Design of multivariable linear‐quadratic controllers using transfer functions

TLDR: In this paper, a discrete equivalent of the Wiener-Hopf equation is used to design a linear-quadratic (LQ) controller using a spectral factorization approach.

Abstract: Optimal linear-quadratic (LQ) controller design is usually associated with state space techniques. However, when one has measurements of the outputs to be controlled, there are many advantages to designing these LQ controllers using input-output transfer function models. The design procedure leads to a discrete equivalent of the Wiener-Hopf equation, which can be solved using a spectral factorization approach. In this paper the design procedure is presented and various interpretations of the resulting controllers are discussed. In particular, the controllers are shown to be of the internal model controller (IMC) form, and the Wiener-Hopf procedure is shown to be a powerful way of selecting approximate model inverses and filters that yield good performance and robustness characteristics. The approach treats the problem of simultaneous disturbance rejection and set-point tracking, and it easily handles nonsquare systems. The design approach, its performance/robustness trade-offs, and the structure of the resulting controllers are demonstrated using models for several processes, including a two-input/one-output sheet forming process, a (3 × 3) multivariable level control problem, and a (2 × 2) multivariable catalytic reactor.