Robustly and strongly stabilizing low order controller design for infinite dimensional systems

Abstract

This thesis deals with the robust stabilization of in nite dimensional systems by stable and low order controllers. The close relation between the Nevanlinna-Pick interpolation problem and the robust stabilization is well known in the literature. In order to utilize this relation, we propose a new optimal solution strategy for the Nevanlinna-Pick interpolation problem. Di erently from the known suboptimal solutions, our method includes no mappings or transformations, it directly solves the problem in the right half plane. We additionally propose a method via suboptimal solutions of an associated Nevanlinna-Pick interpolation problem to robustly and strongly stabilize a set of plants which include the linearized models of two well known under actuated robots around their upright equilibrium points. In the literature, it is shown that the robust stabilization of an in nite dimensional system by stable controllers can be reduced to a bounded unit interpolation problem. In order to use this approach to design a nite dimensional controller, we propose a predetermined structure for the solution of the bounded unit interpolation problem. Aforementioned structure reduces the problem to a classical Nevanlinna-Pick interpolation problem which can be solved by the optimal solution strategy of this thesis. Finally, by combining the nite dimensional solutions of the bounded unit interpolation problem with the nite dimensional approximation techniques, we propose a method to design nite dimensional and stable controllers to robustly stabilize a given plant. Since time delay systems are one of the best examples of in nite dimensional systems, we provide numerical examples of various time delay systems for each proposed method.