PaperlayerView exact revision on arXiv

Source-linked AI summary

Cooperative Control of Linear Multi-Agent Systems via Distributed Output Regulation and Transient Synchronization

Georg Seyboth, Wei Ren, Frank Allgöwer

arXiv:1406.0085v1eess.SYcs.SYmath.DS

TL;DR

Cooperative output regulation covers reference tracking and disturbance rejection, but conventional formulations do not explicitly account for synchronization during transients. The paper adds distributed coupling based on agents’ transient state components to control synchronization error and support cooperative responses to local disturbances. In the helicopter example, the authors report small transient synchronization error and cooperative reaction to local disturbances.

  • Problem

    Cooperative output regulation does not explicitly take state or output synchronization during transient phases into account.

    Source: B. Contribution · p. 2

  • Method

    The paper adds distributed coupling based on agents’ transient state components to regulate synchronization error, with an LMI design method for coupling gains in identical-agent networks.

    Source: B. Contribution · p. 2, §V · p. 9

  • Results

    In the non-identical helicopter example, the elevation-angle synchronization error remains very small during transients, and the group reacts cooperatively to local disturbances.

    Source: §VI · p. 14

  • Takeaways & Limitations

    The proposed coupling allows synchronization-error dynamics to be tuned for a desired exponential decay rate and supports cooperative responses to disturbances affecting individual agents.

    Source: §VII · p. 14

Abstract

from arXiv · show

A wide range of multi-agent coordination problems including reference tracking and disturbance rejection requirements can be formulated as a cooperative output regulation problem. The general framework captures typical problems such as output synchronization, leader-follower synchronization, and many more. In the present paper, we propose a novel distributed regulator for groups of identical and non-identical linear agents. We consider global external signals affecting all agents and local external signals affecting only individual agents in the group. Both signal types may contain references and disturbances. Our main contribution is a novel coupling among the agents based on their transient state components or estimates thereof in the output feedback case. This coupling achieves transient synchronization in order to improve the cooperative behavior of the group in transient phases and guarantee a desired decay rate of the synchronization error. This leads to a cooperative reaction of the group on local disturbances acting on individual agents. The effectiveness of the proposed distributed regulator is illustrated by a vehicle platooning example and a coordination example for a group of four non-identical 3-DoF helicopter models.

I. INTRODUCTION

The paper extends cooperative output regulation beyond autonomous consensus and leader-follower tracking to agents facing external references and disturbances. It introduces global and local exosystems and transient-state coupling to regulate tracking and disturbance rejection while controlling synchronization dynamics.

I. INTRODUCTION

  • Distributed control is motivated by local neighbor-to-neighbor information exchange, with scalability, flexibility, and robustness as stated advantages.Source: §I · p. 1

A. Related Work

  • Prior consensus studies focus on autonomous state or output synchronization, while leader-follower control uses a leader to steer group motion through external references.Source: A. Related Work · p. 1
  • Cooperative output regulation handles common-reference tracking and local disturbance rejection, but does not explicitly impose state or output synchronization during transients.Source: A. Related Work · p. 1, B. Contribution · p. 2

B. Contribution

  • The paper introduces global and local exosystems, derives distributed-problem solvability conditions from an overall output-regulation formulation, and proposes output-feedback coupling based on agents’ transient states.The coupling design targets synchronization-error decay and a cooperative response to external disturbances; an LMI method places synchronization-error poles in a specified region for identical agents.Source: B. Contribution · p. 2

C. Outline

  • The paper illustrates its design in vehicle platooning and a four-helicopter example with non-identical 3-DoF models.Source: B. Contribution · p. 2

II. PRELIMINARIES; A. Agent Models

  • The paper’s setup uses linear state-space agent models and defines matrix and graph notation, including the H∞ norm and Kronecker product.Source: A. Agent Models · p. 3, §II · p. 3

B. External Signals

The setup distinguishes a global signal affecting every agent from local signals affecting individual agents. Each signal is modeled as a generalized disturbance that may include references and disturbances.

B. External Signals

  • Global signals affect all agents, while local signals affect individual agents; either type may contain references and disturbances.The global signal is modeled as a solution of an autonomous linear system.Source: B. External Signals · p. 3
  • The formulation assumes the external signals belong to a known family and allows the global signal to be treated as a pure reference with disturbances assigned to local signals.Source: B. External Signals · p. 3

C. Information Structure

Agents communicate over a directed graph and exchange information only with direct neighbors, without forwarding through multi-hop paths.

C. Information Structure

  • A directed edge represents possible information flow, and each agent can transmit local measurements or controller states to its direct neighbors.Source: C. Information Structure · p. 3
  • The communication graph is called connected when it contains a directed spanning tree; then its Laplacian has one zero eigenvalue and all other eigenvalues have positive real parts.Source: C. Information Structure · p. 3
  • The adjacency matrix records directed links, while the Laplacian is defined as diag(AG1)−AG.Source: C. Information Structure · p. 3

D. The Distributed Output Regulation Problem

The distributed output-regulation problem combines asymptotic reference tracking and disturbance rejection with a focus on cooperative transient behavior.

D. The Distributed Output Regulation Problem

E. The Overall Output Regulation Problem

The cooperative problem is assembled as a single overall output-regulation problem, whose solvability is necessary for a distributed solution. Its regulator equation is solvable exactly when the corresponding local regulator equations are solvable.

E. The Overall Output Regulation Problem

  • The distributed problem can be solved only if the overall output-regulation problem has a centralized solution.The overall problem combines all agents into one generalized plant and one exosystem.Source: E. The Overall Output Regulation Problem · p. 4
  • Under stabilizability and detectability conditions, a centralized solution exists if and only if the regulator equation is solvable.The exosystem spectrum is assumed to lie in the closed left half-plane.Source: E. The Overall Output Regulation Problem · p. 4
  • The overall regulator equation is solvable if and only if the local regulator equations are solvable.The proof partitions the overall equations by agent and establishes the correspondence in both directions.Source: E. The Overall Output Regulation Problem · pp. 4–5

F. List of Assumptions

The assumptions require stabilizable agent dynamics, detectability, solvable local regulator equations, and a directed connected communication graph rooted at agent 1. Agent 1 has direct access to the global signal, while agents cooperate to estimate it.

F. List of Assumptions

  • Each agent pair (A_k, B_k) must be stabilizable, and the relevant agent pairs must be detectable.The detectability condition on the overall pair follows when at least one agent's pair is detectable; that agent must be the spanning-tree root.Source: F. List of Assumptions · p. 5
  • Agent 1 must directly access the global signal d_g, and the communication graph must be directed, connected, and rooted at agent 1.The assumptions identify agent 1 as the informed agent; the detectability requirement can be relaxed to agent 1 for ease of presentation.Source: F. List of Assumptions · p. 5
  • The local regulator equations must be solvable for every agent, a condition equivalent to solvability of the overall regulator equation.The paper states that this assumption is necessarily satisfied when a centralized regulator exists.Source: F. List of Assumptions · p. 5
  • The communication assumptions are required for constructing the distributed regulator, and the graph must contain a spanning tree rooted at agent 1.Cooperation is needed for agents to obtain an estimate of the global signal.Source: F. List of Assumptions · p. 5

G. Synchronization with Pole Placement Constraints

The section designs static diffusive coupling gains for identical agents using pole-placement constraints on synchronization-error modes. LMI regions let the design specify performance such as a minimum decay rate.

G. Synchronization with Pole Placement Constraints

  • The design procedure incorporates synchronization-error pole constraints, including a desired decay rate, through a convex LMI region.The section considers identical agents with static diffusive coupling and presents pole placement as a performance-constrained gain design.Source: G. Synchronization with Pole Placement… · pp. 5–6
  • The LMI-based gain K ensures state synchronization while placing all synchronization-error poles inside a chosen region R.The construction uses K = ZY^-1 when the stated LMIs are satisfied.Source: G. Synchronization with Pole Placement… · p. 6
  • A transformed network state separates the synchronized component from the synchronization error, reducing the design to simultaneous stabilization of N−1 diagonal modes.The error component converges to zero when the modes associated with nonzero Laplacian eigenvalues are stabilized.Source: G. Synchronization with Pole Placement… · p. 6
  • Using a real common Lyapunov matrix can make the pole-placement design conservative, particularly for complex Laplacian eigenvalues.The remark identifies both the real-matrix requirement and the use of a common matrix across modes as sources of conservatism.Source: G. Synchronization with Pole Placement… · p. 6
  • The illustrated LMI regions include a half-plane and an intersection region whose pole placement guarantees a minimum decay rate, damping ratio, and maximum undamped frequency.For S(γ,r,θ), the stated specifications are γ, ζ = cos(θ), and ω_d = r sin(θ).Source: G. Synchronization with Pole Placement… · p. 6

IV. THE DISTRIBUTED REGULATOR

Theorem 3 constructs a distributed regulator under Assumptions 1–5, using transient-state coupling and local controller and observer design. The proof establishes convergence of the regulation errors, and the design estimates only global and agent-specific generalized disturbances.

IV. THE DISTRIBUTED REGULATOR

  • Theorem 3 constructs a distributed regulator that solves Problem 1 when Assumptions 1–5 hold.The construction selects stabilizing feedback gains, coupling, and observers for the agents.Source: §IV · pp. 6–7
  • The regulation error e_k converges to zero for every agent and all initial conditions, so the distributed regulator solves the problem.The proof also establishes convergence of the transient state components; when initial external signals are zero, the agent states converge to zero.Source: §IV · pp. 7–8
  • Each agent estimates the global generalized disturbance and only its own local generalized disturbance, rather than reconstructing every signal in the group.The paper contrasts this structure with a prior formulation that requires each agent to reconstruct the full signal vector and yields higher-order dynamic controllers.Source: §IV · p. 8
  • The section illustrates the regulator on a five-vehicle platoon with a sinusoidal disturbance at agent 2 and constant disturbances at the other vehicles.The platoon includes a virtual leader that generates the reference signal.Source: §IV · p. 8

1) The velocities vk of all vehicles converge to the velocity commanded by the virtual leader:

Example 1 formulates platooning as distributed output regulation: vehicles track the virtual leader’s velocity and converge to desired relative distances. Both tested gain choices achieve asymptotic tracking and disturbance rejection, but the group does not respond cooperatively to local disturbances during transients.

Example 1

  • Each vehicle’s velocity converges to the virtual leader’s commanded velocity, while its relative distance converges to the desired constant r_k.The velocity and distance targets are stated in the regulation objectives.Source: 1) The velocities vk of all vehicles… · p. 8
  • Vehicles communicate with their predecessor and follower, and local disturbances and references are combined into local generalized disturbance signals.The platooning problem is formulated as distributed output regulation.Source: 1) The velocities vk of all vehicles… · p. 8
  • Only agent 1 receives the global reference, while all agents receive local references and states and use local disturbance estimators.The local disturbance estimators reconstruct w_k.Source: 1) The velocities vk of all vehicles… · p. 8
  • Both gain choices track the references and reject disturbances asymptotically in the distributed regulation problem.The simulations use local exogenous input signals shown in Fig. 3 and results shown in Fig. 4.Source: 1) The velocities vk of all vehicles… · p. 8, Figure 4 · p. 9, Figure 3 · p. 9
  • Local disturbance rejection is not cooperative: more aggressive gains speed rejection, but other vehicles lack disturbance information and cannot explicitly prioritize inter-vehicle distances during transients.The text says maintaining desired inter-vehicle distances is more important than maintaining desired velocity for this platooning example.Source: 1) The velocities vk of all vehicles… · p. 9

V. IMPROVING THE TRANSIENT BEHAVIOR: IDENTICAL AGENTS

The section extends distributed output regulation with coupling between agents’ transient state components, aiming to synchronize their regulation errors at a chosen decay rate. The design is presented first for identical agent dynamics, then implemented with observers and illustrated in vehicle platooning.

V. IMPROVING THE TRANSIENT BEHAVIOR: IDENTICAL AGENTS

  • The added coupling uses agents’ transient state components to stabilize synchronization errors without replacing the underlying output regulation objective.Under Assumption 6 and without observer errors, agreement of transient components corresponds to agreement of regulation errors.Source: §V · pp. 9–10
  • The identical-dynamics assumption still allows non-identical measurement maps, local exosystems, and generalized disturbance input matrices.The paper notes that this assumption is relaxed in Section VI.Source: §V · p. 9
  • The full-information design selects F so A−BF is Hurwitz and chooses H to achieve the required decay rate for the coupled synchronization dynamics.The paper presents the full-information case first to clarify the main idea and support the output-feedback proof.Source: §V · pp. 9–10
  • The regulator solves the output regulation problem while enforcing exponential synchronization of regulation errors at a chosen decay rate, supporting cooperative responses to local disturbances.The full-information result establishes a decay rate of at least γ; the output-feedback result also guarantees rate γ under its stated assumptions.Source: §V · p. 10
  • The output-feedback implementation uses observers and requires the corresponding systems to be exponentially stable at rate η greater than the target synchronization rate γ.Under these conditions, the synchronization errors decay exponentially at rate γ.Source: §V · p. 10
  • In Example 2, the platoon rejects local disturbances on inter-vehicle distances more efficiently than Example 1 and maintains small synchronization errors.Different choices of F and H let the design emphasize disturbance rejection or synchronization, depending on requirements.Source: §V · p. 11, Figure 5 · p. 11

VI. IMPROVING THE TRANSIENT BEHAVIOR: NON-IDENTICAL AGENTS

For non-identical agents, the paper models each system as a perturbation of a common nominal plant and designs transient-state coupling to target disturbance attenuation and synchronization decay. A four-helicopter example reports improved transient cooperation, while the robustness guarantee does not ensure the target decay rate under perturbations.

Nominal model and uncertainty

  • Non-identical agents are represented as stable additive perturbations of a common nominal plant; the method excludes differing state dimensions and is presented for full information.The paper says the output-feedback implementation can be carried out analogously to Theorem 4.Source: §VI · p. 11

Coupling-gain design

  • The gain-design target combines an H∞ bound on the network disturbance-to-synchronization output with decay rate γ for synchronization modes.The synthesis uses one controller gain across the transformed subsystems; the uncontrolled synchronous mode bounds achievable H∞ performance.Source: §VI · pp. 12–13
  • Theorem 5 uses LMIs to select H for exponential stability with a specified decay rate; the small-gain condition ensures stability under bounded agent perturbations.The design requires η < 1/η∆ and sets H=ZY^-1.Source: §VI · pp. 12–13

Robustness limitation

  • Perturbations preserve robust stability and nominal performance, but can couple synchronous and error modes, so synchronization may converge more slowly than the designed rate.Theorem 5 does not guarantee robust pole placement of the synchronization-error modes at the target rate in the presence of perturbations.Source: §VI · p. 13

Example 3

  • Example 3 uses four parameter-varying Quanser helicopters tracking travel and elevation references while rejecting local rotor-force disturbances over an undirected cycle.The four helicopters have different parameter values; the simulated disturbance magnitude ranges from −2.5 N to 2.5 N per helicopter.Source: §VI · p. 13
  • With four non-identical helicopters, the extended regulator considerably improved cooperation over Fig. 8, keeping transient elevation-angle synchronization small and producing cooperative responses to local disturbances.The uncertainty bound was η∆=0.3129; the feasible design used η=0.95(η∆)^−1 and achieved a target decay rate γ=3.Source: §VI · p. 14

VII. CONCLUSION

The paper presents a distributed regulator that adds transient-state coupling to cooperative output regulation, allowing synchronization error decay to be specified. Its examples illustrate improved transient cooperation and cooperative responses to disturbances acting on individual agents.

Conclusion

  • The proposed transient-state coupling allows a desired exponential decay rate for synchronization error and improves group behavior during transients.The conclusion describes cooperative output regulation as a framework for a wide range of multi-agent coordination problems.Source: §VII · p. 14, Figure 10 · p. 14
  • Under the novel control law, the group can react cooperatively to external disturbances acting on individual agents.The paper illustrates the regulator with vehicle platooning and a four-helicopter coordination example.Source: §VII · p. 14
Loading 1406.0085v1…