Source-linked AI summary
Dynamic Average Consensus under Limited Control Authority and Privacy Requirements
Solmaz S. Kia, Jorge Cortes, Sonia Martinez
TL;DR
The paper develops a distributed algorithm for tracking dynamic-input averages in directed networks. It characterizes convergence and steady-state error, including cases with zero error, and extends the design to limited control authority and privacy preservation.
Problem
The paper addresses distributed tracking of the time-varying average of signals using only information available across the network.
Method
The paper proposes a distributed algorithm and an extension addressing limited control authority and privacy preservation requirements.
Results
The algorithm yields steady-state-error tracking, with error and convergence rate depending on design parameters and zero error for special input classes; it retains convergence properties under time-varying topologies.
Takeaways & Limitations
The framework supports dynamic average tracking while accommodating limited control authority and privacy preservation against internal and external adversaries.
Takeaways & Limitations
The conclusions identify numerous avenues of research as open for future work, including discrete-time implementations with the considered features.
Abstract
from arXiv · showhide
This paper introduces a novel continuous-time dynamic average consensus algorithm for networks whose interaction is described by a strongly connected and weight-balanced directed graph. The proposed distributed algorithm allows agents to track the average of their dynamic inputs with some steady-state error whose size can be controlled using a design parameter. This steady-state error vanishes for special classes of input signals. We analyze the asymptotic correctness of the algorithm under time-varying interaction topologies and characterize the requirements on the stepsize for discrete-time implementations. We show that our algorithm naturally preserves the privacy of the local input of each agent. Building on this analysis, we synthesize an extension of the algorithm that allows individual agents to control their own rate of convergence towards agreement and handle saturation bounds on the driving command. Finally, we show that the proposed extension additionally preserves the privacy of the transient response of the agreement states and the final agreement value from internal and external adversaries. Numerical examples illustrate the results.
1 Introduction
The paper addresses dynamic average consensus for time-varying inputs using neighbor information, with emphasis on convergence control, actuation limits, and privacy. It proposes algorithms with characterized tracking error, time-varying-topology guarantees, and privacy properties.
- Problem setting: Dynamic average consensus requires agents to track the time-varying average of distributed signals using only neighbor information.The problem supports applications involving dynamic information collected by multiple agents.
- Research gap: Prior approaches address static or dynamic consensus but commonly omit individual convergence-rate constraints, bounded control authority, or privacy issues.Some existing methods also rely on fixed or undirected topologies, specialized inputs, or bounded higher-order differences.
- Core algorithm: The proposed continuous-time algorithm tracks the network-wide average on strongly connected, weight-balanced digraphs with a characterized steady-state error and convergence behavior.The error can vanish for static inputs and for dynamic inputs differing by a constant, while correctness extends to suitable time-varying topologies.
- Analysis: The analysis covers time-varying interaction topologies, initialization errors, convergence-rate tuning, and discrete-time implementation requirements.The paper establishes correctness for weight-balanced topologies that are infinitely often jointly strongly connected.
- Limited control authority: An extension with local first-order filters lets agents tune their convergence rates independently and handle bounded control commands without changing the ultimate tracking-error bound.Under bounded inputs with bounded relative growth, the extension retains the original algorithm’s correctness guarantees.
- Privacy: The algorithms preserve local-input privacy, while the filtered extension protects agreement-state trajectories and the final agreement value against specified adversaries.The privacy analysis considers internal and external adversaries with different levels of graph, parameter, initialization, and message-history knowledge.
2 Preliminaries
The preliminaries define the notation, linear-system concepts, graph properties, and switching-network model used throughout the paper.
- Notation: The paper defines vector, matrix, norm, saturation, aggregation, and essential-supremum notation for network variables and inputs.
- Linear systems: A stable linear system’s convergence rate is characterized through exponential decay and, for linear time-invariant systems, the least negative eigenvalue real part.
- Graph theory: Directed graphs are represented by weighted adjacency and Laplacian matrices, with strong connectivity and weight balance determining key spectral properties.
- Time-varying interactions: The network model allows piecewise-constant switching among uniformly bounded interaction topologies without chattering.
- Time-varying interactions: For trajectory-independent switching, uniform asymptotic stability is equivalent to exponential stability, and joint strong connectivity is defined over time intervals.
3 Problem statement
The paper formulates dynamic average consensus for distributed agents and extends it to controllable convergence rates, bounded commands, and privacy preservation.
- Dynamic average consensus: Each agent has single-integrator dynamics, an agreement state x_i, and a driving command c_i, while communicating only with its out-neighbors.
- Dynamic average consensus: The primary problem is for agents on a strongly connected, weight-balanced digraph to asymptotically track the average of time-varying inputs.
- Control authority: The controllable-rate problem lets each agent converge at its own desired rate, supporting agents with limited control authority and arrival-time scheduling.
- Control authority: The limited-authority problem imposes bounded driving commands through saturation functions.
- Privacy: The privacy problem requires local inputs, agreement values, and agreement-state trajectories to remain unrevealed or unreconstructible to adversaries.
4 Dynamic average consensus
The proposed dynamic average consensus algorithm tracks averages on directed networks with bounded, tunable error and exact convergence for special inputs, while extending to switching and discrete-time settings.
- Continuous-time algorithm: The continuous-time algorithm solves dynamic average consensus with steady-state error for arbitrary time-varying inputs, whose size is controlled by a design parameter.
- Fixed topology: Over strongly connected, weight-balanced digraphs, the underlying dynamical system is stable and converges exponentially from arbitrary initial conditions.
- Continuous-time algorithm: For static inputs and inputs differing only by static offsets, the steady-state error is zero and convergence is exponential.
- Switching topologies: For admissible time-varying topologies, the transformed trajectories converge asymptotically, including consensus under a spanning-tree condition.
- Discrete time: The discrete-time implementation has nonzero steady-state error, tracks the input average within an O(β^-1) neighborhood, and requires stepsize characterization.
- Privacy: The local input u_i is never communicated directly, supporting privacy of individual inputs in the distributed implementation.
5 Dynamic average consensus with controllable rate of con-
The extension separates information and motion phases so agents can tune convergence rates and accommodate command saturation without worsening the tracking-error bound.
- Controllable convergence rate: A time-varying local gain θ_i lets each agent adjust its convergence rate while the information phase retains a network-wide common rate.
- Controllable convergence rate: The algorithm solves the controllable-rate problem on strongly connected, weight-balanced graphs for inputs with essentially bounded projected derivatives.
- Controllable convergence rate: Each agent’s convergence rate is bounded through its local gain, while α and β tune the overall rate.
- Controllable convergence rate: The local first-order motion filter changes convergence rate without adversely affecting the tracking-error bound.
- Limited control authority: With saturation, the algorithm retains the same error bounds as the unsaturated case when command limits exceed the relevant input-derivative bounds.
- Limited control authority: Under the saturation conditions, each agreement state asymptotically approaches its motion-phase state after the latter becomes bounded and its derivative falls below the command limit.
6 Dynamic average consensus with privacy preservation
The algorithm preserves privacy of local inputs against external and internal adversaries, with stronger guarantees under a nonzero initial input derivative. Its extension additionally hides the final agreement value and agreement-state trajectories while retaining the tracking-error guarantee.
- Local-input privacy: External adversaries cannot reconstruct any agent’s input, while internal adversaries cannot reconstruct the inputs of other agents.The result applies even when external adversaries observe communication histories and internal adversaries possess network information available through their position.
- Local-input privacy: Privileged adversaries cannot reconstruct agent i’s dynamic input when its derivative is nonzero over an initial interval.The unknown initial condition prevents unique recovery of the input from the resulting ordinary differential equation.
- Privacy-preserving extension: The unextended algorithm does not satisfy all privacy requirements, motivating an extension with common and agent-specific signals.The common signal conceals the final agreement value, while the local signal and initial condition can remain hidden from other agents.
- Privacy-preserving extension: The extension prevents external adversaries from obtaining the final agreement value and prevents adversaries from reconstructing agreement-state trajectories when an initial condition or local parameter is unknown.Its agreement-state equation is local, with components set by the individual agent.
- Overall guarantee: Under the hypotheses of Lemma 5.1, the extension preserves the ultimate tracking-error bound while satisfying the stated privacy requirements.The result covers input privacy, final-value privacy against external adversaries, and trajectory privacy under the stated conditions.
7 Simulations
Simulations examine time-varying topologies, discrete-time tracking, and bounded driving commands. The algorithm achieves zero steady-state error for converging common inputs, bounded error under weight-balanced switching, and perfect discrete-time tracking after some time; the extension better handles saturation.
- 7.1 Networks with time-varying interaction topologies: When the switching signal belongs to S_admis, the agreement states remain bounded.This boundedness is observed across the time-varying-topology examples.
- 7.1 Networks with time-varying interaction topologies: In Case 1, inputs converging to a common function yield convergence to the average with zero steady-state error.The switching-network version of Lemma 4.3 provides this conclusion.
- 7.1 Networks with time-varying interaction topologies: In Case 2, the algorithm guarantees only bounded steady-state error during periods when the network is weight-balanced.The error grows during such periods but remains bounded; once strong connectivity returns, tracking of the network-wide average resumes.
- 7.2 Discrete-time implementation: The discrete-time algorithm achieves perfect tracking after some time in the sampled-input example.The simulation uses α = β = 1 and communication bandwidth 2 Hertz, with δ = 0.5 seconds.
- 7.3 Limited control authority: Larger β reduces tracking error but produces larger driving commands that violate saturation bounds.Under the stated requirements, the extended algorithm’s ultimate tracking resembles the unsaturated response, unlike the original algorithm.
8 Conclusions
The paper develops dynamic average-consensus algorithms for strongly connected, weight-balanced digraphs, extending them to time-varying topologies, discrete-time implementations, limited control authority, and privacy preservation.
- The proposed distributed algorithm lets agents track the network average of dynamic inputs with a steady-state error.The error and convergence rate depend on design parameters.
- The steady-state error is zero for special classes of input signals.
- The algorithm retains its convergence properties under time-varying topologies and supports discrete-time implementations.
- Extensions address limited control authority and privacy preservation requirements against internal and external adversaries.
- Future work includes discrete-time implementations with the considered features, algorithms without a priori weight-balanced topologies, and applications to distributed estimation and map merging.
A Proof of the results of Section 4.3
The appendix proves discrete-time convergence by analyzing the transition matrix spectrum under a stepsize restriction and showing that the relevant inputs vanish or remain bounded.
- Spectral conditions: The stepsize must satisfy δ ∈ (0, min{α^-1, β^-1(dout_max)^-1}) for the non-consensus eigenvalues to lie strictly inside the unit circle.
- Spectral conditions: For a strongly connected, weight-balanced digraph, the transition matrix has one eigenvalue at 1 and all remaining eigenvalues inside the unit circle.This establishes semiconvergence of the matrix.
- Convergence arguments: The proof also establishes decay of the homogeneous transition term and convergence of the associated matrix sums under the admissible stepsize.
- Convergence arguments: Under the relevant input conditions, the transformed system has a vanishing input and converges to the equilibrium of its zero-system.
- Convergence arguments: The resulting state converges globally asymptotically to the average of the input signals for every agent.
B Supporting material for the proof of Lemma 5.2
The supporting results establish asymptotic tracking for scalar systems with bounded, eventually slowly varying inputs and boundedness under input-dependent stability conditions.
- Supporting stability results: For a system with an essentially bounded driving signal, the transformed error converges asymptotically to zero from any initial condition.
- Supporting stability results: Input-dependent Lyapunov analysis establishes the sign conditions needed for negative definiteness outside the bounded-error region.
- Tracking result: If u and its derivative are essentially bounded and |u̇(t)| is eventually below a finite bound, then x(t) converges to u(t).
- Supporting stability results: Input-to-state stability and boundedness of βu + u̇ imply bounded trajectories for every finite initial condition.