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

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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\varepsilon$</tex-math></inline-formula> . Three fundamental and yet challenging questions are addressed: 1) how to design a distributed controller to guarantee synchronization of coupled two-timescale agents for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\varepsilon \in (0,\varepsilon ^{*})$</tex-math></inline-formula> , where the stability bound <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\varepsilon ^{*}$</tex-math></inline-formula> has to be determined?; 2) how to enlarge the stability bound <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\varepsilon ^{*}$</tex-math></inline-formula> for a given feedback gain matrix?; and 3) how to cope with the situation when the singular perturbation parameter exceeds the prescribed stability bound <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\varepsilon ^{*}$</tex-math></inline-formula> ? 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\varepsilon ^{*}$</tex-math></inline-formula> 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\varepsilon ^{*}$</tex-math></inline-formula> . Then, an integrated algorithm is developed to design a distributed controller with an even larger stability bound <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\varepsilon ^{*}$</tex-math></inline-formula> . 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.