Unified Necessary and Sufficient Conditions for the Robust Stability of Interconnected Sector-Bounded Systems

Abstract

Classical conditions for ensuring the robust stability of a linear system in feedback with a sector-bounded nonlinearity include small gain, circle, passivity, and conicity theorems. In this work, we present a similar stability condition, but expressed in terms of relations defined on a general semi-inner product space. This increased generality leads to a clean result that can be specialized in a variety of ways. First, we show how to recover both sufficient and necessary-and-sufficient versions of the aforementioned classical results. Second, we show that suitably choosing the semi-inner product space leads to a new necessary and sufficient condition for weighted stability, which is in turn sufficient for exponential stability. Finally, in the spirit of classical robust stability analysis, we provide linear matrix inequalities that allow for the efficient verification of the conditions of our theorem.

Generalized Necessary and Sufficient Robust Boundedness Results for Feedback Systems

“…An earlier version of Theorem 1 [8] used the S-lemma and therefore assumed linearity and was non-constructive.…”

“…354] and using a technique similar to that used in [15]. Details of this approach may be found in [7,8].…”

“…The operators G and Φ may be nonlinear or even unbounded. The requirements (8) and (10) involve the truncated T norms, so they are defined even for unbounded signals. 8 Critically, the conclusion (11) states that when u ∈ L 2 , we have y ∈ L 2 , which is a statement about input-output stability.…”

“…By Proposition 25, we have y T ≤ γ u T . Apply Theorem 1 as before to obtain (8). From the way N is constructed in the proof of Theorem 1 (Step 5 in Appendix A.1), N is independent of T and therefore the same N may be used for all T .…”