LMI-Based Lower Bound Analysis of the Best Achievable H∞ Performance for SISO Systems

Abstract

: In this paper, we study H ∞ performance limitation analysis for continuous-time SISO systems using LMIs. By starting from an LMI that characterizes a necessary and sufficient condition for the existence of desired controllers achieving a prescribed H ∞ performance level, we represent lower bounds of the best H ∞ performance achievable by any LTI controller in terms of the unstable zeros and the unstable poles of a given plant. The transfer functions to be investigated include the sensitivity function (1 + PK) −1 , the complementary sensitivity function (1 + PK) −1 PK, and(1 + PK) −1 P, the first and the second of which are well investigated in the literature. As a main result, we derive lower bounds of the best achievable H ∞ performance with respect to (1 + PK) −1 P assuming that the plant has unstable zeros.More precisely, we characterize a lower bound in closed-form by means of the first non-zero coefficient of the Taylor expansion of the plant P(s) around its unstable zero.

Reduction of SISO H-infinity Output Feedback Control Problem

“…Furthermore, this study unifies some of the existing work [8,9,10]. The work [8] obtained a lower bound of the H ∞ performance limitations of (1 + P K) −1 P , where P and K are transfer functions of a SISO linear time-invariant system and a controller, respectively. This lower bound was obtained from a detailed analysis of the resulting LMI problem.…”

“…In this study, we extend the analysis obtained in [10] and provide the performance limitation for a more general SISO H ∞ output feedback control problem. The analysis in [8] for the dual problems can be regarded as facial reduction.…”