Controller Synthesis for Maximizing Disturbance Bounds in Linear Uncertain Systems

Abstract

A recently developed design methodology [1]-[4] for maximizing the disturbance bound for SISO systems with no uncertainty is extended to SISO systems with structured uncertainty. In particular, we consider systems of the formYs=Gus,αUs+Gws,αWswhere U(s) is the control input, W(s) is the disturbance, Y(s) is the output and a is a vector characterizing the bounded uncertain parameters. The objective is to synthesize a controller Gc(s) to (i) maximize the disturbance bound (ii) simultaneously satisfy hard magnitude constraints on the control, output and intermediate states and (iii) satisfy bandwidth constraints. The methodology proposed is based on pointwise design in the frequency domain. The plant uncertainty is characterized by templates that depend on the frequency. Since the magnitude and phase vary with both frequency and the uncertain parameters, a Nichols chart is utilized to represent the plant templates. These templates are generated by bounding the uncertain plant by the so-called Kharitonov polynomials. Controller synthesis is accomplished by loop shaping so that the nominal loop transfer function stays within the acceptable boundaries in the Nichols chart corresponding to each frequency. In order to generate the allowable design regions, the state/control constraints and bandwidth limitations are incorporated into a set of target transfer functions. These target transfer functions are chosen on the basis of a unit step distrubance. Selecting the targets is particularly easy if only disturbance rejection is considered. The entire design process utilizes classical control theory and has much in common with QFT [5]. The design procedure is illustrated by an example.