Dual LMI approach to H_∞ performance limitation analysis of sensitivity and complementary sensitivity functions

In this paper, we study a dual-LMI-based approach to H_∞ performance limitation analysis of SISO systems. The scope includes the analysis of the sensitivity function S = (1 + PK)^-1 and the complementary sensitivity function T = (1 + PK)^-1 PK where P and K stand for the plant and the controller, respectively. The H_∞ performance limitations for these transfer functions are well investigated, and exact closed-form performance bounds are already known for the cases where the plant has the sole unstable zero (i.e., non-minimum phase zero) of degree one or the sole unstable pole of degree one. The goal of this paper is to show that such exact bounds can be reproduced by a dual LMI approach. To this end, we study a Lagrange dual of the standard SDP that is usually used to design H_∞ optimal controllers by numerical computation. By characterizing the structure of dual feasible solutions in terms of unstable zeros and unstable poles of the plant, we clarify that we can construct an optimal solution for the dual SDP analytically. It follows that we obtain exact H_∞ performance bounds that are consistent with the known results.