H∞ 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_∞ 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) = (x₁s+x₂)/(s+x₃)) 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_∞ norm constraints. The problem is solved by converting the H_∞ 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 x₃ by solving linear equations. The union of the resulting stability regions yields the set of all x₁ and x₂ which simultaneously satisfy the H_∞ condition and closed loop stability for a fixed x₃. The entire three dimensional set meeting specifications is obtained by sweeping x₃ over the stabilizing range.