Structured Lyapunov functions for synchronization of identical affine-in-control agents—Unified approach

• Structured Lyapunov functions for distributed synchronization of linear systems.
• Lyapunov analysis of nonlinear problems on reducible graphs having a spanning tree.
• Structured Lyapunov functions are applied to affine-in-control systems; global results.
• Cooperative stability conditions for affine-in-control single-agents.
• Crucial contraction properties of single-agents needed for cooperative stability.

This paper brings structured Lyapunov functions guaranteeing cooperative state synchronization of identical agents. Versatile synchronizing region methods for identical linear systems motivate the structure of proposed Lyapunov functions. The obtained structured functions are applied to cooperative synchronization problems for affine-in-control nonlinear agents. For irreducible graphs a virtual leader is used to analyze synchronization. For reducible graphs a combination of cooperative tracking and irreducible graph cooperative synchronization is used to address cooperative dynamics by Lyapunov methods. This provides a connection between the synchronizing region analysis, incremental stability and Lyapunov cooperative stability conditions. A class of affine-in-control systems is singled out based on their contraction properties that allow for cooperative stability via the presented Lyapunov designs.