Polynomial control: Past, present, and future

Polynomial techniques have made important contributions to systems and control theory. Engineers in industry often find polynomial and frequency domain methods easier to use than state equation-based techniques. Control theorists show that results obtained in isolation using either approach are in fact closely related. Polynomial system description provides input-output models for linear systems with rational transfer functions. These models display two important system properties, namely poles and zeros, in a transparent manner. A performance specification in terms of polynomials is natural in many situations; see pole allocation techniques. A specific control system design technique, called polynomial equation approach, was developed in the 1960s and 1970s. The distinguishing feature of this technique is a reduction of controller synthesis to a solution of linear polynomial equations of a specific (Diophantine or Bezout) type. In most cases, control systems are designed to be stable and meet additional specifications, such as optimality and robustness. It is therefore natural to design the systems step by step: stabilization first, then the additional specifications each at a time. For this it is obviously necessary to have any and all solutions of the current step available before proceeding any further. This motivates the need for a parametrization of all controllers that stabilize a given plant. In fact this result has become a key tool for the sequential design paradigm. The additional specifications are met by selecting an appropriate parameter. This is simple, systematic, and transparent. However, the strategy suffers from an excessive grow of the controller order. This article is a guided tour through the polynomial control system design. The origins of the parametrization of stabilizing controllers, called Youla-Kucera parametrization, are explained. Standard results on reference tracking, disturbance elimination, pole placement, deadbeat control, H, control, l(1) control and robust stabilization are summarized. New and exciting applications of the Youla-Kucera parametrization are then discussed: stabilization subject to input constraints, output overshoot reduction, and fixed-order stabilizing controller design. Copyright (c) 2006 John Wiley & Sons, Ltd.