A Study of Optimality in the H1 Loop-shaping Design Method

This dissertation deals with the problem of characterizing the controllers which are potentially optimal in the robustness optimization for (weighted) normalized coprime factor/gap metric uncertainty, which is the basis for the H1 Loop Shaping Design Method. This dissertation considers single-input-single-output systems. Given a plant P and a stabilizing controller C, we ask if C can be obtained from the optimization procedure for some choice of weighting function. We derive necessary and su cient conditions for optimality which involve right half plane pole/zero counts and a certain winding number test based on the Nyquist diagram of PC. It turns out that the conditions for optimality do not require the explicit computation of a potential weighting function. The results give a characterization of this class of H1 optimal designs in the language of classical control. We then examine some optimality properties of the optimal robustness problem for weighted normalized coprime factor/gap metric uncertainty. We give some lower and upper bounds of the open loop shape jPCj and some closed-loop performance objectives when the optimality is achieved. Furthermore, we show that certain weighted mixed sensitivity problem under some assumptions is equivalent to optimal robustness problem for weighted normalized coprime factor/gap metric uncertainty. Finally, we consider the issue whether the classical compensator can be an H1 optimal controller in the H1 loop-shaping design. We show by example that it is di cult or sometimes impossible to use suitable weighting function to obtain a classical compensator directly from H1 loop-shaping design.