⚡ electrical engineering

Scale-Free delta-Level Coherent Output Synchronization of Multi-Agent Systems with Adaptive Protocols and Bounded Disturbances

This paper proposes an adaptive, scale-free framework for multi-agent systems that ensures coherent output synchronization within a prescribed threshold $\delta$ despite bounded disturbances, without requiring prior knowledge of the network size, communication topology, or disturbance magnitude.

Original authors: Anton A. Stoorvogel, Ali Saberi, Donya Nojavanzadeh

Published 2026-02-12

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CC BY 4.0

Original authors: Anton A. Stoorvogel, Ali Saberi, Donya Nojavanzadeh

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 you are a conductor of a massive orchestra, but there’s a catch: the musicians are in different rooms, they can only hear snippets of what their neighbors are playing, and there is a constant, unpredictable amount of background noise—like a construction crew working outside the building.

Your goal is to make sure the entire orchestra stays "in sync" (coherent) so that the music sounds like one unified piece, rather than a chaotic mess. Even more importantly, you want to guarantee that the "messiness" never exceeds a certain level, no matter how many musicians you add or how loud the construction noise gets.

This paper, "Scale-Free $\delta$ -Level Coherent Output Synchronization," provides the mathematical "sheet music" to achieve exactly that.

Here is the breakdown of how it works using everyday concepts:

1. The Problem: The "Chaos vs. Control" Struggle

In most systems (like a swarm of drones or a power grid), two things fight each other:

The Disturbance: External "noise" (wind hitting a drone, a surge in a power line) that pushes agents away from each other.
The Tuning Problem: Usually, to keep things quiet, a leader has to know exactly how loud the noise is to set the right volume for the control signals. If the noise gets louder than expected, the system breaks.

2. The Solution: The "Smart Volume Knob" (Adaptive Protocols)

The researchers developed a way for each agent (each musician) to have its own "Smart Volume Knob."

Instead of a central conductor telling everyone what to do, each agent looks at its neighbors and asks: "How much are we disagreeing right now?"

If the disagreement is small, the agent stays quiet to save energy.
If the disagreement starts to creep toward a specific limit (which the researchers call $\delta$ ), the agent automatically turns up its "corrective" effort.

The magic is that the agents don't need to know how loud the noise is or how many other agents are in the network. They only need to know their own "instrument" (their internal model).

3. Two Ways to Play: "Soloists" vs. "Collaborators"

The paper offers two different strategies for the orchestra:

The Noncollaborative Protocol (The "Soloist" Approach):
Imagine each musician only listens to the output (the sound) of their neighbors. They don't talk to each other; they just try to match the sound they hear. This is harder because you have less information, so the researchers had to make stricter assumptions about the "instruments" to make it work.
- Analogy: A group of dancers in a dark room, trying to stay in sync just by feeling the vibrations of the floor.
The Collaborative Protocol (The "Teamwork" Approach):
Here, the musicians share a bit more information—not just the sound, but also their "intentions" (their internal states). This extra bit of communication makes the system much more flexible. It allows the agents to be more complex and handles much tougher "instruments" more easily.
- Analogy: Dancers who can also see each other's shadows, making it much easier to stay in rhythm.

4. Why is this "Scale-Free"?

This is the most impressive part. In many mathematical models, if you go from 10 agents to 10,000 agents, the math "breaks" or requires much more power.

This paper is "Scale-Free," meaning the rules they wrote work just as well for a tiny group of 5 drones as they do for a massive swarm of 121 (or even a million). The "Smart Volume Knob" scales perfectly. Whether the orchestra has 5 people or 5,000, the level of "messiness" stays below that pre-set limit ( $\delta$ ).

Summary for the Layperson

The paper provides a mathematical recipe for keeping a large group of moving parts (like robots or machines) working in perfect harmony. It ensures that even if the environment is noisy and unpredictable, the group will stay "in sync" within a strictly controlled margin of error, without needing a central boss or prior knowledge of how much trouble the environment will cause.

This paper, titled "Scale-Free $\delta$ -Level Coherent Output Synchronization of Multi-Agent Systems with Adaptive Protocols and Bounded Disturbances," addresses a fundamental challenge in cooperative control: how to ensure a group of agents maintains a specific level of output agreement (coherency) when subjected to external disturbances, without knowing the network size or the magnitude of those disturbances.

Below is a detailed technical summary of the work.

1. The Problem: $\delta$ -Level Coherent Synchronization

In Multi-Agent Systems (MAS), the goal is typically to achieve synchronization (agreement on a state or output). Traditional approaches, such as $H_\infty$ or $H_2$ almost synchronization, have two major practical drawbacks:

Tuning Dependency: The performance depends heavily on the communication graph structure.
Disturbance Dependency: The synchronization error scales linearly with the disturbance magnitude. To guarantee a specific error bound, the controller must know the maximum possible disturbance beforehand.

The Research Objective: The authors seek to solve the Scale-Free $\delta$ -Level Coherent Output Synchronization problem. Specifically, for a given threshold $\delta > 0$ , they aim to design protocols such that the "coherency level" (the norm of the disagreement between an agent and its neighbors) remains below $\delta$ for all time after a transient period, regardless of:

The number of agents ( $N$ ).
The communication topology (graph $G$ ).
The magnitude of the bounded disturbances ( $w_i$ ).

2. Methodology

The paper proposes two distinct classes of adaptive nonlinear protocols:

A. Noncollaborative Protocols (Problem 1)

These protocols rely only on local output information ( $\zeta_i$ ) and the agent's own model.

Assumptions: The agents must be minimum-phase, left-invertible, and the disturbances must lie within the range of the input matrix ( $Im \, E \subset Im \, B$ ).
Design: The authors use a coordinate transformation to separate the dynamics into a stable part and a part controlled by an adaptive gain. They employ an Algebraic Riccati Equation (ARE) to design the feedback and an adaptive law ( $\rho_i$ ) that increases the control gain until the disagreement falls below the threshold $d$ (related to $\delta$ ).

B. Collaborative Protocols (Problem 2)

These protocols allow for "extra" communication, where agents exchange information about their internal protocol states ( $\tilde{\zeta}_i$ ).

Assumptions: These are significantly relaxed. The requirement $Im \, E \subset Im \, B$ is removed, and the system only needs to be right-invertible and minimum-phase.
Design: The methodology uses an adaptive observer to estimate states and an adaptive state-feedback mechanism. Two adaptive gains are used: one ( $\rho_i$ ) to handle the observer error and another ( $\alpha_i$ ) to handle the control effort. The $\alpha_i$ gain is designed to scale the control law to ensure the output error is minimized.

3. Key Contributions

Scale-Free Framework: The protocols are "one-shot" designed. Once the agent model is known, the same protocol works for a network of 5 agents or 121 agents, and for any graph (connected, disconnected, directed, or undirected).
Agnosticism to Disturbances: Unlike $H_\infty$ methods, these protocols do not require knowledge of the disturbance bound. The adaptive gains automatically "search" for the necessary strength to suppress the disturbance to the level $\delta$ .
Protocol Versatility: The paper provides a hierarchy of solutions, from highly restrictive noncollaborative protocols to more flexible collaborative protocols that can handle a wider range of agent dynamics.

4. Results and Proof Highlights

Theoretical Guarantees: Through Lyapunov stability analysis and the use of the Gagliardo-Nirenberg inequality, the authors prove that the adaptive gains $\rho_i$ and $\alpha_i$ remain bounded and that the synchronization error converges to the $\delta$ -level.
Handling Disconnected Graphs: A significant result is the ability to handle graphs with multiple "basic bi-components" (disconnected sub-networks). The protocol ensures coherency within each component.
Numerical Validation: The authors provide extensive simulations (Figures 4–33) demonstrating:
- Scalability: Performance is consistent as $N$ increases from 5 to 121.
- Robustness: The system handles various noise patterns (sinusoidal, sawtooth, etc.).
- Topology Independence: Success is shown on Vicsek fractal graphs, Circulant graphs, and disconnected graphs.

5. Significance

This work is significant for the deployment of large-scale autonomous systems (e.g., drone swarms, autonomous vehicle platoons, or distributed sensor networks). In real-world scenarios, the exact number of agents in a network may change, and the environmental noise is often unpredictable. By providing a scale-free and disturbance-agnostic control framework, this paper offers a mathematically rigorous way to guarantee performance bounds in highly uncertain and dynamic environments.