Guaranteed performance control of switched linear systems: A differential-Riccati-equation-based approach

This paper studies a quadratic performance certificate in terms of an upper bound and a lower bound for switched linear systems over all admissible dwell-time switching signals. The infinite-horizon performance is reformulated into the summation of the finite-horizon performance for each subsystem over switching intervals that can be calculated by solving a family of differential Riccati equations. The time-varying positive definite solutions are exploited to construct a novel piecewise quadratic Lyapunov function for the closed-loop system, which guarantees that the Lyapunov function is decreasing at and between the switching instants. This decreasing property results in consistency between the piecewise quadratic Lyapunov function for switched systems and standard quadratic Lyapunov function for non-switched systems, based on which an upper bound of the quadratic performance is obtained by referring to the techniques for non-switched systems. In addition, a lower bound is established by solving a family of algebraic Riccati equations. The effectiveness of the guaranteed performance control method is illustrated by making use of a numerical example.