Switched systems with transient unsustainable modes: Stability and feasibility

Abstract

Externally switched systems have unique characteristics that make it is especially challenging to ensure their stability and feasibility. These challenges are exacerbated when the system contains modes with uncontrollable dynamics, tight input constraints, and/or unstable dynamics. While past results have addressed stability and feasibility of switched systems, little has been done, thus far, for constrained, switched systems with uncontrollable modes. This work examines linear systems in this context with unknown switching signals that have upper and lower bounds on the duration the system will dwell in a mode and constraints on which modes can be switched between. These constraints create a finite number of different possible switching signal statuses, which depend on the current mode and the time spent dwelling in that mode. At design time, state constraints corresponding to each possible status are computed and, at run-time, are activated based on the switching signal status. If these time-varying state constraints respect certain properties, persistent feasibility, convergence, and exponential stability are established under all possible switching signals. This is later extended to be a sequence of state constraints for use in model predictive control. These results are then demonstrated on two numerical examples.