Fixed-order Controller Design for State Space Polytopic Systems by Convex Optimization

In this paper, a new method for fixed-order controller design of systems with polytopic uncertainty in their state space representation is proposed. The approach uses the strictly positive realness (SPRness) of some transfer functions, as a tool to decouple the controller parameters and the Lyapunov matrices and represent the stability conditions and the performance criteria by a set of linear matrix inequalities. The quality of this convex approximation depends on the choice of a central state matrix. It is shown that this central matrix can be computed from a set of initial fixed-order controllers computed for each vertex of the polytope. The stability of the closed-loop polytopic system is guaranteed by a linear parameter dependent Lyapunov matrix. The results are extended to fixed-order H infinity controller design for SISO systems.