J -lossless and extended J -lossless factorizations approach for @-domain H ∞ control

Abstract

The paper addresses problems of numerical conditioning of the discrete-time H X control design based on J -lossless factorizations of the chain-scattering representation of a plant. It is demonstrated that for a sufficiently small sampling period forward-shift operator techniques may become ill-conditioned and numerical robustness and reliability of design procedures can be significantly improved by utilizing a so-called i operator form of the source problem. State space models of all i -domain controllers are given. The solutions are obtained by considering two coupled i -domain algebraic Riccati equations that can be solved by applying a generalized eigenproblem formulation. Numerical conditioning of these solutions can effectively be evaluated by using a suitable relative condition number. An extended J -lossless factorization for generalized plants with zeros on the stability boundary circle is also given. A so-called zero compensator technique adapted to the i -domain models is utilized to cancel such zeros and some attempt is made to simplify the resulting controller. Two numerical examples are given to illustrate the properties of the approach. The first example deals with a simple problem of mixed sensitivity H X design. The second example concerns a method for robust pole placement design for a plant with zero located on the boundary of the stability region.