On Feedback Stabilization of Linear Systems

Constant gain fixed order controllers for linear systems are considered. A necessary and sufficient condition for the stabilizability of such a system by a chosen order controller is established. When fixed order controllers are designed to be optimal with respect to a quadratic cost, the usual gradient search algorithm requires a starting value which is a stabilizing controller, and such starting values can be hard to find. A method is proposed and studied for obtaining starting values by a homotopy on the parameter σ when the system matrix A is replaced by A+σ = 1. Properties of the optimizing feedback control as a function of σ are given.