H<inf>∞</inf> performance limitations analysis for SISO systems: A dual LMI approach

Abstract

In a very recent study by the first author and his colleagues, a novel LMI-based approach has been proposed to the best achievable H ∞ performance limitations analysis for continuous-time SISO systems. Denoting by P and K a plant and a controller, respectively, and assuming that the plant P has an unstable zero, it was shown that a lower bound of the best achievable H ∞ performance with respect to the transfer function (1+ P K) −1 P can be given analytically in terms of the real part of the unstable zero and the first non-zero coefficient of the Taylor expansion of P around the unstable zero. The goal of this paper is to show that, if the plant P has no unstable zeros except for the sole real unstable zero of degree one, then the lower bound shown there is exact. The exactness proof relies on the detailed analysis on the Lagrange dual problem of the SDP characterizing H ∞ optimal controllers. More precisely, we show that we can construct an optimal dual solution proving the exactness analytically in terms of the state-space matrices of the plant P and the unstable zero. This analytical expression of the dual optimal solution is also a main result of this study.

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. The exactness of the lower bound was proved in [10] by using a property in the dual problem. This technique was also used in [9], which deals with the H ∞ performance limitations of sensitivity and complementary sensitivity functions for a SISO linear time-invariant system.…”

LMI-Based Lower Bound Analysis of the Best Achievable H&infin; Performance for SISO Systems

“…This tightness is also observed in our extensive numerical experiments on various plants with only one unstable zero z ∈ R + , posing the question of establishing a possible exactness proof. Indeed, very recently, we have succeeded in proving the exactness by a dual LMI approach [18]. On the other hand, for the cases (ii), (iii) and (iv) where the plant has multiple zeros (including complex conjugate ones and duplicated ones), we see from Table 1 that the lower bounds in Theorem 3 are not necessarily tight.…”

“…The other notations are standard. This paper is a refined version of the one presented in [10]. In the present paper we included a complete proof of Lemma 3 which is missing in [10].…”

“…This paper is a refined version of the one presented in [10]. In the present paper we included a complete proof of Lemma 3 which is missing in [10]. As clarified below, Lemma 3 plays an important role in deriving lower bounds of the best achievable H ∞ performance with respect to (1 + PK) −1 P particularly when the plant P has duplicated (higher degree) unstable zeros.…”