H∞ controllers for unstable distributed plants

This paper presents a simple formula for two block H^∞ optimal and suboptimal controllers for unstable SISO distributed plants, with rational weighting functions. The controller is expressed in terms of (i) inner and outer parts of the plant, (ii) a finite dimensional spectral factor obtained from the weighting functions, and (iii) a rational function satisfying certain interpolation conditions. Under certain genericity assumptions, this rational function is of dimension less than or equal to n₁+l-1 (n₁+l in the suboptimal case), where l is the number of unstable poles of the plant and n₁ is the order of the sensitivity weighting function. There are 2(n₁+l) (2(n₁+l+1) in the suboptimal case) linear equations, which determine this rational function. These linear equations can be written directly from the structure of the controller. A delay system example is also given