Pietro DeLellis

Brief paper: Novel decentralized adaptive strategies for the synchronization of complex networks

This paper is concerned with the analysis of the synchronization of networks of nonlinear oscillators through an innovative local adaptive approach. In particular, time-varying feedback coupling gains are considered, whose gradient is a function of the local synchronization error over each edge in the network. It is shown that, under appropriate conditions, the strategy is indeed successful in guaranteeing the achievement of a common synchronous evolution for all oscillators in the network. The theoretical derivation is complemented by its validation on a set of representative examples.

Distributed Adaptive Control of Synchronization in Complex Networks

This technical note studies distributed adaptive control of synchronization in complex networks. An effective distributed adaptive strategy to tune the coupling weights of a network is designed based on local information of node dynamics. The analysis is then extended to the case where only a small fraction of coupling weights can be adjusted. A general criterion is derived and it is found that synchronization can be reached if the subgraph consisting of the edges and nodes corresponding to the updated coupling weights is connected. Finally, simulation examples are given to illustrate the theoretical analysis.

On QUAD, Lipschitz, and Contracting Vector Fields for Consensus and Synchronization of Networks

In this paper, a relationship is discussed between three common assumptions made in the literature to prove local or global asymptotic stability of the synchronization manifold in networks of coupled nonlinear dynamical systems. In such networks, each node, when uncoupled, is described by a nonlinear ordinary differential equation of the form ẋ = <i>f</i> (<i>x</i>,<i>t</i>) . In this paper, we establish links between the QUAD condition on <i>f</i> (<i>x</i>, <i>t</i>), i.e.,(<i>x</i>-<i>y</i>)^T[<i>f</i>(<i>x</i>, <i>t</i>)-<i>f</i>(<i>y</i>, <i>t</i>)] - (<i>x</i>-<i>y</i>)^T Δ(<i>x</i>-<i>y</i>) ≤-ω(<i>x</i>-<i>y</i>)<i>T</i>(<i>x</i>-<i>y</i>) for some arbitrary Δ and ω, and contraction theory. We then investigate the relationship between the assumption of <i>f</i> being Lipschitz and the QUAD condition. We show the usefulness of the links highlighted in this paper to obtain proofs of asymptotic synchronization in networks of identical nonlinear oscillators and illustrate the results via numerical simulations on some representative examples.

Synchronization and control of complex networks via contraction, adaptation and evolution

Complex networked systems abound in Nature and Technology. They consist of a multitude of interacting agents communicating with each other over a web of complex interconnections. Flocks of birds, platoon of cooperating robots, swirling fishes in the Ocean are all examples whose intricate dynamics can be modeled in terms of three essential ingredients: (i) a mathematical description of the dynamical behavior of each of the agents in the network; (ii) an interaction (or coupling) protocol used by agents to communicate with each other and (iii) a graph describing the network of interconnections between neighboring agents. These three elements are actually mapped onto the mathematical model usually considered in the literature to describe a complex network which uses appropriate equations to describe the node dynamics, the coupling protocol and the network topology.

Pinning control of complex networks via edge snapping

In this paper, we propose a hierarchy of novel decentralized adaptive pinning strategies for controlled synchronization of complex networks. This hierarchy addresses the fundamental need of selecting the sites to pin through a fully decentralized approach based on edge snapping. Specifically, we present three different strategies of increasing complexity which use a combination of network evolution and adaptation of the coupling and control gains. Theoretical results are complemented by extensive numerical investigations of the performance of the proposed strategies on a set of testbed examples.

Analysis and stability of consensus in networked control systems

Abstract In this paper, the consensus problem in networks of integrators is investigated. After recalling the classical diffusive protocol, we present in a unified framework some results on the rate of convergence previously presented in the literature. Then, we introduce two switching communication protocols, one based on a switching coupling law between neighboring nodes, the other on the conditional activation of links in the network. We show that the former protocol induces the monotonicity of each system in the network, enhancing the speed of convergence to consensus. Moreover, adopting this novel protocol, we are able to control the network, steering the nodes’ dynamics to a desired consensus value. The aim of the latter protocol is instead to select adaptively the activation of the edges of the network, in accordance with the dynamics of the network. After showing the effectiveness of both approaches through numerical simulations, the stability properties of these protocols are discussed.

Collective behaviour across animal species

We posit a new geometric perspective to define, detect, and classify inherent patterns of collective behaviour across a variety of animal species. We show that machine learning techniques, and specifically the isometric mapping algorithm, allow the identification and interpretation of different types of collective behaviour in five social animal species. These results offer a first glimpse at the transformative potential of machine learning for ethology, similar to its impact on robotics, where it enabled robots to recognize objects and navigate the environment.

Convergence and synchronization in heterogeneous networks of smooth and piecewise smooth systems

This paper presents a framework for the study of convergence in networks where the nodes’ dynamics may be both piecewise smooth and/or nonidentical. Sufficient conditions are derived for global convergence of all node trajectories towards the same bounded region in the synchronization error space. The analysis is based on the use of set-valued Lyapunov functions and bounds are derived on the minimum coupling strength required to make all nodes in the network converge towards each other. We also provide an estimate of the asymptotic bound on the mismatch between the node state trajectories. The analysis is performed both for linear and nonlinear coupling protocols. The theoretical analysis is extensively illustrated and validated via its application to a set of representative numerical examples.

Criteria for stochastic pinning control of networks of chaotic maps

This paper investigates the controllability of discrete-time networks of coupled chaotic maps through stochastic pinning. In this control scheme, the network dynamics are steered towards a desired trajectory through a feedback control input that is applied stochastically to the network nodes. The network controllability is studied by analyzing the local mean square stability of the error dynamics with respect to the desired trajectory. Through the analysis of the spectral properties of salient matrices, a toolbox of conditions for controllability are obtained, in terms of the dynamics of the individual maps, algebraic properties of the network, and the probability distribution of the pinning control. We demonstrate the use of these conditions in the design of a stochastic pinning control strategy for networks of Chirikov standard maps. To elucidate the applicability of the approach, we consider different network topologies and compare five different stochastic pinning strategies through extensive numerical simulations.

Fully adaptive pinning control of complex networks

In this paper we propose a novel adaptive pinning control strategy for synchronization of complex networks. The novelty of this strategy is the fully decentralized adaptive selection of the coupling and control gains. After giving a sketch of the stability proof, we validate the analysis with numerical simulations on a testbed example.