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Containment Control of Linear Multi-Agent Systems with Multiple Leaders of Bounded Inputs Using Distributed Continuous Controllers
Zhongkui Li, Zhisheng Duan, Wei Ren, Gang Feng
TL;DR
The paper addresses containment control for multi-agent systems with general linear dynamics and multiple leaders whose inputs may be nonzero and time varying. It develops static and adaptive distributed continuous controllers based on neighboring relative states. Under stated graph conditions, containment error is uniformly ultimately bounded with an arbitrarily small static-controller bound, while the adaptive design avoids global-information requirements.
Problem
Containment control must handle general linear agent dynamics and multiple leaders with possibly nonzero, time-varying inputs, beyond prior zero-input or restricted-dynamics settings.
Method
The paper designs distributed static and adaptive continuous containment controllers from neighboring relative states, with observer-based extensions for local output information.
Results
Containment error is uniformly ultimately bounded under an undirected follower subgraph and leader reachability to every follower, with the static-controller bound made arbitrarily small.
Takeaways & Limitations
The adaptive controller can be implemented by each follower in a fully distributed fashion without requiring global information.
Takeaways & Limitations
The static controller requires Laplacian eigenvalues and upper bounds on leaders’ control inputs, while discontinuous alternatives can cause chattering.
Abstract
from arXiv · showhide
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.
1 Introduction
Containment control addresses multi-agent networks with multiple leaders by driving followers into the geometric space spanned by the leaders. This paper extends prior work to general linear dynamics and possibly nonzero, time-varying leader inputs, while developing continuous and fully distributed controllers.
- Motivation: Containment control drives followers into the geometric space spanned by multiple leaders, unlike leader-follower tracking with only one leader.Applications include keeping follower vehicles within a moving safety area formed by leader vehicles.
- Limitations of prior work: Earlier containment results primarily restrict agent dynamics to single or double integrators or second-order Euler-Lagrange systems.These restrictions may be limiting in some circumstances.
- Discontinuous control: A discontinuous distributed controller achieves asymptotic containment under an undirected follower subgraph and a directed leader path to every follower.The discontinuous design can produce undesirable chattering in implementation.
- Continuous control: A static continuous controller instead makes the containment error uniformly ultimately bounded, with an arbitrarily small upper bound.Its design requires Laplacian eigenvalues and upper bounds on leaders’ control inputs.
- Adaptive and output-feedback extensions: The proposed adaptive continuous controller removes those global-information requirements and can be designed and implemented by each follower fully distributedly.The paper also discusses observer-based extensions when only local output information is available, under stabilizability and detectability.
- Problem: Prior general-linear containment control assumes zero leader inputs, whereas this paper allows multiple leaders with possibly nonzero and time-varying inputs.Nonzero leader actions may be needed to regulate trajectories, avoid obstacles, or form a desirable safety area.
2 Mathematical Preliminaries
The preliminaries define matrix notation, graph terminology, and Laplacian properties used to analyze multi-agent containment. They specify undirected graphs, directed paths, spanning trees, and the eigenvalue structure of graph Laplacians.
- Matrix notation: The notation includes transpose, identity matrices, all-ones vectors, block-diagonal matrices, positive definiteness, Kronecker products, vector norms, and matrix eigenvalues.A matrix is Hurwitz when all eigenvalues have strictly negative real parts.
- Graph theory: A directed graph consists of nodes and ordered edges, with the first node the parent and the second node the child.A node is a neighbor of another when the corresponding directed edge identifies it as the parent.
- Graph theory: An undirected graph has reciprocal edges, and a path is a sequence of ordered edges linking two nodes.A subgraph selects subsets of the original graph’s nodes and edges.
- Graph theory: A directed spanning tree has a root with no parent and directed paths from that root to every other node.This property connects graph structure to reachability from a designated root.
- Laplacian properties: The Laplacian has zero as an eigenvalue with the all-ones vector as a right eigenvector, while its nonzero eigenvalues have positive real parts.Zero is simple exactly when the graph has a directed spanning tree.
3 Problem Formulation and Discontinuous Containment Controllers
The paper formulates containment for general linear agents with multiple leaders and bounded, possibly time-varying leader inputs. It designs a distributed controller from neighboring relative states and proves asymptotic containment under graph and stabilizability conditions.
- Graph assumptions: The analysis assumes an undirected follower subgraph and a directed path from at least one leader to every follower.Under these conditions, the relevant Laplacian block is positive definite and its associated matrix has the required nonnegative weighting properties.
- Problem formulation: Unlike earlier work assuming zero leader inputs, this formulation allows leaders’ inputs to be nonzero and time varying, subject to boundedness.The bound is expressed as ∥u_i∥≤γ_i for each leader.
- Problem formulation: The network contains M followers and N−M leaders with general continuous-time linear dynamics, and followers should converge to the leaders’ convex hull.Leaders have no neighbors, while followers have at least one neighbor.
- Controller design: A distributed static controller uses neighboring relative states, a linear feedback term, and a nonlinear term to suppress the effect of nonzero leader inputs.The controller parameters include positive coupling gains c1 and c2, feedback gain K, adjacency weights a_ij, and the nonlinear function ĝ.
- Controller design: Under the stated assumptions and an LMI-based gain choice, the containment error ξ asymptotically converges to zero.ξ=0 characterizes the follower states as the Laplacian-weighted combination of leader states, so convergence solves containment.
- Controller properties: The nonlinear function is nonsmooth, so the resulting controller is discontinuous and solutions are interpreted in the Filippov sense.The passage identifies this as a well-posedness issue for the closed-loop dynamics.
4 Continuous State Feedback Containment Controllers
The paper replaces a discontinuous containment controller with continuous static and adaptive designs. The static design yields arbitrarily small ultimate error but requires global information, while the adaptive design removes those requirements and remains uniformly ultimately bounded.
- 4.1 Static Continuous Containment Controllers: Boundary-layer approximation converts the discontinuous controller into a continuous static containment controller.The nonlinear function is a saturation function with boundary-layer width κ.
- 4.1 Static Continuous Containment Controllers: The continuous static controller makes the containment error uniformly ultimately bounded and exponentially convergent to a residual set.The convergence rate is no less than exp(−αt).
- 4.1 Static Continuous Containment Controllers: Choosing a sufficiently small boundary-layer width κ makes the containment error arbitrarily small, but continuous control does not guarantee asymptotic stability.The residual set depends on the communication graph, follower count, leader-input bounds, and κ.
- 4.2 Adaptive Continuous Containment Controllers: Designing the static controller requires the minimal eigenvalue of L1 and upper bounds on the leaders’ control inputs.These quantities are described as global or impractical for followers to know explicitly.
- 4.2 Adaptive Continuous Containment Controllers: The adaptive controller updates each follower’s coupling gain using neighboring relative states, enabling fully distributed implementation without global information.Its feedback gains are selected using a solution to an LMI, and each follower maintains a time-varying coupling gain.
- 4.2 Adaptive Continuous Containment Controllers: Under the adaptive controller, both containment error and adaptive coupling gains are uniformly ultimately bounded, with exponential convergence to a residual set under the stated parameter condition.The residual set decreases as κ and ϕi decrease.
5 Continuous Output Feedback Containment Controllers
This section extends containment control to settings where agents access only local outputs rather than states. An observer-based continuous controller yields uniformly ultimately bounded containment error under the stated graph and stability conditions.
- Output-feedback extension: The output-feedback extension addresses cases where neighboring agents can access outputs but not full states.The section motivates observer-based control for limited information availability.
- Controller design: A distributed observer-based containment controller with fixed coupling gains is proposed.The controller uses local observer estimates of follower and leader states.
- Boundedness result: Under the LMI condition and stated assumptions, the containment error ζ is uniformly ultimately bounded.The boundedness follows from the Lyapunov analysis of the observer-based closed-loop system.
- Scope and features: The proposed controllers apply to general linear dynamics with multiple leaders having bounded, possibly nonzero inputs, while remaining continuous.Their continuity avoids the chattering associated with discontinuous controllers, and the adaptive version requires no global information.
6 Simulation Examples
A simulation with eight agents, two leaders, and six followers illustrates the adaptive controller. The trajectories indicate containment, while the follower coupling gains remain bounded.
- Setup: The simulation uses an eight-agent network with nodes 7 and 8 as leaders and the remaining agents as followers.The communication graph is specified in Figure 1.
- Setup: The leaders use bounded time-varying inputs u7 = K7x7 + 4 sin(2t) and u8 = K8x8 + 2 cos(t).The example explicitly notes that both leader inputs are bounded.
- Controller: The adaptive controller (31) is used to solve the containment control problem.The selected example parameters include κ = 0.1, ϕi = 0.005, and τi = 5 for i = 2, · · · , 7.
- Results: The simulated state trajectories indicate that the containment control problem is solved.Figure 2 depicts the trajectories of the leaders and followers.
- Results: The coupling gains di associated with the followers are clearly bounded.Their trajectories are shown in Figure 3.
7 Conclusion
The paper develops distributed static and adaptive continuous containment controllers for general linear multi-agent systems with multiple leaders and bounded inputs. Under the stated graph conditions, the containment error is uniformly ultimately bounded, while general directed graphs remain future work.
- Problem and scope: The paper addresses containment control for general linear multi-agent systems with multiple leaders whose inputs may be nonzero and time varying.This broadens the setting beyond restricted integrator dynamics and zero leader inputs.
- Controllers: Distributed static and adaptive continuous controllers based on neighboring agents’ relative states and estimates are designed.The adaptive controller can be implemented fully distributed without global information.
- Guarantee: Under an undirected follower subgraph and leader reachability to every follower, the containment error is uniformly ultimately bounded.A sufficient controller-existence condition is that each agent is stabilizable and detectable.
- Scope boundary: Distributed containment control for general directed communication graphs is identified as a future research topic.The conclusion does not claim that the presented results cover this setting.