On the Stability Margin of Networked Dynamical Systems

Abstract

This paper is concerned with the stability (gain and phase) margin of networked dynamical systems, e.g., vehicles in formation, each of which has access to the state of its neighbors and subsequently uses a state feedback gain $F$ for a certain global objective such as attitude synchronization. Here, the network topology is directed and described by a generalized Laplacian matrix $L$ . An individual dynamical system can adopt its own state feedback control law such as a linear-quadratic-regulator controller for an ample stability margin, but it may lose the stability margin to a great extent when the same control strategy utilizing relative state information is used after being interconnected with other dynamical systems. This paper reveals and elaborates upon the following four facts: First, the stability margin after interconnection is quantified via the minimum singular value of a frequency-dependent matrix made up of $F$ and $L$ ; Second, the stability margin of a networked dynamical system having a pole at the origin is at most the inverse of the zero-eigenvalue sensitivity of $L$ ; Third, there exists an upper bound of the stability margin that has a computational merit, and asymptotically converges to the exact margin with respect to network size, probability of link existence, and control gain in a random network setting; and finally, $L$ can be designed to maximize the stability margin. Numerical examples are provided to demonstrate the elaboration.