Quadratic State Feedback Stabilizability via Controller Switching

Abstract

In this chapter, we consider the problem of quadratic stabilizability of a given system via state feedback controller switching. The switched controller is defined by a finite collection of given continuous-time controllers called the basic controllers. Our stabilizability problem is to design a suitable rule for switching from one basic controller to another. We introduce the concept of quadratic state feedback stabilizability via controller switching. Roughly speaking, a system is said to be quadratically stabilizable via controller switching if there exists a switched controller such that the closed-loop system is stable with a quadratic Lyapunov function. We derive necessary and sufficient conditions for quadratic state feedback stabilizability via controller switching for both asynchronously switched controller systems, where controller switching is determined by the value of the plant state and for the more practically important problem of synchronously switched controller systems, where the controller can only be switched at prespecified switching times. We also propose algorithms to determine switching sequences that ensure quadratic stability of the corresponding closed-loop system.