On sufficient and necessary conditions of compositional input-to-state safety for nonlinear systems

This paper is concerned with studying the problem of compositional input-to-state safety (ISSf) using input-to-state safe barrier functions (ISSf-BFs) for nonlinear systems given as an interconnection of subsystems. A sufficient condition is proposed to verify ISSf for nonlinear interconnected systems, which allows that only some subsystems are ISSf and the others are not. The idea behind the sufficient condition is to establish ISSf of nonlinear interconnected systems based on ISSf of some lower dimensional subsystems, in conjunction with a general type of interconnection between subsystems. Also, a necessary condition is derived by using additively separable ISSf-BFs, which provides a tool for analyzing ISSf of interconnected subsystems. Meanwhile, the necessary condition indicates that the ISSf problem of the overall nonlinear system can be divided into decoupled subproblems, which is dealt with less computational effort. In particular, a sufficient and necessary condition of compositional ISSf is obtained. Finally, three examples are presented to demonstrate the effectiveness of the theoretical results.