Robustness Analysis of Uncertain and Nonlinear Systems

Abstract: Control design is often done based on simplified models. After design it is necessary to verify that the real closed loop system behaves well. This is mostly done by experiments and simulations. Theoretical analysis is an important complement to this that can help to verify critical cases. Structural information about uncertainties, time-variations, nonlinearities, and signals can be described by integral quadratic constraints. The information provided by these constraints can be used to reduce conservatism in analysis of robust stability and robust performance. This thesis treats several aspects of this method for robustness analysis. It is shown how the Popov criterion can be used in combination with other integral quadratic constraints. A new Popov criterion for systems with slowly time-varying polytopic uncertainty is obtained as a result of this. A corresponding result for systems with parametric uncertainty is also derived. The robustness analysis is in practice a problem of finding the most appropriate integral quadratic constraint. This can be formulated as a convex but infinite-dimensional optimization problem. The thesis introduces a flexible format for computations over finite-dimensional subspaces. The restricted optimization problem can generally be formulated in terms of linear matrix inequalities. Duality theory is used to obtain bounds on the computational conservatism. A class of problems is identified for which the dual is particularly attractive.