Singularly Perturbed Systems Divide Control

This paper deals with the reduced-order finite frequency H∞filtering problem for singularly perturbed systems. Necessary and sufficient conditions for the existence of solutions to continuous-time and discrete-time problems are derived in terms of linear matrix inequalities. Furthermore, a formula for all the desired reduced-order filters is given when these conditions are feasible. In order to alleviate ill-conditioned LMIs resulting from the interaction of slow and fast dynamic, the sufficient conditions which are independent of the singularly perturbationεare given. Since the separation of states into slow and fast ones is not involved, the results are applicable to not only standard, but also non-standard singularly perturbed systems.For the general linear systems, necessary and sufficient conditions for the existence of a stabilizing state feedback controller are given based on the generalized KYP lemma, and use the results to study singularly perturbed systems, a composite state feedback controller is constructed, which preserves the stability and positive real property.This paper also studies the model matching problem for SISO singularly perturbed systems in the (semi)finite frequency ranges. The necessary and sufficient conditions for the existence of an H∞sub-optimal controller are derived based on the generalized KYP lemma, moreover, the controller is also singularly perturbed. It is further shown that this controller is obtained through designing its slow and fast parts, and then the composite controller is established.Finally, based on the completely parametric approach to eigenstructure assignment, the problem of finite frequency H∞control for singularly perturbed systems with pole assignment constraint is studied. The purpose is to design a state feedback controller such that the eigenvalues of the closed-loop state matrix are at desired locations, and a prescribed H∞performance level is also achieved in the given low and high frequency ranges. It is further shown that the problem of full-order system could be break down into fast and slow subproblems, through designing state feedback controllers to solve the fast and slow subproblems, respectively, a composite controller is constructed on the basis of the above two controllers.