An ISS small gain theorem for general networks

Abstract

We provide a generalized version of the nonlinear small-gain theorem for the case of more than two coupled input-to-state stable (ISS) systems. For this result the interconnection gains are described in a nonlinear gain matrix and the small-gain condition requires bounds on the image of this gain matrix. The condition may be interpreted as a nonlinear generalization of the requirement that the spectral radius of the gain matrix is less than one. We give some interpretations of the condition in special cases covering two subsystems, linear gains, linear systems and an associated artificial dynamical system.Keywords Interconnected systems -input-to-state stability -small-gain theorem -large-scale systems -monotone maps MSC-classification: 93C10 (Primary) 34D05, 90B10, 93D09, 93D30 (Secondary)

The converging-input converging-state property for Lur’e systems

“…Indeed, it is possible that, for fixed ξ , the right-hand side of (4.27) is converging to 1 as z → 0 or z → ∞. Therefore, rather than comparing Corollary 4.11 with classical small-gain theorems [11,13,14,23,45], it is more appropriate to view it in the context of "modern" nonlinear ISS small-gain results, see for example [10,21,36,43].…”

Numerical ISS controller design via a dynamic game approach

“…Remark 13. Moreover, we intend to address the extension of our analysis to event based ISpS feedback laws and the proposed computational approach for ISpS controllers will be used as a building block for a distributed feedback design for large networks of systems based on the small gain arguments from [2], [3].…”

“…Besides yielding a theoretically sound concept for the quantitative and qualitative analysis of stability of nonlinear systems under perturbations, one of its particular features is the possibility to analyze the stability of interconnected systems by means of analyzing low dimensional subsystems via ISS small gain arguments, see, e.g., [16], [2], [3]. The latter particularly allows for a rigorous approach to decentralized controller design by designing input-to-state stabilizing controllers.…”

Application of the LISS Lyapunov-Krasovskii small-gain theorem to autonomously controlled production networks with time-delays

“…, n in the Definition 2.4, then V i is called an ISS LyapunovKrasovskii functional of the ith subsystem. The (global) small-gain condition (SGC) is denoted as More details about the global SGC can be found in [7], [14]. Note that (7) is equivalent to the cycle condition (see [14], Lemma 2.3.14): for all (k 1 , ..., k p ) ∈ {1, ..., n} p , where…”

“…Taking a network of many subsystems into account, it was proved in [6], that the whole network is stable provided that each subsystem has an ISS Lyapunov-Razumikhin function or ISS Lyapunov-Krasovskii functional and that the property of the small-gain condition in matrix form (see [7]) is satisfied. The used matrix in this condition describes the interconnection structure of the network.…”