On a Class of Controlled Invariant Sets Marian V. Iordache and Panos J. Antsaklis Abstract— In this paper we introduce a new class of controlled invariant sets, called controllable invariant sets. Intuitively, a controllable invariant set has the property that from any “large enough” connected region of the set it is possible to reach any such other region of the set, regardless of disturbances. Disturbances are assumed to be bounded. The range of the control inputs is assumed to be given and is allowed to be bounded. The main result of the paper is a nonrecursive approach for the computation of controllable invariants. The other results of the paper deal with properties of the proposed method and of controllable invariance. The results of the paper assume hybrid system modes with linear discrete-time dynamics. I. I NTRODUCTION Controlled invariant sets have been used in the hybrid systems literature for the solution to the safety problem (e.g. [7]). This paper introduces a new class of controlled invariant sets, called controllable invariant sets. The context is that of mode dynamics of hybrid systems with control inputs and bounded disturbances. The controllable invariant sets are defined as follows. Given a closed neighborhood of the origin Ω, let Ω x denote the neighborhood Ω around x (i.e. Ω x = {y : y − x ∈ Ω}). Then J is a controllable invariant set if Ω x is a controlled invariant at all points x of J , and for all points x 1 and x 2 of J it is possible to reach Ω x2 from any point of Ω x1 , regardless of disturbances. Thus, this definition implies a certain reversibility, meaning that as long as we keep the state within J , it is always possible to return it to the initial condition (i.e., initial neighborhood). Such a reversibility fits most engineering systems. Note that in general the controllable invariant sets are proper subsets of a maximum controlled invariant set. In practice, one factor that may cause a controlled invariant set to be not controllable is the bounded range of the control inputs. In this paper we approach the computation of the con- trollable invariant sets for linear discrete-time dynamics and rectangular neighborhoods Ω. The computational ap- proach is nearly optimal, in the sense that the controllable invariant set J that is obtained is an open set whose closure J contains all controllable invariant sets with the same neighborhood type Ω as J . The computation involves linear programming and projections (also known as Fourier- Motzkin eliminations). Related approaches have been used in [5], [10] for predecessor operator computations, in [3], [11] for the computation of the maximal controlled invariant The authors are with the Department of Electrical Engineering, Uni- versity of Notre Dame, IN 46556, USA. E-mail: iordache.1, antsak- lis.1@nd.edu. The authors gratefully acknowledge the partial support of the Lockheed Martin Corporation, of the National Science Foundation (NSF ECS99- 12458), and of DARPA/IXO-NEST Program (AF-F30602-01-2-0526). set, and also in other contexts, e.g. [4], [2]. Note that some model uncertainties could be incorporated in this framework [6]. To our knowledge, the entire material of this paper is new. The paper is organized as follows. After presenting our notation and definitions in section II, a motivation is presented in section III. The motivation shows the rele- vance of the controllable invariant sets to a hybrid system abstraction problem. Then, the computation is approached in section IV. The approach is formally proved in the same section. Section IV includes also an investigation of the properties of the computational approach and of the controllable invariant sets, in general. II. DEFINITIONS This is our notation. Given a hybrid system of set of modes Q, we denote by Inv(q) the invariant set of the mode q ∈ Q. Also, let X denote the domain of the continuous state variable x. In this paper we assume that the dynamics of each mode q can be described by x(t + 1) = A(q)x(t)+ B(q)u(t)+ E(q)d(t) (1) where u is the control input and d is the disturbance, which will be assumed bounded. For each mode q, we define the following • The operator Pre represents the predecessor operator. That is, Pre(M ) is the set of continuous states from which M can be robustly reached. In other words, ∀x 0 ∈ Pre(M ) there is a control policy (which may depend on x 0 ) which, no matter of disturbances, leads the continuous state x from x 0 to some x f ∈ M . • I ⊆ Inv(q) is a controlled invariant set if for all x ∈ I there is an admissible control law such that for all subsequent times t: x(t) ∈ I , regardless of the disturbance input. • Let Reach : X →P (P (X )), where for M ⊂ X we have M ∈ Reach(x) if it is possible to robustly reach M starting from x (i.e. no matter of disturbances, it is possible to reach M from x.) 1 In other words Reach(x) is the collection of sets M with the property that it is possible to robustly reach M from x. In this paper, we introduce the following class of con- trolled invariant sets, that we call controllable invariant sets. Let Ω o denote the interior of Ω. Definition 2.1 Given a set Ω ⊂ R n , let Ω x = {z ∈ R n : ∃y ∈ Ω,z = y + x}. For some q ∈ Q we say that I ⊆ Inv(q) is a controllable invariant set if a connected compact set Ω ⊂ R n exists such that 0 ∈ Ω o and 1 P(Y )= {E : E ⊆ Y } denotes the collection of all subsets of Y . Proceeding of the 2004 American Control Conference Boston, Massachusetts June 30 - July 2, 2004 0-7803-8335-4/04/$17.00 ©2004 AACC ThA17.3 2522