A lifting approach. Control Engineering Laboratory. lifted time-invariant system description without causality

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1 State-space solution to the periodic multirate H control problem: A lifting approach Mats F. Sagfors Hannu T. Toivonen Department of Chemical Engineering, Abo Akademi University, FIN{25 Abo, FINLAND msagfors@abo.fi, htoivone@abo.fi Bengt Lennartson Control Engineering Laboratory Chalmers University of Technology S{42 96 Gothenburg, SWEDEN bl@control.chalmers.se Abstract A state-space solution to the H control problem for periodic multirate systems is presented. The solution is based on the lifting method, an equivalent timeinvariant system description is derived for the original periodic multirate problem. The H -optimal controller for the multirate system is expressed in terms of to algebraic Riccati equations. The so-called causality constraints are represented by a set of positive deniteness conditions coupling criteria. Introduction Multirate systems appear in digital control systems hen dierent parts of the control system operate ith dierent rates. Due to the rapid development of computer technology facilitating the implementation of such systems, multirate control problems have received a considerable interest lately. The development of a theory for multirate systems started in the late 95's ith the ork by Kranc [5], Kalman Bertram [3] Friedl []. More recent ork on multirate systems is concerned ith e.g. system analysis stability [2, 2, 8, 9, 2], optimal control of multirate systems ith a quadratic cost function [, 4, 5, 7, 6, 7, 2, 25], H control of multirate systems [6, 25, 26]. The lifting technique is a stard procedure that is commonly applied to periodic multirate systems to represent the periodic system by a time-invariant, lifted system, e.g. [6, 2, 2, 25, 26]. The lifted system has as its inputs outputs the inputs outputs during one period of the periodic system. The main dif- culty encountered in this approach is the causality constraint, hich has limited the applicability of stard state-space solutions on the lifted system description. Previous solutions to the multirate H problem have therefore been derived through constrained model matching [6, 25, 26]. In this ork, a state-space solution based on the lifting technique is derived for the periodic multirate H problem. The solution consists of to stard discrete algebraic Riccati equations, a set of matrix positive deniteness criteria a set of associated coupling conditions. An interesting feature of the result is that the algebraic Riccati equations are exactly those associated ith the H problem for the lifted time-invariant system description ithout causality constraints. The causality constraint appears in the solution as a set of positive deniteness criteria coupling conditions. These conditions are easy to check, as they are analogous to the criteria appearing in the stard discrete-time solution. Thus, stard discrete-time solution methods can be applied. 2 Problem statement Consider a linear time-invariant nite-dimensional discrete-time plant described by x(t ) = Ax(t) B (t) B 2 m(t); x() = z(t) = C x(t) D 2 m(t); t = ; ; 2 : : : () x(t) 2 R n is the state vector, (t) 2 R m is the disturbance, m(t) 2 R m2 is the control signal z(t) 2 R p is the controlled output. The measurements used for feedback are assumed to be available periodically, such that the measurement signal y() is given by y(t) = C 2 (t)x(t) D 2 (t)(t); (2) It is assumed that the measurement function is periodic ith the period N, i.e. C 2 (t) = C 2 (t N); D 2 (t) = D 2 (t N). The outputs may be sampled ith different rates. The signal y(t) contains only those measurements hich are available at time t, i.e. it has a periodically time-varying dimension. The updating of the control signals is assumed to be generated by the folloing N-periodic mechanism, cf. [8]: x m (t ) = [I? H(t)]x m (t) H u (t)u(t) m(t) = [I? H(t)]x m (t) H u (t)u(t); (3) H(t) := diagf (t); 2 (t); : : : ; m2 (t)g is an N- periodic matrix function hich selects the control signals to be updated at time instant t. The functions i (t) take the value if the i:th input is updated at time instant t, zero otherise. The control signal u(t) contains only those signals of m(t) updated at time instant t. The matrix H u (t) is therefore a submatrix of H(t), hich consists of the nonzero columns of H(t). It is assumed that the hold function is synchronized at t =, i.e. H(kN) = I.

2 Remark 2. The time-varying dimensions of the signals y() u() guarantee the existence of matrix inverses in the solution to the control problem. For an alternative ay to guarantee nonsingularity, see [7]. The system dened by () { (3) is periodic ith the period N. By extending the state-vector, the system can be represented as G : x e (t ) = A e (t)x e (t) B (t) B 2e (t)u(t) z(t) = C (t)x e (t) D 2 (t)u(t) y(t) = C 2e (t)x e (t) D 2 (t)(t) x e (t) := [x (t) x m(t)] A B2 (I? H(t)) A e (t) := I? H(t) (4) B ; B := ; C (t) := [ C D 2 (I? H(t)) ] ; D 2 (t) := D 2 H u (t); B2 H B 2e (t) := u (t) ; C H u (t) 2e (t) := [ C 2 (t) ] : We need the folloing denitions, cf. [22], Denition 2. Let Ae (; ) denote the state-transition matrix of the system G. Then G is said to be exponentially stable if there exist c ; c 2 > such that k Ae (t ; t)k c e?c2(t?t) ; for all t t: (5) Denition 2.2 The system G is said to be stabilizable (respectively, detectable) if there exists a bounded function K(t) (respectively, L(t)) such that AeB 2eK (respectively, AeLC 2e ) satises (5). The folloing assumptions are made in the paper: A The system G is stabilizable detectable. A2 The triple (C ; A; B ) is stabilizable detectable. A3 The matrices D 2 D 2 D 2 (t)d2(t) are nonsingular. A4 D 2 C = B D 2(t) = for all t. The rst to assumptions are necessary for the existence of a stabilizing solution. The other assumptions lead to technical simplications. They are quite stard but can be relaxed. The set of admissible controllers is assumed to consist of exponentially stabilizing discrete causal control las u = Ky such that u(t) is a function of present past measurements fy(t); y(t? ); : : :g only. The folloing orst-case performance measure induced by the disturbance 2 l 2 (; ) ill be considered, kzk l2 J (K) := sup : (6) 6= kk l2 The multirate H control problem is dened as follos: Find an admissible controller K, if one exists, hich achieves a specied attenuation level >, such that J (K) <. 3 Main results In this section, the lifting technique is applied to represent the periodic multirate H control problem as an equivalent time-invariant H problem. A controller can then be designed for the time-invariant lifted system, but the requirement that the controller for the lifted system should represent a causal controller imposes restrictions on the controller structure. Due to this causality constraint, stard controller synthesis methods have not been directly applicable to the problem of designing controllers for periodic systems via lifting. Various special methods have therefore been studied for the problem of designing optimal controllers hich satisfy a causality constraint [6, 2, 25, 26]. In this section, a to-riccati equation solution is presented to the H problem for lifted systems ith controller causality constraints. Separate solutions to the periodic full-information, state-feedback, estimation, output feedback problems are given. 3. The multirate full-information statefeedback problems Consider the N-periodic discrete system in (4). Consider the problem of nding a causal stabilizing N- periodic full-information controller u = K fi x (7) such the closed-loop system has l 2 -induced norm (6) from to z less than. Introduce the lifting operator L N : l 2! l 2 such that (L N v)(k) = [v (kn) v (kn ) v (kn N? )] dene the lifted signals := L N, u := L N u, z := L N z. Assuming that H(kN) = I, the associated time-invariant lifted system is given by x(k ) = Ax(k) B (k) B2 u(k); z(k) = C x(k) D (k) D 2 u(k); k = ; ; : : :, x(k) := x(kn), A := A N B := [ A N? B A N?2 B B ] B 2 := [ B 2 (N; ) B 2 (N; ) B 2 (N; N? ) ] C := C C A (8) (9). C A N? D := [ D (p; q)]; p; q = ; ; : : :; N? D 2 := [ D2 (p; q)]; p; q = ; ; : : :; N? X p?q B 2 (p; q):= A p?q?l B 2 H (ql; q)h u (q); p > q l= H (; ) is the state transition matrix of the holdfunction (3) I; p = q H (p; q) := [I? H(p? )] : : :[I? H(q)]; p > q ; p q D (p; q) := C A p?q? B ; p > q

3 D 2 (p; q) := 8 >< >: ; p < q D 2 H u (q); p = q C B 2 (p; q) D 2 H (p ; q )H u (q); p > q. Notice that the lifted system (8) has an n-dimensional state vector. This is due to the assumption H(kN) = I, i.e., that all inputs are updated at time instants kn. If the state of the lifted system ere dened at time instants only part of the inputs are subject to change, then the state of the lifted system ould include those elements of x m hich are not updated at these time instants, cf. the augmented state in (4). From the construction of the lifted system (8) it follos that an N-periodic full-information controller K fi for (4) can be characterized as a time-invariant fullinformation controller K L := [K x K ] : l 2 l 2! l 2 for (8), such that u(k) = ([ K x K ] x )(k) := () (K fi [x ] )(kn). (K fi [x ] )(kn N? ) The controller (7) stabilizes the periodic system (4) if only if the controller () stabilizes (8), the l 2 -induced norm of the closed-loop system (4), (7) is equal to the H norm of the closed loop (8), (). The problem of constructing a stabilizing periodic fullinformation controller for (4) hich achieves a speci- ed l 2 -induced norm bound can thus be reduced to an H -optimal full-information control problem for the time-invariant lifted system (8). Hoever, the equivalent H problem dened for (8) is not a stard H problem, because the requirement that (7) should be causal imposes corresponding causality constraints on the structure of the lifted controller in (). In particular, partitioning K in compliance ith u u, the direct coupling term of K should be loer block triangular. The folloing theorem gives a state-space solution to the causally constrained lifted full-information H problem associated ith the multirate H -optimal full-information problem. Theorem 3. (Full-information) Consider the N- periodic system (4) the associated lifted system (8). There exists a stabilizing full-information controller K L described by () such that K fi is causal such that the closed-loop system (8), () has H norm less than if only if there exists a symmetric positive semidenite solution S to the discrete algebraic Riccati equation E S := M S := B B 2 B B 2 S = A S A C C? E S M? S E S () D S A D C 2 S B B 2? 2 I D D D D 2 D 2D D 2D 2 such that is stable A? [ B B2 ] M? S E S (2) 2 I? B S e(i ) [I B 2e (i)k(i)] B > ; (3) i = N? ; : : : ;, K(i) :=?[D 2 (i)d 2(i) B 2e (i)s e(i )B 2e (i)]? B 2e(i)S e (i ) (4) the matrices fs e ()g are dened recursively according to S e (i) = A e(i)s e (i )A e (i) e(i)c (i)?e? S (i)m S (i)e S(i) (5) S S e (N) := ; i = N? ; : : :; E S (i) := B B 2e (i) S e (i)a e (i) B B M S (i) := B 2e (i)? S e (i) B 2e (i) 2 I D 2 (i)d 2(i) Moreover, a stabilizing causal N-periodic full-information controller, u = K fi [x ], hich achieves the norm bound is given by u(kn i) = K(i)[A e (i)x e (kn i) B (kn i)]; i = ; ; : : :; N? ; k = ; ; : : : (6) Proof: By stard discrete H control theory, a necessary condition for the existence of a full-information controller K L hich achieves the performance bound for the system (8) is that the discrete algebraic Riccati equation () has a stabilizing symmetric positive semidenite solution. In this case, ith x() =, e have the expansion [] kzk 2 l 2? 2 k k 2 l 2 = L k ( (k); u(k)) := u X k= L k ( (k); u(k)) (7)? u? u (k)m S? u? u (k) (k) :=?M? S E S x(k) Moreover, the contribution to the quadratic cost (7) from the time instant k can be expressed as z (k)z(k)? 2 (k) (k) = L k ( (k); u(k))x (k)sx(k)?x (k )Sx(k ) (8) From the denitions of the signals z(k) (k), the system equation (4), hich describes the system behavior beteen the discrete time instants kn

4 kn N, e have similarly the expansion associated ith the Riccati dierence equation (5) z (k)z(k)? 2 (k) (k) := (9) N? X i= X N? = i= [z (kn i)z(kn i)? 2 (kn i)(kn i)] I i ((kn i); u(kn i))x e(kn)s e ()x e (kn) I i (; u) :=?x e(kn N)S e (N)x e (kn N)?? u? u M S (i) u? u u (kn i) :=?M S (i)? E S (i)x e (kn i): Recursive evaluation of S e () from S e (N) to S e () gives, cf. [9, 23], S e () = A e S e (N) Ae C C? E Se M? Se E Se E Se := M Se := B B 2e B B 2e S e (N) Ae S e (N)? 2 I D B B 2e D D 2e C D D D 2e D 2eD D 2eD 2e the matrices for the lifted system associated ith (4) are dened in analogy ith (9). By the assumption H(kN) = I, the matrices of the lifted system are given by A B B2 A e = ; B = ; B2e = C = [ ] ; D = D ; D2e = D2 ; denotes an irrelevant block. This implies, together ith () (5), that S e () = S e (N). The result of the theorem then follos from (7), (8), (9), the denitions of x x e, by stard discrete H control theory [3,, 24]. 2 Remark 3. Notice that the solution of the causally constrained H problem is obtained in terms of the same algebraic Riccati equation () hich is associated ith the lifted system (8) the H problem ithout causality constraints. The constraints appear only in the positive deniteness conditions (3) in the controller structure (6). Remark 3.2 Theorem 3. gives a causal N-periodic controller hich achieves the specied l 2 -induced norm bound for the system (4). The corresponding lifted time-invariant controller K L in () can be recovered by applying lifting to the periodic controller (6). It is easy to see that the lifted version of (6) is a static full-information controller of the form K L = [K x K ], K has zeros above the main block diagonal. Theorem 3. gives a solution to the periodic fullinformation problem. A periodic state-feedback problem can be solved similarly { the lifted constrained fullinformation controller K L = [K x K ] then has a K ith zeros on above the main block diagonal, the positive deniteness conditions (3) are replaced by stronger conditions. The state-feedback problem is applicable e.g. to periodic sampled-data systems [6, 25], strict causality from to u is required. Theorem 3.2 (State-feedback) Consider the N- periodic system (4) the associated lifted system (8). There exists a causal, linear stabilizing multirate statefeedback controller u = K sf x such that the the lifted closed-loop system has H norm less than if only if there exists a symmetric positive semidenite solution S to the discrete algebraic Riccati equation () such that (2) is stable 2 I? B S e(i )B > ; i = N? ; : : : ; the matrices fs e ()g are dened recursively according to (5). Moreover, a stabilizing causal N- periodic state-feedback controller, u = K sf x, hich achieves the norm bound (6) is given by i = ; ; : : :; N? ; u(kn i) = K? (i)x e (kn i); k = ; ; : : :, K? (i):=?[d 2 (i)d 2(i) B 2e (i)s? e (i )B 2e(i)]? B 2e (i)s? e (i )A e (i) S? e (i ) := S e(i )[I??2 B B S e(i )]? : 3.2 A multirate estimation problem An analogous result dual to the full-information feedback solution is obtained for the multirate estimation problem. Dene the nite-dimensional N-periodic discrete-time system associated ith the multirate system (), (2) x(t ) = Ax(t) B (t) z(t) = C x(t) y(t) = C 2 (t)x(t) D 2 (t)(t) (2) The multirate H -optimal estimation problem consists of nding an N-periodic stable causal estimator hich achieves the performance bound ^z = Fy (2) kz? ^zk l2 sup < (22) 6= kk l2 The time-invariant system obtained by applying lifting to (2) over the period N of the sampling function is given by x(k ) = Ax(k) B (k) z(k) = C x(k) D (k) y(k) = C2 x(k) D2 (k) (23) x(k) := x(kn) the lifted signals z are dened as in (8), y := L N y. The system matrices of the lifted system are obtained analogously to (9).

5 The N-periodic estimator described by (2) is equivalent to a time-invariant lifted estimator F L : l 2! l 2 described by ^z(k) = (F L y)(k) := it achieves the performance bound 3 7 (Fy)(kN). 5 (24) (Fy)(kN N? ) kz? ^zk l2 sup < (25) 6= k k l2 for the system (23) if only if F achieves the bound (22) for the system (2). The multirate H -optimal estimation problem can thus be described in terms of a time-invariant H -optimal estimation problem for the lifted system (23) subject to a causality constraint imposed by the causality of (2) the structure of (24). More specically, the direct coupling term of F L is required to have a loer block-triangular structure. The folloing theorem gives a state-space solution to the causally constrained lifted H estimation problem associated ith the multirate H -optimal estimation problem. Theorem 3.3 (Multirate estimation) Consider the N-periodic system (2) the associated lifted system described by equation (23). There exists a stable estimator F L described by (24) such that F is causal such that the H performance bound (25) is achieved if only if the there exists a symmetric positive semidenite solution Q to the discrete algebraic Riccati equation Q = AQ A B B? E Q M? Q E Q (26) E Q := AQ C 2 B D D 2 M Q := C 2 Q [ C such that A? E Q M? Q C 2 ]? 2 I D D D 2 D is stable, C 2 D D 2 D 2D : 2 2 I? C [I L(i)C 2 (i)]q(i)c > ; (27) i = ; ; : : :; N?, L(i) is given by L(i) :=?Q(i)C 2(i)[D 2 (i)d 2(i) C 2 (i)q(i)c 2(i)]? the matrices fq(i)g are dened recursively according to Q(i ) = AQ(i)A B B?E Q(i)M? Q (i)e Q (i) Q() = Q; i = ; ; : : :; N? E Q (i) := AQ(i) [ C C2(i) ] M Q (i) := C 2 (i) Q(i) C 2 (i)? 2 I D 2 (i)d 2(i) : Moreover, a stable causal N-periodic estimator F hich achieves the norm bound is given by ^x(kn i ) = A^x(kN i) (28)?L(i )[y(kn i )? C 2 (i )A^x(kN i)] ^z(kn i) = C ^x(kn i); i = ; ; : : :; N? ; k = ; ; : : : Proof: The proof is analogous to the proof of Theorem Remark 3.3 In duality ith Theorem 3.2, a strictly causal estimator can be constructed, the criteria (27) are replaced by stronger conditions. In analogy ith the full-information state-feedback problems, the time-invariant lifted estimator F L associated ith (28) hich achieves the performance bound (25) can be determined by applying lifting to the periodic estimator (28). 3.3 Multirate output-feedback problem The causally constrained lifted state-feedback controller estimator results of Theorems can be combined to provide a lifting-based solution of the multirate output feedback problem. Theorem 3.4 (Output feedback) Consider the multirate system described by (){(3). There exists a causal periodic controller u = Ky hich stabilizes the system hich achieves the performance bound J (K) < if only if the folloing conditions are satised: (a) The conditions of Theorem 3. are satised, (b) the conditions of Theorem 3.3 are satised, (c) [S(i)Q(i)] < 2, i = ; : : : ; N, S(i) is given by the partition (cf. (5)) S() S () S() := S S 2 () e () Moreover, hen conditions (a){(c) hold, a stabilizing N-periodic multirate controller hich achieves the H performance bound is given by ^x e (kn i ) = A t (i)^x e (kn i) B 2e (i)u(kn i) In L Z (i ) y(kn i )?C 2 (i )[A t (i)^x e (kn i) B 2e (i)u(kn i)] u(kn i) = K(i)A t (i)^x e (kn i) (29) i = ; ; : : :; N? ; k = ; ; : : : K(i) is given by (4), A t (i) := A e (i) B W (i) W (i) := 2 I? B S e(i )[I B 2e (i)k(i)]b? B S e(i )[I B 2e (i)k(i)]a e (i) L Z (i) := Z(i)C 2(i)[D 2 (i)d 2(i) C 2 (i)z(i)c 2(i)]? Z(i) := Q(i)(I??2 S(i)Q(i))? :

6 Proof: The proof follos from Theorems stard discrete-time H theory [3, ]. 2 Remark 3.4 Theorem 3.4 gives a periodic controller hich achieves the H performance bound for the multirate system. The corresponding time-invariant lifted controller can be determined by lifting the periodic controller (29) over the period N. Remark 3.5 The set of all stabilizing controllers satisfying the bound (6) can be constructed from (29) using a linear fractional transformation, cf. [4]. 4 Conclusion A state-space solution to the multirate H problem has been derived using the lifting technique. This paper provides a closed solution in terms of algebraic Riccati equations to the equivalent lifted time-invariant H problem ith causality constraints. The solution is expressed in terms of a pair of discrete-time algebraic Riccati-equations together ith a set of matrix positive-deniteness conditions a set of coupling constraints. The state-space solution presented here using the lifting approach can be shon to be equivalent to a direct solution in terms of periodic Riccati equations [23]. A particular feature of the algebraic Riccati equations is that they are the same as those associated ith the unconstrained lifted discrete-time H -problem. The causality constraint of the multirate problem appear here as a set of additional positive-deniteness constraints a set of coupling restrictions. References [] H. Al-Rahmani G. F. Franklin. A ne optimal multirate control of linear periodic timeinvariant systems. IEEE Trans. Autom. Contr., 35(4):46{45, 99. [2] M. Araki K. Yamamoto. Multivariable multirate sampled-data systems: State-space description, transfer characteristics, nyquist criterion. IEEE Trans. Autom. Contr., AC-3(2):45{54, 986. [3] T. Basar P. Bernhard. H -optimal control related minimax design problems. A dynamic game approach. Birkhauser, Boston, 2 edition, 995. [4] M. C. Berg, N. Amit, J. D. Poell. Multirate digital control system design. IEEE Trans. Autom. Contr., 33(2):39{5, 988. [5] J. R. Broussard D. P. Glasson. Optimal multirate ight control design. Proc. Joint Autom. Cont. Conf., pages WP{E, 98. [6] T. Chen L. Qiu. H design of general multirate sampled-data control systems. Automatica, 3(7):39{52, 994. [7] P. Colaneri G. De Nicolao. Multirate LQG control of continuous-time stochastic systems. Automatica, 3:59{596, 995. [8] P. Colaneri, R. Scattolini, N. Schiavoni. LQG optimal control of multirate sampled-data systems. IEEE Trans. Autom. Contr., 37(5):675{682, 992. [9] A. Feuer G. C. Goodin. Sampling in Digital Signal Processing Control. Birkhauser, Boston, USA, 996. [] B. Friedl. Sampled-data control systems containing periodically varying members. In st IFAC World Congress, pages 36{367, Mosco, 96. [] M. Green D. J. N. Limebeer. Linear Robust Control. Prentice Hall, Engleood Clis, N. J., 995. [2] H. Ito, H. Ohmori, A. Sano. Stability analysis of multirate sampled-data control systems. IMA J. of Math. Contr. Information, :34{354, 994. [3] R. E. Kalman J. E. Bertram. A unied approach to the theory of sampling systems. J. Franklin Inst., 267:45{436, 959. [4] H. Katayama A. Ichikaa. H -control ith output feedback for time-varying discrete systems. Int. J. Control, 63(6):67{78, 996. [5] G. M. Kranc. Input{output analysis of multirate feedback systems. IRE Trans. Autom. Cont., 3:2{28, 957. [6] B. Lennartson. Periodic solutions of Riccati equations applied to multirate sampling. Int. J. Control, 48(3):25{42, 988. [7] B. Lennartson. Combining infrequent indirect measurements by estimation control. Ind. Eng. Chem. Res., 28:653{658, 989. [8] O. Lindgarde B. Lennartson. Performance robust frequency response for multirate sampleddata systems. In Amer. Control Conference, Albuquerque, Ne Mexico, June 997. [9] S. Longhi. Structural properties of multirate sampled-data systems. IEEE Trans. Autom. Contr., 39(3):692{696, 994. [2] D. G. Meyer. A parametrization of stabilizing controllers for multirate sampled-data systems. IEEE Trans. Autom. Contr., 35(2):233{236, 99. [2] L. Qiu T. Chen. H 2 -optimal design of multirate sampled-data systems. IEEE Trans. Autom. Contr., 39(2):256{25, 994. [22] R. Ravi, P. Khargonekar, K. D. Minto, C. N. Nett. Controller parametrization for timevarying multirate plants. IEEE Trans. Autom. Contr., 35():259{262, 99. [23] M. F. Sagfors, H. T. Toivonen, B. Lennartson. H control of multirate sampled-data systems: A Riccati equation solution. In Conf. Decision Control, 997. [24] A. A. Stoorvogel. The H Control Problem. A State Space Approach. Prentice Hall, 992. [25] P. G. Voulgaris B. Bamieh. Optimal H H 2 control of hybrid multirate systems. Systems & Control Letters, 2:249{26, 993. [26] P. G. Voulgaris, M. A. Dahleh, L. S. Valavani. H H 2 optimal controllers for periodic multirate systems. Automatica, 3(2):25{263, 994.