On Maximal Robustly Positively Invariant Sets

Set invariance plays a fundamental role in the analysis and design of control systems for constrained systems, since if the initial state is contained inside an invariant set, all future states will stay within the set and hence will satisfy the imposed system constraints, (Blanchini, 1999). In literature, two types of convex sets are essentially used as candidate invariant sets: ellipsoidal and polyhedral sets. The use of ellipsoidal sets has the advantage that the complexity is fixed, (Kurzhanski and Varaiya, 2000), (Kurzhanski and Varaiya, 2002). However, they have a rather restricted shape, which may be very conservative in typical problems. In this paper we will focus only on polyhedral sets in conjunction with linear dynamics. The construction of maximal robustly positively invariant set for linear time-invariant (LTI) systems was studied in literature in different contexts, see for example the study in (Kolmanovsky and Gilbert, 1998). The method, proposed in this early studies constructs an invariant set by iteratively adding additional constraints until invariance is obtained. However, the iterative number is unknown in advance and can be very large. In this paper we provide a novel method for constructing maximal robustly positively invariant sets for LTI systems that does not suffer from these drawbacks. Based on forward reachable sets, the method provides additional insight for a better understanding of the properties of the maximal robustly positively invariant sets. We will also discuss a method for computing an a priori lower bound relevant to the proposed method. From literature, only the work in (Rakovic et al., 2004) proposed a method for determining an upper bound of the number of steps in the iterative construction of the maximal invariant sets. The method presented in the current paper offers a slight improvement for this upper bound. The following notation will be used throughout the paper.N , {0,1,2, . . .} denotes the set of nonnegative integers, N+ denotes the set N \0 andNs , {0,1,2, . . . ,s − 1}. Whenever time is unspecified, a variablex stands forx(k) for somek ∈ N. For someε > 0 we denoteBp(ε) = {x ∈ Rn : ‖x‖p ≤ ε}, where‖x‖p is the p−norm of the vector x = [x1 x2 . . .xn] , i.e. ‖x‖p = (|x1| p + |x2| + . . .+ |xn|) 1 p . Given two setsX1 ⊂ Rn and X2 ⊂ Rn, the Minkowski sum of the setsX1 andX2 is defined by X1⊕X2 , {x1+x2| x1 ∈ X1,x2 ∈ X2}. The Pontryagin difference of the set X1 with respect toX2 is defined by X1⊖X2 = {x| x+ x2 ∈ X1, for all x2 ∈ X2}. The setX1 is a proper subset of the set X2 if and only if X1 lies strictly insideX2. A C-set is a convex and compact set containing the origin as an interior point. A polyhedron, or a polyhedral set, is the intersection of a finite number of half spaces. A polytope is a closed and bounded polyhedral set. The paper is organized as follows. Section 2 deals with a general framework of robustly positively invariant sets. Section 3 is concerned with the minimal robustly positively invariant set while Section 4 is concerned with the maximal robustly constraintadmissible set. Section 5 is dedicated to the problem of computing an a priori lower bound. The simulation