An LQ sub-optimal stabilizing feedback law for switched linear systems

Abstract

The aim of this paper is the design of a stabilizing feedback law for continuous time linear switched system based on the optimization of a quadratic criterion. The main result provides a control Lyapunov function and a feedback switching law leading to sub optimal solutions. As the Lyapunov function defines a tight upper bound on the value function of the optimization problem, the sub optimality is guaranteed. Practically, the switching law is easy to apply and the design procedure is effective if there exists at least a controllable convex combination of the subsystems.

Dynamic output feedback for switched linear systems based on a LQG design

“…In [21] and in order to meet some performance requirements, we designed a state feedback switching law (i.e.…”

“…In order to explain this last point, let us outline the design proposed in [21]. First, let us consider the following Riccati equation:…”

“…In recent papers [21], a switched LQ design has been developed to meet some performance requirements based on the minimization of a switched quadratic cost function. It has been shown that the provided stabilizing feedback law derived from a control Lyapunov function (not necessarily convex), approaches the optimal one in a sense precise in section 2.…”

“…So, its design must guarantee both observation and control goals, whatever the values of the unknown estimation error. Thus, a direct application of the results in [21] is impossible since it implies that the estimation error must be known.…”

“…In Section 2, the problem statement is given and we recall the main result of paper [21]. This allows to explain to the reader why the provided feedback law not only stabilizes the switched system but also approaches the optimal solution.…”