@article{277a4805f18b427392814e021df2cd68, title = "Gain margin and Lyapunov analysis of discrete-time network synchronization via Riccati design", abstract = "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. {\textcopyright} 2023 John Wiley & Sons Ltd.", keywords = "directed graph, gain margin, Lyapunov function, multi-agent network, Riccati equation", author = "Liangyin Zhang and Chen, {Michael Z. Q.} and Yun Zou and Guanrong Chen", year = "2023", month = sep, day = "4", doi = "10.1002/rnc.6974", language = "English", journal = "International Journal of Robust and Nonlinear Control", issn = "1049-8923", publisher = "Wiley Blackwell", }