On a robust controller design using a minimax optimization approach

Abstract

An extension of the Linear Quadratic Regulator, and Linear Quadratic Gaussian methodology is proposed to design a robust controller from the viewpoint of eigenvalue sensitivity. The resulting system is optimal under the worst case of norm-bounded initial condition uncertainty, norm-bounded deterministic noise, and norm-bounded parameter uncertainty. The uncertainties are modelled such that we can formulate the problem as a zero-sum, two-sided minimax game. The minimax solution is proved to be a saddle point solution under either the positivity of control-disturbance authority, or the strict concavity of the open loop system. The existence of the solution is proven by using the successive approximation technique. The theory considers both finite dimensional and infinite dimensional systems. By using the trace operator of the product of two Hilbert-Schmidt operators, all the matrix theory in the finite dimensional minimax control theory can be extended to distributed parameter systems. A finite dimensional approximation scheme also is developed. It is proven that the minimax solution of the infinite dimensional problem can exactly be specified by a convergent sequence of minimax solutions of finite dimensional approximations of the problem. Examples that provide verification of the theory presented in this dissertation include (1) scalar minimax controller design, (2) finite dimensional output feedback minimax robust controller design, and (3) minimax controller design of a parabolic heat problem by spline approximation. The contribution of the dissertation is (1) to broaden the application of minimax control theory and make it a straightforward engineering tool for robust controller design by using eigenvalue sensitivity approach and (2) to show the feasibility of finite dimensional approximation of the worst case distributed parameter system robust control.