[Paper Review] Distributed Control of Multi-agent Systems with Unknown Time-varying Gains: A Novel Indirect Framework for Prescribed Performance

Zeqiang Li, Yujuan Wang+1 more|arXiv (Cornell University)|Jul 12, 2022

Adaptive Control of Nonlinear Systems4 citations

TL;DR

This paper proposes a novel indirect distributed control framework for multi-agent systems with unknown time-varying gains and non-identical control directions. By combining a fully distributed robust filter for trajectory estimation and a new adaptive backstepping protocol using a generalized Nussbaum function, the method ensures prescribed performance with arbitrary convergence rate and residual set size, while guaranteeing semi-global uniform ultimate boundedness of all signals.

ABSTRACT

In this paper, a new yet indirect performance guaranteed framework is established to address the distributed tracking control problem for networked uncertain nonlinear strict-feedback systems with unknown time-varying gains under a directed interaction topology. The proposed framework involves two steps: In the first one, a fully distributed robust filter is constructed to estimate the desired trajectory for each agent with guaranteed observation performance that allows the directions among the agents to be non-identical. In the second one, by establishing a novel lemma regarding Nussbaum function, a new adaptive control protocol is developed for each agent based on backstepping technique, which not only steers the output to asymptotically track the corresponding estimated signal with arbitrarily prescribed transient performance, but also largely extends the scope of application since the unknown control gains are allowed to be time-varying and even state-dependent. In such an indirect way, the underlying problem is tackled with the output tracking error converging into an arbitrarily pre-assigned residual set exhibiting an arbitrarily pre-defined convergence rate. Besides, all the internal signals are ensured to be semi-globally ultimately uniformly bounded (SGUUB). Finally, simulation results are provided to illustrate the effectiveness of the co-designed scheme.

Motivation & Objective

• To address the distributed tracking control problem in uncertain nonlinear multi-agent systems with unknown time-varying control gains and non-identical control directions under a directed topology.
• To overcome the limitations of prior works that assume constant or known time-varying gains, or identical control directions.
• To achieve prescribed performance control—specifically, arbitrary pre-assigned convergence rate and residual set size for the output tracking error—without requiring prior knowledge of control directions.
• To extend the applicability of adaptive backstepping control to systems with state-dependent and time-varying gains under general directed communication graphs.

Proposed method

• Designs a fully distributed robust filter to estimate the desired trajectory for each agent, ensuring observation performance even with non-identical interaction directions.
• Introduces a novel lemma on Nussbaum functions to handle unknown and time-varying control gains, including state-dependent cases.
• Develops an adaptive backstepping control protocol for each agent that ensures the output tracks the estimated trajectory with prescribed transient performance.
• Employs a Lyapunov-based stability analysis to prove semi-global uniform ultimate boundedness (SGUUB) of all closed-loop signals.
• Uses a chain of first-order low-pass filters to bound the tracking error dynamics and derive performance bounds.
• Integrates the filter and control protocol into a two-step indirect framework to decouple the challenges of gain uncertainty and performance specification.

Experimental results

Research questions

• RQ1 How can distributed tracking be achieved in multi-agent systems with unknown time-varying gains and non-identical control directions?
• RQ2 Can prescribed performance control be achieved without prior knowledge of control direction or gain dynamics?
• RQ3 What is the role of a generalized Nussbaum function in handling time-varying and state-dependent gains in a distributed setting?
• RQ4 How can the convergence rate and residual set size of the tracking error be arbitrarily pre-assigned in the presence of uncertain gains?
• RQ5 What conditions ensure the semi-global uniform ultimate boundedness of all internal signals under the proposed control scheme?

Key findings

• The proposed framework ensures that the output tracking error converges into an arbitrarily pre-assigned residual set with an arbitrarily pre-defined convergence rate.
• The method extends the scope of application by allowing unknown time-varying and even state-dependent control gains, which are not restricted to constant or known time-varying forms.
• The use of a novel Nussbaum function lemma enables stability analysis under non-identical control directions, overcoming limitations in prior works requiring identical or partially known directions.
• All internal signals, including estimation and control errors, are proven to be semi-globally ultimately uniformly bounded (SGUUB).
• The simulation results validate the effectiveness of the co-designed scheme in achieving prescribed performance under complex uncertainty and directed communication topologies.
• The bound on the steady-state tracking error is quantitatively derived as $ rac{ ho_{ty}}{ ext{min}(m{L}+m{B}) imes ext{scaling factor}} $, with explicit dependence on filter parameters and network topology.