Wiener-Hopf design of optimal decoupling one-degree-of-freedom controllers

Abstract

This reasonably self-contained paper is directed to the design of a multivariable optimal one-degree-of-freedom feedback loop which incorporates a decoupling controller in the forward path. The criterion for optimality is a quadratic-cost functional that penalizes both tracking error and saturation. The controllers on which optimization is based are general enough to allow for non-unity feedback and rectangular plant transfer matrices possessing normal row rank. Nevertheless, for the sake of brevity and clarity, attention is focused mainly on the square case. Earlier treatments of the problem have employed multi-degree-of-freedom controllers. The solution we present for the one-degree-of-freedom case is considerably more difficult to obtain, especially when saturation is taken into account. Explicit formulas are derived for the set of all decoupling controllers yielding finite cost, as well as those that are optimal. It is shown that these controllers are strictly-proper under conditions usually prevailing in practice. Four fully worked examples serve to illustrate many important numerical aspects of the theory and all major proofs are transferred to the Appendix.