Robust Quadratic Stability of Two Interconnected Systems with Respect to the Structure of a Lyapunov Function

In this paper we consider a system formed by the interconnection of two linear time-invariant systems as the simplest example of networked dynamical systems. The interconnection between a plant and a controller, which is present in virtually any controlled system, is also an example of such a network. We show that if such a network exhibits certain robust stability properties, then there necessarily exists a specifically structured Lyapunov function that certifies the stability of the network. More precisely, the robustness property we consider is stability robustness (quadratic stability) with respect to an uncertainty in the interconnection channels between the systems. This interconnection uncertainty also accounts for the case when the two systems are disconnected. The considered class of Lyapunov functions is characterized by an additive structure in the sense that the Lyapunov function for the network is composed as a sum of two quadratic terms, where each term is a function of a state vector of one system only. Previously, it was shown in the literature that the existence of such structured Lyapunov functions implies the robustness of the network. The main contribution of this paper is a proof of the converse statement: if the network composed of two systems is robustly stable (in the sense of quadratic stability), then it necessarily admits a Lyapunov function with an additive structure.