Guaranteed cost control for linear periodically time-varying systems with structured uncertainty and a generalized H2 objective

This paper considers the problem of finding a controller for a discrete time linear periodically time-varying system with structured parametric uncertainty which achieves robust ℓ2ℓ2 semi-norm performance (the equivalent of robust H2H2 norm performance for time-varying systems). This problem is shown to be a generalization of a guaranteed cost control problem when the structure of the uncertainty is neglected. To reduce the conservatism when the uncertainty structure is considered, static uncertainty scalings such as those used in the D-K iteration heuristic for μμ-synthesis are introduced. Although the problem of simultaneously optimizing the controller and uncertainty scalings in non-convex, it is shown that it lends itself to a solution methodology conceptually similar to D-K iteration in which each step is a convex optimization. To demonstrate the effectiveness of the developed methodology, the control of a set of hard disk drives with multirate sampling characteristics and uncertain parameters is considered and a controller is designed which minimizes the worst-case ℓ2ℓ2 semi-norm performance of the system. It is then shown that the resulting robust controller achieves worst-case ℓ2ℓ2 semi-norm performance which is comparable to the best achievable ℓ2ℓ2 semi-norm performance for the nominal system.