Proc. 9th IFAC/IFORS/IMACS/IFIP/ Symposium on Large Scale Systems: Theory and Applications (LSS 2001), 2001, pp

Transcription

1 INVARIANT POLYHEDRA AND CONTROL OF LARGE SCALE SYSTEMS Jean-Claude Hennet LAAS - CNRS 7, Avenue du Colonel Roche, Toulouse Cedex 4, FRANCE PHONE: (+33) , FAX: (+33) , hennet@laas.fr Eug^enio B. Castelan LCMI/DAS/UFSC, Florianopolis (S.C) - Brazil, Fax +55 (48) , eugenio@lcmi.ufsc.br Abstract The study addresses the problem of controlling a linear discrete time system subject to bounded additive disturbances, stepwise-in-time constraints and bounds on performance requirements. The objective is to propose a design technique which is practical and ecient, but not necessarily optimal. Positively invariant polyhedral domains are constructed by eigenstructure assignment. The design problem is then solved locally by constructing a domain of initial states which is both positively invariant and included in the domain of satisfactory performance. An index of the size of the domain is maximized by Linear Programming. keywords Positive Invariance, Constrained Control, Disturbance Attenuation, Linear Programming, Spectral Assignment 1 Introduction In the recent years, many studies have been devoted to the control of systems subject to stepwise-in-time constraints on their input, state or output variables. Within this framework, the methods based on viability theory (Aubin, 1991) and on the construction of positively invariant domains have proved to be particularly ecient when coupled with numerical techniques such as LMI (Linear Matrix Inequalities) for the quadratic case (Queinnec et al., 1999) and LP (Linear Programming) for polyhedral domains (Hennet, 1995). In addition, these techniques allow to integrate the presence of bounded structured uncertainties and additive noise. In particular, the problem of optimal attenuation of bounded disturbances has been treated by several authors through the construction of positively invariant domains (Blanchini and Sznaier, 1995), (Shamma, 1996). In spite of its ability to integrate constraints and disturbances, which are almost always present in real systems, the positive invariance approach has not yet been much developped in applications. This is probably due to the lack of a global methodology oriented to the solving of real problems in an ecient (but not necessary optimal) way. The purpose of this paper is precisely to describe a simple methodology for the control of large scale systems subject to constraints and additive disturbances. This methodology is based on spectral assignment as the basic design technique for achieving performance requirements. It is shown that eigenstructure assignment can also guarantee the existence of positively invariant polyhedral domains of the state space which can be selected as feasibility and stability domains in presence of constraints and disturbances. Linear Programming is also used as a complementary tool to achieve constraints satisfaction and disturbance 1

2 attenuation requirements, while maximizing a linear index related to the size of the invariant domain to be selected. The second section of this paper describes some principles of construction of a domain of satisfactory performance for a real system. The third section presents a method for constructing positively invariant polyhedral domains with respect to a linear discrete-time system with bounded additive disturbances. The fourth section proposes an algorithm to include a positively invariant polyhedron in the domain of satisfactory performance. The nal result takes the form of a state feedback control law and of an associated admissible domain in the state space. 2 Domain of Satisfactory Performance Consider the discrete-time linear system described by the following state equation : x k+1 = Ax k + B 2 u k + B 1 w k : (1) At any time period k 2 N, x k 2 < n is the state vector, u k 2 < m the control input vector, w k 2 < q the disturbance input vector, which is random and takes its value in a closed and bounded polyhedral set in < q : w k 2 R[L; ] = fw 2 < q jlw g (2) with L 2 < lq ; l > q; 2 < q+. By convention, inequalities between vectors are componentwise. The system is subject to linear constraints - on its state vector S x x k x 8k 2 N (3) - on its control input vector: S u u k u 8k 2 N : (4) Several stepwise-in-time linear performance objectives can be selected with a tolerance margin with respect to the optimal achievable value. They can be represented as combined constraints on state and control vectors, in the form: Z s x k + Z u u k : 8k 2 N (5) Such a reformulation of objectives as constraints is classical to obtain compromise solutions in multiple criteria decision making (Yu, 1994). It is also used in control through the guaranteed cost approach, which produces stepwise-in-time linear constraints when applied to `1 performance indices (Salapaka and Dahleh, 2000). Assuming that the set of constraints (3), (4), (5) denes a nonempty set in < n+m, it is possible to select a feasible extended state (x ; u ) as the target state. Through a change of variables for x k and u k, the target is selected as the new origin, in the interior of the domain of constraints. System (1) and the constraints are then reformulated, and the problem is interpreted as a regulation problem toward the (zero state, zero control) target. Then, if system (1) is controlled by a stabilizing state feedback law: u k = F x k 8k 2 N (6) the set of constraints (3), (4), (5) denes a polyhedral set of the state space, R[Q; ] < n : R[Q; ] = fx 2 < n j Qx :g (7) with Q 2 < qn, 2 < q, 0. The domain R[Q; ] is called the domain of satisfactory performance of system (1) under control (6). 2

3 3 Positive Invariance with Disturbance Attenuation Consider now the autonomous discrete-time linear system x k+1 = A 0 x k + B 1 w k (8) with x k and w k 2 R[L; ] dened as for model (1). A polyhedral set in the state space, R[G; ] < n is dened s follows: R[G; ] = fx 2 < n j Gx :g (9) with G 2 < gn, 2 < g. By denition, the set R[G; ] is positively invariant with respect to system (8) if and only if any trajectory fx k g generated by (8) from any initial state x 0 2 R[G; ] is entirely contained in R[G; ]. An analytical characterization of this property for linear systems with bounded additive disturbances is provided by the following theorem. Theorem 1 (Hennet and Dorea, 1994) A necessary and sucient condition for positive invariance of R[G; ] with respect to system (8) and for any disturbance vector w k in R[L; ], is the existence of two nonnegative matrices H 2 < gg and M 2 < gl such that : HG = GA 0 (10) ML = GB 1 (11) H + M : (12) Particular positive invariance relations are obtained for symmetrical polyhedral sets. Consider a symmetrical polyhedral set S( ; ) < n dened by: S( ; ) = fx 2 < n j x g: And suppose that the domain of disturbances is also symmetrical: w k 2 S(; ) = fw 2 < q j w g (13) The following corollary can be derived from theorem 1 by a technique similar to the one used for the undisturbed case (Bitsoris, 1988): Corollary 1 A necessary and sucient condition for positive invariance of S( ; ) with respect to system (8) and for any disturbance vector w k in S(; ) is the existence of two matrices H and M such that : H = A 0 (14) M = B 1 (15) jhj + jm j : (16) where jhj and jmj are the absolute values of matrices H and M (by denition, jhj ij = jh ij j and jmj ij = jm ij j). It can be noted that if the polyhedral set R[G; ] (resp. S( ; )) is closed and bounded in < n and contains the zero state (condition > 0, resp. > 0), then its positive invariance implies Lyapunov stability of system (8) since any trajectory is contained in R[G; ] (resp. S( ; )), which is a stability domain. As a consequence, a possible technique for regulation of the controlled system (1) is to construct a state feedback u k = F x k such that a closed and bounded polyhedral set R[G; ] containing the zero state in its interior is positively invariant with respect to the closed-loop system (8) in which A 0 = A + B 2 F (see e.g. (Hennet, 1995)). In some control problems under linear state constraints, it is possible to select the domain of constraints as a candidate domain R(G; ) for closed-loop positive invariance. The design technique then consists in solving the set of linear equalities and inequalities (10), (11), (12) for G and given, with the entries of 3

4 matrices H, F and M as unknown variables. Clearly, such a design technique is not always successful and cannot be directly applied to constraints on control variables. This is the reason why a more systematic scheme is rather proposed in this paper. A feedback matrix F and an associated positively invariant domain R(G; ) will be directly constructed under the constraint of inclusion of the polyhedral set R(G; ) in the domain of satisfactory performance R(Q; ). 4 Regulator Design with Positive Invariance Properties The choice of a candidate stability domain is a basic step in the application of the positive invariance methodology to solve a regulator design problem. To avoid a trial/error method with potential risks of slow convergence or even failure, it is desirable to construct candidate positively invariant domains in a systematic way. The only technique that we know able to solve this problem is eigenstructure assignment. It is well known that if the pair (A; B 2 ) of system (1) is controllable, then it is possible to assign a symmetric set of n complex numbers as the closed-loop spectrum by state feedback, with some degrees of freedom in the choice of the eigenvectors associated. In this study, full controllability of the pair (A; B 2 ) is not required but the system has to be stabilizable and all the uncontrollable poles have to lie in the spectral region which will be described in the sequel. Consider the desired closed-loop spectrum (A 0 ), the real Jordan form, J, of the closed-loop state matrix A 0 = A+BF, and V the matrix of generalized real eigenvectors of the closed-loop system. Then, the following matrix equation is satised : JV 1 = V 1 A 0 (17) The rows of matrix V 1 form a set of left generalized real eigenvectors of matrix A 0. Consider rst the disturbance-free system : The spectrum (A 0 ) can be chosen so as to satisfy the following theorem. x k+1 = A 0 x k (18) Theorem 2 : Spectral Conditions (Bitsoris, 1988) A sucient condition for the existence of a simplicial symmetrical invariant polytope S( ; ), with 2 < nn ; 2 < n + for system (18) is that all the eigenvalues of A 0 (real and complex), denoted i + j i, are such that: j i j + j i j < 1 (19) The spectral domain dened by (19) is represented on Fig.1. I R Figure 1: The spectral domain 4

5 It is not dicult to show that the spectral condition of Theorem 2 implies the existence of positive vectors such that : jjj : (20) From Corollary 1, conditions (17) and (20) characterize positive invariance of S( ; ) with = V 1 and H = J relatively to system (18). Suppose now that condition (19) on eigenvalue i = i + j i is replaced by the tighter condition : j i j + j i j < 1 i (21) with 0 i 1. Let i represent the real directions of left eigenvectors of A 0 associated to i, and let J i denote the corresponding rows of J. Then, by construction, there exist positive vectors 2 < n such that the following relations are satised: i A 0 = J i (22) jj i j (1 i ): (23) If the spectrum of A 0 lies in the domain of Fig.1, then to each row J l of matrix J can be associated a contraction factor l, 0 l 1 such that for some positive vector, A 0 = J (24) jjj diag(1 l ): (25) Taking now into account the additive disturbance vector w k 2 S(; ), the following property can be derived. Theorem 3 A sucient condition for the existence of a simplicial symmetrical polytope S( ; ), with = V 1 ; 2 < n + positively invariant with respect to system (8) is the choice of an eigenstructure assignment for which the real Jordan matrix J satises (24), (25) with 0 l 1 for l = 1; :::; n such that there exists a matrix M which satises: M = B 1 (26) jmj diag( l ) (27) where the positive vector satises (23). This theorem can be shown by constructing the positively invariant domain which plays the role of a candidate stability domain in this study : S( ; ) with = V 1 : (28) It can be noted that the best disturbance attenuation is obtained for l = 1, that is in the eigensubspace of the closed-loop pole located in 0. According to conditions (26) and (27), such an assignment is particularly interesting if the directions of disturbances, dened by the column-vectors of matrix B 1, belong to the eigensubspace associated to the eigenvalue 0. More generally, the maximal value of i in (23) determines the maximal disturbance attenuation factor in the directions of the eigenvectors associated to the eigenvalues i which satisfy (21). The spectral assignment is summarized by the choice of the real Jordan form, J, of the closed-loop system. In particular, the real Jordan block associated to a simple complex eigenvalue, i + j i, is : i i : i i Classically (Chen, 1984), the matrices of real state directions, V and input directions W associated to the spectral assignment are selected so as to satisfy: AV V J = BW (29) 5

6 In particular, a candidate state direction v i in the transmission subspace associated to a real eigenvalue i can be obtained by arbitrarily selecting w i 2 < m and solving in v i the following equation: [ i I A B] vi w i = 0: (30) For large scale systems, the eigenvalue assignment problem can be solved very easily by arbitrarily selecting one or several input directions w i for each selected closed-loop eigenvalue i and solving block by block the linear relations (29). The limit of this technique is that the robustness of the assignment is not maximized and can only be tested afterwards. In practice, it is often important to limit the control eorts by not changing the location of the poles which are already in the region dened by (21) and to maximize the robustness of the assignment. In particular, a classical index for evaluating the robustness of the assignment is the condition number k(v ) = kv k 2 kv 1 k 2 (Kautsky et al., 1985). Once matrices V and W have been computed, a state feedback gain matrix achieving the desired eigenvalue assignment is dened by: F = W V 1 (31) 5 The Constrained Control Scheme The positive invariance approach is able to provide local solutions to constrained control problems by imposing the positive invariance of a domain in the state space satisfying the following conditions (Vassilaki et al., 1988) : 1. The zero state lies in the interior of, 2. is positively invariant with respect to the controlled system, 3. R[Q; ], where R[Q; ]] describes the polyhedron of constraints. An extension of Farkas' lemma provides a set of necessary and sucient conditions on G; Q; ; under which: R[G; ] R[Q; ]: (32) Theorem 4 (Hennet, 1989) The system Qx is satised by any point of the non-empty convex polyhedral set dened by the system Gx if and only if there exists a matrix U with non-negative coecients satisfying conditions: U G = Q (33) U : (34) By virtue of this theorem, the adressed design problem can be solved by the proposed methodology if the conditions of Theorem 3 are satised and if it is possible to impose inclusion of the positively invariant polyhedron S( ; ) (with = V 1 ) in the polyhedral domain of satisfactory performance, R[Q; ]. By application of theorem 4, a necessary and sucient condition for inclusion of S( ; ) in R[Q; ] is the existence of a matrix U 2 < nq such that: The choice = V 1 then imposes: and condition (36) becomes: U = Q (35) ju j (36) U = QV (37) jqv j (38) As previously mentioned, the components of vector determine the size of the invariant domain S( ; ). In the proposed scheme, this domain plays the role of the set of admissible initial states. Any system trajectory starting in this domain remains in it and, by consequence, in the satisfactory performance domain. Therefore, 6

7 it is important to maximize some positive combination of the components of in the form of the linear criterion : C = nx i=1 c i i with all c i > 0 (39) The problem of optimal sizing of S( ; ) with positive invariance and inclusion in the domain of satisfactory performance is thus solved through resolution of the Linear Program with criterion (39) and constraints (23), (27), (38). The cases when the LP is not feasible correspond to the impossibility to construct a positive vector such that S( ; ) is positively invariant, contains the image of the disturbance in the state domain (conditions (23), (27)) and is contained in the domain of satisfactory performance (condition (38)). In these cases, several techniques may be used to achieve feasibility : modify the eigenstructure assignment, enlarge the satisfactory domain or even, if it is possible, reduce the amplitude of disturbances. The proposed methodology has been applied to a production planning problem in random stationary demand conditions. A closed-loop policy has been proposed. The choice of zero as the multiple closedloop eigenvalue has been interpreted in terms of model predictive control (Hennet and Barthes, 1998), (Hennet, 2000). 6 Conclusion The methodology proposed in this paper applies to large scale discrete time linear models. The use of such models is frequent in production, logistics or storage systems. Most of these systems are also characterized by stepwise-in-time constraints and by the presence of additive disturbances. In the context of the eigenstructure assignment approach to regulator design, conditions have been stated to guarantee a sucient attenuation of the eects of the disturbances and to construct state domains which remain positively invariant with respect to the system trajectories. The positively invariant domains included in the domain of constraints, and more generally in the domain of satisfactory performance, can be considered as domains of admissible initial states with respect to this control scheme. A Linear Programming formulation has been proposed to maximize the size of the domain of admissible states in certain directions. At this stage, the constrained control problem has been solved with a relatively low computational burden, but only for the initial states which belong to the invariant domain which has been constructed. A remaining problem is to use a dual-mode control approach to rst attract into this invariant domain some possible initial states which do not belong to it. This problem has been solved in the particular case of multistage manufacturing systems by proposing a combined open-loop and closed-loop production planning scheme (Hennet, 2000). However, the generic design of a dual-mode controller attracting the system state in a positively invariant domain is still an open problem. References Aubin, J. P. (1991). Viability Theory. Birkhauser. Boston. Bitsoris, G. (1988). Positively invariant polyhedral sets of discrete-time linear systems. International Journal of Control 47, 1713{26. Blanchini, F. and M. Sznaier (1995). Persistent disturbance rejection via static-state feedback. IEEE Transactions on Automatic Control 40(6), 1127{1131. Chen, C.T. (1984). Linear System Theory and Design. Holt, Rinehart and Winston. Hennet, J-C. (1989). Une extension du lemme de farkas et son application au probleme de regulation lineaire sous contraintes. C.R. Ac. Sciences, t.308, I, pp Hennet, J.-C. and C.E.T. Dorea (1994). Invariant regulators for linear systems under combined input and state constraints. In: Proceedings of the 33th IEEE Conference on Decision and Control. Vol. 2. Lake Buena Vista, Florida. pp. 1030{

8 Hennet, J.C. (1995). Discrete-time Linear Systems in Control and Dynamic Systems, vol.71, C.T.Leondes Ed., pp Academic Press. Hennet, J.C. (2000). A combined open-loop and closed-loop production planning scheme. In: Proceedings of the IFAC-MIM 2000 Symposium, Rio, Patras, Greece. pp. 268{273. Hennet, J.C. and I. Barthes (1998). Closed-loop planning of multi-level production under resource constraints. In: Proceedings of the IFAC Symposium INCOM'98, Nancy, (France). pp. 485{490. Kautsky, J., N.K. Nichols and P. Van Dooren (1985). Robust pole assignment in linear state feedback. International Journal of Control 41(5), 1129{1155. Queinnec, I., S. Tarbouriech and H.X.De Araujo (1999). H2 disturbance attenuation by dynamic output feedback of a nitrication process submitted to actuator limitations. In: 38th IEEE Conference on Decision and Control. Phoenix, Arizona, USA. pp. 1433{1438. Salapaka, M.V. and M. Dahleh (2000). Multiple Objective Control Synthesis LNCIS 252. Springer. Shamma, J. S. (1996). Optimization of the l 1 -induced norm under full state feedback. IEEE Transactions on Automatic Control 41(4), 533{544. Vassilaki, M., J.C. Hennet and G. Bitsoris (1988). Feedback control of linear discrete-time systems under state and control constraints. Int. Journal of Control, vol 47, pp Yu, P.L. (1994). Multiple criteria decision making. In: Handbooks in Operations Research and Management Science - Op timization, vol. 1, G.L. Nemhauser, A.H.G. Rinnooy Kan, M.J. Todd Eds.. North- Holland. pp. 663{699. 8