A small gain condition for interconnections of ISS systems with mixed ISS characterizations

Abstract

We consider interconnected nonlinear systems with external inputs, where each of the subsystems is assumed to be input-to-state stable (ISS). Sufficient conditions of small gain type are provided guaranteeing that the interconnection is ISS with respect to the external input. To this end we extend recently obtained small gain theorems to a more general type of interconnections. The small gain theorem provided here is applicable to situations where the ISS conditions are formulated differently for each subsystem and are either given in the maximization or the summation sense. Furthermore it is shown that the conditions are compatible in the sense that it is always possible to transform sum formulations to maximum formulations without destroying a given small gain condition. An example shows the advantages of our results in comparison with the known ones.

A cyclic small-gain condition and an equivalent matrix-like criterion for iISS networks

“…For the maximum formulation (4) of iISS subsystems, this paper develops necessary conditions as well as an iISS small-gain criterion which is a sufficient condition for the stability of iISS networks. This paper also demonstrates that the small-gain criterion is equivalently expressed by a matrix-like condition generalizing an ISS result [6], [4]. The allowable number of non-ISS subsystems for stability of the network is discussed in terms of some necessary conditions.…”

“…The ISS small-gain theorem establishes stability of interconnected systems if "large" nonlinear gain of one subsystem is compensated by "small" nonlinear gain of the other subsystem [15], [29]. Even if the number of subsystems is more than two, the idea still remains valid [5], [16], [18], [6], [4], [17], [30]. Due to the conservation principle underlying natural dynamics of systems, saturated energy decrease in one part is balanced by saturated energy increase in another part.…”

“…which is the weighted maximum of Lyapunov functions V i of individual subsystems [14], [16], [18], [6], [4]. Weights are W i .…”

Numerical construction of LISS Lyapunov functions under a small-gain condition

“…Many available small-gain results state that ISS for the overall system follows from the existence of a so-called -path with respect to μ [4][5][6][7][8]21]. Furthermore, an ISS Lyapunov function for the interconnected system can be constructed using this path and the ISS Lyapunov functions of the subsystems.…”

“…Standard examples of such functions are summation and maximization, but in [8] some examples are provided that also other types of aggregation functions may be useful depending on the system under consideration. We would like to point out that the particular relation of the maximization and summation formulation of small-gain conditions is analyzed in [5]. In [16] the authors study interconnections where small-gain conditions are satisfied after certain transient periods and derive stability results.…”