The computation of the viability kernel for switched systems

Abstract

The computation of the viability kernel provides the guarantee for the security evolution of the systems. In this paper, we focus on the computation of the viability kernel for discrete-time and continuous-time switched systems. A connection between the backward reachable set and the viability kernel for switched systems is established. The methods of computing the viability kernel for switched systems are constructed by using this connection. First, a method of computing the viability kernel for discrete-time switched systems is proposed. Then, taking into account the special structure of switched linear systems, a simple algorithm that is easy to implement is developed. Moreover, the methods of dealing with the discrete systems are extended to the continuous systems, and the algorithms of computing the viability kernel for continuous-time switched systems and switched linear systems are proposed. Finally, examples are listed to illustrate the effectiveness of the main results.

On the computation of the robust viability kernel for switched systems

“…The method we have proposed is the development and improvement of the literature [20]. Compared to results of [20], we consider the viability for switched systems with arbitrary switching rule and bounded disturbances. The method proposed in Algorithm 2 has three advantages.…”

“…Numerical methods of computing the viability kernel for dynamical systems with mixed constraints have been widely investigated from different perspectives; see for instance [17,18] and the references therein. Moreover, the methods of computing the viability kernel for hybrid systems were developed and some results are reported in [19][20][21][22][23]. The computation of the viability kernel is implemented based on the reachable set in [20].…”

“…Moreover, the methods of computing the viability kernel for hybrid systems were developed and some results are reported in [19][20][21][22][23]. The computation of the viability kernel is implemented based on the reachable set in [20]. Aubin and Lygeros [21] have introduced a notion of hybrid strategy and proved the convergence of the iterative algorithm by the non-smooth theory.…”

“…To the best of our knowledge, the viability kernel of switched systems has received little attention. Although method of computing the viability kernel for the switched systems has been proposed in [20], it should be noted that this work has not consider the influence of disturbances on systems. In fact, the effect of disturbances is a common issue in the analysis and synthesis of switched systems.…”

“…Thus, considering the viability kernel for switched systems with disturbances is meaningful and important. We study this problem in the paper based on the results of [20]. In addition, viability kernel obtained from [20], in principle, is an outer approximation of the real one.…”