Qualitative theory of non-smooth dynamical systems

Bernardo, Mario di; Champneys, Alan R; Budd, Chris; Kowalczyk, Piotr

doi:10.1007/978-1-84628-708-4_2

Cited by 116 publications

(169 citation statements)

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“…MRAC is a powerful approach, yet simple to implement, which can guarantee robustness to parameter variations, noise and unmodelled dynamics. Given the increasing interest in the analysis and control of switched and hybrid systems [2], it is natural to seek an extension of the MRAC approach to Piecewise Affine (PWA) plants. Examples of MRAC schemes for PWA systems can be found in [3,4,5].…”

Section: Introductionmentioning

confidence: 99%

Model reference adaptive control of discrete‐time piecewise linear systems

Bernardo

Montanaro

Olm

et al. 2012

Intl J Robust & Nonlinear

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SUMMARYThis article presents a switched MRAC controller for discrete-time piecewise linear systems. In the spirit of the work by Landau [1], the proof of asymptotic stability of the corresponding closed-loop error is carried out recasting its dynamics as a feedback system and showing the appropriate passivity features of the feedforward and the feedback paths. The challenge lies on the fact that both loops can be piecewise linear. Numerical results show excellent performance of the proposed controller even in the face of sudden variations of the plant parameters.

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Section: Introductionmentioning

confidence: 99%

Model reference adaptive control of discrete‐time piecewise linear systems

Bernardo

Montanaro

Olm

et al. 2012

Intl J Robust & Nonlinear

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“…Non-smooth systems are characterized by a discontinuity in at least of of the variables. They arise in various applications [Bernardo et al, 2008]. Examples include electronic circuits [di Bernardo et al, 1998;Banarjee and Verghese, 2001], hybrid control systems [Ye et al, 1998;Cassandras et al, 2001], and biological networks [Battogtokh et al, 2006;Aihara and Suzuki, 2010;Izhikevich, 2007].…”

Section: Non-smooth Systemsmentioning

confidence: 99%

Controlling Synchronization Patterns in Complex Networks

Lehnert¹

2016

Springer Theses

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In this thesis, I consider the control of synchronization in delay-coupled complex networks. As one main focus, several applications to neural networks will be discussed. In the first part, I focus on the stability of synchronization in complex networks meaning that the control is realized by considering the stability of synchrony in dependence on the parameters. In the second part, adaptive control of synchronization is studied. To this end, adaptive control algorithms are developed that tune the system parameters such that the desired control goal is reached.Besides zero-lag synchrony -a state where all nodes follow the same dynamics without a phase lag -groups and cluster states are considered, i.e., states where the network consists of several groups where the nodes within one group are in zero-lag synchrony and, in the case of cluster synchrony, with a constant phase lag between the clusters.The stability of synchronization can be accessed via the master stability function [Pecora and Carroll, 1998]. This convenient tool allows for treating the node dynamics and the network topology in two separate steps, and, thus, allows for a quite general treatment of different network topologies. In this thesis, I will discuss the generalization of the master stability function to group and cluster states and to non-smooth systems.The master stability function can be used to investigate synchrony in neural networks. Neurons are excitable systems where type-I and type-II excitability can be distinguished. Here, the stability of synchrony for both types in complex networks with excitatory and inhibitory links is investigated on two generic models, namely the normal form of the saddle-node infinite period bifurcation and the FitzHugh-Nagumo system. Furthermore, synchronization in systems with heterogeneous delays or node parameters is studied.In situations where parameters are unknown or drift, adaptive control methods are useful since they allow for an automatically realized adaption of the control parameters. A convenient adaptive method is the speed-gradient method that minimizes a predefined goal function [Fradkov, 1979[Fradkov, , 2007. I first apply this method in the control of an unstable focus and an unstable periodic orbit embedded in the Rössler attractor. Furthermore, I show that clusters states in networks of delayed coupled Stuart-Landau oscillators can be controlled by adaptively tuning the phase of the complex coupling strength or by adapting the topology. The first method is particularly simple because only one parameter has to be adapted, while the second method is more reliable in the sense that its success is widely independent of the choice of the control parameters and the initial conditions. ZUSAMMENFASSUNGIn dieser Arbeit wird die Kontrolle von Synchronisation in komplexen Netzwerken mit retardierten Kopplungen untersucht. Dabei werden verschiedene Anwendungen auf neuronale Netzwerke diskutiert. Der erste Teil der Arbeit beschäftigt sich mit der Stabilität von Synchronisation in komplexen Ne...

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“…Many examples are shown in the survey books. [10][11][12] These systems are described by piecewise smooth functions, either in continuous or in discrete time, and the study of their dynamics leads to a new kind of bifurcations, due to the switching manifolds.…”

Section: Introductionmentioning

confidence: 99%

Border collision bifurcations in a two-dimensional piecewise smooth map from a simple switching circuit

Gardini

Fournier-Prunaret

Chargé

2011

Chaos: An Interdisciplinary Journal of Nonlinear Science

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A unified approach to attractor reconstruction Chaos: An Interdisciplinary Journal of Nonlinear Science 17, 013110 (2007); 10.1063/1.2430294 Border collision bifurcations in a two-dimensional piecewise smooth map from a simple switching circuit Laura Gardini, 1,a) Daniè le Fournier-Prunaret, 2,b) and Pascal Chargé 2,c) In recent years, the study of chaotic and complex phenomena in electronic circuits has been widely developed due to the increasing number of applications. In these studies, associated with the use of chaotic sequences, chaos is required to be robust (not occurring only in a set of zero measure and persistent to perturbations of the system). These properties are not easy to be proved, and numerical simulations are often used. In this work, we consider a simple electronic switching circuit, proposed as chaos generator. The object of our study is to determine the ranges of the parameters at which the dynamics are chaotic, rigorously proving that chaos is robust. This is obtained showing that the model can be studied via a two-dimensional piecewise smooth map in triangular form and associated with a one-dimensional piecewise linear map. The bifurcations in the parameter space are determined analytically. These are the border collision bifurcation curves, the degenerate flip bifurcations, which only are allowed to occur to destabilize the stable cycles, and the homoclinic bifurcations occurring in cyclical chaotic regions leading to chaos in 1-piece.

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Qualitative theory of non-smooth dynamical systems

Cited by 116 publications

References 0 publications

Model reference adaptive control of discrete‐time piecewise linear systems

Model reference adaptive control of discrete‐time piecewise linear systems

Controlling Synchronization Patterns in Complex Networks

Border collision bifurcations in a two-dimensional piecewise smooth map from a simple switching circuit

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