[Paper Review] Containment Control of Linear Multi-Agent Systems with Multiple Leaders of Bounded Inputs Using Distributed Continuous Controllers

Zhongkui Li, Zhisheng Duan+2 more|arXiv (Cornell University)|Dec 28, 2013

Distributed Control Multi-Agent Systems34 references3 citations

TL;DR

This paper proposes distributed continuous controllers for containment control in linear multi-agent systems with multiple leaders whose inputs are bounded and time-varying. Using relative state information, a static continuous controller ensures uniformly ultimately bounded containment error with adjustable upper bound, while an adaptive version enables fully distributed design without global knowledge, overcoming chattering and prior assumptions on eigenvalues or input bounds.

ABSTRACT

This paper considers the containment control problem for multi-agent systems with general linear dynamics and multiple leaders whose control inputs are possibly nonzero and time varying. Based on the relative states of neighboring agents, a distributed static continuous controller is designed, under which the containment error is uniformly ultimately bounded and the upper bound of the containment error can be made arbitrarily small, if the subgraph associated with the followers is undirected and for each follower there exists at least one leader that has a directed path to that follower. It is noted that the design of the static controller requires the knowledge of the eigenvalues of the Laplacian matrix and the upper bounds of the leaders' control inputs. In order to remove these requirements, a distributed adaptive continuous controller is further proposed, which can be designed and implemented by each follower in a fully distributed fashion. Extensions to the case where only local output information is available are discussed.

Motivation & Objective

• To address containment control in multi-agent systems with general linear dynamics and multiple leaders whose control inputs are nonzero and time-varying.
• To eliminate chattering in existing discontinuous controllers by designing continuous alternatives.
• To remove reliance on global knowledge—such as eigenvalues of the Laplacian matrix and upper bounds of leader inputs—by proposing a fully distributed adaptive controller.
• To extend the framework to cases where only local output information is available, using distributed observers.
• To establish conditions under which containment control is achievable using stabilizable and detectable agents.

Proposed method

• Designs a distributed static continuous controller based on relative states of neighboring agents, ensuring uniform ultimate boundedness of the containment error.
• Uses the boundary layer concept to smooth a discontinuous controller into a continuous one, reducing chattering while maintaining bounded error.
• Introduces a distributed adaptive continuous controller that eliminates the need for global information by enabling each follower to tune its control gains locally.
• Employs linear matrix inequalities (LMIs) to design controller gains, with stability proven via Lyapunov-based analysis and Schur complement lemma.
• Proposes observer-based controllers using relative state estimates when only local outputs are available, ensuring stability under detectability and stabilizability conditions.
• Validates the approach through simulation with eight agents, including two leaders with sinusoidal inputs, demonstrating convergence and bounded coupling gains.

Experimental results

Research questions

• RQ1 Can containment control be achieved in multi-agent systems with general linear dynamics and multiple leaders whose inputs are nonzero and time-varying?
• RQ2 How can chattering in discontinuous controllers be eliminated while maintaining containment performance?
• RQ3 Can a fully distributed adaptive controller be designed without requiring knowledge of eigenvalues of the Laplacian matrix or bounds on leader inputs?
• RQ4 What conditions ensure the existence of distributed observers when only local output information is available?
• RQ5 How can the upper bound of the containment error be made arbitrarily small using continuous controllers?

Key findings

• The containment error is uniformly ultimately bounded under the proposed static continuous controller, with the upper bound adjustable by design parameters.
• The upper bound of the containment error can be made arbitrarily small by tuning the controller gain, provided the follower subgraph is undirected and leaders have directed paths to all followers.
• The adaptive continuous controller enables fully distributed implementation without requiring global information such as eigenvalues or input bounds.
• Simulation results confirm that the adaptive controller achieves containment with bounded coupling gains and convergent state trajectories, even under time-varying leader inputs.
• The LMI-based design yields feasible controller gains, with numerical solutions for K and Γ obtained via the Sedumi toolbox.
• The observer-based controller ensures containment when only local outputs are available, provided each agent is stabilizable and detectable.