ROBUST CONSTRAINED REGULATORS FOR UNCERTAIN LINEAR SYSTEMS

Transcription

1 ROBUST CONSTRAINED REGULATORS FOR UNCERTAIN LINEAR SYSTEMS Jean-Claude HENNET Eugênio B. CASTELAN Abstract The purpose of this paper is to combine several control requirements in the same regulator design problem. These requirements are mainly stabilization and respect of control constraints for an uncertain linear system. The originality of the approach stands from the use of polyhedral norms as Lyapunov functions for the closedloop systems. The concept of polyhedral stabilizability of uncertain systems is introduced and used for the design of robust constrained stabilizing controllers. 1. Introduction The use of induced norms for expressing the desired properties of a controlled system has become widely used, in particular for stability and robustness purposes ([14]), ([15]). Contractivity of certain polyhedral norms can also be used as a means to guarantee satisfaction of state and/or control constraints all along a system trajectory. When the contractivity is only local, as it is usually the case in the presence of control constraints, it characterizes the existence of positively invariant domains, which are closed with respect to the system evolution. Outside of this domain, the system behaviour usually becomes non-linear and its stability has to be investigated. This study analyses the case of uncertain linear systems subject to control constraints. Uncertainties on matrices A and B are supposed bounded, and the uncertain system is supposed polyhedral stabilizable. Namely, it is assumed that there exists a simplicial symmetrical polytope which can be made positively invariant for all possible plants under the same linear state feedback. Such a polytope, denoted S(P, 1 n ) is constructed from the polyhedral domain of constraints, S(F, 1 m ), which is made positively invariant for the nominal system. As shown in some previous works [10], positive invariance of unbounded polyhedra such as S(F, 1 m ) cannot generally apply to uncertain systems, while positive invariance of S(P, 1 n ) may be obtained in some bounded domains of parametric uncertainties. A saturated feedback law is then introduced to restore control constraints satisfaction outside of S(F, 1 m ). It then becomes possible, using the selected polyhedral norm as a Lyapunov function, to construct a larger domain of the state space in which robust constrained stability is obtained. Laboratoire d Automatique et d Analyse des systèmes du C.N.R.S, 7, Avenue du Colonel Roche, Toulouse cédex, FRANCE, hennet@laas.fr Laboratório de Controle e Microinformática (LCMI/EEL/UFSC), Florianópolis, S.C., BRAZIL, eugenio@lcmi.ufsc.br 1

2 2 Preliminaries Consider an uncertain discrete time linear plant with uncertainties on matrices A and B: x k+1 = A(q)x k + B(q)u k ; x k R n, u k R m. (1) Vector q parametrizes the uncertainties ; it is supposed to belong to a compact set Q. The state vector, x k is supposed fully measured and all the pairs (A(q), B(q)) are supposed controllable. The considered class of state feedback regulators is described by : u k = F x k with F R m n. (2) Under such a control law, the uncertain closed-loop system takes the form : x k+1 = [A(q) + B(q)F ]x k. (3) Before defining the polyhedral stabilizability property, let us first recall that the L norm of a vector x R n is : x = max i=1,...,n x i, and that the L norm of a matrix M R s n (not necessarily a square matrix) is : Mx M = max. i=1,...,n x Definition 1 : Polyhedral stabilizability by constant state-feedback System (1) is polyhedral stabilizable by constant state-feedback if there exists a state feedback gain matrix F R m n and a non singular matrix P R n n such that, for any q Q, the norm ν P defined x R n by ν P (x) = P x (4) is contractive all along the trajectories of (3). This property is equivalent to the contractivity for (3) of the unit ball of R n, characterized by : P x k+1 1 x k R n such that P x k 1. (5) The property of polyhedral stabilizability can be seen as an analogous, for L instead of L 2 norms, of the classical quadratic stabilizability property, introduced by B.R.Barmish [1] and studied, in particular, by J.C. Geromel et al. [7] to obtain convex formulations of many robust control problems. A necessary and sufficient condition for the contractivity property (5) to hold true is that matrix H(q) defined by H(q) = P [A(q) + B(q)F ]P 1 (6) satisfies: H(q) 1. (7) From a classical property of L norms, property (7) is equivalent to: 1 n H(q)x 1 n x ; 1 n x 1 n (8)

3 where 1 n is the vector of R n with all its components equal to 1. Definition 2 : Polyhedral stability margin If system (1) is polyhedral stabilizable with the pair (F, P ) satisfying (6) and (7), the associated stability margin of (1) is ɛ ; 0 ɛ 1, defined by : ɛ = min q Q (1 H(q) ). (9) In the sequel, inequality (7) will be supposed strictly satisfied. It corresponds to : 0 < ɛ 1, (10) for a particular construction of P ) that will be used to guarantee the existence of a domain of states for which stability and control constraints satisfaction are always satisfied. Control constraints take the frequently encountered form of saturations on the components of the control vector : 1 m u k 1 m k N (11) Such constraints can be re-written: u k 1 k N (12) Using (2), the domain of control constraints can be represented in the state space R n by the polyhedron S(F, 1 m ) defined by : S(F, 1 m ) = {x R n ; 1 m F x 1 m }. (13) 3. Construction of an Invariant Regulator for the Nominal System Consider the nominal system associated with the a-priori expected value of vector q in (1): x k+1 = Ax k + Bu k. (14) Under a state feedback regulator (2), the closed-loop nominal system takes the form: x k+1 = A 0 x k with A 0 = A + BF. (15) For a deterministic plant such as (14), positive invariance has become a classical technique for constructing state feedback regulators (2) that both satisfy the constraints all along the system trajectory and stabilize the system from any initial state in a domain Ω [8], [6], [16], [3], [9]. Definition : Positive Invariance [9] The domain Ω R n n is positively invariant with respect to system (15) if and only if: x k Ω = x k+r = A r 0x k Ω k N, r N. If the pair (A,B) is stabilizable, it is always possible [13] to construct a non-empty domain Ω R n containing a neighbourhood of the zero state and such that :

4 (1) Ω S(F, 1 m ), (2) Ω is a positively invariant domain of (15), (3) System (15) is stable for any x 0 Ω. For the particular choice Ω = S(F, 1 m ), the following result is established in [11]: Proposition 1 There exists a state feedback gain matrix F R m n which meets requirements (1), (2), (3) of constrained invariant regulation with Ω = S(F, 1 m ) if and only if (A,B) is stabilizable and the unstable subspace of A has dimension r with r m. Under this condition, there exists a matrix Σ R m m such that: ΣF = F (A + BF ) (16) Σ 1. (17) Furthermore, if all the stable eigenvalues of A, denoted λ i = µ i + jσ i for i = 1,..., n r, are simple and satisfy the spectral condition: µ i + σ i 1, (18) there exists[ a matrix ] D R (n m) n such that : D P = R F n n is non-singular, S(P, 1 n ) is a positively invariant symmetrical polytope of (15). Then again, from the classical positive invariance conditions [2], [9]), [14], this condition is equivalent to the existence of a matrix H R n n such that : HP = P (A + BF ) (19) H 1 n 1 n. (20) The above relations (19) and (20) are clearly equivalent to relations (6), (7) for the nominal value of q. They show that system (15) is polyhedral stable. But they are associated with a particular choice of matrix P for obtaining positive [ ] invariance of S(F, 1 m ) and positive D invariance of a symmetrical polytope of the type S(, 1 F n ). This choice is possible under the above mentionned conditions. Note, however, that the extra conditions put on the stable eigenvalues of A are sufficient, but not necessary. They can be used for an easy construction of matrix P as a matrix of left real (generalized) eigenvectors of A 0 for a choice of matrix H under the real Jordan form [11]. 4 Polyhedral stabilizability of the uncertain system From Proposition 1, existence of a matrix Σ R m m satisfying relation (16) is a necessary condition for positive invariance of S(F, 1 m ) with respect to the nominal system (15). It is well known [17], [5] that this condition is equivalent to the invariance of the subspace KerF

5 relatively to (15). Then, clearly, positive invariance of S(F, 1 m ) for the uncertain system (3) would require the existence of a set of matrices (Σ(q) R m m, q Q), such that : Σ(q)F = F (A(q) + B(q)F ). (21) This existence condition can be possibly satisfied only under rather strong structural constraints on matrix F and on the family of matrices (A(q), q Q). Condition (21) is actually equivalent to the invariance of the subspace KerF with respect to (3) for any q Q. And this would require : F A(q)x = 0 x ; F x = 0 ; q Q. (22) To obtain a control scheme robust with respect to bounded unstructured uncertainties on A, it would not be realistic to suppose (22). Instead, it is more appropriate to extend to the uncertain case invariance relation (20) rather than (16), since KerP =. So, the analysis of robust stability under control constraints can be based on the following assumption: Assumption 1 : Polyhedral stabilizability The uncertain system [ (1)] is supposed polyhedral stabilizable with respect to the polyhedral D norm ν P, with P =. The associated polyhedral stability margin ɛ is supposed strictly F positive (10). Assumption 1 is justified by the fact that if the nominal system (14) admits S(P, 1 n ) as a positively invariant polytope, with H = P 1 (A + BF )P satisfying H = 1 γ, ɛ < γ 1, then there exists a family of matrices (A(q), B(q)) for which this assumption holds true. It is not difficult to construct a parametric neighbourhood of the nominal matrices (A,B) for which S(P, 1 n ) remains a positively invariant polytope. Indeed, for A = A(q) A and B = B(q) B satisfying P ( A + BF )P 1 γ ɛ, (23) it is easy to derive : P (A + A + (B + B)F )P 1 ɛ. This boundedness condition on uncertainties (23) applies to the case of totally unstructured uncertainties on A and B, treated in [10]. Assumption 1 allows to consider larger classes of uncertainties, including cases when uncertain parameters of A(q) and B(q) are dependent and distributed according to a special structure. Under Assumption 1, the control u k = F x k would stabilize any instance of system (1) with a polyhedral stability margin at least equal to ɛ if this control law were feasible. But actually there is no guarantee that it satisfies constraints (11) from any initial state belonging to S(F, 1 m ) since this polyhedron is not, in general, positively invariant with respect to the uncertain system (1). The proposed solution to this problem is the use of a saturated control instead of the linear feedback u k = F x k. Under such a non-linear control law, constraints are automatically satisfied, but the stability of the scheme has to be investigated.

6 5. The saturated regulator Suppose that the initial state of the system is in S(F, 1 m ) but not in S(P, 1 n ) and that the uncertain model with linear feedback (3) is valid as long as x k S(F, 1 m ). Model (3) is supposed to satisfy the polyhedral stabilizability assumption of the preceding section. But if the state vector trajectory happens to move outside of S(F, 1 m ), a saturated control is applied, such that the control law takes the following form (for i = 1,..., m) [4] : u i = (sat(f x)) i = { 1, if (F x k ) i > 1 (F x) i, if (F x k ) i 1 The corresponding closed-loop dynamical system can be represented by : (24) x k+1 = A(q)x k + B(q) sat(f x k ), x k R n (25) System (25) is a locally linear dynamical system. The dynamics of the closed-loop system follow the linear model (3) only for states belonging to S(F, 1 m ). Outside of S(F, 1 m ), the dynamic behaviour of system (25) is non-linear, due to action of saturated inputs. The construction of saturated control laws which globally stabilize a system is possible only for asymptotically or critically stable open-loop linear system [4]. In this work, the open-loop uncertain system is not supposed asymptotically stable. Here, it has simply been assumed in section 3 that the dimension of the unstable subspace of A, denoted r is less than or equal to the number of controls, m : r m. As stated in Proposition 1, this condition is necessary and sufficient to obtain positive invariance of S(F, 1 m ) for the nominal system (14), Now, the problem of robust constrained control of uncertain linear systems can be solved by showing the existence of a domain D of the state space including a neighbourhood of the zero state for which the closed-loop uncertain system (25) is stable. Proposition 2 Assumption 1 on polyhedral stabilizability of the uncertain linear system (1) is a sufficient condition for the saturated control law (24) to solve the robust constrained regulation problem in domains D defined by : D = {S(P, α1 n ); 0 < α α max with α max = 1 + π 1 + π ɛ and π = max q Q P B(q). (26) Proof To show that result, it is necessary to recall a basic result on positively invariant domains : If S(P, 1 n ) is a positively invariant polytope of system (3), S(P, α1 n ) is also a positively invariant polytope of system (3) for any strictly positive value of the scalar α. This homothesis property is easy to show from the positive invariance characteristic relations (6) and (7) [12]. From Assumption 1, it is clear S(P, 1 n ) is an admissible domain of initial states for the robust constrained [ control ] problem. Indeed, this domain is positively invariant for system (3), D and from P =, P x F 1 = F x 1. Therefore, for α = 1, systems (25) and (3) are identical and S(P, 1 n ) belongs to D. And, from the homothesis property mentionned

7 above, positive invariance of S(P, α1 n ) is similarly established for α 1. The saturated control law (24) will now be used to increase the upper bound of α. For any state in R n, satisfaction of control constraints (11) all along the trajectories of system (25) is automatically verified by the choice of the saturated control law (24). Under this control law, only the robust stability issue for (25) has to be investigated. Under Assumption 1, the norm ν P is contractive along any trajectory of the uncertain linear system (3). To show stability of system (25) in a domain D of the state space, it suffices to show that the polyhedral norm ν P is a Lyapunov function of system (25) for any state in D. Suppose that at time k, the state vector satisfies, for some value of α greater than 1, ν P (x k ) = P x k α. Let z k+1 = [A(q) + B(q)F ]x k. Then, from Assumption 1, matrix H(q) defined by (6) satisfies H(q) 1 ɛ, and therefore, System (25) can be rewritten as follows: ν P (z k+1 ) = P z k+1 α(1 ɛ). (27) x k+1 = (A(q) + B(q)D k F )x k (28) where the time-varying matrix D k = D(x k ) is : D k = diag { β i k } (29) Its diagonal elements β i k = β i x k are defined as follows: { β i k = 1 if (F x k ) i 1 βk i = 1 (F x k ) i if (F x k ) i > 1, for i = 1,..., m (30) The domain of application of saturations is limited by the following condition : 0 < β min β i k 1 for i = 1,..., m and k = 0, 1,... (31) From (28), system (25) can also be written as: x k+1 = [A(q) + B(q)F B(q)(I m D k )F ]x k (32) Then, Using (27), it yields: P x k+1 = P [A(q) + B(q)F B(q)(I m D k )F ]x k. (33) P x k+1 α(1 ɛ) + P B(q)(I m D k )F x k. (34) Set α max = 1 β min. Then, (I m D k ) 0 and I m D k α max 1 α max

8 and P x k+1 α max (1 ɛ) + α max 1 α max P B(q)F x k. Under condition F x k α, the condition of positive invariance of S(P, α1 n ) for any α such that 0 < α α max is guaranteed if : β min = 1 ɛ 1 + π or equivalently α max = 1 + π 1 + π ɛ. (35) 6. Concluding Remarks Some robustness properties for constrained control by state feedback have been derived from the polyhedral stabilizability approach. The simple assumption of norm-bounded uncertainties on the parameters of (A,B) has lead to general results which can be refined and improved if more knowledge is available about the structure of uncertainties. In particular, experimental results show that in general, the saturation control law (25) can be successfully applied to domains S(P, α1 n ) with α > α max. The apparent conservativeness of some assumptions and results in this paper has two reasons. The first one is the request for dealing with unpredicted uncertainties in any possible direction, due for instance to unmodelled dynamics. The second reason is the search for an easily understandable description of the polyhedral stabilizability approach, which is not very frequently encountered in the control literature. In particular, only systems satisfying the conditions of Proposition 1 have been considered. However, the case r > m has also been solved by constructing, for the nominal system, positively invariant domains Ω strictly included in S(F, 1 m ). And the proposed robust scheme can be extended to this case. Also, the method mentionned in this paper for solving positive invariance relations is through eigenstructure assignment, with matrices Σ in (16) and H in (19) under the real Jordan form. But this is not the only possible technique (see e.g. [16]). And the advantage of direct schemes is in the possible optimization of the most relevant performance indices. The problem can then be solved by mathematical programming techniques, with positive invariance relations simply taken as constraints (e.g. in [15]). References [1] B.R.Barmish Necessary and sufficient conditions for quadratic stabilizability of uncertain systems, J. Optim. Theory Appl. Vol 46, pp , [2] G.Bitsoris : Positively invariant polyhedral sets of discrete-time linear systems. Int. Journal of Control, vol 47 (1988), pp [3] G.Bitsoris, M.Vassilaki : The Linear Constrained Regulation Problem for Discrete-time Systems, 11th IFAC World Congress, Tallinn, 1990, vol 2, pp [4] C. Burgat and S. Tarbouriech: Global stability of linear systems with saturated controls, Int. Journal of Systems Science, Vol. 23, No 1, 1992, pp

9 [5] E.B. Castelan and J.C. Hennet, Eigenstructure assignment for state-constrained linear continuous-time systems, Automatica, vol.28, No.3, pp , [6] J. Chéganças and C. Burgat : Régulateur P-invariant avec contraintes sur les commandes, Congrès AFCET-Automatique, Toulouse, [7] J.C. Geromel, P.L.D. Peres and J. Bernussou, On a Convex Parameter Space Method for Linear Control Design of Uncertain Systems, SIAM Journal on Control and Optimization, vol. 29, No 2, pp , March [8] P.Gutman and P. Hagander, A New Design of Constrained Controllers for Linear Systems, IEEE Trans. Autom. Control, vol. 30, No.1, 1985, pp [9] J.C. Hennet and J.P. Beziat, Invariant Regulators for a class of constrained linear systems, Automatica,, vol. 27, No.3, pp , 1991 ; also in IFAC, Tallinn, vol. 2, pp , [10] J.C. Hennet and E.B.Castelan, Robust invariant controllers for constrained linear systems, 1992 American Control Conference, Chicago Vol.2, pp , [11] J.C. Hennet and E.B.Castelan, Constrained Control of Unstable Multivariable Linear Systems 1993 European Control Conference, Groningen, Netherlands, Vol.4, pp , [12] J.C. Hennet and J.B. Lasserre, Construction of Positively Invariant Polytopes for Stable Linear Systems, 12th IFAC World Congress, Sydney, 1993, Vol.9, pp [13] R.E.Kalman and J.E.Bertram : Control systems analysis and design via the second method of Lyapunov,Trans. A.S.M.E, D 82, 1960, pp [14] H.Kiendl, J.Adamy, P.Stelzner Vector Norms as Lyapunov functions for linear systems. IEEE Trans. Autom. Control, vol. 47, No.6, 1992, pp [15] M.Sznaier, Z.Benzaid Robust control of systems under mixed time/frequency domain constraints via convex optimization IEEE CDC, Tucson, 1992, pp [16] M.Vassilaki - J.C.Hennet - G.Bitsoris : Feedback control of linear discrete-time systems under state and control constraints, Int. Journal of Control, vol 47 (1988), pp [17] W.M.Wonham : Linear multivariable control - A geometric approach. Springer-Verlag 1985.