Inverse optimality problem for singularly perturbed systems

The inverse linear quadratic optimal problem for singularly perturbed system is considered in this paper. First a state feedback controller is given, which is ε-dependent, such that the closed-loop system is asymptotically stable and extended strictly positive real (ESPR) in terms of Linear Matrix Inequality (LMI). Then, a sufficient condition is presented that the state feedback controller is the optimal controller to make a certain performance index achieve minimum. And the weight matrices of the performance index are derived to be an expression based on positive real lemma. In order to solve the inverse optimal control problem for the system, an algorithm to the minimization problem with the LMI constraints is proposed, in which an optimal controller and the weight matrices of the linear quadratic performance index can be obtained. Finally, one numerical example is provided to demonstrate the effectiveness and correctness of the proposed results.