ChronoSync Protocol: Distributed Time Synchronization

• ChronoSync is a decentralized chronometer synchronization protocol that aligns software clocks of multi-agent systems through consensus-based controllers and hybrid system modeling.
• It employs a Luenberger-style observer and Lyapunov-based stability analysis to guarantee exponential convergence of both clock synchronization and drift estimation even in the presence of bounded perturbations.
• The protocol is practically applicable to distributed sensor networks, autonomous vehicles, and robotic teams, ensuring robust operation under asynchronous and intermittent communication.

ChronoSync is a decentralized chronometer synchronization protocol designed for multi-agent systems, enabling agents with independently drifting and environmentally perturbed hardware clocks to achieve consensus on a shared software clock with a configurable common drift. Synchronization is accomplished via a consensus-based controller, hybrid-system formulation, and Lyapunov-based stability analysis. The protocol guarantees not only practical synchronization of software clocks but also online estimation of each agent's unknown hardware clock drift, with resilience to asynchronous, intermittent, and directed communication patterns and robustness to bounded disturbances (Zegers et al., 6 Apr 2025).

1. Agent and Network Modeling

Each agent $p \in V$ maintains two clocks: a hardware clock $\theta_p(t)$ subject to environmental perturbations, and a software clock $\vartheta_p(t)$ manipulated via a control input. The evolution of these clocks is formalized as:

• Hardware clock:

$\dot \theta_p(t) \in a_p + \delta_p B,$

where $a_p > 0$ denotes agent $p$'s (unknown) natural drift, $\delta_p \in [0, a_p)$ bounds all environmental perturbations, and $B = \{d \in \mathbb{R}: |d| \leq 1\}$.

• Software clock:

$\dot \vartheta_p(t) \in a_p + \delta_p B + u_p(t),$

where $u_p(t)$ is the steerable control correcting software time.

Agent communication occurs over a connected, undirected, static graph $G = (V, E)$, with adjacency matrix $A$ and Laplacian $L = \Delta - A$. The consensus operations, including projection and disagreement coordinates, utilize the orthonormal basis $V \in \mathbb{R}^{N \times (N-1)}$ and corresponding diagonal $D$, yielding $L = V D V^\top$ and $S = I - (1_N 1_N^\top)/N = V V^\top$.

Agents broadcast software clock samples to neighbors based on their own asynchronously operated timers $\tau_p$. The timers evolve according to

$$\dot \tau_p \in -b_p + \delta_p B, \quad \tau_p \in [0, T_2^p],$$

and upon reaching zero, trigger broadcasts and timer resets $\tau_p^+ \in [T_1^p, T_2^p]$ for $0 < T_1^p \leq T_2^p$. Broadcast intervals are thus bounded by the inequalities:

$$T_1^p / b_{p, \max} \leq t_{k+1}^p - t_k^p \leq T_2^p / b_{p, \min}.$$

2. Decentralized Consensus Protocol

ChronoSync's core mechanism is a consensus-based steering law for adjusting software clock rates. Each agent $p$ maintains:

• Its current software time $\vartheta_p(t)$,
• The most recent broadcast time estimate $\hat \vartheta_q$ for each neighbor $q$,
• An estimate $\hat a_p$ of its own hardware clock drift $a_p$.

Defining a user-configurable reference drift $a^*$, the decentralized update takes the form:

$$u_p(t) = a^* - \hat a_p(t) + k_u \sum_{q \in N_p} \left( \hat \vartheta_q(t) - \hat \vartheta_p(t) \right),$$

where $k_u > 0$ is the consensus gain. The $a^* - \hat a_p$ component steers the software clock rate towards the desired common drift $a^*$, while the consensus sum reduces local disagreement.

Unknown hardware clock drifts $a_p$ are estimated online using a Luenberger-style observer:

$$\begin{aligned} \dot{\hat a}_p &= k_a [\theta_p - \hat \theta_p], \ \dot{\hat \theta}_p &= \hat a_p + k_\theta [\theta_p - \hat \theta_p], \end{aligned}$$

with positive gains $k_a$, $k_\theta$. The error dynamics are:

$$\dot{\tilde a}_p = -k_a \tilde \theta_p, \qquad \dot{\tilde \theta}_p \in \tilde a_p - k_\theta \tilde \theta_p + \delta_p B,$$

guaranteeing exponential convergence of both drift and clock estimates despite bounded disturbances.

3. Hybrid System Formulation

The ensemble of agents and their synchronization protocol are modeled as a hybrid system. The state vector aggregates the disagreement coordinates $\eta = V^\top \vartheta \in \mathbb{R}^{N-1}$, local software-error $$\tilde \vartheta = \vartheta - \hat \vartheta \in \mathbb{R}^N$$, drift estimation error $\tilde a \in \mathbb{R}^N$, hardware clock estimation error $\tilde \theta \in \mathbb{R}^N$, and timers $\tau \in \mathbb{R}^N$.

The hybrid system's dynamics are:

• Flow set: $$C = \{\xi : \forall p,\, 0 \leq \tau_p \leq T_2^p\}$$,
• Jump set: $D = \bigcup_p \{\xi : \tau_p = 0\}$.

The combined flow and jump evolution is:

$$\begin{aligned} \dot \xi \in F(\xi) &= \begin{pmatrix} \dot \eta \ \dot{\tilde \vartheta} \ \dot{\tilde a} \ \dot{\tilde \theta} \ \dot \tau \end{pmatrix} = \begin{pmatrix} V^\top \tilde a + k_u V^\top L \tilde \vartheta - k_u D \eta + V^\top (d_p)_{p \in V} \ \tilde a + k_u L \tilde \vartheta - k_u V D \eta + (d_p)_{p \in V} \ -k_a \tilde \theta \ \tilde a - k_\theta \tilde \theta + (d_p)_{p \in V} \ -b + P_\tau \end{pmatrix}, \end{aligned}$$

where $(d_p)_{p \in V}$ denotes the bounded disturbances and $P_\tau$ arises from timer perturbations.

At a timer crossing ($\tau_p = 0$),

$$\begin{aligned} &\eta^+ = \eta, \,\, \tilde a^+ = \tilde a, \,\, \tilde \theta^+ = \tilde \theta, \ &\tau_p^+ \in [T_1^p, T_2^p], \,\, \tilde \vartheta_p^+ = 0,\,\, \tilde \vartheta_q^+ = \tilde \vartheta_q \,\, \forall q \neq p. \end{aligned}$$

4. Stability and Convergence Properties

Synchronization objectives and estimation guarantees are established via a Lyapunov analysis. Consider the candidate function:

$$V(\xi) = z^\top P(\tau) z,\,\,\, z = (\eta, \tilde \vartheta, \tilde a, \tilde \theta),$$

where $P(\tau)$ is block-diagonal with $P_1 \succ 0$, $P_3 \succ 0$, and $$P_2(\tau) = \mathrm{diag}(P_{2,p} e^{\sigma \tau_p})$$. The Lyapunov function admits quadratic bounds: $\alpha_1 \|z\|^2 \leq V \leq \alpha_2 \|z\|^2$.

During flows, the function decreases up to a disturbance-driven offset:

$$\dot V \leq -\bar \mu V + (\|P(T_2)\|/\kappa)\, \delta_{\max}^2,$$

with $\bar \mu = (\mu - \kappa \|P(T_2)\|)/\alpha_2 > 0$. At any broadcast-induced jump, $V$ does not increase:

$$\Delta V = -\tilde \vartheta_p^\top P_{2,p} e^{\sigma \tau_p} \tilde \vartheta_p \leq 0.$$

Combining these effects yields the global practical exponential stability (GPES) estimate: For any solution $\varphi$ and $(t, j) \in \mathrm{dom} \, \varphi$,

$$\| \varphi(t, j) \|_A \leq \kappa_1 e^{-\alpha (t + j)} \| \varphi(0, 0) \|_A + \kappa_2,$$

with GPES attractor $A = \{ z = 0 \}$.

5. Performance, Practical Considerations, and Parameter Effects

The protocol guarantees global practical exponential convergence of both clock synchronization (i.e., $\eta \to 0$) and drift estimation errors $(\tilde a \to 0$, $\tilde\theta \to 0)$, even under bounded but unknown clock perturbations $\delta_p$. The convergence rate $\alpha$ and ultimate synchronization error $\kappa_2$ are explicit functions of the largest perturbation $\delta_{\max}$ and design parameters $(k_u, k_a, k_\theta, \sigma, T_1^p, T_2^p, P)$.

A representative simulation with $N=12$ agents using $\nu = 0.06$ s tolerance, $a^*=1$ Hz, $\delta_p = 20$ ppm, $k_u = 0.72$, $k_a = 4.2$, $k_\theta = 3$, $\sigma = 35$, $T_1^p = 0.05$ s, $T_2^p = 0.1$ s for all $p$, and initial disagreement of $\|\eta(0)\| \approx 0.5$ s, demonstrates:

Quantity	Convergence Behavior	Value/Bound
Software time disagreement	$\|\eta(t)\| < 8 \times 10^{-6}$ by $t \approx 80$ s	$\ll \nu = 0.06$ s
Software clock drifts	$\dot\vartheta_p \to 1 \pm 2.27 \times 10^{-5}$	Fig. 2 in (Zegers et al., 6 Apr 2025)
Drift estimator error	$\hat a_p \to a_p$ within $\pm 3.06 \times 10^{-6}$	Fig. 3 in (Zegers et al., 6 Apr 2025)
Hardware clock estimate error	$\tilde\theta_p \to \pm 1.18 \times 10^{-6}$	Fig. 4 in (Zegers et al., 6 Apr 2025)
Timer trajectories	$\tau_p(t) \in [0.05, 0.10]$ for all $p$	Fig. 6 in (Zegers et al., 6 Apr 2025)

Software clocks rapidly synchronize, and both drift estimates and hardware clock estimates converge within tight margins. The protocol is robust to environmental disturbances, agent heterogeneity, and asynchronous, intermittent communication windows.

6. Applications and Significance

ChronoSync addresses decentralized time-base alignment in settings where agents' clocks are individually perturbed and no global reference is available. Affected applications include distributed sensor networks, cooperative robotic teams, autonomous vehicle fleets, and any other systems requiring precise, resilient, and autonomous time synchronization without centralized control or pervasive connectivity.

Significance lies in the combination of fully distributed operation, closed-form dynamics for design, explicit disturbance and parameter dependence for performance calibration, and proven guarantees of GPES for both synchronization and bias estimation. All objectives are attained under both asynchronous and directed event-driven communication, making ChronoSync applicable to a wide range of practical multi-agent scenarios with adversarial or stochastic environmental noise.

7. Limitations and Directions for Future Research

ChronoSync currently assumes a static, connected, undirected communication graph for its formal analysis, although communication between agents may nonetheless be directed and intermittent due to autonomous timer-driven broadcasts. A plausible implication is that extensions to time-varying or partially connected topologies could further broaden practical utility. Environmental perturbations are required to be bounded, with robustness scaling characterized explicitly by the ultimate error $\kappa_2$. Additional investigation into relaxation of this boundedness, stronger disturbance rejection, or integration with time-varying hybrid network models represents plausible directions for continued research, as does experimental validation beyond simulation (Zegers et al., 6 Apr 2025).