Optimal control with regional pole constraints via the mapping theory

The design of optimal linear, time-invariant systems with regional pole constraint is studied. The problem is to find the static state feedback controller to ensure that all closed-loop system poles lie inside the desired region H and, meanwhile, to minimise a multiobjective performance index. The desired region H could be represented by several inequalities. For some special cases, H may consist of several disjoint subregions. The performance index consists of two parts. One part is used to penalise the sustained error, and the other part is used to guarantee that the optimal solution will not occur on the boundary of the admissible controller set and to improve the robustness property of the closed-loop system. The necessary and sufficient condition for the existence of the admissible controller is found. The necessary condition that the optimal control law must be satisfied is derived. Furthermore, the robustness analysis of the regional pole restriction under unstructured perturbation is studied. Based on the Gersgorin's theorem, a new method is presented, which calculates allowable bounds of the unstructured perturbation, so that all the perturbed poles shall still remain inside some regions.