Contraction Analysis for a Class of NonDifferentiable Systems with Applications to Stability and Network Synchronization

Bernardo, Mario di; Liuzza, Davide; Russo, Giovanni

doi:10.1137/120883001

Cited by 38 publications

(32 citation statements)

References 26 publications

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“…As a representative example, we consider a network of time-switched systems of the form [44] As shown in Fig.12.6, the theoretical predictions are confirmed by the numerical simulations obtained considering T = 2 and random initial conditions.…”

Section: Recent Extensionssupporting

confidence: 56%

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Convergence, Consensus and Synchronization of Complex Networks via Contraction Theory

Bernardo

Fiore

Russo

et al. 2015

Complex Systems and Networks

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This chapter reviews several approaches to study convergence of networks of nonlinear dynamical systems based on the use of contraction theory. Rather than studying the properties of the collective asymptotic solution of interest, the strategy focuses on finding sufficient conditions for any pair of trajectories of two agents in the network to converge towards each other. The key tool is the study, in an appropriate metric, of the matrix measure of the agents' or network Jacobian. The effectiveness of the proposed approach is illustrated via a set of representative examples. IntroductionThe problem of steering the collective behaviour of a network of dynamical agents towards a desired common target solution is a fundamental problem in network science and control theory [1][2][3]. A typical problem is that of achieving consensus or synchronization in a network of linear or nonlinear systems. Here, the key challenge, once the coupling function among the nodes has been selected, is to prove convergence of all nodes towards the desired common asymptotic behaviour. This convergence problem is usually solved locally by means of the Master Stability Function Approach (MSF) [4] or globally via Lyapunov stability theory [5].Global convergence is extremely useful particularly when the agents are subject to high level of noise or perturbations as it is often the case in applications; think for example of the recent application of synchronization to biological systems [6].

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Section: Recent Extensionssupporting

confidence: 56%

“…Specifically, contraction of Carathéodory systems was discussed in [44]. It was shown that the switched system of interest is contracting if a unique matrix measure exists such that each of the system modes is contracting.…”

Section: Recent Extensionsmentioning

confidence: 99%

Convergence, Consensus and Synchronization of Complex Networks via Contraction Theory

Bernardo

Fiore

Russo

et al. 2015

Complex Systems and Networks

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“…To illustrate the strategies presented above we use an example adapted from [14]. Specifically, we consider a switched (or blinking) network of the forṁ…”

Section: Networkmentioning

confidence: 99%

“…x m . Following similar arguments in [13], [14], the problem of proving convergence of network (8) is equivalent to that of guaranteeing that the following linear systems:…”

Section: Networkmentioning

confidence: 99%

“…Contraction analysis was highlighted as a promising approach to study incremental exponential asymptotic stability of nonlinear systems [8], [9], [10], [11], [12]. It has been recently extended in [13], [14] to some classes of switched systems [15], [16]. In particular, as shown in [8], [17], contraction can be studied through the matrix measure, also known as logarithmic norm, of the Jacobian of the system under investigation.…”

Section: Introductionmentioning

confidence: 99%

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Switched adaptive strategies for contraction and incremental stability of Carathéodory systems using multiple norms

Bernardo

2015

2015 54th IEEE Conference on Decision and Control (CDC)

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In this paper, we present switched adaptive strategies to incrementally stabilise a class of Carathéodory nonlinear systems. Using the concept of switched matrix measures based on the use of multiple norms, and the transaction coefficients between these norms, sufficient conditions are presented to prove incremental stability of a switched system. Furthermore, based on these sufficient conditions, two strategies are proposed to incrementally stabilise a switched system of interest: one based on monitoring the switching signal in a continuous way, the other on monitoring it only at selected discrete-time instants. We prove that in both situations, the switched system can be effectively stabilised. As a representative application, we illustrate the efficiency of these strategies by synchronizing a network system with blinking (switching) graph structures.

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Loss of coordination in complex directed networks: An incremental approach based on matrix measures

Russo

2017

Intl J Robust & Nonlinear

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SummaryThis paper is concerned with the problem of understanding the dynamical mechanisms responsible for the loss of a desired behavior in complex directed networks. Specifically, we give 2 sufficient conditions for the desynchronization of a directed complex network, which are based on the use of matrix measures. The effectiveness of our approach is shown on 2 applications that are directly motivated by our research in smart cities and systems/synthetic biology.

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Contraction Analysis for a Class of NonDifferentiable Systems with Applications to Stability and Network Synchronization

Cited by 38 publications

References 26 publications

Convergence, Consensus and Synchronization of Complex Networks via Contraction Theory

Convergence, Consensus and Synchronization of Complex Networks via Contraction Theory

Switched adaptive strategies for contraction and incremental stability of Carathéodory systems using multiple norms

Loss of coordination in complex directed networks: An incremental approach based on matrix measures

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