A controller design algorithm for a finite number of linear systems

In this paper, an algorithm is presented for the simultaneous stabilization of a set of linear time invariant (LTI) single-input and multi-input systems. Using the notion of quadratic stabilizability, a simple construction of a stabilizing controller is obtained. The class of LTI system x = Ax + bu, where A belongs to a set {A(a)}, is considered. It is shown that this class of system is simultaneously stabilizable if {A(a)} is simultaneously transformable into controllable companion form. The modifications of these results are suggested to accommodate the perturbation of matrix b. We also characterize a class of systems that are simultaneously transformable into companion form. In addition to the above results, it has also been proved that a collection of LTI systems is stabilizable by a single controller if they are simultaneously transformable into Hessenberg form. This class of system is shown to be larger than the class of uncertain systems that satisfies matching conditions.