A Passivity-Agnostic Framework for Distributed Adaptive Synchronization under Unknown Leader Dynamics

Abstract

We present a passivity-agnostic framework for distributed adaptive synchronization under position-only communication, bounded disturbances, and unknown leader dynamics. By passivity-agnostic we mean the design does not require the closed loop system to be strictly positive real (SPR) a priori: it certifies SPR when present and recovers it by frequency shaping when absent. Followers are heterogeneous second-order systems with unknown (possibly unstable) dynamics. In the SPR regime, a structured reparameterization yields gradient-based adaptive error dynamics; Lyapunov analysis guarantees global asymptotic synchronization in the disturbance-free case, exact rejection of constant disturbances, and bounded responses to time-varying disturbances, with parameter convergence under persistent excitation. In the non-SPR regime, frequency shaping recovers effective passivity of the unshaped transfer function, enabling the same stability guarantees via standard passivity/Lyapunov arguments using Meyer-Kalman-Yakubovich (MKY) Lemma. Simulations across star, cyclic, path, and arbitrary graphs demonstrate scalable synchronization, robust tracking, and parameter adaptation under multiple disturbance profiles, confirming that the frequency-shaped non-SPR designs match the performance of the SPR case.

Figures (8)

• Figure 1: An example of the graph $\mathcal{G}$ and two induced graphs $\mathcal{G}_m$ and $\mathcal{G}_0$ with $m = 5$, illustrating the decoupling of $\mathcal{G}$ and the assignment of $w_{ij}$—not actual communication.
• Figure 2: Block diagram of \ref{['das:SPR:dynamics']}–\ref{['das:SPR:Wu']}. Solid lines depict the nominal loop. The dashed “Ideal” region indicates preprocessing $\mathbf{W}_u^{-1}$ acting on $\bar{z}$ (cf. \ref{['das:SPR:ref:u2']}).
• Figure 3: Four network topologies are considered adhering to Remark \ref{['Rem:Threshold']}. The blue circles with a number in each topology indicate the followers while the arrows show the communication network. In general, the red arrows express to which follower the leader trajectory is delivered $(\mathcal{G}_0)$ while the black arrows are the links among the followers $(\mathcal{G}_m)$. More specifically, the red dashed arrows in cyclic and arbitrary network mean that we perform in-depth analysis in various scenarios while the green arrows in arbitrary network shows the exchange information from various levels of followers from the leader $x_0$.
• Figure 4: The performance output and the convergence for various $\bar{\delta}_k$.
• Figure 5: The performance outputs for four network topologies using $\bar{\delta}_3$

Theorems & Definitions (13)

• Remark 1: Network Connectivity Conditions
• Remark 2: Information Flow
• Definition 1
• Remark 3: Routh--Hurwitz Gain
• Lemma 1
• proof
• Remark 4: Position-only Exchange
• Remark 5: Why reparameterization?
• Theorem 1
• proof