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 tomaximize the stability margin. Numerical examples are provided to demonstrate the elaboration.