Reachable Set Estimation of Switched Systems With All Subsystems Unstable

Reachable set estimation of switched systems consisting entirely of unstable subsystems poses a challenge that remains unsolved, particularly when employing Lyapunov methods. In this work, a lemma for bounding the Lyapunov functions of switched continuous-time systems with dwell time confined to a bounded interval is proposed. For switched systems with all subsystems unstable, the main difficulty in estimating the reachable set is that conventional constant symmetric positive definite matrices in the quadratic Lyapunov functions (referred to as Lyapunov matrices in this work) will lead to infeasible conditions due to parameter restrictions. To obtain implementable reachable set estimation conditions, our novelty is constructing time-varying Lyapunov matrix functions to develop sufficient conditions for enclosing the reachable set of all unstable subsystems. The core idea is to use the derivatives of matrix functions over a finite time domain to compensate for restrictions on parameters. Furthermore, a linear interpolation approach is utilized to enable time-varying Lyapunov matrix functions. Superior conservatism reduction features are showcased by numerical simulation results.