H{sub}∞ Design With First Order Controllers

摘要

This paper considers the problem of determining the complete set of first order controller parameters for which the frequency weighted H{sub}∞norm of some closed loop transfer function is less than a specified constant and the closed loop system is stable. The results apply to single-input single-output, linear, time invariant plants of arbitrary order. The problem of determining all first order controllers (C(s) = (x{sub}1 s+x{sub}2)/(s+x{sub}3)) which stabilize such a plant has been recently solved in [10]. In this paper, these results are extended to determine the subset of controllers which also satisfy various robustness and performance specifications which can be formulated as specific H{sub}∞ norm constraints. The problem is solved by converting the H{sub}∞ problem into the simultaneous stabilization of the closed loop characteristic polynomial and a family of related complex polynomials. The stability boundary of each of these polynomials can be computed explicitly for fixed x{sub}3 by solving linear equations. The union of the resulting stability regions yields the set of all x{sub}1 and x{sub}2 which simultaneously satisfy the H{sub}∞ condition and closed loop stability for a fixed x{sub}3. The entire three dimensional set meeting specifications is obtained by sweeping x{sub}3 over the stabilizing range.