X(s)Z(s) is left biproper there exists a biproper L(s):=[Ll(s) .. L2(s)] such that X(s)Z(s)[L (s):L2 (s)] = [I: 0]. 1 Using this equation the set of open loop diagon:ilizers of Z(s) can be characterized easily as in Ozguler and Eldem (1989). For a given full row rank transfer matrix Z(s) and a given pxp strictly proper diagonal matrix A(s), the set of open loop A-diagonalizers OLD(Z,A) is defined as OLD (Z,A): = {M(s): Z(s)M(s)=A}. The following result, which we include for the sake of completeness, is a different version of Lemma 1 in Ozguler and Eldem (1989). Lemma L OLD(Z,A) is nonempty iff X(s)A(s) is proper. Proof The simple proof is omitted here. The above result basically implies that the infinite zero orders of the decoupled system is bounded below by a(X(s»ci which are the essential orders of the system to be decoupled. In feedback realization of the set of OLD's the following feedback control laws with constant precompensation is considered: i) Dynamic state feedback: u(s)=-F(s)x(s)+Gv(s), ii) Constant state feedback: u(s)= -Fx(s)+Gv(s), iii) Dynamic output feedback: u(s) =-Zc(s)y(s) + Gv(s). iv) Constant output feedback: u(s)=-Zc yes) + Gv(s). In the above, F(s) and Zc(s) are proper and F and Zc are constant compensators in the feedback path. G is a full column rank constant precompensator. Remark L The solvability conditions for general dynamic state feedback decoupling are well known in the literature, Hautus and Heynmann (1983), Dion and Commault (1988) and Ozguler and Eldem (1989). The constant state feedback case, also known as the Morgan's problem (Morgan (1964» is solved only recently by Descusse, Lafay and Malabre (1988). The general versions of dynamic and constant output feedback decoupling has not been solved yet. The solution for restricted dynamic output feedback is given by Eldem and Ozguler (1989). The solution to dynamic output feedback decoupling with internal

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stability, where the initial transfer matrix is square and nonsingular, is due to Hammer and Khargonekar (1984). 3. PRELIMINARY RESULTS For a given pxm, full row rank, strictly proper transfer matrix Z(s) and a strictly proper diagonal matrix A (X(s)A proper), OLD (Z,A) can be characterized easily as

where Ll and L2 are as defined previously and N is proper. Lemma 2. Let M be in OLD (Z,A). Then M can be realized by dynamic state feedback iff it is right biproper. Proof: The simple proof is omitted here. Remark 2. The above lemma implies that dynamic state feedback decoupling problem is equivalent to finding a right bip~oper element of OLD(Z,A). When A=diag {s-ne,l} this condition reduces to the condition given, for instance, in Dion and Commault (1988). It is also clear from the above result that only right biproper open loop diagonalizers admit feedback realizations. Therefore, only POLD(Z,A) right biproper subset of OLD(Z,A) is considered in the rest of the paper. Lemma 3. Let M be in POLD(Z,A). Then, M is realizable by constant state feedback iff the first m columns of a basis of the static left kernel of

(P(S»- (si-A)-l BM] [

(4)

K(si-Ar1BM

are linearly independent. Here F(s) is an arbitrary dynamic state feedback realization of M (which always exists as M is right biproper) and K is a constant matrix the rows of which span the left kernel of BMo' Proof: Let (F,G) be a constant state feedback realization of M. Then, M=[I+F(sI-ArlBrlG which implies that F(sI-ArlBM=-M- as G=M o ' For any dynamic state feedback realization F(s), it is also true that F(s)(sI-Ar l BM=-M-. Then, [F-F(s)](sI-Ar1BM=O. As s(sI-Arl BM is right biproper we have [F-F(s)]o [S(SI-ArlBM]o = [FF(s)]o [BM]o=O. Consequently, F=[F(s)]o +LK for some constant matrix L. This implies that [LK-(F(s»)"J (sI-ArlBM=O, i.e.,

[-I : L]

(F(s»-(sl-A)-l

BM]

is nonzero for each i. Here. A.- l denotes the ith diagonal element of K1 and (\i denotes the ith column. Proof: If the hypothesis holds, there exists a constant vector Ni such that

5)

;: 0 (;,

K(sI-Ar'BM For the converse. note that the above equation holds fo some constant matrix L. Define F as F=[F(s)]o +LK. Then,

[ (X(s»c/-

«J\-lr X(s)ZM1,o)cj (7)

F(sI-ArlBM = {[F(s)]o + LK} (sI-Ar l BM = {F(s)-(F(s)r + LK} (sI-A)-! BM =F(s) (S]-A)-l BM ;:M-

is proper. Let No:=[Nl' ....• N ]. Then. in view of the above expression it folio$; that

Consequently, M= [I + F(sI-A) -lBr1G. Lemma 4. Let M be in POLD(Z,A). Then M is

realizable by dynamic (constant) output feedback iff M-A- 1 is proper (constant). Proof" If M can be realized by dynamic (constant) output feedback. then there exists a proper Zc (constant Zcl and a constant G such that M=(I+ZcZr G. which implies that ZcZM=-M-. Since ZM=A, it follows that M-A -1 is proper (constant). For sufficiency let Zc:=-M-A- , then ZcA=ZcZM=-M-. Thus, (I + ZcZ)M=Mo =G, i.e., M==(I+ZcZr1G.

is proper. Since ZM 1 =A and ZM 2 =0 we have

X(s)Z(Ml - M l .o - M 2•oNo )A- l

which is also proper for any N-. Now let M2 -1 be a left biproper left inverse of M2 such that M - 1 2 M1 =0 (left inverse exists as M2 is right biproper). Choose N- as

4. MAIN RESULTS The problem to be treated in this section can be formalized as follows: Definition 1. Dynamic output feedback Adecoupling (A-DDOF): Given a pxm, full row rank. strictly proper transfer matrix Z and a diagonal. nonsingular. strictly proper A, find a proper Zc and a full column rank constant G such that Z(I+ZcZr1G=A; or equivalently: Find a proper N such that M:=M 1 +M 2 N admits a dynamic output feedback realization, Le., (M 1 +M2NrA-1 is proper and (Ml +M 2 N)o has full column rank where M1:=LlX(s)A and M 2:=L2· Constant output feedback A-decoupling (ADCOF) has a similar definition (proper Z is replaced by constant Zc and (M1 +M2NrA-'l is constant). Theorem 1. A-DDOF is solvable iff the first row of a basis of the static kernel of -1

+

Using equations (8) and (9) it follows that

is proper. Since the first expression above is biproper. defining N as N:==N o +N-. implies that (Ml +M2NrA-l is proper. Equation (11) also implies that

where Yes) is proper. Since X(s) is strictly polynomial and nonsingular, then the right hand side of the above equation is nonsingular. Therefore, Ml 0 +M2 oNo has full column rank. Thus, if Z and G are defined as Zc:=(M1 +M2~YA-1 and G:=M l 0 +M 2,oN o ' then (Zc,G) is a solution of A-DDOF. For necessity let (Zc,G) be a solution of ADD OF. Then. Zc=(M1 +M2NrA-1 for some proper matrix N. Thus,(M - +M N- +M -N )"A- 1 l 2 2 o is proper. Premultiplying this expression by X(s)Z we obtain

+

[(X(s»ci-«~ ) X(s)ZM 1•o )Ci

(8)

(6)

:«~-lr X(s)ZM2 •o )+]

18

X(s)Z (M1- + M2N- + M2- No )A- 1 = X(s) - X(s) ZM1,o A -1 - X(s)ZM2,oNoA- 1

A-diagonalizers of a given strictly proper transfer . matrix Z The solutions are obtained via the feedback realizations of these open loop diagonalizers. REFERENCES -Descusse, J., Lafay, J. F. and Malabre, M. (1988) "Solution to Morgan 's problem", IEEE Trans. Auto. Contr., vo!. AC-33, no. 8, pp. 732-739. -Descusse, J., Lafay, J. F. and Maiabre, M. (1983) "On the structure at infinity o[ linear block decouplable systems", IEEE Trans. Auto. Contr., vo!. AC-28, pp. 1115-1118. -Dion, J. M. and Commault, C. (1988) " Minimal delay decoupling problem: Feedback implementation with stability" SIAM J. Contr. Optimiz., vo!. 20, no. 1, pp.66-82. -Dion, J. M., Torres, J. A and Commault, C. (1990) " New [eedback invariants and the block decoupling problem" Int. J. Control, vo!. 51, no. 1, pp. 219-236. -Eldem, V. and Ozguler, A B. (1989) "A solution to the diagonalization problem by constant precompensator and dynamic output [eedback", IEEE Trans. Auto. Contr., vo!. AC-34, no. 10, pp. 1061-1067. -Fomey, G. D. (1975) "Minimal basis o[ rational vector spaces with applications to multivariable linear systems", SIAM J. Contr. Optimiz., vo!. 13., pp.493-520. -Hammer, J. and Khargonekar, P. P. (1984) "Decoupling o[ linear systems by dynamical output [eedback", Math. System Theory, vo!. 17., no 2., pp. 135-157. -Hautus, M. L. J. and Heynmann, M. (1983) "Linear [eedback decoup1infS> trans[er function analysis", IEEE trans. Auto. Contr., vo!. AC-28, pp. 823-832. -Kucera, V. (1983) "Block decoupling by dynamic compensation with internal properness and stability", Problems of Control and Information Theory, vo!. 12., no. 6., pp. 379-389. -Morgan, B. S. (1964) "The synthesis o[ linear multivariable systems by state [eedback" J. A. C. c., vo!. 64., pp. 468-472. -Ozguler, A B. and Eldem, V. (1989) "The set o[ open loop block diagonalizers o[ trans[er matrices" Int. J. Control, vo!. 49., no. 1, pp. 161-168. -Wolovich, W. A and Falb, P. L. (1976) "Invariants and canonical [om2s under dynamic compensation", SIAM J. Contr. Optimiz., vo!. 14, pp. 996-1008.

which is also proper. Therefore,

for each i, which concludes the proof. Note that the richness of the set of solutions of A-DDOF is only due to the richness of the subset of POLD(Z,A) which admit dynamic output feedback realization. Before presenting the solution of A-DCOF, a characterization of the set of all solutions of A-DDOF is given in the seque!. Towards this end let (Ze* ,G *) be a solution of ADD OF and for some Rroper N* we have * -1 G * =M +M N *. (I+ZZe) 2 1 Theorem 2. (Ze,G) is a solution of A-DDOF iff there exist a strictly proper matrix Ne and a constant matrix Nr such that

Zc =Z; -(M2Nc + M; Nr)A -1, G =G· +M2,oN,

Proof The proof is omitted here. The solution of A-DCOF is now in order. Theorem 3. Let (Zc * ,G * ) be a solution of ADD OF. Then, A-DCOF is solvable iff the first row of a basis of the intersections of the static kernels of

and

is nonzero for each i. Proo[: The proof is omitted here. S. CONCLUSIONS In this work A-decoupling problems, where the closed loop transfer matrix A is specified, are considered. The problems are formulated by using the right biproper set POLD(Z,A) of open loop

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