H ∞ Control with Synchronous Controller Switching

In this chapter, we consider H∞ control problems that employ synchronous state and output feedback controller switching. These problems are of interest in their own right but will also be used in the following chapters as a theoretical tool in robust stabilizability problems. As in the previous chapters, the controller is defined by a collection of given controllers called basic controllers. Then, our control strategy is a rule for switching from one basic controller to another. The control goal is to achieve a level of performance defined by an integral performance index similar to the requirement in standard H∞ control theory (e.g., see [16, 54, 59]). The switching rule is computed by solving a Riccati differential equation of the game type and a discrete-time dynamic programming equation. Riccati differential equations of the type considered in this chapter have been widely studied in the theory of H∞ control, and there exist reliable methods for obtaining solutions. The solution to discrete-time dynamic programming equations has been the subject of much research in the field of optimal control theory. Furthermore, many methods of obtaining numerical solutions have been proposed for specific optimal control problems. The main results of this chapter were originally presented in the papers [106–109, 128].