Almost Optimal Linear Quadratic Control Using Stable Switched Controllers

Abstract

This chapter is concerned with the output feedback stabilization and optimal control of a linear time-invariant system via a stable controller. It is well known that standard methods for output feedback controller design, such as the LQG method, may lead to unstable controllers even if the original plant is stable. However, many control engineering practitioners regard the use of an unstable controller as highly undesirable. This has motivated a number of authors to consider the problem of strong stabilization. This problem involves finding an output feedback controller to stabilize a system such that the controller itself is also stable. For the case in which one restricts attention to linear time-invariant controllers, a necessary and sufficient condition for strong stabilizability has been obtained in terms of a certain parity interlacing condition; see [146, 161]. This parity interlacing condition is much stronger than the conditions of stabilizability and detectability that are the conditions to simply stabilize the system (e.g., see [55]). Furthermore, the dimension of the required linear time-invariant controller may be arbitrarily large (e.g., see [133]).