Lyapunov, Adaptive, and Optimal Design Techniques for Cooperative Systems on Directed Communication Graphs

TLDR

A Lyapunov equation for digraphs shows that cooperative control results for undirected or balanced digraphs extend to strongly connected digraphs. The paper introduces three design techniques—Lyapunov, neural adaptive, and LQR-based optimal—for cooperative control of multiagent systems on directed graphs. Neural adaptive control addresses tracking of nonlinear networked systems with unknown dynamics, while the LQR-based optimal design applies to linear and two-mass–spring systems, enabling topology-independent gain selection. The methods achieve successful cooperative tracking in first- and high-order nonlinear systems, as demonstrated on coupled Lagrangian and FitzHugh–Nagumo models, and provide an unbounded synchronization region for linear systems with decoupled gain design.

Abstract

This paper presents three design techniques for cooperative control of multiagent systems on directed graphs, namely, Lyapunov design, neural adaptive design, and linear quadratic regulator (LQR)-based optimal design. Using a carefully constructed Lyapunov equation for digraphs, it is shown that many results of cooperative control on undirected graphs or balanced digraphs can be extended to strongly connected digraphs. Neural adaptive control technique is adopted to solve the cooperative tracking problems of networked nonlinear systems with unknown dynamics and disturbances. Results for both first-order and high-order nonlinear systems are given. Two examples, i.e., cooperative tracking control of coupled Lagrangian systems and modified FitzHugh–Nagumo models, justify the feasibility of the proposed neural adaptive control technique. For cooperative tracking control of the general linear systems, which include integrator dynamics as special cases, it is shown that the control gain design can be decoupled from the topology of the graphs, by using the LQR-based optimal control technique. Moreover, the synchronization region is unbounded, which is a desired property of the controller. The proposed optimal control method is applied to cooperative tracking control of two-mass–spring systems, which are well-known models for vibration in many mechanical systems.