⚡ electrical engineering

Robust synchronization for multi-agent systems governed by PDEs with observable and unobservable disturbances

This paper proposes a robust synchronization control scheme for multi-agent systems governed by parabolic PDEs that utilizes a boundary disturbance observer and distributed controllers to achieve exponential tracking and input-to-state stability in the presence of both observable boundary and unobservable domain disturbances.

Original authors: Yongchun Bi, Jun Zheng, Guchuan Zhu, Jiye Zhang

Published 2026-05-20

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CC BY 4.0

Original authors: Yongchun Bi, Jun Zheng, Guchuan Zhu, Jiye Zhang

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 fleet of autonomous drones flying in formation, or a group of robots moving in perfect unison. In the real world, these systems are rarely perfect. They face wind gusts, sensor glitches, and unexpected mechanical hiccups. This paper tackles a very specific and difficult version of this problem: how to keep a group of agents (like robots or drones) synchronized when they are governed by complex, continuous physics (like heat spreading or fluid flow) and are bombarded by two very different types of "noise."

Here is the breakdown of the paper's solution using everyday analogies.

The Setting: A Symphony of Heat

Think of each agent in the system not as a simple robot, but as a long, thin metal rod. The "state" of the rod is its temperature at every point along its length. The physics governing this is a Parabolic Partial Differential Equation (PDE). In simple terms, this just means the temperature at one spot depends on the temperature of its neighbors, and it changes over time like heat diffusing through a metal bar.

The goal is for all these rods to follow a specific "leader" temperature pattern (the reference trajectory) and to match each other perfectly (synchronization).

The Problem: Two Kinds of Noise

The researchers identified two distinct types of troublemakers disrupting this harmony:

The "Visible" Noise (Observable Disturbances): Imagine someone is actively blowing hot or cold air directly onto the ends of the rods. The system can "see" this happening at the boundary. It's like a known variable; the system knows, "Hey, someone is blowing air on the left end."
The "Invisible" Noise (Unobservable Disturbances): This is the tricky part. Imagine invisible heat sources appearing randomly inside the metal rod, or the actuators (the devices controlling the ends) having tiny, unpredictable glitches. The system has no idea these are happening. It doesn't know their size, their speed, or their pattern. It's like trying to balance a broom on your hand while someone is secretly shaking the floor beneath you.

Most previous research could only handle one type of noise or assumed the noise was predictable. This paper asks: What if we have both the visible wind on the ends AND the invisible shaking inside, all at the same time?

The Solution: A Smart Team Strategy

The authors designed a two-part strategy to solve this:

1. The "Detective" (Disturbance Observer)

To handle the visible noise at the ends, the team built a "detective" for each agent.

How it works: The detective runs a parallel simulation of the rod. It compares what the rod should be doing (based on the control inputs) with what it is actually doing.
The Trick: Even though there is invisible noise shaking the inside of the rod, the detective is smart enough to isolate the specific signal coming from the ends. It effectively says, "The difference between my prediction and reality at the edge must be caused by that visible wind."
The Result: The system estimates the wind and immediately applies a counter-force to cancel it out. It's like noise-canceling headphones, but for the ends of the rods.

2. The "Chain Reaction" (Distributed Control)

To keep the rods synchronized, they don't all talk to a central computer (which would be slow and require too much data). Instead, they use a directed cycle.

The Analogy: Imagine a line of people passing a message. Person 1 listens to Person N, Person 2 listens to Person 1, and so on.
The Efficiency: Each agent only needs to listen to one neighbor. This drastically reduces the communication load.
The Robustness: Even if the invisible noise is shaking the rods, the control law ensures that if one agent drifts slightly, the neighbor pulls it back. The system is designed so that small errors don't explode; they stay contained.

The Results: Why It Matters

The paper proves mathematically (using advanced tools like "Generalized Lyapunov Methods," which are like energy-checking tools for complex systems) that this approach works:

When the invisible noise is gone: The system locks onto the target perfectly and stays there, with errors vanishing exponentially fast (like a ball rolling to the bottom of a bowl).
When the invisible noise is present: The system doesn't crash. Instead, the errors stay bounded. This means the rods might wiggle a little bit because of the invisible shaking, but they won't go crazy. The worse the shaking, the more they wiggle, but the relationship is predictable and safe.
Input-to-State Stability (ISS): This is the technical term for the "boundedness" mentioned above. It guarantees that the system's reaction is always proportional to the size of the disturbance. If the invisible noise is small, the error is small. If the noise is huge, the error is larger, but still controlled.

The Simulation: Putting It to the Test

The authors ran computer simulations with 5 agents (rods).

They introduced visible wind at the ends and invisible shaking inside.
They showed that the "detective" successfully estimated and canceled the wind.
They showed that even with the invisible shaking, the agents stayed close to the leader and close to each other.
They demonstrated that as they reduced the intensity of the invisible shaking, the errors got smaller, proving the system is robust.

Summary

In essence, this paper presents a robust control strategy for a group of complex physical systems. It solves the problem of keeping them in sync when they are being attacked by known boundary disturbances (which it actively cancels out) and unknown internal disturbances (which it tolerates without failing). It achieves this with minimal communication, making it a highly efficient and reliable method for real-world applications where perfect information is never available.

Technical Summary: Robust Synchronization for Multi-Agent Systems Governed by PDEs with Observable and Unobservable Disturbances

Problem Statement
This paper addresses the robust synchronization problem for multi-agent systems (MASs) governed by parabolic partial differential equations (PDEs). The specific challenge lies in a complex disturbance scenario involving three distinct types of perturbations simultaneously:

Observable Dirichlet boundary disturbances: Disturbances acting at the Dirichlet boundary ( $x=0$ ) which can be measured or estimated.
Unobservable in-domain disturbances: Disturbances $f^{(i)}(x,t)$ occurring within the spatial domain with no prior information regarding their magnitude, dynamics, or structure.
Unobservable implementation disturbances: Disturbances $d^{(i)}_0(t)$ and $d^{(i)}_1(t)$ affecting the boundary control inputs at both Dirichlet and Robin boundaries, arising from practical implementation issues such as quantization errors, noise, or time delays.

The control objective is to design distributed controllers using only the reference signal and local output information from a single neighboring agent. The system must achieve:

Estimation and rejection of the observable Dirichlet boundary disturbances.
Exponential synchronization and reference tracking in the absence of unobservable disturbances.
Bounded tracking and synchronization errors, along with Input-to-State Stability (ISS), in the presence of unobservable disturbances.

Methodology
The authors propose a distributed control scheme based on the following core components:

Disturbance Observer Design: For each agent, a disturbance observer is constructed to estimate the observable Dirichlet boundary disturbance $q^{(i)}(t)$ . This observer utilizes boundary output measurements ( $y^{(i)}(t)$ and $y^{(i)}_v(t)$ ) and is designed to ensure the robustness of the observer error system even when unobservable in-domain and boundary disturbances are present.
Distributed Control Law: The control inputs $U^{(i)}_0(t)$ $U_{0}^{(i)} (t)$ and $U^{(i)}_1(t)$ $U_{1}^{(i)} (t)$ are designed using a directed cycle topology. Agent 1 receives information from Agent $N$ $N$ , and Agent $i$ $i$ receives information from Agent $i-1$ $i - 1$ .
- The Dirichlet control input $U^{(i)}_0(t)$ combines the reference signal $r(t)$ and the estimated disturbance $-\hat{q}^{(i)}(0,t)$ to achieve tracking.
- The Robin control input $U^{(i)}_1(t)$ utilizes the boundary state of the neighboring agent to coordinate synchronization.
Stability Analysis Framework:
- Well-posedness: The existence and uniqueness of solutions for the closed-loop system are established using the lifting technique and semigroup theory.
- Generalized Lyapunov Method (GLM): To handle the nonhomogeneous Dirichlet boundary terms introduced by disturbances without requiring derivatives of the disturbances, the authors employ a generalized Lyapunov method combined with a recursion technique.
- Max-Norm Estimates: Unlike previous works often confined to Hilbert spaces, this analysis derives robustness estimates in the continuous functional space $C([0,1])$ (max-norm), providing point-wise characterizations of performance.

Key Contributions and Results
The paper presents four main theoretical results:

Disturbance Estimation: The estimation error $q^{(i)}(t) - \hat{q}^{(i)}(0,t)$ is proven to converge exponentially to zero if in-domain disturbances are absent. In the presence of unobservable in-domain disturbances, the estimation error remains bounded.
Robust Tracking: The tracking error system is shown to be robust. In the absence of unobservable disturbances, the system achieves exponential tracking. When unobservable disturbances are present, the tracking error remains bounded, with the bound scaling with the magnitude of the disturbances.
Robust Synchronization: The synchronization error between agents is proven to be robust. Similar to tracking, exponential synchronization is achieved without unobservable disturbances, while bounded synchronization is maintained when they are present. The analysis explicitly accounts for the coupling effects of the directed cycle topology.
Input-to-State Stability (ISS): The closed-loop system, comprising the plant, reference, observer, and auxiliary systems, is proven to be Input-to-State Stable (ISS) with respect to all disturbances (observable and unobservable) and the reference signal. This provides a quantitative framework for the system's robustness.

Significance and Claims
The paper claims to address a gap in existing literature where most studies focus on either fully observable disturbances or specific types of unobservable disturbances (e.g., in-domain or Neumann/Robin boundaries). This work is notable for:

Handling Mixed Disturbances: It is the first to consider the simultaneous presence of observable Dirichlet boundary disturbances and unobservable disturbances (both in-domain and at Dirichlet-Robin boundaries) in PDE-based MAS synchronization.
Overcoming Analytical Difficulties: The use of the Generalized Lyapunov Method allows the authors to handle Dirichlet boundary disturbances without needing the derivatives of the disturbances, a common limitation in classical Lyapunov analysis for such systems.
Stronger Performance Characterization: By deriving estimates in the max-norm (continuous space) rather than just $L^2$ (Hilbert space), the paper provides stronger, point-wise guarantees on the estimation, tracking, and synchronization performance.
Practical Relevance: The proposed scheme relies on minimal communication (directed cycle) and local information, making it suitable for practical engineering applications where communication bandwidth is limited and disturbances are inevitable.

Numerical simulations with $N=5$ agents validate the theoretical findings, demonstrating effective disturbance estimation, rejection, and robust synchronization under various disturbance amplitudes. The authors conclude that their framework offers a systematic approach to robust synchronization in complex PDE-based multi-agent environments.