Coordination of passive systems under quantized measurements

In this paper we investigate a passivity approach to collective coordination and synchronization problems in the presence of quantized measurements and show that coordination tasks can be achieved in a practical sense for a large class of passive systems.

💡 Research Summary

The paper addresses the problem of coordinating and synchronizing a network of passive nonlinear systems when only quantized relative measurements are available. The authors adopt a passivity‑based design that was originally developed for continuous‑time systems with exact measurements and extend it to the quantized setting, where each communication event is triggered only when a measured signal crosses a quantization boundary. This leads to an asynchronous, event‑driven information exchange that greatly reduces communication load compared with periodic sampling schemes.

The network is modeled as N agents connected over an undirected, connected graph G = (V, E). Each agent i has state ξ_i ∈ ℝ^{n_i}, input u_i ∈ ℝ^p, and output w_i ∈ ℝ^p described by
  \dot ξ_i = f_i(ξ_i) + g_i(ξ_i) u_i, w_i = h_i(ξ_i) + v_i.
The functions f_i, g_i, h_i are locally Lipschitz, satisfy f_i(0)=0, g_i(0) full column rank, and h_i(0)=0. A continuously differentiable storage function S_i(ξ_i) exists such that the system is strictly passive (or passive) with respect to the input‑output pair (u_i, y_i = h_i(ξ_i)).

For the coordination task, the output w_i is interpreted as velocity, and the position x_i(t) = ∫_0^t w_i(τ) dτ + x_i(0) is introduced. The goal is twofold: (i) all agents must track a common, possibly time‑varying reference velocity v(t); (ii) the relative positions (or integrated velocities) z_k = (D^T ⊗ I_p) x (or w) must converge to prescribed sets A_k (often A_k = {0}, i.e., agreement).

In the original unquantized framework, a potential function P_k(z_k) ≥ 0 is assigned to each edge k, with minima exactly at A_k, and the control law is
  u_i = – Σ_{k∈E_i} d_{ik} ψ_k(z_k), ψ_k = ∇P_k.
This law uses only locally available relative measurements and guarantees that the closed‑loop system (including the auxiliary dynamics \dot z = (D ⊗ I_p) ψ(z) – u) is passive from \dot x to –u.

The paper replaces the exact relative signals z_k by their quantized versions q(z_k). The uniform quantizer is defined as q(r) = Δ·⌊r/Δ + ½⌋, where Δ > 0 is the resolution. The quantized control becomes
  u_i = – Σ_{k∈E_i} d_{ik} ψ_k(q(z_k)).
Because q is discontinuous, the resulting closed‑loop dynamics are modeled as a differential inclusion using Clarke’s generalized gradient ∂q. The authors employ nonsmooth analysis and tools from set‑valued Lyapunov theory to handle the discontinuities.

The main theoretical contribution is a Lyapunov‑type inequality for the composite storage function V(z) = Σ_k P_k(q(z_k)). One shows that
  \dot V ≤ –α‖z‖² + βΔ,
with α > 0 depending on the strict passivity margins and the curvature of the potentials, and β proportional to the Lipschitz constants of ψ_k. Consequently, for any given Δ, the trajectories converge to a bounded neighborhood of the desired set A = Π_k A_k, whose radius shrinks linearly with Δ. This is termed “practical consensus” or “practical synchronization”: exact convergence is impossible under quantization, but the error can be made arbitrarily small by refining the quantizer.

The second part of the paper treats output synchronization of identical linear passive agents. Assuming each agent satisfies A^T P + P A ≤ 0 and B^T P = C for some positive definite P, the authors design a quantized diffusive coupling u_i = – Σ_{j∈N_i} ψ(q(w_i – w_j)). Using a Lyapunov function based on the quadratic form ξ^T (I_N ⊗ P) ξ, they derive an inequality analogous to the coordination case, showing that the state differences converge to a set whose size is O(Δ). Thus, even with asynchronous, quantized measurements, the network achieves approximate synchronization.

The paper emphasizes the asynchronous event‑triggered nature of the communication: a broadcast occurs only when a relative measurement crosses a quantization threshold. This reduces bandwidth usage dramatically compared with periodic gossip or sampled‑data approaches, which often require synchronized clocks and high sampling rates. The authors also discuss practical issues such as chattering (rapid toggling near a quantization boundary) and suggest using hysteresis quantizers or imposing a minimum inter‑event time to mitigate it.

Finally, the authors outline several avenues for future work: (1) non‑uniform or adaptive quantizers, (2) time‑varying communication graphs, (3) robustness to disturbances and model uncertainties, and (4) experimental validation on robotic swarms or power‑grid applications.

In summary, the paper provides a rigorous passivity‑based framework that extends continuous‑time coordination and synchronization results to the realistic scenario where only quantized, asynchronously exchanged measurements are available. By leveraging nonsmooth control theory, it demonstrates that practical convergence can be guaranteed for a broad class of passive nonlinear agents, offering a valuable tool for the design of large‑scale, bandwidth‑constrained networked control systems.