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Robust time-varying formation design for multi-agent systems with disturbances: Extended-state-observer method
Le Wang, Jianxiang Xi, Ming He, Guangbin Liu
TL;DR
Robust time-varying formation control for second-order multi-agent systems with external disturbances remains insufficiently addressed by prior time-varying methods. The paper uses an extended-state observer to estimate disturbance compensation and derives formation-center and gain-design results. It analyzes tracking and robustness properties of the resulting system.
Problem
Prior time-varying formation methods did not address disturbance rejection, while disturbances can drive multi-agent systems to oscillations or divergences.
Source: §1 · p. 2Method
An extended-state observer estimates disturbance compensation for a neighboring-information-based protocol, with gain criteria derived using formation feasibility and algebraic Riccati equation techniques.
Source: §1 · pp. 2–3Results
The paper derives an explicit formation-center expression that determines impacts of disturbance compensation and presents sufficient design conditions with tracking and robustness analyses.
Source: §1 · p. 3Takeaways & Limitations
The formation-center expression describes how disturbance compensation affects the motion mode of the whole formation.
Source: §1 · p. 3The formulation considers second-order multi-agent systems with unknown bounded external disturbances.
Source: §2 · p. 3
Abstract
from arXiv · showhide
Robust time-varying formation design problems for second-order multi-agent systems subjected to external disturbances are investigated. Firstly, by constructing an extended state observer, the disturbance compensation is estimated, which is a critical term in the proposed robust time-varying formation control protocol. Then, an explicit expression of the formation center function is determined and impacts of disturbance compensations on the formation center function are presented. With the formation feasibility conditions, robust time-varying formation design criteria are derived to determine the gain matrix of the formation control protocol by utilizing the algebraic Riccati equation technique. Furthermore, the tracking performance and the robustness property of multi-agent systems are analyzed. Finally, the numerical simulation is provided to illustrate the effectiveness of theoretical results.
1. INTRODUCTION
The paper addresses robust time-varying formation control for disturbed second-order multi-agent systems, a gap left by prior time-varying methods that did not address disturbance rejection. It proposes ESO-based compensation and derives formation-center and design results.
1. INTRODUCTION
- Environmental disturbances can drive multi-agent systems to oscillations or divergences, making disturbance rejection with closed-loop stability a practical concern.The introduction gives atmospheric forces on quadrotors as an example and reviews prior disturbance-rejection approaches for time-invariant formations.Source: §1 · p. 2
- Prior time-varying formation studies did not address disturbance rejection, while methods for time-invariant formations could not be applied directly to time-varying configurations.The paper frames robust time-varying formation design with unknown external disturbances as insufficiently investigated.Source: §1 · p. 2
- An extended-state observer estimates disturbance compensation for a robust time-varying formation protocol using relative neighboring information.The paper separates formation-agreement dynamics, used to derive the formation center, from disagreement dynamics describing relative agent motion.Source: §1 · p. 2
- The design criteria determine the protocol gain through algebraic Riccati equation techniques together with formation-feasibility conditions.The paper also analyzes tracking performance and closed-loop robustness stability.Source: §1 · p. 3
- The formation-center expression captures how disturbance compensation affects the collective motion, alongside the proposed real-time compensation and robustness analysis.The paper contrasts this with prior time-varying formation results that did not determine the impact of disturbance compensation on the formation center.Source: §1 · p. 3
2. PROBLEM DESCRIPTION
The problem is to control second-order agents with unknown bounded disturbances toward a desired time-varying formation. The formulation uses a disturbance-compensating protocol and specifies formation reachability, graph structure, and design objectives.
2. PROBLEM DESCRIPTION
- Each agent's position derivative is its velocity, while its velocity dynamics include damping, control input, and an unknown bounded external disturbance.The damping constants are αp and αv.Source: §2 · p. 3
- The interaction graph is a directed graph with a Laplacian, and a spanning tree implies a single zero eigenvalue with the remaining eigenvalues having positive real parts.The stated graph condition provides the spectral property used in the setup.Source: §2 · p. 4
- The proposed protocol uses relative neighboring information and adds robust disturbance compensation determined by an extended state observer.The gain matrix Ku is part of the protocol, and the observer includes an intermediate variable with bandwidth constants.Source: §2 · p. 4
- Robust time-varying formation reachability requires bounded closed-loop states and eventual proximity of each agent to its desired formation offset from a shared center function.The definition bounds that error by ε after a finite time tε.Source: §2 · p. 4
- The control objective is robust time-varying formation for second-order multi-agent systems under external disturbances, including an explicit formation-center expression, gain design, and tracking and robustness analysis.The agents interact over a directed graph.Source: §2 · p. 4
3. FORMATION CENTER FUNCTIONS
The section derives the formation center function and examines how time-varying formations and disturbance compensation affect it. It separates the formation’s shared motion from relative formation error and identifies conditions under which the center candidate yields formation reachability.
3. FORMATION CENTER FUNCTIONS
- The coordinate transformation splits the formation error into κ(t), the first transformed component, and ϕ(t), the remaining components.The construction uses U^-1, with U chosen to transform the Laplacian into Jordan canonical form.Source: §3 · p. 5
- The transformed dynamics separate the formation’s macroscopic motion from relative movement among agents.The section identifies the agreement and disagreement subsystems with these respective roles.Source: §3 · p. 5
- Theorem 1 gives an explicit condition on the formation center function when the system reaches the desired formation.The section also examines how the time-varying formation and disturbance compensation affect that function.Source: §3 · p. 5
- If ϕ(t) stays within ε/εN, each agent’s state approaches its desired formation offset from κ(t) within ε, making κ(t) a candidate center.The bound holds after some finite tε.Source: §3 · p. 6
4. ROBUST TIME-VARYING FORMATION DESIGN
The design procedure checks formation feasibility, selects a control gain using an algebraic Riccati equation, and chooses observer bandwidths for disturbance estimation. Under the stated feasibility condition, the resulting protocol reaches the robust time-varying formation and supports analysis of tracking and robustness.
4. ROBUST TIME-VARYING FORMATION DESIGN
- The section analyzes the protocol’s tracking performance and robustness stability after presenting the design algorithm.Source: §4 · pp. 7–8
- The algorithm first checks feasibility; if condition (19) fails, it stops because the expected formation cannot be reached by protocol (2).Source: §4 · p. 7
- The procedure solves an algebraic Riccati equation for a positive definite matrix and uses it to determine the control gain.The design also chooses sufficiently large ESO bandwidth constants to estimate robust disturbance compensation.Source: §4 · p. 7
- For bounded initial states, satisfying formation feasibility condition (19) ensures the designed protocol reaches the robust time-varying formation.The theorem establishes this guarantee for the protocol produced by the design algorithm.Source: §4 · p. 8
2 P, then differentiating V (t) along the trajectories of (21) yields
The analysis establishes convergence of the disagreement dynamics and boundedness of the closed-loop states, yielding robust time-varying formation reachability under the stated conditions.
Stability and reachability
- The Lyapunov derivative is bounded above by a negative term, supporting the stability argument for the disagreement dynamics.The derivation uses the ordering of the nonzero eigenvalue real parts.Source: §2 · p. 8
- The disagreement modes converge asymptotically, making each modal matrix and the resulting closed-loop matrix Hurwitz.This follows from the convergence of η_k(t) for k = 2, …, N.Source: §2 · p. 8
Boundedness conditions
- Under bounded ∥ū ⊗ θ2∥∞ and sufficiently large σ_i, the proof establishes bounded observer, compensation, and control states.The boundedness argument covers the states of the robust disturbance compensation and the control protocol.Source: §2 · pp. 9–10
Stability and reachability
- The system is robust time-varying formation-reachable: all closed-loop states are bounded and the formation error eventually falls below any given positive bound.The proof states that a finite time exists after which the error remains within the chosen bound.Source: §2 · p. 10
5. NUMERICAL SIMULATIONS
A six-agent simulation illustrates robust time-varying formation under external disturbances, with the agents forming a rotating regular pentagon in position and velocity.
Simulation setup
- The example uses six second-order agents in three-dimensional space and a 0–1 weighted digraph as the interaction topology.The simulation specifies the agents’ initial states and positions and velocities along the X, Y, and Z axes.Source: §5 · p. 10, Figure 1 · p. 10
- The ESO bandwidth uses σ_i = 10, and the selected control gain is Ku = [1.0654, 1.8576] ⊗ I3.The gain is determined from Theorem 2.Source: §5 · p. 11
Simulation visualizations
- The simulation plots formation-center position and velocity curves over 20 seconds, marking initial states with circles and final states with squares.Figures 2 and 3 show agent and center trajectories at selected times.Source: §5 · p. 11, Figure 2 · p. 12, Figure 3 · p. 12
Simulation results
- The disturbed six-agent system achieves robust time-varying formation, with a rotating regular pentagon in both position and velocity states.The reported simulation results cover Figures 2–5.Source: §5 · p. 11, Figure 2 · p. 12, Figure 3 · p. 12
6. CONCLUSIONS
The paper presents robust time-varying formation design for disturbed second-order multi-agent systems with directed topologies, and establishes reachability under suitable gain and ESO bandwidth choices.
Conclusion
- With suitable gain and ESO bandwidth choices, the systems can reach the expected robust time-varying formation.The conclusion states this as a proven property of the design.Source: §6 · p. 13
- The proposed protocol uses relative neighboring information and an ESO to estimate and compensate for external disturbances.The paper also derives an explicit formation-center function and determines how disturbance compensation and time-varying formation affect its motion mode.Source: §6 · p. 13
- The design conditions use an algebraic Riccati equation together with formation-feasibility conditions, while tracking performance and robustness stability are analyzed.Source: §6 · p. 13