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Consensus conditions of continuous-time multi-agent systems with time-delays and measurement noises
Xiaofeng Zong, Tao Li, Ji-Feng Zhang
TL;DR
The paper studies stochastic consensus in continuous-time multi-agent systems with time-delays and measurement noise, where mean-square and almost-sure convergence need not imply each other. It analyzes additive and multiplicative noise using delay-equation and Lyapunov methods, deriving consensus conditions and showing that exponentially convergent weak consensus implies strong consensus for the multiplicative-noise case.
Problem
The paper addresses stochastic consensus under time-delays and measurement noise, where mean-square and almost-sure convergence may not imply each other.
Source: §1 · pp. 1–2, §2 · p. 4Method
The analysis transforms the closed-loop system to a stochastic delay equation and uses differential resolvent, martingale-convergence, and degenerate Lyapunov methods.
Source: §1 · p. 2, §6 · pp. 12–13Results
For multiplicative noise, stochastic weak consensus with exponential convergence rate implies stochastic strong consensus, and sufficient strong-consensus conditions are established for a strongly connected undirected graph.
Source: §1 · p. 3Takeaways & Limitations
The paper provides sufficient conditions for weak and strong stochastic consensus with additive noise and for strong consensus with multiplicative noise.
Source: §6 · pp. 12–13The model assumes independent Gaussian white noises; the authors note that this independence assumption is conservative.
Source: §2 · p. 4
Abstract
from arXiv · showhide
This work is concerned with stochastic consensus conditions of multi-agent systems with both time-delays and measurement noises. For the case of additive noises, we develop some necessary conditions and sufficient conditions for stochastic weak consensus by estimating the differential resolvent function for delay equations. By the martingale convergence theorem, we obtain necessary conditions and sufficient conditions for stochastic strong consensus. For the case of multiplicative noises, we consider two kinds of time-delays, appeared in the measurement term and the noise term, respectively. We first show that stochastic weak consensus with the exponential convergence rate implies stochastic strong consensus. Then by constructing degenerate Lyapunov functional, we find the sufficient consensus conditions and show that stochastic consensus can be achieved by carefully choosing the control gain according to the noise intensities and the time-delay in the measurement term.
1 Introduction
The introduction identifies gaps in stochastic consensus analysis when time-delays and measurement noises coexist, especially for weak and almost-sure consensus and multiplicative noise. It presents methods and conditions for additive- and multiplicative-noise cases.
1 Introduction
- Earlier continuous-time work had not addressed mean-square weak consensus or almost-sure consensus with delays and noise, and multiplicative noise coupled with delayed states was little studied.The paper targets both additive and multiplicative measurement-noise cases.Source: §1 · p. 2
- For additive noise, the paper establishes explicit necessary and sufficient conditions for stochastic consensus under general digraphs.The introduction identifies differential-resolvent analysis for weak consensus and martingale convergence for strong consensus.Source: §1 · p. 2
- With a spanning tree and fixed delay and noise intensity, mean-square weak consensus can be achieved using a gain c(t) with integral diverging and c(t) tending to zero.The supplied condition specifies ∫c(t)dt = ∞ and lim c(t) = 0.Source: §1 · pp. 2–3
- For additive noise, a gain satisfying the stated square-integrability condition yields mean-square and almost-sure strong consensus with a spanning tree, relaxing earlier graph-balance and delay-bound restrictions.The introduction says the mean-square strong-consensus result relaxes restrictions in Liu et al. (2011a).Source: §1 · p. 3
- For multiplicative noise, exponentially convergent stochastic weak consensus implies strong consensus; sufficient strong-consensus conditions are given for strongly connected undirected graphs with fixed measurement delay and bounded noise intensity.The control gain is chosen according to the delay and noise-intensity bound.Source: §1 · p. 3
- The paper distinguishes the noise cases: additive-noise mean-square weak consensus need not imply almost-sure weak consensus, while multiplicative-noise weak consensus with exponential rate implies strong consensus.It also states that multiplicative-noise stochastic consensus does not necessarily depend on delay in the noise term.Source: §1 · p. 3
2 Problem formulation
The problem formulation models agents on a directed graph whose neighbor measurements may be delayed and noisy. It distinguishes mean-square and almost-sure weak and strong consensus, then frames control-gain design as the route to achieving consensus.
2 Problem formulation
- The directed graph encodes direct information flow from agent j to agent i, and its Laplacian is defined as L = D − A.The degree matrix D uses each agent’s number of incoming neighbors.Source: §2 · p. 3
- Neighbor relative-state measurements may have separate delays in the measurement term and in the state-dependent noise intensity.The formulation names these delays τ1 and τ2, respectively.Source: §2 · pp. 3–4
- Measurement noises are assumed to be independent Gaussian white noises, an independence assumption the authors describe as conservative.They note that reducing this conservatism would require further analysis.Source: §2 · p. 4
- The noise model includes additive intensities independent of agent states and multiplicative intensities depending on relative states; the control gain is designed to obtain either consensus criterion.The noise may act through delayed relative states in the noise term.Source: §2 · p. 4
- Mean-square weak consensus requires pairwise state differences to vanish in expected squared norm; strong consensus additionally requires convergence to a shared random vector.The definitions also identify average-consensus when that vector’s expectation equals the initial average.Source: §2 · p. 4
- Almost-sure weak consensus requires pairwise state differences to vanish with probability one; strong consensus additionally requires convergence to a shared finite random vector.The average-consensus variant requires that vector’s expectation to equal the initial average.Source: §2 · p. 4
- Mean-square and almost-sure convergence generally do not imply one another, while probability-one behavior can be more relevant when only one network trajectory is observed.The text notes that mean-square analysis is generally easier because taking expectation yields a deterministic system.Source: §2 · p. 4
3 Networks with time-delay and additive noises
This section develops consensus conditions for additive-noise networks with measurement delays on general digraphs. It addresses weak and strong stochastic consensus using differential-resolvent decay estimates and martingale convergence, and also states a deterministic-consensus result when the additive measurement noises vanish.
3 Networks with time-delay and additive noises
- A differential-resolvent decay estimate enables the consensus analysis despite the lack of an explicit resolvent expression under time-delay.The estimate supports the subsequent mean-square and almost-sure consensus conditions.Source: §3 · p. 5
3.1 Mean square consensus
- With (C1), mean-square weak consensus requires a spanning tree and (C4′), while a spanning tree with (C1) and (C4) suffices.The conditions are stated for the system under Assumptions 2.1 and 3.1 and the theorem’s delay-eigenvalue bound.Source: §3.1 · p. 5
- Mean-square weak consensus follows from a spanning tree and (C1),(C3); for decreasing c(t), these conditions are also necessary.This concise corollary makes (C2) unnecessary for mean-square weak consensus.Source: §3.1 · p. 7
- Mean-square strong consensus follows from a spanning tree and (C1)–(C2), and requires a spanning tree and (C2) under (C1).The consensus limit is a common Gaussian random variable.Source: §3.1 · p. 7
- For any given delay, the gain can be designed to guarantee mean-square weak and strong consensus on a graph with a spanning tree, without a delay-bound restriction.The analysis also provides necessary and sufficient conditions for mean-square weak consensus on general digraphs.Source: §3.1 · pp. 7–8
3.2 Almost sure consensus
- Almost-sure weak consensus follows from a spanning tree and (C5), and requires a spanning tree; for undirected graphs, it also requires (C5′).The gain conditions encode a decay rate that cannot be too large; mean-square weak consensus need not imply almost-sure weak consensus.Source: §3.2 · p. 8
- Mean-square weak consensus need not imply almost-sure weak consensus: with c(t) = log^-1(4 + t), the latter fails.For this example, the cited passage states that c(t) satisfies (C1) and (C3), while the relevant limit equals 1.Source: §3.2 · p. 8
- Under the stated assumptions, almost-sure strong consensus holds if and only if G contains a spanning tree and condition (C2) holds.The theorem also assumes (C1) and the stated spectral bound involving c̄, t0, τ1, and the nonzero eigenvalues of the Laplacian.Source: §3.2 · p. 9
- For any fixed delay τ1, a suitable gain ensures almost-sure weak or strong consensus; gains satisfying (C1)–(C3) ensure strong consensus for any τ1.The gain conditions for weak and strong consensus are, respectively, (C1) with (C5), or (C2) with the stated spectral bound.Source: §3.2 · p. 9
- When all measurement-noise intensities vanish, a spanning tree, (C1), and the stated spectral bound suffice for deterministic consensus.The theorem assumes σji = 0 for all agents i and j.Source: §3.2 · p. 9
4 Networks with time-delays and multiplicative noises
For multiplicative noise, the section establishes that exponential weak consensus yields strong consensus and gives sufficient control-gain conditions for mean-square and almost-sure consensus. It also distinguishes the effect of measurement delay from noise-term delay: the latter can slow convergence without changing the stated consensus guarantee.
4 Networks with time-delays and multiplicative noises
- Exponential-rate mean-square or almost-sure weak consensus implies the corresponding strong consensus under the stated assumptions.The section uses this result to reduce the strong-consensus analysis to establishing weak consensus with an exponential rate.Source: §4 · p. 9
- A degenerate Lyapunov functional is used to derive the sufficient consensus conditions for multiplicative-noise systems.The proof constructs the functional for the disagreement state and applies the resulting estimate in Theorem 4.2.Source: §4 · p. 9
- With an undirected connected graph and a control gain satisfying the theorem’s sufficient condition, the agents reach mean-square and almost-sure strong consensus at exponential rates bounded by γτ2 and γτ2/2.The sufficient condition is stated in Theorem 4.2; the supplied passages do not provide its complete formula.Source: §4 · pp. 9–11
- For linear multiplicative noise with 2τ2 ≥ τ1, mean-square average-consensus also requires an upper bound on the control gain.The necessary bound is given in terms of the noise intensity and network size.Source: §4 · p. 10
- The noise-term delay τ2 does not enter the sufficient consensus condition, but increasing τ2 decreases the convergence-rate bound and can prolong consensus.The stated rate parameter decreases with τ2 and tends to zero as τ2 tends to infinity.Source: §4 · p. 11
- When the noise intensity is zero, a fixed gain in the stated deterministic range ensures consensus.The passage identifies this as consistent with the paper’s deterministic consensus result.Source: §4 · p. 11
5 Simulation examples
The simulations illustrate strong and weak consensus in the additive-noise case and examine how the two delays affect multiplicative-noise consensus. In particular, the noise-term delay can lengthen convergence, while the measurement delay calls for a smaller gain in the illustrated case.
5 Simulation examples
- With additive noise, c(t)=(1+t)^−1 and τ1=0.2 yield almost-sure strong consensus among the four agents.The trajectories tend to a common value.Source: §5 · p. 11, Figure 1 · p. 11
- With additive noise, c(t)=(1+t)^−1/3 violates condition (C2) and produces almost-sure weak consensus without convergence to a common value.The agents tend to get together over time, illustrating the distinction between weak and strong consensus.Source: §5 · p. 11, Figure 2 · p. 11
- With τ1=0.2 and k=0.12, varying the noise-term delay τ2 from 0 to 10 or 100 does not change whether consensus is achieved, but may prolong its arrival.The simulations compare these delay settings in Figs. 4–6.Source: §5 · p. 12, Figure 4 · p. 12, Figure 5 · p. 12
- At τ1=3.5, the illustrated gain k=0.12 does not achieve almost-sure consensus, whereas k=0.013 satisfies the sufficient condition and achieves it.The text associates the unsuccessful case with Fig. 7 and the successful lower-gain case with Fig. 8.Source: §5 · p. 12, Figure 7 · p. 12
6 Conclusion
The paper develops stochastic consensus conditions for high-dimensional multi-agent systems with time-delays and additive or multiplicative measurement noises. It uses different proof tools for the two noise cases and identifies assumptions that constrain the multiplicative-noise results.
6 Conclusion
- For additive noises, the semi-decoupled approach and differential resolvent function establish sufficient conditions for stochastic weak consensus, while the martingale convergence theorem yields stochastic strong consensus.The conclusion identifies both tools and the corresponding consensus results.Source: §6 · pp. 12–13
- Fig. 8 shows the four agents' states with multiplicative noises at k = 0.013, τ1 = 3.5, and τ2 = 0.Source: Figure 8 · p. 13
- For multiplicative noises, a degenerate Lyapunov functional establishes sufficient conditions for mean square and almost sure strong consensus.The analysis considers delays in the measurement and noise terms.Source: §6 · p. 13
- The multiplicative-noise analysis assumes a fixed undirected graph and equal time-delays across channels, leaving the more general case for future work.The authors identify relaxing these assumptions as a direction for further research.Source: §6 · p. 13