The computation of the viability kernel for switched systems

Jianfeng Lv,Yan Gao

doi:10.1186/s13662-018-1756-y

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.

Highlights

The problem of viability [1] concerns the dynamic evolutions governed by complex systems under uncertainty that are found in many domains involving living beings, from biological evolution to economics [2], from environmental sciences to financial markets [3, 4], from control theory to cognitive sciences [5,6,7]
In order to discuss the method of computing the viability kernel conveniently, we present some important features of switched systems
We develop an efficient algorithm of computing the backward reachable set for system (3.5) on the constraint set K

Summary

Introduction

The problem of viability [1] concerns the dynamic evolutions governed by complex systems under uncertainty that are found in many domains involving living beings, from biological evolution to economics [2], from environmental sciences to financial markets [3, 4], from control theory to cognitive sciences [5,6,7]. It has traditionally been computed using the viability kernel algorithm [13] and level set approach [14] These methods require gridding the state space, and their time and memory complexity grow exponentially with the state dimension. Maidens in [17] proposed an algorithm of computing the viability kernel by using backward reachable set for dynamical. Haimovich et al in [25] developed a method of computing the invariant set for continuous-time switched linear systems with disturbances and arbitrary switching. Because of the large amount of computation and considerable time consumption, the implementation of the algorithms is difficult To overcome these limitations, we restrict our attention to switched linear systems. We compute the viability kernel for continuous-time switched systems and propose an algorithm for switched linear systems in Sect.

Preliminaries

Example 2 Consider the following continuous-time switched linear system:

Conclusions