A spectral characterization of H∞-optimal feedback performance and its efficient computation

Abstract

We study the spectral properties of a ‘Toeplitz + Hankel’ operator which arises in the context of the mixed-sensitivity H ∞-optimization problem and whose largest eigenvalue characterizes the optimal achievable performance ϵ 0. The existence of such an operator was first shown by Verma and Jonckheere [26], who also'noted the potential numerical advantage of computing eo through its eigenvalue characterization rather than through the ϵ-iteration. Here, we investigate this operator in detail, with the objective of efficiency computing its spectrum. We define an ‘adjoint’ linear-quadratic problem that involves the same ‘Toeplitz + Hankel’ operator, as shown by Jonckheere and Silverman [13–16]. Consequently, a finite polynomial algorithm allows ϵ 0 to be characterized as simply as the largest root of a polynomial. Finally, a computationally more attractive state space algorithm emerges from the H t8 /LQ relationship. This algorithm yields a very good accuracy evaluation of the performance ϵ 0 by solving just one algebraic Riccati equation. Thorough exploitation of this algorithm results in a drastic computation reduction with respect to the standard e-iteration.