Synchronization Control of Networked Two-Timescale Dynamic Agents: A Singular Perturbation Approach

Kun Liang, Wangli He, Feng Qian, Guanrong Chen

Abstract: This article is intended to solve the synchronization control problem for a group of agents with two-timescale characteristic, described by singularly perturbed systems (SPSs) with a small singular perturbation parameter $\varepsilon$. Three fundamental and yet challenging questions are addressed: 1) how to design a distributed controller to guarantee synchronization of coupled two-timescale agents for $\varepsilon \in (0,\varepsilon ^{*})$, where the stability bound $\varepsilon ^{*}$ has to be determined?; 2) how to enlarge the stability bound $\varepsilon ^{*}$ for a given feedback gain matrix?; and 3) how to cope with the situation when the singular perturbation parameter exceeds the prescribed stability bound $\varepsilon ^{*}$? First, a decoupled method is applied to transfer the synchronization problem of networked two-timescale dynamic agents to the stability problem of SPSs associated with the eigenvalues of the network Laplacian matrix. Second, based on a specially constructed Lyapunov function, sufficient conditions are derived for simultaneously stabilizing the decoupled systems and computing $\varepsilon ^{*}$ by utilizing a particle swarm optimization (PSO)-assisted method. Third, with the derived feedback gain matrix, some criteria are established to further enlarge the stability bound $\varepsilon ^{*}$. Then, an integrated algorithm is developed to design a distributed controller with an even larger stability bound $\varepsilon ^{*}$. Fourth, a novel network design method, incorporated with the concept of synchronization region, is proposed as a powerful solution to deal with the case when the feedback gain matrix is incapable of stabilizing the system. Finally, three examples are given to verify the theoretical results.