String stability of interconnected systems

TLDR

String stability denotes uniform boundedness of all states in an infinite interconnection of nonlinear systems when initial states are uniformly bounded, and it is known that subsystems with input‑output gains less than one guarantee input‑output stability. The authors derive sufficient weak‑coupling conditions that guarantee asymptotic string stability for a class of interconnected nonlinear systems. Under the weak‑coupling conditions, string‑stable systems remain stable under small structural or singular perturbations, and with parameter mismatch, the conditions guarantee uniform boundedness of all subsystem states under a gradient‑based adaptation law, leading to asymptotic convergence to zero.

Abstract

Introduces the notion of string stability of a countably infinite interconnection of a class of nonlinear systems. Intuitively, string stability implies uniform boundedness of all the states of the interconnected system for all time if the initial states of the interconnected system are uniformly bounded. It is well known that the input output gain of all the subsystems less than unity guarantees that the interconnected system is input-output stable. The authors derive sufficient ("weak coupling") conditions which guarantee the asymptotic string stability of a class of interconnected systems. Under the same "weak coupling" conditions, string-stable interconnected systems remain string stable in the presence of small structural/singular perturbations. In the presence of parameter mismatch, these "weak coupling" conditions ensure that the states of all the subsystems are all uniformly bounded when a gradient-based parameter adaptation law is used and that the states of all the systems go to zero asymptotically.