H/sub /spl infin// design with first order controllers

This paper considers the problem of determining the complete set of first order controller parameters for which the frequency weighted H/sub /spl infin// 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 [R.N. Rantaris et al., 2002]. 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 /spl infin// norm constraints. The problem is solved by converting the H/sub /spl infin// 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 /spl infin// 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.