⚡ electrical engineering

Continuous-Time Decentralized Online Estimation With Additive Noises

This paper proposes a continuous-time decentralized online estimation algorithm for unknown parameters over fixed digraphs with additive communication noises, proving its mean square convergence under a stochastic spatial-temporal persistence of excitation condition by analyzing the stability of the resulting non-autonomous linear stochastic differential equations.

Original authors: Xiaozheng Fu, Yan Chen, Tao Li

Published 2026-07-01

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CC BY 4.0

Original authors: Xiaozheng Fu, Yan Chen, Tao Li

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a group of friends trying to guess the location of a hidden treasure (the "unknown parameter"). They are scattered across a city, and they can only talk to their immediate neighbors. They don't have a central boss telling them the answer; they have to figure it out together by sharing what they see.

This paper is about how these friends can successfully find the treasure even when two things go wrong:

Their eyes are shaky: Sometimes, the map they look at is blurry or changes randomly (random measurement matrices).
Their walkie-talkies are noisy: When they whisper their guesses to each other, static and interference get in the way (additive communication noises).

Here is a breakdown of what the authors did, using simple analogies:

The Problem: A Noisy, Shaky Team

In the real world, sensors (like radar or microphones) aren't perfect. They might glitch, and the signals they send to each other get distorted by "static."

The "Shaky Eyes": The paper assumes the data each person gets isn't just a fixed number; it's like looking at the treasure through a window that randomly changes its shape.
The "Static": When friends pass notes, the notes get scribbled on by static. The paper focuses on "additive noise," which means the static is a constant background hum, regardless of how loud or quiet the friend is talking.

The Solution: A Continuous Conversation

Instead of checking in once an hour (discrete time), these friends are in a constant, flowing conversation (continuous time). They use a specific recipe (an algorithm) to update their guess:

The "New Clue" Step: They look at their own shaky map and adjust their guess based on what they just saw.
The "Group Hug" Step: They listen to their neighbors, average their guesses, and adjust their own guess to match the group, even though the message is a bit garbled by static.

The Big Challenge: Math Without a Map

Usually, when mathematicians try to prove a system will work, they look for a perfect, clean formula (an "analytical solution") to describe the outcome.

The Analogy: Imagine trying to predict the path of a leaf blowing in a storm where the wind direction changes randomly every second. There is no single clean line you can draw to show exactly where the leaf will go.
The Paper's Trick: Since they couldn't find a perfect formula, the authors used a "numerical approximation" method. Think of this as taking a series of very fast, tiny snapshots of the leaf's path. By stitching these snapshots together, they could prove that, over time, the leaf (the error in their guess) will eventually settle down and stop moving, even with the stormy wind.

The Key Findings

The authors proved that the group will eventually find the treasure (converge to the correct answer) if they follow two main rules:

Keep Talking Enough (Persistence of Excitation): Even if the maps are shaky, the friends must keep looking at the treasure often enough and from enough different angles. If they stare at the same blurry spot for too long, they won't learn. The paper calls this "stochastic spatial-temporal persistence of excitation." In plain English: "Keep the data coming in from enough different sources so the randomness averages out."
Turn Down the Volume Slowly (Algorithm Gains): The friends need to adjust how much they trust new information versus what they already know.
- At the start, they should trust new clues a lot (high gain).
- As time goes on, they should trust the "static" less and let their collective guess settle down. The paper shows that if they turn down the volume on new information at just the right speed (mathematically, like $1/\sqrt{t}$ ), the noise won't stop them from finding the truth.

The Special Case: The "Switching" Map

The paper also looked at a scenario where the "shaky eyes" follow a specific pattern, like a light switch flipping on and off randomly (a Markov chain). They proved that even with this flipping behavior, as long as the switch flips fast enough and the group keeps talking, they will still find the treasure.

The Bottom Line

This paper provides a mathematical guarantee that a team of decentralized agents (like sensors or robots) can successfully estimate a hidden value together, even if:

Their individual sensors are unreliable and random.
Their communication lines are full of static.
They are constantly updating their guesses in real-time.

They did this by turning a messy, real-world problem into a math problem about "stochastic differential equations" (equations that describe systems with random noise) and proving that, with the right settings, the chaos eventually settles into a clear answer.

Technical Summary: Continuous-Time Decentralized Online Estimation With Additive Noises

Problem Formulation
This paper addresses the problem of decentralized online estimation for an unknown parameter vector $\theta$ within a multi-agent system operating over a fixed directed graph (digraph). Unlike previous works that often assume ideal communication or deterministic measurement matrices, this study considers a realistic scenario where two distinct sources of uncertainty coexist:

Random Measurement Matrices: Each node $i$ obtains a linear measurement $dz_i(t) = H_i(t)\theta dt + \text{noise}$ , where the measurement matrices $H_i(t)$ are random and time-varying. In a specific case, these matrices are modeled as containing Markov chains.
Additive Communication Noises: When agents exchange state estimates with their neighbors to reach consensus, the communication channels are subject to additive white noise (modeled as Brownian motions). The intensity of this noise is independent of the agent states.

The objective is to design a continuous-time decentralized algorithm where each node updates its estimate using a local innovation term (processing new measurements) and a consensus term (weighted sum of neighbor estimates), ensuring that the estimation error converges to zero in the mean square sense.

Methodology
The authors transform the convergence analysis of the proposed estimation algorithm into the stability analysis of a non-autonomous linear stochastic differential equation (SDE) with random time-varying coefficients and additive noise. The core methodological steps include:

Algorithm Construction: A continuous-time decentralized cooperative online estimation algorithm is formulated. The error dynamics are derived, resulting in an SDE of the form $dx(t) = A(t)x(t)dt + D(t)dw(t)$, where $A(t)$ represents the drift coefficient (dependent on measurement matrices and graph topology) and $D(t)$ represents the diffusion coefficient (dependent on noise intensity and gains).
Numerical Approximation Approach: Since analytical solutions for SDEs with random time-varying coefficients are generally unavailable, the authors employ a numerical approximation method. They construct:
- A Discrete-Time $\Delta$ -Numerical Approximate Solution (DTNAS).
- A Continuous-Time $\Delta$ -Numerical Approximate Solution (CTNAS).
Stability Analysis: The study establishes the asymptotic stability of the trivial solution for these SDEs. The authors prove that the mean square asymptotic stability of the true solution is equivalent to that of the numerical approximate solutions under specific conditions. This involves deriving sufficient conditions for the stability of the DTNAS and CTNAS, and then bounding the difference between the true solution and the CTNAS.
Persistence of Excitation: A key analytical tool is the development of a "stochastic spatial-temporal persistence of excitation" condition. This condition requires that the conditional expectation of the integral of the drift coefficient over a fixed-length interval has an induced matrix measure that is upper bounded by a sequence whose summation diverges to negative infinity.

Key Contributions

Handling Coexisting Uncertainties: The paper addresses the simultaneous presence of random measurement matrices and additive communication noises, a scenario where existing Lyapunov function methods and standard numerical approximations for SDEs with deterministic coefficients are insufficient.
Stability of Random Time-Varying SDEs: The authors develop a theoretical framework for the mean square asymptotic stability of non-autonomous SDEs with random time-varying coefficients. They prove that if the drift and diffusion coefficients are bounded by square-integrable functions and satisfy specific decay and excitation conditions, the solution is mean square asymptotically stable.
Algorithm Gain Design: Based on the stability results, the paper provides conditions for designing algorithm gains ( $\alpha(t)$ and $\beta(t)$ ). Specifically, if the measurement matrices and the communication graph satisfy the stochastic spatial-temporal persistence of excitation condition, the gains can be designed (typically decaying as $O((t+1)^{-1/2-\epsilon})$ ) to guarantee mean square convergence.
Markovian Switching Case: The study investigates a special case where the measurement matrices contain Markov chains with strongly 1-exponential ergodicity. It is shown that under these conditions, the algorithm converges in mean square.

Results

Theoretical Proofs: The paper proves that under Assumption 1 (boundedness and independence of coefficients) and Condition 1 (stochastic persistence of excitation and decay rates), the solution to the error SDE is mean square asymptotically stable.
Convergence Rates: The analysis provides explicit bounds on the mean square error, showing that convergence is achieved if the drift coefficient decays at a rate of $O((t+1)^{-1/2-\epsilon_1})$ and the diffusion coefficient decays at $O((t+1)^{-1/2-\epsilon_2})$ , with $\epsilon_1, \epsilon_2 > 0$ .
Numerical Example: A simulation involving 10 nodes on a balanced fixed digraph is presented. The unknown parameter is a 3-dimensional vector. The measurement matrices are modeled using independent Markov chains. The results demonstrate that:
- The algorithm converges to the true parameter.
- Higher noise intensity leads to slower convergence rates of the mean square errors.
- Larger algorithm gains (within the stable decay range) lead to faster convergence rates.

Significance and Claims
The paper claims to advance the field of decentralized estimation by providing a rigorous stability analysis for continuous-time algorithms operating under realistic, non-ideal conditions (random measurements and noisy communications). By transforming the estimation problem into the stability analysis of SDEs with random time-varying coefficients, the authors bridge a gap in the literature where such equations were previously difficult to analyze directly.

The significance lies in the demonstration that mean square convergence can be guaranteed even when measurement matrices are random and expectations are unknown, provided a stochastic persistence of excitation condition is met. The authors modestly note that while this work focuses on additive noises and model-based algorithms, future work could explore multiplicative noises, time-varying signal tracking, and data-driven model-free approaches.