In this paper we present new decidability results, wider decidable classes of linear hybrid systems with linear vector field of the form x=Ax+Bu at each discrete location. Combination of the systematic treatment for the decidability problems of the linear differential systems x=Ax+Bu by H. Anai et al [4] and the framework dealing with decidability problems of hybrid systems based on the mathematical logic and model theory introduced by G. Lafferriere et al [13, 14] enables these decidability results. Moreover, we also discuss some extensions of these decidability results.

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In this paper, we investigate some qualitative properties for time-controlled switched systems consisting of several linear discrete-time subsystems. First, we study exponential stability of the switched system with commutation property, stable combination and average dwell time. When all subsystem matrices are commutative pairwise and there exists a stable combination of unstable subsystem matrices, we propose a class of stabilizing switching laws where Schur stable subsystems (if exist) are activated arbitrarily while unstable ones are activated in sequence with their duration time periods satisfying a specified ratio. For more general switched system whose subsystem matrices are not commutative pairwise, we show that the switched system is exponentially stable if the average dwell time is chosen sufficiently large and the total activation time ratio between Schur stable and unstable subsystems is not smaller than a specified constant. Secondly, we use an average dwell time approach incorporated with a piecewise Lyapunov function to study the L₂ gain of the switched system. We show that when all subsystems are Schur stable and achieve an L₂ gain smaller than a positive scalar γ0, (1) if all subsystems have a common Lyapunov function in the sense of L₂ gain, then the switched system achieves the same L₂ gain γ0 under arbitrary switching; (2) if there does not exist a common Lyapunov function, then the switched system under an average dwell time scheme achieves a weighted γ0 gain 70, and the weighted L₂ gain approaches normal L₂ gain if the average dwell time is chosen sufficiently large.

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We consider the two cases to address the stabilization problem of a class of multi-modal hybrid systems with binary-switches. In one case it is reduced to the stabilization problem similar to that for the class of usual continuous linear systems under some condition. In the other case, an approach to find a dynamic state feedback controller that renders the closed loop system well-posed as well as globally asymptotically stable at the equiribrium state is proposed.

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This paper considers the impulsive behavior of a switching system interconnection described by the implicit representation. The switching interconnection consists of two system interconnections which are switched at a certain instant. We define the concatenability of the behaviors of these interconnections, and derive a necessary and sufficient condition for this concatenability. If the behaviors are concatenable, then every trajectory before the switching instant can be switched to an appropriate trajectory without causing any impulses. Moreover, the regular feedback structure of the interconnection after the switching instant guarantees that the concatenability condition is always satisfied.

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This paper shows necessary and sufficient conditions for the existence of state feedback in hybrid systems described by hybrid automata with forcible events. We introduce two semantics of controlled hybrid automata using transition systems. The conditions are based on the transition systems. It is also shown that the conditions are preserved under bisimilar relations so that synthesis methods of state feedback in discrete event systems can be applied in the hybrid systems if the bisimilar relations are finitary.

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This paper deals with analysis of worst-case L₂ gain in terms of bumpy responses due to disturbance for systems with controller switching. Control systems often have bumpy responses caused by switching controllers. Those responses are harmful, since they not only degrade control performance but also may damage plants, actuators and so on. Several methods have been proposed to attenuate bumpy responses. However, most of those methods lack performance and/or robustness guarantee against disturbance and/or uncertainty. In particular, no exact definitions of bumpy responses have been formulated. This paper first defines bumpy responses to be dealt with. Based on the definition, we propose a method to analyze the responses by using LMIs. The method can be applied to the worst-case Hankel norm analysis. The effectiveness of the proposed method is examined by using a numerical example.

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