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## Publicatie

# A Study of Partial Synchronization in Networks of Delay-Coupled Systems

### Boek - Dissertatie

This thesis studies partial synchronization in networks of delay-coupled systems. Partial synchronization refers to the phenomenon that coupled systems can be grouped into clusters such that synchrony is only observed within each cluster. The coupling considered here contains a time-delay. A pattern of partial synchronization can be characterized by a partial synchronization manifold which is a linear invariant subspace of the state space of the network dynamics, corresponding to partially synchronous motion. The existence and asymptotic stability of such a manifold are required to observe partial synchronization in simulations and experiments. The latter guarantees that the manifold attracts the neighboring solutions. Therefore, the partial synchronization problem can be tackled in two steps: 1) identify partial synchronization manifolds; 2) analyze the stability of the partial synchronization manifolds. This thesis aims to provide an efficient method to characterize all partial synchronization manifolds and tractable stability conditions of partial synchronization manifolds. In return, the method and stability conditions provide insights into the relation between the dynamics of individual systems, coupling parameters, time-delays, and partial synchronization.First, a method is proposed to identify all partial synchronization manifolds in delay-coupled networks, using the information on network structure and coupling type. The method relies on a recently proposed existence criterion for partial synchronization manifolds. The criterion is expressed as a row-sum check on the ordered adjacency matrix (which represents the network structure). In this thesis, the criterion is extended so that the method is also applicable to networks with different systems, and the computational efficiency is significantly improved for the case where only the transmitted signals in coupling contain a time-delay. In addition, the method allows to automatically decompose the network dynamics into synchronization error dynamics describing the deviations of the system states within the clusters, and dynamics on manifolds describing the network dynamics after synchronization. The method is implemented as a publicly available software tool.Second, the stability of the partial synchronization manifolds is analyzed by using the second method of Lyapunov. The stability analysis of partial synchronization manifolds is recast as the stability analysis of an equilibrium (or zero solution) of the synchronization error dynamics. It is shown that the linearized error system can be interpreted as a linear parameter-varying (LPV) time-delay system. By assessing the stability of this LPV time-delay system using a Lyapunov-Krasovskii functional (LKF), the sufficient conditions for the local stability of partial synchronization manifolds are derived and expressed in the form of linear matrix inequalities (LMIs), whose feasibility can be assessed by using standard numerical tools.Third, the partial synchronization in a practical sense is considered. From an application perspective, the conditions for partial synchronization are too stringent, as inevitable perturbations or uncertainties (considered as additive to the nominal setting) destroy the zero equilibrium of the error system. Consequently, exact partial synchronization, where the states of the systems within each cluster are perfectly synchronized, cannot be achieved. Instead, an approximate synchronization, practical partial synchronization, may be observed, where the states of the systems within each cluster converge to each other up to some tolerance, and this tolerance tends to zero if (the size of) the perturbations tend to zero. In this case, the error system in the analysis is viewed as a non-autonomous time-delay system with a bounded additive perturbation. By assessing the practical stability of this error system using LKFs, the conditions for practical partial synchronization is derived and again formulated in term of LMIs. Besides, an explicit relation between the size of perturbation and the bound of the synchronization error is provided. All these conditions (for both exact and practical synchronization) induce restrictions on the coupling strength and time-delay. They can be used to check whether for a given set of parameters partial synchronization occurs and shed a light on the parameter dependence.Fourth, experiments on a network of electronic neurons are performed to demonstrate the obtained theoretical results. Each electronic neuron is the implementation of a neuron model in a circuit board. A cross correlation-based notion of practical synchronization is introduced to quantify the synchronization between the neurons. The experiments confirm the theory well though the theoretical results tend to be conservative. The synchronization regions in network parameter space determined by the theoretical conditions are a qualitative match of those obtained via experiments, although the former are smaller than the latter due to the conservatism of the theory.Finally, a control design is presented to achieve vehicle platooning which is a real-life application of controlled network synchronization. The design, grounded in model predictive control (MPC), employs a cooperative adaptive cruise control (CACC) strategy to achieve vehicle platooning. The CACC strategy uses vehicle-to-vehicle communication to allow for inter-vehicle data exchange such that short inter-vehicle distances can be reached while guaranteeing safety. To improve the cohesion of the platoon when encountering disturbance, a bi-directional network topology is used. It is shown that compared to the commonly used unidirectional topology, the bi-directional topology allows the vehicles to maintain shorter distances.

Jaar van publicatie:2021

Toegankelijkheid:Open