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Multi-Agent Consensus With Relative-State-Dependent Measurement Noises
Tao Li, Fuke Wu, Ji-Feng Zhang
TL;DR
The paper addresses distributed consensus under measurement noises whose intensity depends on agents’ relative states. Using stochastic differential equations and small consensus-gain theorems, it derives conditions for mean-square and almost-sure consensus, performance rates, and steady-state error. For homogeneous channels, the mean-square gain condition is necessary and sufficient and does not depend on the specific network topology.
Problem
Existing models often assume noise intensity is time-invariant and state-independent, whereas relative-state-dependent noises couple uncertainty evolution with agent-state evolution.
Method
The paper analyzes the resulting multiplicative-noise stochastic differential equations and develops small consensus-gain theorems for undirected multi-agent networks.
Results
For homogeneous channels, a necessary and sufficient mean-square consensus condition on the control gain is obtained, while sufficient conditions also quantify convergence rate and steady-state error.
Takeaways & Limitations
The allowable control gain depends on the number of nodes and noise coefficient rather than the specific network topology in the homogeneous case.
Abstract
from arXiv · showhide
In this note, the distributed consensus corrupted by relative-state-dependent measurement noises is considered. Each agent can measure or receive its neighbors' state information with random noises, whose intensity is a vector function of agents' relative states. By investigating the structure of this interaction and the tools of stochastic differential equations, we develop several small consensus gain theorems to give sufficient conditions in terms of the control gain, the number of agents and the noise intensity function to ensure mean square (m. s.) and almost sure (a. s.) consensus and quantify the convergence rate and the steady-state error. Especially, for the case with homogeneous communication and control channels, a necessary and sufficient condition to ensure m. s. consensus on the control gain is given and it is shown that the control gain is independent of the specific network topology, but only depends on the number of nodes and the noise coefficient constant. For symmetric measurement models, the almost sure convergence rate is estimated by the Iterated Logarithm Law of Brownian motions.
I. INTRODUCTION
The note studies consensus when measurement-noise intensity depends on agents’ relative states, creating coupled, time-varying uncertainty dynamics. It develops small-gain conditions for mean-square and almost-sure consensus, including convergence and steady-state guarantees.
- Motivation: Relative-state-dependent noises make network uncertainty evolve together with agents’ states, complicating distributed control design and closed-loop analysis.Examples include logarithmic quantization and analog fading channels.
- Model distinction: The model extends prior work by using a vector-valued noise intensity function of agents’ relative states rather than state-independent or merely time-varying noise intensity.The resulting consensus-error dynamics form a stochastic differential equation with multiplicative noises.
- Main results: Small control gains provide sufficient conditions for asymptotically unbiased mean-square and almost-sure average consensus, with performance related to gain, noise, and topology parameters.The results quantify both mean-square steady-state error and convergence rate.
- Main results: For independent homogeneous channels, 0 < k < N/[(N −1)σ2] is necessary and sufficient for mean-square consensus.This condition concerns the control gain when noise intensity grows at rate σ.
- Almost-sure convergence: Almost-sure consensus requires a weaker condition than mean-square consensus for homogeneous linear-growth noise intensity functions.For symmetric noise intensity functions, the almost-sure convergence rate is estimated using the Iterated Logarithm Law of Brownian motions.
II. PROBLEM FORMULATION
The problem formulation models high-dimensional first-order agents communicating over an undirected graph with noisy relative-state measurements. It asks when consensus occurs and how gain, noise intensity, and topology determine performance.
- Agent and network model: Each agent is a high-dimensional first-order system with state and control vectors in R^n, and the network information flow is represented by an undirected graph.The graph uses an adjacency matrix and Laplacian to encode neighbor interactions.
- Noisy measurements: Agent i receives neighbor j’s state through a noisy measurement yji(t), where the measurement noise is associated with relative-state information.The neighbor set Ni determines which agents contribute measurements.
- Noise assumptions: The noise intensity function fji maps R^n to R^n and satisfies ∥fji(x)∥ ≤ σ̄∥x∥.This linear-growth bound ties noise magnitude to the measured relative state.
- Noise assumptions: The integrated measurement-noise processes are independent Brownian motions, giving the model a stochastic-differential formulation.The noise processes satisfy ∫_0^t ξji(s)ds = wji(t).
- Control objectives: The distributed protocol uses a control gain matrix K and neighbor-relative measurements to determine each agent’s input.The formulation asks for consensus conditions and performance relationships involving K, noise intensity, and graph parameters.
III. MEAN SQUARE AND ALMOST SURE CONSENSUS
The analysis transforms consensus errors into disagreement coordinates and derives gain conditions ensuring mean-square and almost-sure average consensus. It also relates steady-state error and convergence to gain and network structure.
- Consensus guarantees: A positive-definiteness condition on the transformed drift-diffusion matrix yields asymptotically unbiased mean-square and almost-sure average consensus.The limiting random vector has expectation equal to the initial network average.
- Stability relationship: Mean-square and almost-sure exponential stability need not imply each other generally, although linear-growth drift and diffusion allow moment exponential stability to imply almost-sure exponential stability.The paper uses this relationship to distinguish the strength of the consensus conditions.
- Small-gain theorem: For K = kI_n, a connected graph and 0 < k < N/[ (N−1)σ2 ] ensure asymptotically unbiased mean-square and almost-sure average consensus.The theorem gives a small-consensus-gain condition under the stated noise-growth assumptions.
- Homogeneous channels: For homogeneous control channels, the gain condition can be selected independently of the specific network topology and is inversely related to the noise-intensity growth rate.The relevant dependence is on the number of nodes and the noise bound rather than the particular graph structure.
IV. LINEAR NOISE INTENSITY FUNCTION
For linear relative-state-dependent noise, the paper derives mean-square and almost-sure consensus conditions, quantifies performance, and distinguishes their control-gain requirements. Homogeneous models yield topology-independent mean-square thresholds, while smaller gains reduce steady-state error at the cost of slower convergence.
- Linear noise intensity functions reduce the measurement-noise model to noise proportional to each agents’ relative state.The section assumes f_ji(x)=Σ_ji x, including homogeneous scalar models f_ji(x)=σ_ji x.
- A connected graph with 0 < k < N/[barσ^2(N−1)] guarantees asymptotically unbiased mean-square average consensus for heterogeneous scalar growth rates.Theorem 4.2 states the corresponding condition using the maximum noise coefficient barσ; Corollary 4.1 gives a necessary-and-sufficient condition for homogeneous σ.
- For homogeneous communication and control channels, the mean-square gain condition is necessary and sufficient and depends on the agent count and noise coefficient rather than network topology.The admissible gain must be positive and sufficiently small; its upper bound is inversely proportional to the squared noise-growth rate.
- Reducing k lowers the mean-square steady-state error, but k approaching zero slows convergence; topology design can instead improve synchronizability through λ2(L)/λN(L).The reported error bound depends on control gain and topology, and the gain optimizing mean-square convergence is also discussed.
- Almost-sure consensus requires a weaker gain condition than mean-square consensus in relative-state-dependent noise models.For homogeneous models, k + k^2σ^2/2 > 0 can permit positive gains and sufficiently negative gains, whereas mean-square consensus requires a small positive gain.
- Relative-state-dependent noise can help almost-sure consensus, although it harms mean-square consensus when the noise level exceeds the threshold N/[k(N−1)].The paper contrasts the almost-sure condition k + k^2σ^2/2 > 0 with the mean-square restriction 0 < kσ^2(N−1)/N < 1.
V. CONCLUDING REMARKS
The note establishes consensus conditions and quantitative behavior for high-dimensional agents with relative-state-dependent measurement noises. In homogeneous linear settings, mean-square consensus has an exact gain range, while almost-sure consensus requires weaker conditions and future work extends the scope.
- Main results: 0 < k < N/[(N −1)σ2] is necessary and sufficient for mean-square consensus with homogeneous linear noise intensity and control channels.Here σ is the growth rate of the noise intensity function, and the condition is independent of the specific network topology.
- Main results: Almost-sure consensus requires weaker control-gain conditions than mean-square consensus in these multi-agent networks.The conclusion specifically contrasts the two consensus notions for relative-state-dependent measurement noises.
- Future research: Future work includes discrete-time models, random link failures, time delays, and distributed tracking with relative-state-dependent measurement noises.
APPENDIX
The appendix supplies matrix lemmas and stochastic-calculus arguments for analyzing disagreement and average-state dynamics. These tools establish mean-square and almost-sure convergence, characterize the limiting average, and support almost-sure rate estimates based on Brownian-motion laws.
- Matrix identities: The matrix φ satisfies φφT = IN −JN and φTφ = IN−1, providing the disagreement-space identities used in the proofs.
- Average-state convergence: The network average converges in mean square and almost surely to a random variable with finite second-order moment.The resulting protocol is identified as an asymptotically unbiased mean-square average-consensus protocol.
- Almost-sure convergence: The same stochastic framework establishes asymptotically unbiased almost-sure average consensus under the stated theorem conditions.
- Mean-square convergence: Applying Itô calculus and comparison arguments to the disagreement norm yields mean-square convergence when the relevant matrix condition is positive definite.
- Convergence rates: The Law of the Iterated Logarithm of Brownian motions is used to estimate the almost-sure convergence rate.