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

The problem of determining all first order controllers (C(s)=(x
₁
s+x
₂
/s+x
₃
)) which stabilize a given single-input-single-output (SISO) linear time-invariant (LTI) plant of arbitrary order has been recently solved. In this note, 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
_infin
norm constraints. The problem is solved by converting the H
_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
₃
by solving linear equations. The union of the resulting stability regions yields the set of all x
₁
and x
₂
which simultaneously satisfy the H
_infin
condition and closed-loop stability for a fixed x
₃
. The entire three-dimensional set meeting specifications is obtained by sweeping x
₃
over the stabilizing range