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Pole structure assignment via non-regular static state feedback
Automatica 35 (1999) 1549}1555
Brief Paper
Pole structure assignment via non-regular static state feedback夽 J. J. Loiseau *, P. Zagalak, V. Kuc\ era Institut de Recherche en Cyberne& tique de Nantes, UMR 6597 CNRS, Ecole Centrale de Nantes, BP 92101, F-44321, Nantes Cedex 03, France Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, P.O. Box 18, 18208 Praha, Czech Republic Trnka Laboratory for Automatic Control, Faculty of Electrical Engineering, Czech Technical University, Karlov na& m. 13, 12135 Prague, Czech Republic Received 14 April 1997; revised 21 October 1997; received in "nal form 11 March 1999
Abstract The limits of the static state feedback u"Fx#Gv, with G not necessarily nonsingular, in altering the transmission pole structure of the linear system x "Ax#Bu are studied. A necessary and su$cient condition for a list of polynomials to be the denominator polynomials of the Smith}McMillan form of the closed-loop transfer matrix (sI!A!BF)\BG is established. The condition involves the calculation of the in"mal set of the controllability indices that the system can attain through feedback while satisfying the Rosenbrock inequalities. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Linear systems; Static state feedback; Transmission poles; Controllability indices; Lattice of integer partitions; Fixed point algorithm
1. Introduction We consider the linear, "nite-dimensional, timeinvariant system x "Ax#Bu,
(1)
where A31L"L and B31L"K are real matrices. Applying, to system (1), the control law u"Fx#Gv,
(2)
where F31K"L and G31K"N, we get the closed-loop system x "(A#BF)x#BGv.
(3)
As far as control is concerned, we are interested in studying the properties of the closed-loop system (3), and how the choice of F and G can a!ect these properties. In particular one can be interested in studying: E The eigenvalue structure of the closed-loop system (3), that is the set of the invariant factors, say a (s) for G 夽 The original version of the paper was presented at the IFAC Conference on System Structure and Control, Nantes, France, July 5}7, 1995. This paper was recommended for publication in revised form by Associate Editor A.A. Stoorvogel under the direction of Editor R. Tempo. * Corresponding author. Tel.: #33-2-4037-6967; fax #33-2-40376930. E-mail address: [email protected] (J.J. Loiseau)
i"1,2, n, of the matrix A#BF. The eigenvalue structure plays a fundamental role when studying the internal stability of Eq. (3). E The (transmission) pole structure of the closed-loop system (3), that is the set of the invariant factors, say t (s) for i"1,2,p, which appear as the denominG ators of the diagonal elements of the Smith}McMillan form of the closed-loop transfer matrix (sI !A!BF)\BG. The pole structure represents L the transmission dynamics of the closed-loop system (3). These two structures are sizably a!ected by the choice of F and G. In the case where system (1) is controllable, that is if the rank of [BAB2AL\B] is n, the possible polynomials a (s) have been completely described by RosenG brock. This famous result reads as follows. Theorem 1 (Rosenbrock, 1970, Corollary 1 of Theorem 4.2). Given a system (1) and a set of n monic polynomials, say a (s)5a (s)525a (s), where a (s)5a (s) L G G> means that a (s) divides, without remainder, a (s). G> G Assume that (1) is controllable, and let c 5c 525c L denote the list of its controllability indices arranged in non-increasing order, and completed by zeros to the number n (hereafter, these indices will simply be called controllability indices of (A,B)). Then there exists a matrix F31K"L such that the invariant factors of A#BF are a (s), a (s),2, a (s) if and only if the following conditions L
0005-1098/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 9 9 ) 0 0 0 6 5 - 5
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Proposition 1. There exists an integer 0(q4n and a pair A 31O"O, B 31O"N such that
hold: L L dega (s)4 c for i"1,2,2, n, H H HG HG with equality holding for i"1.
(4)
It is to be noted that Rosenbrock was also interested in assigning the transmission pole structure; see Rosenbrock (1970, Theorem 4.2). Theorem 1 has been widely commented in the control literature. Alternative proofs have been proposed by Dickinson (1974), who used a state-space approach, Kuc\ era (1981), who applied the theory of polynomial equations, Flamm (1980), and more recently OG zc7 aldiran (1990), who studied the problem in the geometric framework. Generalizations have been given to the realm of implicit systems, see Kuc\ era and Zagalak (1988), Zagalak and Loiseau (1992), and even to the case where (A, B) is not a controllable pair, see Zaballa (1984), Loiseau and Zagalak (1993). However, no such strong result has been given concerning the assignment of the transmission invariant factors of the closed-loop system. Our present aim is to "ll this gap.
2. Non-regular static state feedback The di$culty of our problem clearly comes from the use of a possibly non-regular feedback; the matrix G in (2) can be singular and even non-square, if p(m. Indeed, if the pair (A#BF, BG) is controllable, that is in particular if (A, B) is controllable and G is square and nonsingular, then the transmission pole structure of (3) coincides with the eigenvalue structure of (3) (Rosenbrock, 1970) and we have a (s)"t (s) for i"1,2,2, m G G a (s)"1 for i"m#1,2, n. G So in that case Theorem 1 answers our question. The non-regular static state feedback arises whenever there are more control inputs than external inputs. Such a feedback is useful in solving the transmission type problems like decoupling, see for instance Zagalak, Lafay, and Herrera-Hernandez (1993), disturbance rejection, or model matching, see Kuc\ era (1991). Then the controllability of system (1) is no longer invariant with respect to a non-regular state feedback (2) so that the eigenvalue structure and the transmission pole structure of the closed-loop system (3) can be di!erent from each other. Under such circumstances Theorem 1 cannot be directly applied. However, a partial picture can be obtained using the well-known results concerning the e!ect of the state feedback (2) on the controllability structure of system (1).
(sI !A!BF)\BG"(sI !A )\B , L O where the matrices sI !A and B are left coprime. O There are several ways, see Kailath (1980), to construct this pair (A , B ), which is in fact the controllable part of (A#BF, BG). Noting that the transmission pole structure of the closed-loop system is given by the invariant factors of A , the application of Theorem 1 to the pair (A , B ) immediately leads to the following corollary of Proposition 1. Corollary 1. Let c 5c 525c , be the controllabil L ity indices of (A ,B ), then the following holds: L L deg t (s)4 c for i"1, 2,2, n, H H HG HG L L deg t (s)" c "q, H H H H where, by convention, deg t (s)"0 for i'p. G
(5) (6)
This result will be useful together with that of Heymann (1976) who described the possible values of the integers c , the controllability indices of (A , B ) as well as G of (A#BF, BG). The following version of the aforementioned result can be found in Loiseau and Zagalak (1994). Theorem 2. Given a system (1) with controllability indices c 5c 525c and a list of n non}negative and L non}increasingly ordered integers c 5c 525 c 50. Then there exist matrices F31K"L and G31K"N L such that the controllability indices of system (3) are c , c ,2, c if and only if L c 4 c H H H AYHXG H AHXG
for i"1, 2,2, n.
(7)
We are now able to state the main result of this section. Proposition 2. Given a system (1) and a set of monic polynomials t (s)5t (s)525t (s). There exists N a static state feedback (2) such that this set gives the transmission pole structure of the closed-loop system (3) if and only if there exists a list of integers c 5c 525c 50 satisfying Eqs. (5)}(7). L The problem now comes down to the search for such a list of integers c , c ,2, c . Let us note that, as n is L "nite, there is a "nite number of lists +c , c ,2, c , of L non-negative integers arranged in non-increasing order and such that L c "q4n. H H
J.J. Loiseau et al. / Automatica 35 (1999) 1549}1555
Thus, Proposition 2 is implicit but one can ultimately compute, via a "nite number of operations, whether or not there exists a list of integers satisfying the requirements (5)}(7). In the following, we shall propose a nicer way to solve the problem. It will require some developments concerning the structural properties of linear systems.
3. Lattice properties of the 5nite lists of integers Given an integer n'0, we de"ne the set L of all the L lists l"+l , l ,2, l , of integers that are arranged in L non-increasing order, l 5l 525l 50, and having L their sums bounded by n, i.e. l #l #2#l 4n. The L relation, say O, which appears in Theorem 1, is called dominance (or majorization) and is de"ned by L L ∀a, b3L : aOb8 a 4 b for i"1, 2,2, n. L H H HG HG This relation is an order on L since it is obviously L re#exive, transitive, and antisymmetric. With every list l3L , one can associate a conjugated L list, denoted by lH, which lies in L and is de"ned by L lH" card+ j " 5i, for i"1, 2,2, n. G H This conjugation is a biunivoque relation. Indeed, it is observed that (lH)H"l, see for instance Brylawski (1973). The conjugated relation denoted as and de"ned by ∀a, b3L : a L
G G b8 a 4 b H H H H for i"1, 2,2, n
is also an order relation. It will be called conjugated dominance and one can observe that b dominates a if and only if bH conjugate-dominates aH, see Loiseau (1988). With each list l3L , we can associate the function L m : +1,2, n,P+0,2, n,, de"ned by J L m (i)" l for i"1, 2,2, n J H HG as well as the function mH : +1,2, n,P+0,2, n,, de"ned J by G mH(i)" l for i"1, 2,2, n. J H H Observe that m is convex and non-increasing. There J exists a one-to-one correspondence between the lists l3L and the convex and non-increasing functions m . L J Indeed, one can recover the list l from the given m J using the formula l "m (i)!m (i#1), for i"1, 2,2, n, G J J where m (n#1)"0. Similarly, mH is concave and nonJ J decreasing, and l "mH(i)!mH(i!1), i"1, 2,2, n, with G J J mH(0)"0. J
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Next, for every x,y3L consider the list z given by L m (i)"max+m (i), m (i), for i"1, 2,2, n (8) X V W and the list w given by mHH(i)"min+mHH(i), mHH(i), for i"1, 2,2, n. (9) U V W The functions m and m are convex and non-increasing, V W which implies that m is convex and non-increasing, too. X Therefore, z is well de"ned. It can be readily seen that z is the minimal element in L dominating both x and y, i.e. L z is the supremum of x and y with respect to the order O. So Eq. (8) de"nes on L the lattice operation &join'; we L write z"sup(x, y). Similarly, since mHH and mHH are conW V cave and non-decreasing functions, mHH is also concave U and non-decreasing, and w is the in"mum of x and y with respect to the order O. The lattice operation of &meet' is de"ned by (9) (m is indeed the convex hull, in the sense of U integer functions, of m and m ); we write w"inf(x, y). V W Summing up all the above observations, we can state the following: Theorem 3. (L ,O, sup, inf ) is a lattice. L If only one of the operations sup and inf is considered on L , then (L ,O, sup) and (L ,O, inf ) are semilattices. L L L In view of Theorem 2, we are now interested in the relation denoted by ¢ and de"ned by ∀a, b3L : a¢b0 a 4 b for i"1, 2,2, n. L H H H ?HXG H @HXG We shall refer to ¢ as the Heymann ordering. Lemma 1. ∀a, b3L , a ¢ bNaOb. L Proof. Let a¢b. Then for i"1, 2,2, n we have L L a4 a4 b" b H H H H HG H ?HX?G H @HX?G HIG where k(i)"1# card+j " b 'a ,. Thus, we can de"ne H G L L k "max k3-, 14k4n a 4 b G H H HG HI for i"1, 2,2, n. (10)
Notice that a 5b 5b G. In view of that we have G IG I L L L L a " a !a 4 b !b G" b , H H G H I H HG> HG HIG HIG> which shows that either k 5k #1 or k "n. It is then G> G G clear that k 5i which establishes the result. 䊐 G In other words, Lemma 1 shows that the order ¢ is a suborder of the order O. An interesting consequence of the mutual relationship between the orders ¢ and O is
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that the set H "+l3L " l¢c,, A L where c3L , is closed under the operation inf, as shown L in Theorem 4. (H ,O,inf) is a subsemilattice of (L ,O,inf ). A L Proof. By Lemma 1, the order O is also an order on H and evidently, H LL . Further, it can be seen that A A L w¢c, where w"inf(x,y) for x,y3H . This comes from A the following inequality, shown in Lemma 2 of Loiseau, Zagalak and Kuc\ era (1995),
w 4max x , y for i"1, 2,2, n. H H H H UHXG H VHXG H WHXG Hence H is a subset of L and is closed under the A L operation inf. Hence, (H ,O,inf ) is a subsemilattice of the A semilattice (L ,O, inf). 䊐 L Example 1. Let n"6, then the lattice L is visualized in Fig. 1. The edges represent the order relation O in such
a way that the larger element lies above the smaller one. Here, for the sake of brevity, the trailing zeros of the six-tuples are suppressed. Let c"+2,2,2,0,0,0,. The semilattice H of the eleA ments of L that are related to c in the Heymann ordering ¢ is shown in Fig. 1 by framing the respective elements of L . By Theorem 2, the elements of H A represent all the sets of the controllability indices the system (1) can attain under the action of the state feedback (2).
4. Main results We shall now come back to the problem of the transmission pole structure assignment. Using the notation introduced in Section 3, it should be clear that the list c in Proposition 2 must satisfy dOc where d is de"ned by d "deg t (s) for i"1, 2,2, p G G d "0 for i"p#1,2, n G and must further satisfy L L c " d H H H H and c ¢c. Lemma 1 gives us a necessary condition for the existence of c , namely dOc. Consider now the set H "+l3L " dOl and l¢c,. A B L This set is non-empty since it contains the list c, for instance. In addition, it is closed under the operation inf since so is H and also the set +l3L "dOl,. Thus, it A L posseses a minimal element h"inf+l3L " dOl and l¢c,. (11) L Using the above notation, Proposition 2 can now be restated in the form of Proposition 3. Under the assumptions of Proposition 2, there exists a static state feedback (2) such that the set t (s)5t (s)525t (s) is the transmission pole struc N ture of the system (3) if and only if L L h" d. H H H H
Fig. 1. The lattice L and its subsemilattice H in Example 1. A
The question now subsisting is that of the e!ective computation of h. For this purpose we shall use the method that consists of characterizing the minimal element of
J.J. Loiseau et al. / Automatica 35 (1999) 1549}1555
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a semilattice as the "xed point of a suitable attractor, as in Wonham (1979) or Wonham and Ramadge (1987), for example. Given the list c"+c , c ,2, c , of the controllability L indices of system (1) and the list d"+d , d ,2, d , of the L degrees of the invariant factors the closed-loop system transfer function is to have, de"ne the operator H : L PL as L L l"+l , l ,2, l ,CHl"sup+l, l, l,2, lL, L where sup is de"ned by (8) and, for i"1,2, n,
max+l ,c G, for j"1, 2,2, i G I lG " H l for j"i#1,2, n H with k being de"ned in the proof of Lemma 1 by Eq. (10). G Without any limitation, we can assume at this level that L c "n, which ensures that k and thus H is wellH H G de"ned for every l3L . We further de"ne the sequence L +hG, de"ned on L as L h"0 (12) hG"sup+d,HhG\, for i51. Lemma 2. (i) H is isotone with respect to the dominance O, i.e. ∀x, y3L ,xOyNHxOHy. L (ii) The lists that are dominated by c in the Heymann ordering are the xxed points of H, i.e. ∀l3L , L l¢ c0l"Hl. (iii) The sequence +hG, is non-decreasing and converges after a xnite, less than n, number of steps to a limit denoted h, that coincides with the minimal element h dexned by Eq. (11), i.e. h"h"inf+l3L " dOl and l¢c,. L Proof. (i) Assume that xOy. It is then clear that k G5k G, V W and that c VG4c WG, where k G and k G are de"ned as in (10), I I V W for i"1, 2,2, n. It implies that either xG"+x ,2, x , x ,2, x ,OxOy, G G G> L if x 5c VG, or G I xG"+c VG,2, c VG, x ,2, x ,OyG, I I G> L if x 4c VG. In either case we have established the claim G I sup+x, x,2, xL,Osup+y, y,2, yL,. (ii) As it was pointed out in the proof of Lemma 1, l ¢ cNl 5c G and k 5i, i"1, 2,2, n, with k being deG G G I "ned as in Eq. (10). It appears that lG"+l ,2, l , l ,2, l ,Ol for i"1, 2,2, n G G G> L and thus Hl"l. Conversely, if l"Hl then lGOl for i"1, 2,2, n. It follows that L L max(l , c G)# l 4l # l , G I H G H HG> HG>
Fig. 2. The subset H
A B
in Example 2.
which shows that l 5c G for i"1, 2,2, n. By the G I de"nition of k , and choosing i"min+j"l 4k, for G H k"1, 2,2, n, we "nally obtain the result: L l " l 4 c 4 c for k"1, 2,2, n. H H H H H JHXI HG H AHXJG H AHXI (iii) Clearly h"0Od"h, and if hG\OhG then, (HhG\)O(HhG) by assertion (i). It follows that hG"sup+d, HhG\,Osup+d, HhG,"hG>. This recurrence establishes that +hG, is non-decreasing. Since +hG, is non-decreasing and de"ned over a "nite set, it is convergent. The limit h satis"es h"sup+d, Hh,"Hh. Thus, h is a "xed point of H and hence it is dominated by c with respect to ¢. It can be "nally veri"ed that, for f3L , if d O f and f¢c, then h"dOf, and, since hG O f L implies HhG O f, one has hG> O f, i50. Thus, h O f and h coincides with the in"mal element h de"ned in Eq. (11). 䊐 Now combining Propositions 2 and 3, and Lemma 2, we can state the "nal result. Theorem 5. Given a system (1) whose list of controllability indices is c, and a set of n monic polynomials t (s)5t (s)525t (s) whose degrees constitute the L list d, then there exists a static state feedback (2) such that
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this set gives the transmission pole structure of the closedloop system (3) if and only if dOc and L L h" d , H H H H where h is the limit of the sequence (12). Example 2. Consider a system (1) of order n"6, the controllability indices of which are given by the list c"+2, 2, 2, 0, 0, 0,. We want to modify system (1) by feedback so as to obtain a system (3) whose transmission pole structure is given by the invariant factors having the degrees given by d"+4, 1, 0, 0, 0, 0,. Thus, the controllable part of system (3) will be of order q"5. To that end we shall construct the subset H of H , A B A which is visualized in Fig. 2 by double framing. The required pole structure can be assigned if and only if H contains an element c satisfying (6). The elements A B that satisfy (6) are connected in Fig. 2 by a bold equiorder line corresponding to q"5. There is one such element, c "+3, 2, 0, 0, 0, 0,, which is the minimal element of H . A B 5. Conclusions We have studied the problem of the transmission pole structure assignment by the non-regular static state feedback (2) in the linear, time-invariant systems (1). We have established a necessary and su$cient condition for a given list of polynomials t (s)5t (s)525t (s) to N give the transmission pole structure of the closed-loop system (3), see Propositions 2 and 3, and we have also found an e$cient way to test the condition to be satis"ed (Theorem 5). To that end, we have studied some features of the set L and its special subsets, with respect to L dominance and the Heymann order (Theorems 3 and 4). We have also introduced the operator H and investigated some of its properties (Lemma 2). The achieved results could help in solving the control problems where the non-regular static state feedback is employed. For example, we believe that the results will be of some importance when trying to solve the so-called Morgan problem; see Zagalak et al. (1993).
Acknowledgements This work was supported in part by the cooperation program No. 1622 between the Academy of Sciences of the Czech Republic and the Centre National de la Recherche Scienti"que, by the Ministry of Education of the Czech Republic under the project VS 97/034, and by the Academy of Sciences of the Czech Republic under the grant 102/97/0861.
References Brylawski, T. (1973). The lattice of integer partitions. Discrete Mathematics, 6, 201}219. Dickinson, B. W. (1974). On the fundamental theorem of linear variable state feedback. IEEE Transactions on Automatic Control, AC-19, 577}579. Flamm, D. S. (1980). A new proof of Rosenbrock's theorem on pole assignment. IEEE Transactions on Automatic Control, AC-25, 1128}1133. Heymann, M. (1976). Controllability indices and feedback simulation. SIAM Journal on Control and Optimization, 14, 769}789. Kailath, T. (1980). Linear systems. Englewood Cli!s, NJ: Prentice-Hall. Kuc\ era, V. (1981). Assigning the invariant factors by feedback. Kybernetika, 17, 118}127. Kuc\ era, V. (1991). Analysis and design of discrete linear control systems. London: Prentice-Hall, and Prague: Academia. Kuc\ era, V., & Zagalak, P. (1988). Fundamental theorem of state feedback for singular systems. Automatica, 24, 653}658. Loiseau, J. J. (1988). Sur la modi"cation de la structure a` l'in"ni par retour d'eH tat statique. SIAM Journal on Control and Optimization, 26, 251}273. Loiseau, J. J., & Zagalak, P. (1993). Eigenstructure assignment in linear and uncontrollable systems. In Proceedings of the International symposium on MTNS (pp. 327}330). Regensburg, Germany. Loiseau, J. J., & Zagalak, P. (1994). On feedback simulation. In Proceedings of the second IEEE Mediterranean symposium (pp. 275}282). Chania, Crete. Loiseau, J. J., Zagalak, P., & Kuc\ era, V. (1995). Pole placement via non-regular static state feedback. In Proceedings of the IFAC conference system structure and control (pp. 275}282). Nantes, France. OG zc7 aldiran, K. (1990). Fundamental theorem of linear state feedback for singular systems. In Proceedings of the 29th IEEE conference on decision and control (pp. 67}72). Honolulu, Hawaii. Rosenbrock, H. H. (1970). State-space and multivariable theory. Wiley, New York. Wonham, W. M. (1979). Linear multivariable control: a geometric approach. New York, Springer. Wonham, W. M., & Ramadge, P. J. (1987). On the supremal controllable sublanguage of a given language. SIAM Journal on Control and Optimization, 25, 637}659. Zaballa, I. (1984). Interlacing inequalities and control theory. Linear Algebra and its Applications, 87, 113}146. Zagalak, P., & Loiseau, J. J. (1992). Invariant factors assignment in linear systems. In Proceedings of the international symposium on implicit and nonlinear systems (pp. 197}204). Ft. Worth, Texas. Zagalak, P., Lafay, J. F., & Herrera-Hernandez, A. N. (1993). The row-by-row decoupling via state feedback: a polynomial approach. Automatica, 29, 1491}1499.
Jean Jacques Loiseau was born in Lorient, France, 1958. He obtained the Ing degree in mechanics and the doctoral degree in Automatic Control, in 1983 and 1986, respectively. He has worked from 1985 to 1990 in ship building, being mainly involved in the control of wind turbines. He reached his present position, in 1991, ChargeH de Recherche of the Centre National de la Recherche Scienti"que, at the Institut de Recherche en CyberneH tique de Nantes of the ED cole Centrale de Nantes. His research interests include the control and analysis of linear systems, time-delay systems, discreteevent systems. In 1996, he was awarded the bronze medal of the CNRS.
J.J. Loiseau et al. / Automatica 35 (1999) 1549}1555 Petr Zagalak was born in Czechoslovakia. He received his master (Ing) degree from the Faculty of Nuclear and Physical Engineering of the Czech Technical University in 1972. Since 1972 he has been with the Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic. During 1972}1981 he worked at the Computer Center of the Institute. In 1982 he received his doctoral (CSc) degree from the Academy. Since that year he has held an appointment as a member of research sta! of the Institute. He has received the Automatica Paper Size in 1990. His current research interests include mainly algebraic methods of control theory, strucuture of linear systems, and numerical aspects of polynomial calculus. Vladim1H r Kuc\ era was born in Prague, Czechoslovakia, in 1943. He studied at the Czech Technical University, Prague, where he obtained an Ing degree in Electrical Engineering in 1966. He received the CSc and DrSc degrees in Automatic Control from the Czechoslovak Academy of Sciences in 1970 and 1979, respectively. The research interest of V. Kuc\ era include linear systems theory and robust control. He has contributed to the theory of the
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Riccati equation, pioneered the use of polynomial equations in control-system design and paved the way to the Youla}Kuc\ era parametrization of all stabilizing controllers for a given plant. He teaches undergraduate and graduate courses on systems and control at the Faculty of Electrical Engineering, Czech Technical University, Prague. Since 1967 V. Kuc\ era has been a member of the Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, Prague. He held visiting positions at National Research Council, Ottawa, Canada (1970}1971), the University of Florida, Gainesville, FL, U.S.A. (1977), Ecole Nationale SupeH rierure de MeH canique, Nantes, France (1981}1982), Australian National University, Canberra, Australia (1984), Teknikum, Uppsala Universitet, Sweden (1989), Instituto PoliteH cnico Nacional, Mexico City (1991), ETH Zurich (1992), PoliteH cnico di Milano (1995), and was Nippon Steel Professor at the Chair of Intelligent Control, Tokyo Institute of Technology, Japan, in 1994. V. Kuc\ era is the author of the three books: Algebraic Theory of Discrete Linear Control (Academia, Prague, 1978), Discrete Linear Control: The Polynomial Equation Approach (Wiley, Chichester, 1979), and Analysis and Design of Discrete Linear Control Systems (Prentince-Hall, London, 1991). He has published 190 research papers. Since 1990, V. Kuc\ era has been the Director of the Insitute of Information Theory and Automation, and since 1995 he has been a Professor of Control Engineering at the Czech Technical Univeristy, Prague. He is the Editor-in-Chief of Kybernetika and serves on the editorial boards of several leading journals. He is a Vice-President of IFAC and Chairman of its Technical Board, a Fellow of the IEEE and a member of the IEEE Control Systems Society Board of Governors, and a Member of the New York Academy of Sciences and the Academy of Engineering of the Czech Republic.
Brief Paper
Pole structure assignment via non-regular static state feedback夽 J. J. Loiseau *, P. Zagalak, V. Kuc\ era Institut de Recherche en Cyberne& tique de Nantes, UMR 6597 CNRS, Ecole Centrale de Nantes, BP 92101, F-44321, Nantes Cedex 03, France Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, P.O. Box 18, 18208 Praha, Czech Republic Trnka Laboratory for Automatic Control, Faculty of Electrical Engineering, Czech Technical University, Karlov na& m. 13, 12135 Prague, Czech Republic Received 14 April 1997; revised 21 October 1997; received in "nal form 11 March 1999
Abstract The limits of the static state feedback u"Fx#Gv, with G not necessarily nonsingular, in altering the transmission pole structure of the linear system x "Ax#Bu are studied. A necessary and su$cient condition for a list of polynomials to be the denominator polynomials of the Smith}McMillan form of the closed-loop transfer matrix (sI!A!BF)\BG is established. The condition involves the calculation of the in"mal set of the controllability indices that the system can attain through feedback while satisfying the Rosenbrock inequalities. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Linear systems; Static state feedback; Transmission poles; Controllability indices; Lattice of integer partitions; Fixed point algorithm
1. Introduction We consider the linear, "nite-dimensional, timeinvariant system x "Ax#Bu,
(1)
where A31L"L and B31L"K are real matrices. Applying, to system (1), the control law u"Fx#Gv,
(2)
where F31K"L and G31K"N, we get the closed-loop system x "(A#BF)x#BGv.
(3)
As far as control is concerned, we are interested in studying the properties of the closed-loop system (3), and how the choice of F and G can a!ect these properties. In particular one can be interested in studying: E The eigenvalue structure of the closed-loop system (3), that is the set of the invariant factors, say a (s) for G 夽 The original version of the paper was presented at the IFAC Conference on System Structure and Control, Nantes, France, July 5}7, 1995. This paper was recommended for publication in revised form by Associate Editor A.A. Stoorvogel under the direction of Editor R. Tempo. * Corresponding author. Tel.: #33-2-4037-6967; fax #33-2-40376930. E-mail address: [email protected] (J.J. Loiseau)
i"1,2, n, of the matrix A#BF. The eigenvalue structure plays a fundamental role when studying the internal stability of Eq. (3). E The (transmission) pole structure of the closed-loop system (3), that is the set of the invariant factors, say t (s) for i"1,2,p, which appear as the denominG ators of the diagonal elements of the Smith}McMillan form of the closed-loop transfer matrix (sI !A!BF)\BG. The pole structure represents L the transmission dynamics of the closed-loop system (3). These two structures are sizably a!ected by the choice of F and G. In the case where system (1) is controllable, that is if the rank of [BAB2AL\B] is n, the possible polynomials a (s) have been completely described by RosenG brock. This famous result reads as follows. Theorem 1 (Rosenbrock, 1970, Corollary 1 of Theorem 4.2). Given a system (1) and a set of n monic polynomials, say a (s)5a (s)525a (s), where a (s)5a (s) L G G> means that a (s) divides, without remainder, a (s). G> G Assume that (1) is controllable, and let c 5c 525c L denote the list of its controllability indices arranged in non-increasing order, and completed by zeros to the number n (hereafter, these indices will simply be called controllability indices of (A,B)). Then there exists a matrix F31K"L such that the invariant factors of A#BF are a (s), a (s),2, a (s) if and only if the following conditions L
0005-1098/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 9 9 ) 0 0 0 6 5 - 5
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Proposition 1. There exists an integer 0(q4n and a pair A 31O"O, B 31O"N such that
hold: L L dega (s)4 c for i"1,2,2, n, H H HG HG with equality holding for i"1.
(4)
It is to be noted that Rosenbrock was also interested in assigning the transmission pole structure; see Rosenbrock (1970, Theorem 4.2). Theorem 1 has been widely commented in the control literature. Alternative proofs have been proposed by Dickinson (1974), who used a state-space approach, Kuc\ era (1981), who applied the theory of polynomial equations, Flamm (1980), and more recently OG zc7 aldiran (1990), who studied the problem in the geometric framework. Generalizations have been given to the realm of implicit systems, see Kuc\ era and Zagalak (1988), Zagalak and Loiseau (1992), and even to the case where (A, B) is not a controllable pair, see Zaballa (1984), Loiseau and Zagalak (1993). However, no such strong result has been given concerning the assignment of the transmission invariant factors of the closed-loop system. Our present aim is to "ll this gap.
2. Non-regular static state feedback The di$culty of our problem clearly comes from the use of a possibly non-regular feedback; the matrix G in (2) can be singular and even non-square, if p(m. Indeed, if the pair (A#BF, BG) is controllable, that is in particular if (A, B) is controllable and G is square and nonsingular, then the transmission pole structure of (3) coincides with the eigenvalue structure of (3) (Rosenbrock, 1970) and we have a (s)"t (s) for i"1,2,2, m G G a (s)"1 for i"m#1,2, n. G So in that case Theorem 1 answers our question. The non-regular static state feedback arises whenever there are more control inputs than external inputs. Such a feedback is useful in solving the transmission type problems like decoupling, see for instance Zagalak, Lafay, and Herrera-Hernandez (1993), disturbance rejection, or model matching, see Kuc\ era (1991). Then the controllability of system (1) is no longer invariant with respect to a non-regular state feedback (2) so that the eigenvalue structure and the transmission pole structure of the closed-loop system (3) can be di!erent from each other. Under such circumstances Theorem 1 cannot be directly applied. However, a partial picture can be obtained using the well-known results concerning the e!ect of the state feedback (2) on the controllability structure of system (1).
(sI !A!BF)\BG"(sI !A )\B , L O where the matrices sI !A and B are left coprime. O There are several ways, see Kailath (1980), to construct this pair (A , B ), which is in fact the controllable part of (A#BF, BG). Noting that the transmission pole structure of the closed-loop system is given by the invariant factors of A , the application of Theorem 1 to the pair (A , B ) immediately leads to the following corollary of Proposition 1. Corollary 1. Let c 5c 525c , be the controllabil L ity indices of (A ,B ), then the following holds: L L deg t (s)4 c for i"1, 2,2, n, H H HG HG L L deg t (s)" c "q, H H H H where, by convention, deg t (s)"0 for i'p. G
(5) (6)
This result will be useful together with that of Heymann (1976) who described the possible values of the integers c , the controllability indices of (A , B ) as well as G of (A#BF, BG). The following version of the aforementioned result can be found in Loiseau and Zagalak (1994). Theorem 2. Given a system (1) with controllability indices c 5c 525c and a list of n non}negative and L non}increasingly ordered integers c 5c 525 c 50. Then there exist matrices F31K"L and G31K"N L such that the controllability indices of system (3) are c , c ,2, c if and only if L c 4 c H H H AYHXG H AHXG
for i"1, 2,2, n.
(7)
We are now able to state the main result of this section. Proposition 2. Given a system (1) and a set of monic polynomials t (s)5t (s)525t (s). There exists N a static state feedback (2) such that this set gives the transmission pole structure of the closed-loop system (3) if and only if there exists a list of integers c 5c 525c 50 satisfying Eqs. (5)}(7). L The problem now comes down to the search for such a list of integers c , c ,2, c . Let us note that, as n is L "nite, there is a "nite number of lists +c , c ,2, c , of L non-negative integers arranged in non-increasing order and such that L c "q4n. H H
J.J. Loiseau et al. / Automatica 35 (1999) 1549}1555
Thus, Proposition 2 is implicit but one can ultimately compute, via a "nite number of operations, whether or not there exists a list of integers satisfying the requirements (5)}(7). In the following, we shall propose a nicer way to solve the problem. It will require some developments concerning the structural properties of linear systems.
3. Lattice properties of the 5nite lists of integers Given an integer n'0, we de"ne the set L of all the L lists l"+l , l ,2, l , of integers that are arranged in L non-increasing order, l 5l 525l 50, and having L their sums bounded by n, i.e. l #l #2#l 4n. The L relation, say O, which appears in Theorem 1, is called dominance (or majorization) and is de"ned by L L ∀a, b3L : aOb8 a 4 b for i"1, 2,2, n. L H H HG HG This relation is an order on L since it is obviously L re#exive, transitive, and antisymmetric. With every list l3L , one can associate a conjugated L list, denoted by lH, which lies in L and is de"ned by L lH" card+ j " 5i, for i"1, 2,2, n. G H This conjugation is a biunivoque relation. Indeed, it is observed that (lH)H"l, see for instance Brylawski (1973). The conjugated relation denoted as and de"ned by ∀a, b3L : a L
G G b8 a 4 b H H H H for i"1, 2,2, n
is also an order relation. It will be called conjugated dominance and one can observe that b dominates a if and only if bH conjugate-dominates aH, see Loiseau (1988). With each list l3L , we can associate the function L m : +1,2, n,P+0,2, n,, de"ned by J L m (i)" l for i"1, 2,2, n J H HG as well as the function mH : +1,2, n,P+0,2, n,, de"ned J by G mH(i)" l for i"1, 2,2, n. J H H Observe that m is convex and non-increasing. There J exists a one-to-one correspondence between the lists l3L and the convex and non-increasing functions m . L J Indeed, one can recover the list l from the given m J using the formula l "m (i)!m (i#1), for i"1, 2,2, n, G J J where m (n#1)"0. Similarly, mH is concave and nonJ J decreasing, and l "mH(i)!mH(i!1), i"1, 2,2, n, with G J J mH(0)"0. J
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Next, for every x,y3L consider the list z given by L m (i)"max+m (i), m (i), for i"1, 2,2, n (8) X V W and the list w given by mHH(i)"min+mHH(i), mHH(i), for i"1, 2,2, n. (9) U V W The functions m and m are convex and non-increasing, V W which implies that m is convex and non-increasing, too. X Therefore, z is well de"ned. It can be readily seen that z is the minimal element in L dominating both x and y, i.e. L z is the supremum of x and y with respect to the order O. So Eq. (8) de"nes on L the lattice operation &join'; we L write z"sup(x, y). Similarly, since mHH and mHH are conW V cave and non-decreasing functions, mHH is also concave U and non-decreasing, and w is the in"mum of x and y with respect to the order O. The lattice operation of &meet' is de"ned by (9) (m is indeed the convex hull, in the sense of U integer functions, of m and m ); we write w"inf(x, y). V W Summing up all the above observations, we can state the following: Theorem 3. (L ,O, sup, inf ) is a lattice. L If only one of the operations sup and inf is considered on L , then (L ,O, sup) and (L ,O, inf ) are semilattices. L L L In view of Theorem 2, we are now interested in the relation denoted by ¢ and de"ned by ∀a, b3L : a¢b0 a 4 b for i"1, 2,2, n. L H H H ?HXG H @HXG We shall refer to ¢ as the Heymann ordering. Lemma 1. ∀a, b3L , a ¢ bNaOb. L Proof. Let a¢b. Then for i"1, 2,2, n we have L L a4 a4 b" b H H H H HG H ?HX?G H @HX?G HIG where k(i)"1# card+j " b 'a ,. Thus, we can de"ne H G L L k "max k3-, 14k4n a 4 b G H H HG HI for i"1, 2,2, n. (10)
Notice that a 5b 5b G. In view of that we have G IG I L L L L a " a !a 4 b !b G" b , H H G H I H HG> HG HIG HIG> which shows that either k 5k #1 or k "n. It is then G> G G clear that k 5i which establishes the result. 䊐 G In other words, Lemma 1 shows that the order ¢ is a suborder of the order O. An interesting consequence of the mutual relationship between the orders ¢ and O is
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that the set H "+l3L " l¢c,, A L where c3L , is closed under the operation inf, as shown L in Theorem 4. (H ,O,inf) is a subsemilattice of (L ,O,inf ). A L Proof. By Lemma 1, the order O is also an order on H and evidently, H LL . Further, it can be seen that A A L w¢c, where w"inf(x,y) for x,y3H . This comes from A the following inequality, shown in Lemma 2 of Loiseau, Zagalak and Kuc\ era (1995),
w 4max x , y for i"1, 2,2, n. H H H H UHXG H VHXG H WHXG Hence H is a subset of L and is closed under the A L operation inf. Hence, (H ,O,inf ) is a subsemilattice of the A semilattice (L ,O, inf). 䊐 L Example 1. Let n"6, then the lattice L is visualized in Fig. 1. The edges represent the order relation O in such
a way that the larger element lies above the smaller one. Here, for the sake of brevity, the trailing zeros of the six-tuples are suppressed. Let c"+2,2,2,0,0,0,. The semilattice H of the eleA ments of L that are related to c in the Heymann ordering ¢ is shown in Fig. 1 by framing the respective elements of L . By Theorem 2, the elements of H A represent all the sets of the controllability indices the system (1) can attain under the action of the state feedback (2).
4. Main results We shall now come back to the problem of the transmission pole structure assignment. Using the notation introduced in Section 3, it should be clear that the list c in Proposition 2 must satisfy dOc where d is de"ned by d "deg t (s) for i"1, 2,2, p G G d "0 for i"p#1,2, n G and must further satisfy L L c " d H H H H and c ¢c. Lemma 1 gives us a necessary condition for the existence of c , namely dOc. Consider now the set H "+l3L " dOl and l¢c,. A B L This set is non-empty since it contains the list c, for instance. In addition, it is closed under the operation inf since so is H and also the set +l3L "dOl,. Thus, it A L posseses a minimal element h"inf+l3L " dOl and l¢c,. (11) L Using the above notation, Proposition 2 can now be restated in the form of Proposition 3. Under the assumptions of Proposition 2, there exists a static state feedback (2) such that the set t (s)5t (s)525t (s) is the transmission pole struc N ture of the system (3) if and only if L L h" d. H H H H
Fig. 1. The lattice L and its subsemilattice H in Example 1. A
The question now subsisting is that of the e!ective computation of h. For this purpose we shall use the method that consists of characterizing the minimal element of
J.J. Loiseau et al. / Automatica 35 (1999) 1549}1555
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a semilattice as the "xed point of a suitable attractor, as in Wonham (1979) or Wonham and Ramadge (1987), for example. Given the list c"+c , c ,2, c , of the controllability L indices of system (1) and the list d"+d , d ,2, d , of the L degrees of the invariant factors the closed-loop system transfer function is to have, de"ne the operator H : L PL as L L l"+l , l ,2, l ,CHl"sup+l, l, l,2, lL, L where sup is de"ned by (8) and, for i"1,2, n,
max+l ,c G, for j"1, 2,2, i G I lG " H l for j"i#1,2, n H with k being de"ned in the proof of Lemma 1 by Eq. (10). G Without any limitation, we can assume at this level that L c "n, which ensures that k and thus H is wellH H G de"ned for every l3L . We further de"ne the sequence L +hG, de"ned on L as L h"0 (12) hG"sup+d,HhG\, for i51. Lemma 2. (i) H is isotone with respect to the dominance O, i.e. ∀x, y3L ,xOyNHxOHy. L (ii) The lists that are dominated by c in the Heymann ordering are the xxed points of H, i.e. ∀l3L , L l¢ c0l"Hl. (iii) The sequence +hG, is non-decreasing and converges after a xnite, less than n, number of steps to a limit denoted h, that coincides with the minimal element h dexned by Eq. (11), i.e. h"h"inf+l3L " dOl and l¢c,. L Proof. (i) Assume that xOy. It is then clear that k G5k G, V W and that c VG4c WG, where k G and k G are de"ned as in (10), I I V W for i"1, 2,2, n. It implies that either xG"+x ,2, x , x ,2, x ,OxOy, G G G> L if x 5c VG, or G I xG"+c VG,2, c VG, x ,2, x ,OyG, I I G> L if x 4c VG. In either case we have established the claim G I sup+x, x,2, xL,Osup+y, y,2, yL,. (ii) As it was pointed out in the proof of Lemma 1, l ¢ cNl 5c G and k 5i, i"1, 2,2, n, with k being deG G G I "ned as in Eq. (10). It appears that lG"+l ,2, l , l ,2, l ,Ol for i"1, 2,2, n G G G> L and thus Hl"l. Conversely, if l"Hl then lGOl for i"1, 2,2, n. It follows that L L max(l , c G)# l 4l # l , G I H G H HG> HG>
Fig. 2. The subset H
A B
in Example 2.
which shows that l 5c G for i"1, 2,2, n. By the G I de"nition of k , and choosing i"min+j"l 4k, for G H k"1, 2,2, n, we "nally obtain the result: L l " l 4 c 4 c for k"1, 2,2, n. H H H H H JHXI HG H AHXJG H AHXI (iii) Clearly h"0Od"h, and if hG\OhG then, (HhG\)O(HhG) by assertion (i). It follows that hG"sup+d, HhG\,Osup+d, HhG,"hG>. This recurrence establishes that +hG, is non-decreasing. Since +hG, is non-decreasing and de"ned over a "nite set, it is convergent. The limit h satis"es h"sup+d, Hh,"Hh. Thus, h is a "xed point of H and hence it is dominated by c with respect to ¢. It can be "nally veri"ed that, for f3L , if d O f and f¢c, then h"dOf, and, since hG O f L implies HhG O f, one has hG> O f, i50. Thus, h O f and h coincides with the in"mal element h de"ned in Eq. (11). 䊐 Now combining Propositions 2 and 3, and Lemma 2, we can state the "nal result. Theorem 5. Given a system (1) whose list of controllability indices is c, and a set of n monic polynomials t (s)5t (s)525t (s) whose degrees constitute the L list d, then there exists a static state feedback (2) such that
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this set gives the transmission pole structure of the closedloop system (3) if and only if dOc and L L h" d , H H H H where h is the limit of the sequence (12). Example 2. Consider a system (1) of order n"6, the controllability indices of which are given by the list c"+2, 2, 2, 0, 0, 0,. We want to modify system (1) by feedback so as to obtain a system (3) whose transmission pole structure is given by the invariant factors having the degrees given by d"+4, 1, 0, 0, 0, 0,. Thus, the controllable part of system (3) will be of order q"5. To that end we shall construct the subset H of H , A B A which is visualized in Fig. 2 by double framing. The required pole structure can be assigned if and only if H contains an element c satisfying (6). The elements A B that satisfy (6) are connected in Fig. 2 by a bold equiorder line corresponding to q"5. There is one such element, c "+3, 2, 0, 0, 0, 0,, which is the minimal element of H . A B 5. Conclusions We have studied the problem of the transmission pole structure assignment by the non-regular static state feedback (2) in the linear, time-invariant systems (1). We have established a necessary and su$cient condition for a given list of polynomials t (s)5t (s)525t (s) to N give the transmission pole structure of the closed-loop system (3), see Propositions 2 and 3, and we have also found an e$cient way to test the condition to be satis"ed (Theorem 5). To that end, we have studied some features of the set L and its special subsets, with respect to L dominance and the Heymann order (Theorems 3 and 4). We have also introduced the operator H and investigated some of its properties (Lemma 2). The achieved results could help in solving the control problems where the non-regular static state feedback is employed. For example, we believe that the results will be of some importance when trying to solve the so-called Morgan problem; see Zagalak et al. (1993).
Acknowledgements This work was supported in part by the cooperation program No. 1622 between the Academy of Sciences of the Czech Republic and the Centre National de la Recherche Scienti"que, by the Ministry of Education of the Czech Republic under the project VS 97/034, and by the Academy of Sciences of the Czech Republic under the grant 102/97/0861.
References Brylawski, T. (1973). The lattice of integer partitions. Discrete Mathematics, 6, 201}219. Dickinson, B. W. (1974). On the fundamental theorem of linear variable state feedback. IEEE Transactions on Automatic Control, AC-19, 577}579. Flamm, D. S. (1980). A new proof of Rosenbrock's theorem on pole assignment. IEEE Transactions on Automatic Control, AC-25, 1128}1133. Heymann, M. (1976). Controllability indices and feedback simulation. SIAM Journal on Control and Optimization, 14, 769}789. Kailath, T. (1980). Linear systems. Englewood Cli!s, NJ: Prentice-Hall. Kuc\ era, V. (1981). Assigning the invariant factors by feedback. Kybernetika, 17, 118}127. Kuc\ era, V. (1991). Analysis and design of discrete linear control systems. London: Prentice-Hall, and Prague: Academia. Kuc\ era, V., & Zagalak, P. (1988). Fundamental theorem of state feedback for singular systems. Automatica, 24, 653}658. Loiseau, J. J. (1988). Sur la modi"cation de la structure a` l'in"ni par retour d'eH tat statique. SIAM Journal on Control and Optimization, 26, 251}273. Loiseau, J. J., & Zagalak, P. (1993). Eigenstructure assignment in linear and uncontrollable systems. In Proceedings of the International symposium on MTNS (pp. 327}330). Regensburg, Germany. Loiseau, J. J., & Zagalak, P. (1994). On feedback simulation. In Proceedings of the second IEEE Mediterranean symposium (pp. 275}282). Chania, Crete. Loiseau, J. J., Zagalak, P., & Kuc\ era, V. (1995). Pole placement via non-regular static state feedback. In Proceedings of the IFAC conference system structure and control (pp. 275}282). Nantes, France. OG zc7 aldiran, K. (1990). Fundamental theorem of linear state feedback for singular systems. In Proceedings of the 29th IEEE conference on decision and control (pp. 67}72). Honolulu, Hawaii. Rosenbrock, H. H. (1970). State-space and multivariable theory. Wiley, New York. Wonham, W. M. (1979). Linear multivariable control: a geometric approach. New York, Springer. Wonham, W. M., & Ramadge, P. J. (1987). On the supremal controllable sublanguage of a given language. SIAM Journal on Control and Optimization, 25, 637}659. Zaballa, I. (1984). Interlacing inequalities and control theory. Linear Algebra and its Applications, 87, 113}146. Zagalak, P., & Loiseau, J. J. (1992). Invariant factors assignment in linear systems. In Proceedings of the international symposium on implicit and nonlinear systems (pp. 197}204). Ft. Worth, Texas. Zagalak, P., Lafay, J. F., & Herrera-Hernandez, A. N. (1993). The row-by-row decoupling via state feedback: a polynomial approach. Automatica, 29, 1491}1499.
Jean Jacques Loiseau was born in Lorient, France, 1958. He obtained the Ing degree in mechanics and the doctoral degree in Automatic Control, in 1983 and 1986, respectively. He has worked from 1985 to 1990 in ship building, being mainly involved in the control of wind turbines. He reached his present position, in 1991, ChargeH de Recherche of the Centre National de la Recherche Scienti"que, at the Institut de Recherche en CyberneH tique de Nantes of the ED cole Centrale de Nantes. His research interests include the control and analysis of linear systems, time-delay systems, discreteevent systems. In 1996, he was awarded the bronze medal of the CNRS.
J.J. Loiseau et al. / Automatica 35 (1999) 1549}1555 Petr Zagalak was born in Czechoslovakia. He received his master (Ing) degree from the Faculty of Nuclear and Physical Engineering of the Czech Technical University in 1972. Since 1972 he has been with the Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic. During 1972}1981 he worked at the Computer Center of the Institute. In 1982 he received his doctoral (CSc) degree from the Academy. Since that year he has held an appointment as a member of research sta! of the Institute. He has received the Automatica Paper Size in 1990. His current research interests include mainly algebraic methods of control theory, strucuture of linear systems, and numerical aspects of polynomial calculus. Vladim1H r Kuc\ era was born in Prague, Czechoslovakia, in 1943. He studied at the Czech Technical University, Prague, where he obtained an Ing degree in Electrical Engineering in 1966. He received the CSc and DrSc degrees in Automatic Control from the Czechoslovak Academy of Sciences in 1970 and 1979, respectively. The research interest of V. Kuc\ era include linear systems theory and robust control. He has contributed to the theory of the
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Riccati equation, pioneered the use of polynomial equations in control-system design and paved the way to the Youla}Kuc\ era parametrization of all stabilizing controllers for a given plant. He teaches undergraduate and graduate courses on systems and control at the Faculty of Electrical Engineering, Czech Technical University, Prague. Since 1967 V. Kuc\ era has been a member of the Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, Prague. He held visiting positions at National Research Council, Ottawa, Canada (1970}1971), the University of Florida, Gainesville, FL, U.S.A. (1977), Ecole Nationale SupeH rierure de MeH canique, Nantes, France (1981}1982), Australian National University, Canberra, Australia (1984), Teknikum, Uppsala Universitet, Sweden (1989), Instituto PoliteH cnico Nacional, Mexico City (1991), ETH Zurich (1992), PoliteH cnico di Milano (1995), and was Nippon Steel Professor at the Chair of Intelligent Control, Tokyo Institute of Technology, Japan, in 1994. V. Kuc\ era is the author of the three books: Algebraic Theory of Discrete Linear Control (Academia, Prague, 1978), Discrete Linear Control: The Polynomial Equation Approach (Wiley, Chichester, 1979), and Analysis and Design of Discrete Linear Control Systems (Prentince-Hall, London, 1991). He has published 190 research papers. Since 1990, V. Kuc\ era has been the Director of the Insitute of Information Theory and Automation, and since 1995 he has been a Professor of Control Engineering at the Czech Technical Univeristy, Prague. He is the Editor-in-Chief of Kybernetika and serves on the editorial boards of several leading journals. He is a Vice-President of IFAC and Chairman of its Technical Board, a Fellow of the IEEE and a member of the IEEE Control Systems Society Board of Governors, and a Member of the New York Academy of Sciences and the Academy of Engineering of the Czech Republic.