Input-to-state stability and interconnections of discontinuous dynamical systems

In this paper we will extend the input-to-state stability (ISS) framework introduced by Sontag to continuous-time discontinuous dynamical systems adopting Filippov’s solution concept and using non-smooth ISS Lyapunov functions. The main motivation for investigating non-smooth ISS Lyapunov functions is the recent focus on “multiple Lyapunov functions” for which feasible computational schemes are available that have proven useful in the stability theory for hybrid systems. This paper proposes an extension of the well-known Filippov’s solution concept, that is appropriate for ‘open’ systems so as to allow interconnections of hybrid systems. It is proven that the existence of a non-smooth ISS Lyapunov function for a discontinuous system implies ISS. In addition, a (small gain) ISS interconnection theorem is derived for two discontinuous dynamical systems that both admit a non-smooth ISS Lyapunov function. This result is constructive in the sense that an explicit ISS Lyapunov function for the interconnected system is given. It is shown how these results can be applied to piecewise linear (PWL) systems. In particular, it is shown how piecewise quadratic ISS Lyapunov functions can be constructed for PWL systems via linear matrix inequalities (LMIs). The theory will be illustrated by an example of an extended ‘flower’ system, which also demonstrates the computational machinery.