Parametrization of All Strictly Causal Stabilizing Controllers of Multidimensional Systems single-input single-output case K. Mori Abstract We give a parametrization of all strictly causal stabilizing controllers of single-input singleoutput multidimensional systems with the structural stability. The approach used is the factorization approach. 1 Introduction The objective of this paper is to present a parametrization method of all strictly causal stabilizing controllers for single-input and single-output multidimensional systems in the framework of the factorization approach[1, 2, 3. A transfer function of the factorization approach is considered as the ratio of two stable causal transfer functions. Further the set of the stable causal transfer functions is considered as a commutative ring. Since the stabilizing controllers are not unique in general, the choice of the stabilizing controllers is important for the resulting closed loop. In the single-input and single-output case, there exist the coprime factorizations of the given stabilizable plant (Corollary 2.1.5 of [4). Thus the stabilizing controllers can be parameterized by the method called Youla- School of Computer Science and Engineering, The University of Aizu, Aizu-Wakamatsu 965-8580, JAPAN Kazuyoshi.MORI@IEEE.ORG
Kučera-parametrization[1, 5, 6, 3, 2, 7. However, it is also known that such parametrization may include a stabilizing controllers which is not causal and may result a direct loop (see Section 2). Thus we present an alternative parametrization of stabilizing controllers, that is the parametrization of all strictly causal stabilizing controllers. This parametrization will include neither any non-causal stabilizing controller nor any direct loop. Fact 4.9 of [8 gives the existence of a strictly causal stabilizing controller of a causal stabilizable plant. On the other hand, this paper gives all strictly causal stabilizing controllers with n-parameters (for n-dimensional systems). 2 Motivation Let us consider the feedback system shown in Figure 1 of the classical discrete-time system. Although this is a 1-dimensional system, the observation here can be applied to multidimensional systems analogously. Let c p Figure 1. Feedback system. A = {f(z) R(z) f(z) has no poles on or inside the unit circle in the complex plane} (1) (z denotes the delay operator), which is the set of stable causal transfer functions. Now we let p = z + 1. Then we have the Bézout identity (z + 1)y + x = 1 over A. Using the Youla-Kučera-parametrization, these y and x can be parametrized as y = r, x = 1 (z + 1)r with a parameter r of A. By letting r = 1, we have y = 1 and x = z. In this case, the stabilizing controller obtained from y and x is 1/z. Unfortunately this is not causal. Otherwise, by letting r = 1, we have y = 1 and x = z + 2. In this case, the stabilizing controller obtained from y and x is 1/(z + 2). This results a direct loop, that is, the current input affects the whole closed feedback system immediately as shown in Figure 2. This loop is normally unsafe even if the total feedback system is stable. To avoid these situation, we will present the parametrization of all strictly causal stabilizing controllers, (i) which does not include any non-causal stabilizing controller and (ii) which does not result any direct loop.
3 Preliminaries We employ the factorization approach [1, 6, 3, 2 and the symbols used in [9 and [10. The reader is referred to Appendix A of [3 for algebraic preliminaries if necessary. For a moment let us consider a general linear systems. Denote by A a commutative ring that is the set of stable causal transfer functions. The total ring of fractions of A is denoted by F; that is, F = {n/d n, d A, d is a nonzero divisor}. This F is considered to be the set of all possible transfer functions. Let Z be a prime ideal of A with Z A. Further, let P = {a/b F a A, b A Z} and P s = {a/b F a Z, b A Z}. A transfer function f is said to be causal (strictly causal) if and only if f is in P (P s ). We here apply the factorization approach to the multidimensional systems with structural stability. To do so, we need to give (i) the set A of stable causal transfer functions and (ii) the ideal Z for the causality; They are given as follows: A = {a/b a, b R[z 1,..., z n, b 0 in U n }, (2) n Z = z i A = {a/b A a, b R[z 1,..., z n, i=1 the constant term of a is zero. }, (3) where U n denotes the closed unit polydisc. Throughout the paper, the plant we consider is causal, single-input and single-output, and its transfer function, which is also called a plant itself, is denoted by p and belongs to P. We consider the feedback system Σ [3, Ch.5, Figure 5.1 shown in Figure 3. In the figure, c denotes a controller and is a transfer function of F. The stabilization problem, considered in this paper, follows the one developed in [1, 3, 2. For details, the reader is referred to [3, 11, 10, 4. Let H(p, c) denote the transfer matrix from [ u 1 u 2 t to [ e 1 e 2 t of the feedback system Σ, that is, [ (1 + pc) 1 p(1 + pc) H(p, c) := 1 c(1 + pc) 1 (1 + pc) 1 F 2 2 (4) provided that 1 + pc is nonzero. We also use H(P, C) and W (P, C) for transfer matrices P and C. For these notations, the reader is referred to [3. We say that the plant p is stabilizable, p is stabilized by c, and c is a stabilizing controller of p if and only if 1 + pc is nonzero and H(p, c) A 2 2. In the definition above, we do not mention the causality of the stabilizing controller. However, it is known that if a causal plant is stabilizable, there always exists a causal stabilizing controller of the plant [11. Because we investigate the set of some kind of stabilizing controllers, we introduce some notations as follows (P P n m ) : S(P ) := {C F m n H(P, C) A (m+n) (m+n) } (5) = the set of all stabilizing controllers of P, SP(P ) := S(P ) P m n (6) = the set of all causal stabilizing controllers of P,
SP s (P ) := S(P ) P m n s (7) = the set of all strictly causal stabilizing controllers of P. We further introduce the set of all causal stabilizing controllers of P including a postcompensator as follows: SP s,post (P ; Ψ) := {ΨC 0 SP s (P ) C 0 P m n }, (8) where Ψ is a diagonal matrix and every entry of Ψ is an element of a minimal number of generators of Z (such as z 1 and z 1 6z 2 z 3 ). For strictly causal plants, we know the following proposition. PROPOSITION 1. (cf. Proposition 1 of [12) Let P be a strictly causal plant. Then any stabilizing controller of P is causal, that is, SP(P ) = S(P ). 4 Main Result The following is the main result of this paper, which leads the parametrization of all strictly causal stabilizing controllers of a multidimensional system. THEOREM 2. Let p 0 be a stabilizable plant of P. Further let P = [ n {}}{ p 0 p 0 p 0 t, (9) and Then, where c i P. Ψ = Diag(z 1, z 2,, z n ). n SP s (p 0 ) = { c i z i [ c 1 c 2 c n SP(ΨP )}, (10) i=1 Once we have Theorem 2, we can obtain the parametrization of all strictly causal stabilizing controllers of a multidimensional system as follows. Because p 0 is stabilizable, SP s (p 0 ) is not empty. Thus SP(ΨP ) is also not empty. Let (N, D) and ( D, Ñ) be right- and left-coprime factorization of ΨP with ΨP = ND 1 = D 1 Ñ, Ỹ N + XD = I 1, ÑY + DX = I n, where Ỹ, X, Y, and X are matrices over A with appropriate sizes (N, Ñ An 1, D, X A 1 1, D, X A n n, Ỹ, Y A1 n ). Then by the Youla-Kučera-parametrization, we have SP(ΨP ) = {( X RÑ) 1 (Ỹ + R D) R A 1 n }. (11)
By combining (10) and (11), we obtain the parametrization of all strictly causal stabilizing controller of p 0 as follows: { n SP s (p 0 ) = c i z i i=1 [ c 1 c 2 c n := {( X RÑ) 1 (Ỹ + R D), R A 1 n}. 5 Stabilizing Controllers with Post-compensators In this section, we investigate the set of stabilizing controllers with a postcompensator. We denote by ζ 1, ζ 2,, ζ n the minimal number of generators of the ideal Z (typical example is (ζ 1, ζ 2,, ζ n ) = (z 1, z 2,, z n )). THEOREM 3. Let P F n m, C F m n, and Ψ = diag(ζ 1,, ζ n ). Then P is causal and H(ΨP, C) is over A if and only if C is causal and H(P, CΨ) is over A. Before proving Theorem 3, we give a lemma. LEMMA 4. Let A be a matrix over P with size n m. Let Ψ be as in Theorem 3. Then A is over A if and only if ΨA is over A. Proof. This is obvious so the proof is omitted. Proof of Theorem 3: Let [ Ha11 H a12 := H(ΨP, C) H a21 H a22 [ (I + ΨP C) 1 ΨP (I + CΨP ) = 1 C(I + ΨP C) 1 (I + CΨP ) 1 and [ Hb11 H b12 := H(P, CΨ) H b21 H b22 [ (I + P CΨ) 1 P (I + CΨP ) = 1 CΨ(I + P CΨ) 1 (I + CΨP ) 1. Only If part. Suppose that P is causal and H(ΨP, C) is over A, that is H aij (1 i, j 2) is over A. Because P is causal, ΨP is strictly causal. By Proposition 1, we see that C is causal. We now show that H(P, CΨ) is over A, that is H bij (1 i, j 2) is over A. H b11. Recall first that H a11 = I ΨP (I + CΨP ) 1 C, and H b11 = I P (I + CΨP ) 1 CΨ. Because H a11 is over A, the matrix ΨP (I + CΨP ) 1 C is also over A.
By applying Lemma 4 twice, we see that the matrix P (I + CΨP ) 1 CΨ is over A. Hence H b11 is over A. H b12. Because H a12 is over A, by Lemma 4, the matrix P (I + CΨP ) 1 is also over A, which is equal to H b12. H b21. H b21 is equal to H a21 Ψ. Because H a21 is over A, by Lemma 4 again, the matrix H b21 is also over A. H b22. Because of H b22 = H a22, H b22 is over A. If part. Suppose that C is causal and H(P, CΨ) is over A. Recall first that [ [ Im O H(A, B) = H(B t, A t ) t Im O O I n O I n. By using this equation, we have [ [ H(ΨC t, P t Im O ) = H(P, CΨ) t Im O O I n O I n. (12) Applying the result of Only if part to H(ΨC t, P t ) in (12), we have that P t is causal and H(C t, P t Ψ) is over A. Hence P is causal and H(ΨP, C) is over A. THEOREM 5. Let P be a plant of P n m. Let Ψ = diag(ζ 1,, ζ n ). Then the following (i) and (ii) hold. (i) SP s,post (P ; Ψ) = {C 0 Ψ C 0 SP(ΨP )}. (13) (ii) Assume that P is stabilizable. Let N, D, Ñ, D, Y, X, Ỹ, X be matrices over A such that ΨP = ND 1 = D 1 N, Ỹ N + XD = I m, ÑY + DX = I n. (14) Then SP s,post (P ; Ψ) ={( X RÑ) 1 (Ỹ + R D)Ψ R A m n } (15) ={(Y + DS)(X NS) 1 Ψ S A m n }. (16) Proof. (i) We prove (13) by showing and. : Let C 0 SP(ΨP ). Because P is causal, by Theorem 3, C 0 Ψ SP(P ). Because every entry of Ψ is in Z, C 0 Ψ is strictly causal, that is, C 0 Ψ SP s (P ). Hence C 0 Ψ SP s,post (P ; Ψ). : Let C SP s,post (P ; Ψ) and C 0 P m n with C = C 0 Ψ. Then H(P, C 0 Ψ) is over A. Because C 0 is causal, by Theorem 3 again, H(ΨP, C 0 ) is over A. Hence C 0 is an element of SP(ΨP ).
(ii) We have N, D, Ñ, D, Y, X, Ỹ, X such that the equations in (14) hold. Hence the Youla-Kučera-parametrization can be applied. By noting that ΨP is strictly causal, we have SP(ΨP ) = {( X RÑ) 1 (Ỹ + R D) R A m n }. (17) By (17) and Theorem 3, we have (15). Analogously (16) can be obtained. 6 Proof of Theorem 2 Before proving Theorem 2, we present a proposition below. PROPOSITION 6. Let p 0 be a plant of P. Further let P = [ n {}}{ p 0 p 0 p 0 t. Suppose that C is a stabilizing controller of P. Then [ n {}}{ C 1 1 1 is a stabilizing controller of p 0. That is, if we decompose C as [ c 1 c 2 c n := C, (18) where c i F, then n i=1 c i is a stabilizing controller of p 0. Proof. Let C be a stabilizing controller of P. Decompose it as in (18). Let us consider W (P, C). This W (P, C) is over A, because C is a stabilizing controller of P. Let Let n V = [ I 2 O 2 (n 1) W (P, C). W 11 = C(I n + P C) 1, W 12 = CP (I 1 + CP ) 1, W 21 = P C(I n + P C) 1, W 22 = P (I 1 + CP ) 1, 1 0 1 0. 1 0 0 1.
that is Let [ W11 W W (P, C) = 12 W 21 W 22 I 1,n = [ n {}}{ 1 1 1 t. and V 11 = W 11 I 1,n, V 12 = W 12, V 21 = [ 1 O 1 (n 1) W 21 I 1,n, V 22 = [ 1 O 1 (u 1) W 22. Now we have and and let Then Further let P = I 1,n P 0 [ p 0 = [ 1 O 1 (n 1) P [ c 0 = CI 1,n. I 1 + CP = [ 1 + c 0 p 0. Then we have V 11 = (1 + c 0 p 0 ) 1 c 0, V 12 = c 0 p 0 (1 + c 0 p 0 ) 1, V 21 = p 0 (1 + c 0 p 0 ) 1 c 0, V 22 = p 0 (1 + c 0 p 0 ) 1. [ V11 V W (p 0, c 0 ) = 12 V 21 V 22 which is over A. That is, C 0 is a stabilizing controller of P 0. Now that we have obtained Proposition 6, we can prove Theorem 2. Proof of Theorem 2: We prove (10) by showing and. : Let C := [ c 1 c 2 c n SP(ΨP ). Then by Theorem 3, CΨ is in SP(P ). Applying Proposition 6, n i=1 c iz i is a stabilizing controller of p 0. The strict causality of n i=1 c iz i is obvious. : Let p 0 be in P. Let c 0 be a strictly causal stabilizing controller of P. Let (n 0, d 0 ) be a coprime factorization of p 0 with y 0 n 0 + x 0 d 0 = 1,,
where y 0 and x 0 are of A with c 0 = y 0 /x 0. Then a right-coprime factorization (N, D) of P can be given as Ỹ N + XD = I 1, where N = [ n 0 n 0 n 0 t A n 1, D = [ d 0 A 1 1, Ỹ = [ y 01 z 1 y 02 z 2 y 0n z n A 1 n, X = [ x 0 A 1 1 such that n y 0 = y 0i z i (y 0i A m n ) i=1 (Note that because c 0 is strictly causal, y 0 can be expressed as above). Let c i be y i /x i and C := [ c 1 c 2 c n. Hence CΨ is in SP(P ). By Theorem 3, C is causal and C is in SP(ΨP ). 7 Conclusion This paper has been given a method to parametrize all strictly causal stabilizing controllers. In this paper, we have focused on the single-input single-output case. The multi-input multi-output case should also be investigated.
z+2 1 z+1 Figure 2. Direct loop. c u 2 p u 1 e 1 y 1 e 2 y 2 Figure 3. Feedback system Σ.
Bibliography [1 C. A. Desoer, R. W. Liu, J. Murray, and R. Saeks, Feedback system design: The fractional representation approach to analysis and synthesis, IEEE Trans. Automat. Contr., vol. AC-25, pp. 399 412, 1980. [2 M. Vidyasagar, H. Schneider, and B. A. Francis, Algebraic and topological aspects of feedback stabilization, IEEE Trans. Automat. Contr., vol. AC-27, pp. 880 894, 1982. [3 M. Vidyasagar, Control System Synthesis: A Factorization Approach, Cambridge, MA: MIT Press, 1985. [4 S. Shankar and V. R. Sule, Algebraic geometric aspects of feedback stabilization, SIAM J. Control and Optim., vol. 30, no. 1, pp. 11 30, 1992. [5 V. Kučera, Stability of discrete linear feedback systems, in Proc. of the IFAC World Congress, 1975, Paper No.44-1. [6 V.R. Raman and R. Liu, A necessary and sufficient condition for feedback stabilization in a factor ring, IEEE Trans. Automat. Contr., vol. AC-29, pp. 941 943, 1984. [7 D.C. Youla, H.A. Jabr, and J.J. Bongiorno, Jr., Modern Wiener-Hopf design of optimal controllers, Part II: The multivariable case, IEEE Trans. Automat. Contr., vol. AC-21, pp. 319 338, 1976. [8 Z. Lin, Output feedback stabilizability and stabilization of linear n-d systems, in Multidimensional Signals, Circuits and Systems, K. Galkowski and J. Wood, Eds. 2001, pp. 59 76, New York, NY: Taylor & Francis. [9 K. Mori, Parameterization of stabilizing controllers over commutative rings with application to multidimensional systems, IEEE Trans. Circuits and Syst. I, vol. 49, pp. 743 752, 2002. [10 V. R. Sule, Feedback stabilization over commutative rings: The matrix case, SIAM J. Control and Optim., vol. 32, no. 6, pp. 1675 1695, 1994. [11 K. Mori and K. Abe, Feedback stabilization over commutative rings: Further study of coordinate-free approach, SIAM J. Control and Optim., vol. 39, no. 6, pp. 1952 1973, 2001.
[12 K. Mori, Elementary proof of controller parametrization without coprime factorizability, IEEE Trans. Automat. Contr., vol. AC-49, pp. 589 592, 2004. [13 J. Cadzow, Discrete-Time Systems: An Introduction with Interdisciplinary Applications, NY, Prentice Hall, 1973.