String stability of interconnected systems

In this paper we introduce 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 I/O gain of all the subsystems less than unity guarantees that the interconnected system is I/O stable. We derive sufficient (“weak coupling”) conditions which guarantee 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 gradient based parameter adaptation law is used and that they go to zero asymptotically