Positive realness and optimality problems for linear systems via dynamic compensation

The inverse linear quadratic (LQ) optimal problem based on dynamic compensation is considered in this paper. First a dynamic compensator with a proper dynamic order is given such that the closed-loop system is asymptotically stable and Extended Strictly Positive Real (ESPR) in terms of Bilinear Matrix Inequality (BMI). In this case, a sufficient condition for the existence of the optimal solution is presented. Then the weight matrices of the linear quadratic performance index are derived to be parameterized expressions. In order to solve the inverse optimal control problem, an algorithm to the minimization problem with the BMI constraint is proposed based on path-following algorithm, in which an optimal dynamic compensator and the weight matrices of the linear quadratic performance index can be obtained. Finally, several numerical examples are provided to demonstrate the effectiveness and feasibility of the proposed results.