Convex Duality Approach to Robust Stabilization of Uncertain Plants

Abstract: In this thesis we are study the problem of designing the controllers that are robust with respect to the parametric uncertainty. In Part I "The Rank-One Problem" we consider the class of systems with restriction that the structure of uncertainty is limited to a vector. In Chapter " Canonical Parametrization of the Dual Problem in Robust Optimization: Non-Rational Case" we extend the class of allowed systems. The main result is the canonical parametrization of all destabilizing uncertainties in the dual problem. The corresponding result in the rational case was previously stated in terms of unstable zero-pole cancellations. For non-rational systems the situation with common zeros is more complicated. The nominal factors can contain a singular component and cannot be treated by unstable cancellations. We have shown that in the general case the common zeros of the plant factors are naturally replaced by a scalar function with the positive winding number. To illustrate the duality principle, the result is applied to a system with delay. By dual parametrization we can easily calculate the optimal uncertainty bound and the optimal controller. Since the optimal controller is not robustly stabilizing in the strong sense,as it is only a limit of suboptimal robustly stabilizing controllers,we have to regularize the limiting controller. In Chapter "Regularization of the Limiting Optimal Controller in Robust Stabilization" we present a method of obtaining the suboptimal controller of lower order that provides the stability margin as close to the optimal one as we wish. The method is illustrated with some scalar examples. In Chapter "Robust Control via Linear Programming" we propose the numerical algorithm for the optimal robust control synthesis. The algorithm proposed is a sequence of the standard linear programming problems of growing dimensions which approximate the initial problem. In the special case, when the uncertainty parameter is real-valued, it is shown that the initial problem can be considered as finite-dimensional in the space of variables. In Part II "Convex Duality: Matrix Case" we generalize the results to the system with matrix uncertainties. We obtain a canonical factorization of a plant with unstructured uncertainty in terms of an unitary matrix function with finite winding number and an outer matrix function. We introduce a metric in the space of factorization and discuss connection with nu-gap metric.