Robustness and fragility of high order controllers: A tutorial

Abstract

In this tutorial paper we begin by providing a brief history of the field of classical robust control, namely, Black's feedback amplifier, the Nyquist criterion and Bode's frequency domain methods. In this classical period there was an underlying emphasis on design for robustness measured via gain and phase margins in the complex plane or in terms of frequency response plots. In the modern period (post 1960) the design emphasis shifted to quadratic optimization, state space models and state feedback and observers. Stability was obtained automatically and Kalman showed that as far state feedback was concerned excellent universal gain and phase margins were obtained as a byproduct of optimization. It took almost twenty years for the community to realize that these margins could disappear under output feedback implementations. The response of researchers, to this realization, was to develop design methods that accounted for plant uncertainty from the outset. This led to the development of H1 control theory. In 1997 the present authors showed that controllers designed by these methods could and frequently would, produce high order controllers which were fragile in the sense that minuscule perturbations of the controller would destabilize the closed loop system. This second failure of quadratic optimization to deliver robustness, provoked a resurgence of interest in low order output feedback and Proportional Integral Derivative (PID) controllers. In the last twenty years the beginnings of a modern approach to PID control has developed incorporating the efficient computation of stabilizing sets, achievable performance in terms of gain and phase margins, multi-objective design and finally exact design of low order multivariable controllers using single-input single-output (SISO) methods. These are briefly introduced in the present paper and elaborated on in the succeeding papers of this session.