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

In this paper, we investigate scale-free delta-level coherent output synchronization for multi-agent systems (MAS) operating under bounded disturbances or noises. We introduce an adaptive scale-free framework designed solely based on the knowledge of agent models and completely agnostic to both the communication topology and the size of the network. We define the level of coherency for each agent as the norm of the weighted sum of the disagreement dynamics with its neighbors. We define each agents coherency level as the norm of a weighted sum of its disagreement dynamics relative to its neighbors. The goal is to ensure that the networks coherency level remains below a prescribed threshold delta, without requiring any a priori knowledge of the disturbance.

💡 Research Summary

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This paper addresses the problem of achieving a prescribed level of output coherence in a network of identical linear agents subject to bounded external disturbances, without requiring any prior knowledge of the communication topology or the size of the network. The authors introduce the notion of “δ‑level coherent output synchronization,” which means that for each agent i the weighted disagreement signal ζ_i(t)=∑j a{ij}(y_i−y_j) satisfies ‖ζ_i(t)‖≤δ after some finite settling time, regardless of the graph structure.

Two families of adaptive protocols are proposed: a non‑collaborative version that uses only the local disagreement ζ_i, and a collaborative version that additionally exchanges an internal protocol state ˜ζ_i=∑j ℓ{ij}x_{i,c} among agents. Both designs are “scale‑free” in the sense that they rely solely on the agent dynamics (A,B,C) and the prescribed δ; no eigenvalue bounds of the Laplacian, no knowledge of the number of agents N, and no disturbance magnitude are required.

Non‑collaborative protocol.
The design assumes a fairly restrictive set of system properties: (A,B) stabilizable, (C,A) detectable, the disturbance matrix E lies in the column space of B, and the triple (A,B,C) is minimum‑phase, left‑invertible, of relative degree one, and left‑invertible. By applying a state transformation (S,T) the plant is partitioned into observable (x₁) and input‑driven (x₂) subsystems. A stabilizing observer gain H₁ is chosen for the observable part, and an algebraic Riccati equation
˜AᵀP+P˜A−P˜B˜BᵀP+I=0
is solved to obtain a positive‑definite matrix P. An adaptive gain ρ_i is defined as a function of the quadratic form ξ̂_iᵀPξ̂_i: if this quantity exceeds a threshold d (chosen based on δ and the norm of CS⁻¹), ρ_i equals ξ̂_iᵀP˜B˜BᵀPξ̂_i; otherwise ρ_i=0. The control input is u_i=−ρ_i˜BᵀPξ̂_i. The authors prove that, under the bounded‑disturbance assumption, the adaptive gain remains bounded, the Lyapunov function V=∑ ξ̂_iᵀPξ̂_i decays to a neighborhood determined by δ, and consequently each ζ_i stays below δ after a finite time.

Collaborative protocol.
In the collaborative setting the agents also share the transformed internal state x_{i,c} through the additional signal ˜ζ_i. This extra information relaxes the stringent assumptions required for the non‑collaborative design; the protocol can be applied to a broader class of linear agents that are merely stabilizable and detectable. The same Riccati‑based adaptive law is employed, but the feedback now incorporates both ζ_i and ˜ζ_i, providing richer information for disturbance attenuation. The resulting theorem guarantees δ‑level coherent output synchronization for any graph in the class G_N, any number of agents, and any bounded disturbance, without any knowledge of the disturbance bound.

Key contributions.

1. Scale‑free design: The protocols are independent of the Laplacian spectrum and the network size, a stark contrast to H∞/H₂‑based almost‑synchronization methods that require graph‑dependent tuning.
2. Disturbance‑agnostic: Only the existence of a finite bound on w_i is needed; the actual bound does not appear in the controller.
3. Two design options: The non‑collaborative scheme offers minimal communication at the cost of strong system assumptions, while the collaborative scheme trades a modest increase in communication for much weaker assumptions.
4. Adaptive gain based on Riccati solution: This provides a systematic way to compute the gain online, ensuring robustness against unknown disturbance magnitudes.

Limitations and future work.
The non‑collaborative protocol’s reliance on minimum‑phase, left‑invertible, and relative‑degree‑one dynamics may limit applicability to many practical systems (e.g., power converters, vehicle platoons). The collaborative protocol, although more flexible, requires exchange of internal states, which raises concerns about communication bandwidth, latency, and packet loss—issues not addressed in the analysis. Moreover, the paper presents only theoretical proofs and lacks extensive simulation or experimental validation, making it difficult to assess performance under realistic network imperfections. Future research directions include extending the framework to handle measurement noise, time‑varying or stochastic disturbances, incorporating communication delays and packet dropouts, and demonstrating the approach on hardware platforms with automatic tuning of the Riccati‑based parameters.

In summary, the work makes a significant step toward topology‑independent, disturbance‑agnostic synchronization of multi‑agent systems by leveraging adaptive, Riccati‑based control laws, and it opens several avenues for practical implementation and further theoretical development.