The concept and model of hybrid systems are introduced. Invariant sets introduced by LaSalle are proposed, and the concept is extended to invariant sets in hybrid systems which include disturbance. It is shown that the existence of invariant sets by arbitrary transition in hybrid systems is determined by the existence of common Lyapunov function in the systems. Based on the Lyapunov function, an efficient transition method is proposed to ensure the existence of invariant sets. An algorithm is concluded to compute the transition mode, and the invariant set can also be computed as a convex problem. The efficiency and correctness of the transition algorithm are demonstrated by an example of hybrid systems. 1. Introduction Hybrid systems and invariant sets are novel topics in recent years. Hybrid systems are systems which include discrete and continuous dynamics. In many applications, hybrid systems have multiple operating modes; each is described by a different dynamic equation [1–3]. Invariant set of hybrid system plays an important role in the many situations when dynamics system is constrained in some way [4, 5]. For hybrid systems, the transition is very important for constructing an invariant set for the systems. For some transitions, the states of the systems may get out of each subsystem’s invariant set. In this paper, an efficient transition method which makes the states of the hybrid systems stay in the invariant sets is proposed. When the system is given an initial set, this method constructs transition regions for the hybrid system to ensure the invariant set existence. Then, the invariant set can be computed as ellipsoid set or polyhedral set [6–9]. Some studies have been done in the area of hybrid systems in recent years. Controller design and invariant set of hybrid systems have been studied [10, 11]. The lecture notes [12] by Lygeros have talked about the basic theory of hybrid system including automata, existence, analysis and synthesis, model checking, and reachability. The set invariance in control provides a survey of the literature on invariant sets and their applications [13], and the stability of mode transitions has been studied in [14]. Research about invariant set of special hybrid systems has been done in [6, 7]. Polyhedral approximation computation of invariant set has been studied in [15]. This paper proposes an efficient approach to compute the transition regions. By the efficient transition based on the Lyapunov function theory, the system will stay in the invariant sets. This method is attractive as the invariant set is
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