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An Algorithm to Find Maximal State Constraint Sets for Discrete-Time Linear Dynamical Systems with Bounded Controls and States PER-OLOF GUTMAN AND MICHAEL CWIKEL Abstract-An algorithm to find a polyhedral approximant to the maximal state constraint set, given state and input bounds, is suggested for linear discrete-time dynamical systems. I. INTRODUCTION In practical control engineering there are always control input and state variable inequality constraints. As pointed out, e.g., in [5] and [I41 remarkably little work has been done toconstruct feedback controllers for such systems and to treat the ensuing questions of admissible sets. By admissible sets we mean state and input constraint sets for which the system can be stabilized with the available inputs within the allowed state set. Some of the solutions suggested in the literature are mentioned in zyxwvutsr [8] The paper [8] extends some of the results of [6] to discrete-time linear systems and presents algorithms for computing admissible sets and a stabilizing feedback controller, thus treating the set admissibility problem of [7]. In some cases the feedback controller yields a stable variable structure closed-Imp system and can thus be seen in the light of. e.g., [lo]. In this note, an algorithm, different from the one of [13], to find approximants to the maximal Q-invariant set w.r.t. G will be given. The results are also found in the conference contribution 191. See [l] for an analysis of the rate of convergence of these approximants. 11. PROBLEM STATEMENT AND OTHER PRELIMINARIES Consider the discrete-time linear dynamical system 1)=Qx(t)+ru(t) (2.1) with x E zyxwvutsrqp R", u E a*, and a, r matrices of appropriate dimensions. The time variable f is restricted to nonnegative integer values. Assume that it is a priori given that u(t)ERCW', v zyxwvutsrq t (2.2a) Manuscript received August 21, 1985; revised September 26, 1986. This work was supported by the Technion-Israel Institute of Technology, the C. F. Lundstrom Mathematics. Stockholm, the Helge Axson Johnson Foundation, and the Per Westling Foundation, the Swedish Board of Technical Development, the Institute of Applied Foundation of Lund University. P.-0. Gutman is with Israel Electrc-Optics Industries. Rehovot, Israel. M. Cwikel is with the Department of hlathematics, Technion-Israel Institute of IEEE Log Number 8612619. Technology, Haifa. Israel. where C l is a compact, convex polyhedral set defined by its vertices wI,. . . , w.&,; containing the origin in its interior I (2.2b) and that the state must remain bounded; (2.3a) G is a compact, convex polyhedral set defined by its vertices g,; . . , gh.; containing the origin in its interior. I (2.3b) Remark 2.1: Conditions (2.2), (2.3) make x = 0 a stationary point with the control u = 0. Remark 2.2: The polyhedral conditions (2.2b), (2.3b) areinserted for algorithmic reasons. Other compact convex constraining sets can be approximated arbitrarily well by polyhedrons. Problem Statement 2.3: Find an initial condition set XEG such that for each x0 E zyxw X there exists { u(t)} p=O E 51 such that if x(O)=x0 then x(?) EG V t and lim X(t)=O. I-X Definition 2.4: A set X is called Q-invariant w.r.t. G ifit satisfies (2.4), (2.5). and furthermore in (2.5). x(t) E X v 1. (2.6) Remark 2.5: Any initial condition set X satisfying (2.4), (2.5) is always contained in an Q-invariant set, e.g., the union of the trajectories specified in (2.5). Problem Statement 2.6: Find Xm,, the maximal Q-invariant set w.r.t. G. Since it is readily shown that the union of any collection of Q-invariant sets. and also the convex hull of any Q-invariant set, are each Q-invariant. it follows that X,, exists and is convex. Henceforth, we will only consider convex Q-invariant sets. Problem Statement 2.7: Find a state feedback control u(t) = u(x(t), t) that accomplishes the stabilization according to (2.5). Optimal control theory can sometimes be used to yield open-loop and, more rarely, closed-loop controls (see, e.g., [ll]). This approach is not used here. In thecasewhere of (2.1) has eigenvalues = 0, there are modes which are pure delays of the inputs. This can easily be seen by jordanizing a. The jordanization also gives linearly transformed con- straint sets X' and Q'. The constraints on the states corresponding to delayed inputs must be compatible with Q'. We will, in some cases, exclude the Jordan-blocks with eigenvalue = 0. In particular, this means that constraints involving linear combinations of old inputs and "proper" dynamical states are forbidden. This leads us to make the followping assumption which, let us note, is satisfied by sampled models of continuous-time systems. Assumption 2.8: The matrix 9 of (2.1) is invertible. In an example we will study the continuous-time double integrator plant whose sampled model, with sampling period = 1 s. is the (subkpace of ii" spanned by the vertices. ' Here and in the sequel, polyhedral interiors and boundaries are taken with respect to 0018-9286/87/0300-02.51$01.00 C 1987 IEEE