Gain margin and Lyapunov analysis of discrete‐time network synchronization via Riccati design

Summary This paper deals with solution analysis and gain margin analysis of a modified algebraic Riccati matrix equation, and the Lyapunov analysis for discrete‐time network synchronization with directed graph topologies. First, the structure of the solution to the Riccati equation associated with a single‐input controllable system is analyzed. The solution matrix entries are represented using unknown closed‐loop pole variables that are solved via a system of scalar quadratic equations. Then, the gain margin is studied for the modified Riccati equation for both multi‐input and single‐input systems. A disc gain margin in the complex plane is obtained using the solution matrix. Finally, the feasibility of the Riccati design for the discrete‐time network synchronization with general directed graphs is solved via the Lyapunov analysis approach and the gain margin approach, respectively. In the design, a network Lyapunov function is constructed using the Kronecker product of two positive definite matrices: one is the graph positive definite matrix solved from a graph Lyapunov matrix inequality involving the graph Laplacian matrix; the other is the dynamical positive definite matrix solved from the modified Riccati equation. The synchronizing conditions are obtained for the two Riccati design approaches, respectively.