All low order H∞ controllers with covariance upper bound

Abstract

This paper obtains a parametrization of the set of all stabilizing controllers of order equal to or less than the plant, which yield a specified H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> norm bound to the closed loop transfer matrix. A Lyapunov based approach to H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> control problems yields a parametrization in terms of the Lyapunov matrix which carries many system properties such as H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> performance, covariance bounds, system entropy at infinity, etc. Since the freedom in the parametrization is explicit in arbitrary matrices of fixed dimensions, the advantage over the existing Q-parametrization is the finiteness of the design parameter space. It is shown that low order H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> suboptimal controllers can be designed by solving two Riccati equations which are uncoupled in one direction. Perspectives on the new parametrization are discussed considering the simple full order controller case. A numerical example of robust control design with an H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> performance criterion is presented for a benchmark problem.