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Communication-aware Synthesis of Safe Controllers for Discrete-Time Linear Multi-Agent Systems with Distributed k-Hop Observation

Yihan Liu, Teng Yan, Meiqi Tian, Bingzhuo Zhong

arXiv:2610.00925v1eess.SYcs.SY

TL;DR

Limited communication leaves remote states unavailable to local controllers, and estimation errors affect closed-loop state evolution through controller gains. The paper develops distributed k-hop observers and jointly synthesizes observers, controllers, and RSI sets while accounting for observer-induced perturbations in safety conditions.

  • Problem

    Under limited inter-agent communication, remote states needed by local feedback must be reconstructed, and the resulting estimation errors couple observer design to safety through controller gains.

    Source: §I · p. 1

  • Method

    The paper develops distributed k-hop observers with uniform error bounds and an LMI-based joint synthesis of observers, local controllers, and RSI sets that accounts for observer-induced perturbations.

    Source: §I · pp. 1–2, §IV · p. 6, §VI · p. 9

  • Results

    The proposed conditions incorporate observer-induced perturbations into local RSI sets and pairwise relative-state safety constraints while jointly synthesizing observers, controllers, and RSI sets.

    Source: §IV · p. 6, §VI · p. 9

  • Takeaways & Limitations

    The framework links communication-aware estimation with safety-guaranteed control for multi-agent systems.

    Source: §VI · p. 9

Abstract

from arXiv · show

This paper studies communication-aware safe control for discrete-time linear multi-agent systems under limited information exchange. The main challenge lies in the coupling between remote-state estimation and safe controller design, since estimation errors affect the state evolution through the controller gains, while the controller design must account for the resulting observer-induced state perturbations to guarantee safety. To address this challenge, a distributed k-hop observer is developed to reconstruct unavailable remote states, and a uniform observer-error bound is derived. The effect of the observer-induced state perturbation on closed-loop safety is accounted for in both the construction of local {\epsilon}i-robust safe invariant (RSI) sets and the enforcement of pairwise relative-state safety constraints. The resulting safety conditions ensure that all agents remain within their local RSI sets while all pairwise relative-state safety constraints are satisfied. A linear matrix inequality (LMI)-based optimization method is developed to jointly synthesize the distributed observers, local controllers, and RSI sets. A case study illustrates the effectiveness of the proposed method.

I. INTRODUCTION

Limited communication leaves remote states unavailable to local controllers, while observer errors can affect safety through controller gains. The paper frames the setting around 1-hop information exchange, k-hop remote agents, and local and pairwise safety requirements.

I. INTRODUCTION

  • Communication-aware estimation has addressed convergence and stability, but observer-induced perturbations have received less attention in safety guarantees.Remote-state estimation errors propagate through controller gains, coupling observer and controller design.Source: §I · p. 1

A. Graph-Theoretic Preliminaries

  • For prescribed k ≥ 2, remote agents are those at graph distance 2 through k; graph distance is the shortest path length between agents.The communication network is an undirected graph whose edges denote direct information exchange.Source: A. Graph-Theoretic Preliminaries · p. 2
  • Each agent directly accesses its own state and input and can exchange them with 1-hop neighbors; transmissions follow graph edges, with negligible within-step delay.Neighbor information may be forwarded within a sampling interval for distributed observer correction.Source: A. Graph-Theoretic Preliminaries · p. 2

B. Interconnected Multi-Agent System

  • Each agent follows discrete-time linear dynamics, with admissible inputs and safety defined by local state constraints and pairwise relative-state constraints.The interconnected system must satisfy its safe set at every time step.Source: B. Interconnected Multi-Agent System · p. 2

C. Observer System

The observer reconstructs remote states that are unavailable through direct 1-hop communication. Each local controller uses its own and neighboring states together with those remote-state estimates.

C. Observer System

  • Because remote states are unavailable through 1-hop communication, distributed k-hop observers reconstruct them for local control.The observer update uses locally available information and a distributed update map.Source: C. Observer System · pp. 2–3

D. Control System

  • The local controller’s information vector combines the agent’s state, its neighbors’ states, and estimates of remote states.Remote-state estimates stand in for states unavailable to the agent directly.Source: D. Control System · p. 3
  • Each agent applies a distributed state-feedback controller using its locally available and estimated k-hop state information.The controller gain is jointly synthesized with the observers.Source: D. Control System · p. 3

E. Problem Formulation

The formulation couples remote-state estimation with control because estimation errors affect state evolution through feedback gains. The observer construction uses consensus and correction information, while its bounded errors feed into the safety design.

E. Problem Formulation

  • Remote-state estimates enter local feedback, and their errors affect closed-loop evolution through controller gains, motivating joint observer-controller safety design.The problem is to synthesize observers, controllers, and a safe invariant set that preserves state safety and input admissibility.Source: E. Problem Formulation · p. 3

III. DISTRIBUTED k-HOP OBSERVER AND ERROR ANALYSIS

  • The observer analysis characterizes estimation errors and the resulting observer-induced state perturbation for the subsequent safety synthesis.The section develops the distributed k-hop observer and analyzes its error evolution.Source: §III · p. 3

A. Observer Construction

  • The observer network requires each estimator component to be connected to true-state correction information, yielding a positive-definite information matrix.Theorem 1 establishes M_j ≻ 0 under the stated graph and pinning conditions.Source: A. Observer Construction · pp. 3–4
  • Theorem 1’s positive-definiteness result rules out uncorrected estimation modes under the stated graph conditions.Each connected component of the estimator subgraph must contain a node with true-state correction information.Source: A. Observer Construction · p. 4
  • The distributed correction mechanism combines estimator-to-estimator information exchange with relayed information associated with the true state or input.The graph-based information structure captures both consensus coupling and true-state pinning.Source: A. Observer Construction · pp. 3–4
  • The input estimator has uniformly bounded error when consecutive input variations are bounded.The stated bound also assumes a bounded initial corrected input-estimation error.Source: A. Observer Construction · p. 4
  • The state observer reconstructs remote states using corrected input estimates and local state-correction information.Its error follows homogeneous observer dynamics and is additionally affected by input-estimation error.Source: A. Observer Construction · p. 5

B. Observer-Error Bound

The observer-error analysis derives a uniform state-estimation error bound from contractive homogeneous error dynamics and bounded input-estimation error. The bound combines a decaying initial-error component with a bounded component and is used to quantify later state perturbations.

B. Observer-Error Bound

  • Assumption 3 requires contractive homogeneous observer-error dynamics, which, together with bounded input-estimation error, supports deriving the uniform bound.The Lyapunov feasibility condition is used to establish the state-estimation error bound.Source: B. Observer-Error Bound · p. 5
  • The bound applies for all time under the stated observer assumptions and the uniform input-estimation error bound.The theorem’s conditions include the observer parameters and input-estimation error bound.Source: B. Observer-Error Bound · p. 5
  • Theorem 3 bounds each agent’s state-estimation error uniformly, with a decaying component from initial error and a bounded component from input-estimation error.The bound is used to quantify observer-induced state perturbations in subsequent analysis.Source: B. Observer-Error Bound · p. 5

C. Observer-Induced State-Perturbation Bound

Remote-state estimation errors enter the implemented dynamics as an additive observer-induced perturbation through the feedback gain. The analysis aggregates remote-agent error bounds and propagates them through the controller gain to bound the resulting physical state perturbation.

C. Observer-Induced State-Perturbation Bound

  • Remote-state estimation error enters the implemented closed-loop physical dynamics as an additive perturbation through the feedback gain K_i.The perturbed dynamics are expressed using the local controller’s augmented state and observer-induced perturbation.Source: C. Observer-Induced State-Perturbation… · p. 5
  • Theorem 4 bounds the observer-induced state perturbation by propagating the stacked remote-state estimation error through the controller gain.The resulting uniform perturbation bound is ε_i = g_iδ_i.Source: C. Observer-Induced State-Perturbation… · p. 6
  • The error bounds pass from individual observed agents to the controller’s stacked remote-state error, then through the state-feedback gain to the physical perturbation bound.The aggregated remote-state error is bounded by δ_i, and the resulting state perturbation by ε_i.Source: C. Observer-Induced State-Perturbation… · p. 6

IV. COMMUNICATION-AWARE SAFE CONTROLLER DESIGN

The section develops a communication-aware safe-controller synthesis framework using the observer-induced state-perturbation bound. It accounts for perturbations in local RSI sets and pairwise relative-state safety constraints, then derives tractable LMI conditions for joint synthesis.

IV. COMMUNICATION-AWARE SAFE CONTROLLER DESIGN

  • The design incorporates the observer-induced state-perturbation bound into communication-aware safe-controller synthesis.The framework accounts for remote-state estimation effects on the closed-loop state.Source: §IV · p. 6
  • The perturbation bound is included in both local ε_i-RSI set construction and pairwise relative-state safety constraints.These are the two safety conditions addressed by the section.Source: §IV · p. 6
  • Tractable LMI conditions jointly synthesize distributed observers and state-feedback controllers under safety constraints.Source: §IV · p. 6

A. Local RSI and Relative-State Safety Conditions

The safety conditions account for observer-induced perturbations in local robust invariance and pairwise relative-state constraints. Tractable conditions use nominal contraction, perturbation accommodation, and a lifted ellipsoidal representation of the augmented-state domain.

A. Local RSI and Relative-State Safety Conditions

  • A local ε_i-RSI set must remain invariant for every admissible augmented state and every observer-induced perturbation bounded by ε_i.The perturbed closed-loop next state must remain in the local safe set.Source: A. Local RSI and Relative-State Safety… · p. 6
  • The robust invariance requirement is decomposed into nominal contraction and accommodation of the observer-induced perturbation.The decomposition provides a basis for subsequent LMI-based synthesis.Source: A. Local RSI and Relative-State Safety… · p. 6
  • The lifted ellipsoidal outer approximation of the augmented-state domain makes it possible to derive a tractable condition for pairwise relative-state safety.The relative dynamics are expressed along the constrained direction, isolating observer perturbations and the neighboring input.Source: A. Local RSI and Relative-State Safety… · p. 6
  • The pairwise safety condition accounts for local closed-loop motion, observer-induced perturbation, and the neighboring agent’s input.The relative dynamics are considered along the constrained relative-state direction.Source: A. Local RSI and Relative-State Safety… · p. 7
  • Theorem 6 gives a sufficient condition for preserving the pairwise relative-state safe set over one sampling step while accounting for observer error and the neighboring agent’s input.Because the admissible augmented-state domain lies within its ellipsoidal outer approximation, the condition also applies to all admissible augmented states.Source: A. Local RSI and Relative-State Safety… · p. 7

B. Joint LMI-Based Observer-Controller Synthesis

The section combines observer-stability, local invariant-set, pairwise safety, and input constraints in a joint LMI-based synthesis problem. Under the stated assumptions, feasibility guarantees an overall safe invariant set that respects the input bounds.

B. Joint LMI-Based Observer-Controller Synthesis

  • The LMI conditions enforce observer stability, nominal RSI contraction, perturbation accommodation, safe-set containment, pairwise relative-state safety, and input limits under remote-state estimation errors.A further constraint bounds controller amplification of estimation error and determines the perturbation bound ε_i.Source: B. Joint LMI-Based Observer-Controller… · p. 7
  • For fixed outer parameters, the observer and controller conditions form one convex synthesis problem because δ_i and ε_i are known before its inner optimization.The decision matrices include observer variables and controller and RSI-related variables.Source: B. Joint LMI-Based Observer-Controller… · pp. 7–8
  • Feasibility under the stated input-variation and observer-initialization conditions guarantees recovered gains satisfy observer stability, local ε_i-RSI, relative-state safety, and input-safety conditions.Theorem 7 also requires the specified fixed design parameters and positive safety margins.Source: B. Joint LMI-Based Observer-Controller… · pp. 7–8
  • The resulting Xsafe is recursively invariant: trajectories initialized within it remain there while every control input satisfies its prescribed bound.The theorem identifies Xsafe as an RSI set for the closed-loop multi-agent system.Source: B. Joint LMI-Based Observer-Controller… · p. 8

V. SIMULATION

A five-agent network is simulated with two-hop observation, and 100 initial conditions are tested over 100 time steps. The reported trajectories satisfy the local RSI, input, and relative-state safety requirements.

V. SIMULATION

  • The case study uses five agents in the communication topology 1−2−3−4, 2−5, 3−5, with observer range k=2.Source: §V · p. 8
  • Every simulated trajectory stays within its local RSI set, all inputs satisfy ∥u_i(t)∥∞≤4, and normalized relative-state safety measures remain below one.The reported figures show the local sets and trajectories, input trajectories, and normalized safety measures, respectively.Source: Figure 3 · p. 8, Figure 4 · p. 9, Figure 5 · p. 9
  • The verification samples 100 initial conditions from local RSI sets satisfying the relative-state constraints and simulates H=100 time steps.All five agents apply their synthesized controllers.Source: §V · p. 9

VI. CONCLUSION

The conclusion describes a distributed k-hop observer with uniform error bounds and safety conditions that account for observer-induced perturbations. It presents joint LMI-based synthesis of observers, controllers, and local RSI sets as a link between communication-aware estimation and safety-guaranteed control.

VI. CONCLUSION

  • The distributed k-hop observer reconstructs unavailable remote states, while uniform error bounds characterize the resulting observer-induced state perturbations.Source: §VI · p. 9
  • An augmented admissible-state domain supports local ε_i-RSI and relative-state safety conditions that account for neighboring and remote states.Source: §VI · p. 9
  • The LMI framework jointly synthesizes observers, safe state-feedback controllers, and local RSI sets while accounting for observer-controller coupling.Source: §VI · p. 9

APPENDIX

The appendix verifies observer properties and checks that the synthesized sets and constraints yield invariant, relative-safe, input-admissible closed-loop behavior.

APPENDIX

  • M_j is positive definite because each connected component has a positive correction node, eliminating nonzero vectors in its nullspace.The proof uses the component-wise condition in Assumption 2.Source: APPENDIX · p. 9
  • The observer-error proofs establish a uniform bound ν_j for all t and derive further bounds using a geometric sum and a conditioning bound.The proof connects the uniform bound to (24) and uses (23) to obtain (25).Source: APPENDIX · p. 9
  • Block-diagonal P_j and Y_j give Λ_o the required distributed structure.The proof then applies the Schur complement to (41).Source: APPENDIX · p. 9
  • The controller inequalities establish local ε_i-RSI sets contained in X_i using contraction and disturbance margins.The proof verifies the stated conditions for the RSI sets, including the disturbance margin for e_i.Source: APPENDIX · pp. 9–10
  • The proof bounds observer-induced perturbations by ε_i and uses a Schur-complement condition to establish the relative-state constraint.It then applies the relative-safety and input constraints recursively for all t ∈ N.Source: APPENDIX · pp. 9–10
  • The input-bound inequalities establish admissible inputs, with the input constraint applied recursively for all t ∈ N.The proof bounds each input component using the controller gain and perturbation terms.Source: APPENDIX · p. 10
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