Robust control under parametric uncertainty: An overview and recent results

Modern Robust Control has had two distinct lines of development: (a) Robustness through quadratic optimization and (b) Robustness under parametric uncertainty. The first approach consists of Kalman's Linear Quadratic Regulator and H∞ optimal control. The second approach is the focus of this overview paper. It provides an account of both analysis as well as synthesis based results. This line of results was sparked by the appearance of Kharitonov's Theorem in the early1980s. This result was rapidly followed by further results on the stability of polytopes of polynomials such as the Edge Theorem and the Generalized Kharitonov Theorem, stability of systems under norm bounded perturbations and the computation of parametric stability margins. Many of these analysis results established extremal testing sets where stability or performance would breakdown. Starting in 1997, when it was established that high order controllers were fragile, attention turned to the synthesis and design of the parameters of low order controllers such as three term controllers and more particularly Proportional-Integral-Derivative (PID) controllers. An extensive theory of design of such systems has developed in the last twenty years. We provide a summary without proofs, of many of these results.