Distributed Adaptive Control of Disturbed Interconnected Systems with High-Order Tuners

TL;DR

This work develops a distributed adaptive control framework for leader–follower multi-agent systems comprising unknown, unstable linear subsystems under disturbances and limited communication. It proposes three distributed tuners, including two high-order tuners, and proves stability and asymptotic leader tracking using the Meyer–Kalman–Yakubovich (MKY) lemma and SPR properties, within a Lyapunov framework. The approach handles time-varying regressors and uses both gradient-based and high-order adaptive laws to update gains, validated on balanced directed graphs with Star-like, Cyclic-like, Path, and Random topologies. Numerical results demonstrate that the modified high-order tuners outperform gradient-based methods in $L_2$ and $L_\infty$ performance metrics, with topology-dependent trends and practical implications for network design and control gain selection.

Abstract

This paper addresses the challenge of network synchronization under limited communication, involving heterogeneous agents with different dynamics and various network topologies, to achieve consensus. We investigate the distributed adaptive control for interconnected unknown linear subsystems with a leader and followers, in the presence of input-output disturbance. We enhance the communication within multi-agent systems to achieve consensus under the leadership's guidance. While the measured variable is similar among the followers, the incoming measurements are weighted and constructed based on their proximity to the leader. We also explore the convergence rates across various balanced topologies (Star-like, Cyclic-like, Path, Random), featuring different numbers of agents, using three distributed algorithms, ranging from first- to high-order tuners to effectively address time-varying regressors. The mathematical foundation is rigorously presented from the network designs of the unknown agents following a leader, to the distributed methods. Moreover, we conduct several numerical simulations across various networks, agents and tuners to evaluate the effects of sparsity in the interaction between subsystems using the $L_2-$norm and $L_\infty-$norm. Some networks exhibit a trend where an increasing number of agents results in smaller errors, although this is not universally the case. Additionally, patterns observed at initial times may not reliably predict overall performance across different networks. Finally, we demonstrate that the proposed modified high-order tuner outperforms its counterparts, and we provide related insights along with our conclusions.

Figures (5)

• Figure 1: An example of an interconnected network of leader ${\ell}$ and $m= 9$ unknown unstable subsystems/followers.
• Figure 2: (a) Scheme proposed in the paper where red color shows the equations (numbers inside brackets), remark, assumption, lemma or theorem associated to the blocks; and (b) The process of constructing the ideas in the paper.
• Figure 3: Four connected (not strongly connected) balanced ($\mathbb{D}=I_m$) networks used in the simulations.
• Figure 4: (a) Performance measures of various $m$ in Figure \ref{['F3']}; and (b) Random graph given by Figure \ref{['F1']} with various tuners.
• Figure 5: Simulation results from network topologies with $m= 9$.

Theorems & Definitions (15)

• Remark 1: Threshold of network
• Remark 2: Communication network
• Lemma 1
• proof
• Remark 3
• Lemma 2
• proof
• Theorem 1
• proof
• Theorem 2