Design methodology for tracking certain and uncertain non-minimum phase systems

Abstract

A design methodology is developed for obtaining tracking controllers for non-minimum phase systems. The discussion centers around a solution for the error equation E(s) = [1- H(s)] Yd(S)'Tracking error is eliminated by making [1-H(s)] orthogonal to Yd(S)-Perfect tracking is achieved and demonstrated for systems that do not have plant uncertainty. Examples are provided to show how to select a feedforward controller which will make [1-H(s)] orthogonal to Yd(S). Feedforward controllers can be selected to perfectly track any combination of exponential, polynomial and sinusoidal input signals. The introduction of plant uncertainty complicates the design process because the perfect-tracking feedforward controller only provides perfect tracking for the nominal system. In the presence of plant uncertainty, robust tracking performance is determined by sensitivity reduction only provided by the feedback controller. Magnitude bounds are developed which guarantee robust tracking for uncertain systems. Phase bounds are also developed which insure robust stability. These bounds are used as guides for designing a feedback controller. QFT is used as the robust design tool because it clearly shows the effect of the non-minimum phase zeros and plant uncertainty on robust tracking performance. Examples are given which show that non-minimum phase systems have inherent performance limitations which are compounded when uncertainty is introduced.