A Probabilistic Approach to Controllability/Reachability Analysis of Hybrid Systems Shun-ichi Azuma Jun-ichi Imura Tokyo Institute of Technology; 2-12-1, O-okayama, Meguro-ku, Tokyo 152-8552, Japan Abstract— This paper proposes a probabilistic approach to solve the controllability and reachability problems of the sampled-data/discrete-time piecewise affine systems. First, an algebraic characterization for the system to be control- lable/reachable is derived. Next, based on the characteri- zation, an approach to determine if the system is control- lable/reachable in a probabilistic sense is proposed based on a randomized algorithm. Finally, it is shown by numerical examples that the proposed approach is useful. I. I NTRODUCTION Although the controllability/reachability problem for the hybrid systems is one of the fundamental and important research topics, it has been often negatively solved; for example, it has been shown that the controllability problem is undecidable [1], and that the piecewise affine (PWA) system is not always controllable even if the subsystem in every mode is controllable in the usual sense [2]. In spite of these theoretical limitations, some results have been obtained based on a deterministic approach; Bemporad et al. have discussed the controllability problem of the discrete- time PWA/MLD systems by specifying in advance the control time period [3]. However, this approach involves the hardness of the combinatorial problem and the computation on polyhedra. So the computation amount exponentially grows with the control time period. From the different points of view, in [4], the authors have derived an easily checkable (necessary and) sufficient condition for a new class of hybrid systems called the sampled-data PWA systems [5] to be controllable; however, the class of the systems to which the condition can be applied is limited. On the other hand, the authors have most recently proposed in [6] an approach based on a probabilistic method [7], [8] for the controllability/reachability analy- sis of both sampled-data PWA systems and discrete-time PWA systems, which determines if the system is control- lable/reachable in a probabilistic sense. By checking if each randomly sampled initial state can be driven to the origin, the algorithm proposed there can determine if the system is not controllable in a deterministic sense, or is controllable in a probabilistic sense. Thus even for the controllability problem to which the existing deterministic approaches cannot give any answer, the proposed algorithm can provide some probabilistic information on the controllability. How- ever, it gives no deterministic judgment on controllability, even if the system is in fact controllable; thus it may not be useful if one likes to guarantee that the system is controllable in a deterministic sense in some control system design. In addition, it is not a polynomial-time algorithm with respect to the control time period. This paper proposes a more sophisticated probabilistic approach to the controllability/reachability analysis for the sampled-data and the discrete-time PWA systems. The algorithm proposed here can determine if the system is controllable in a deterministic sense, or is uncontrollable in a probabilistic sense. In addition, it is a polynomial- time algorithm with respect to the control time period, etc. The key ideas to derive such a useful and efficient algorithm are the following two points: (i) introducing the random sampling of trajectories of the discrete state, and (ii) proposing an efficient algorithm for computing of the set of the initial state driven to the origin. It is stressed here that no polynomial-time approach to the controllability/reachability problem of hybrid systems has been derived so far. Notation: let R, N (N + ), and PC denote the real number field, the set of nonnegative (positive) integers, and the set of all piecewise continuous functions, respectively. Let I n express the n × n identity matrix and let x (i) denote the i-th element of the vector x. Let X 1 -X 2 express the difference set of the sets X 1 and X 2 , let hull(X ) and ri(X ) denote the convex hull and the relative interior of the set X , respectively, and let card(I ) denote the cardinality of the finite set I . The set S given as the form S := {x ∈R n | Ax + b ≤ 0,Cx + d< 0} is called here the polyhedron, where A, C and b, d are some matrices and vectors, respectively. II. SAMPLED- DATA/DISCRETE- TIME PWA SYSTEMS In the hybrid system with logic designed artificially, the discrete transition rule is mostly embedded in the digital device; thus the switching action of the discrete state is determined at each switching time fixed in the digital device. In this paper, taking this fact into account, we focus on the class of the PWA systems with such logical discrete transitions, as shown in Fig. 1. The solution behaviors of the system is expressed by the following model Σ sd , which we call here the sampled-data PWA system model [5]: ˙ x(t)= A I (t) x(t)+ B I (t) u(t)+ a I (t) , I (t)= I (t k ), ∀t ∈ [t k ,t k+1 ), x(t k+1 )= φ(h, I (t k ),x(t k )), I (t k+1 )= I + , if x(t k+1 ) ∈S I + (1) where x ∈R n is the continuous state, I ∈I is the discrete state (it is sometimes called the mode), I := {0, 1,...,M - 1} is the set of the value of the discrete state, M ∈N +