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Mario Di Bernardo - One of the best experts on this subject based on the ideXlab platform.
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observer design for Piecewise Smooth and switched systems via contraction theory
IFAC-PapersOnLine, 2017Co-Authors: Davide Fiore, Mario Di Bernardo, Marco CoraggioAbstract:Abstract The aim of this paper is to present the application of an approach to study contraction theory recently developed for Piecewise Smooth and switched systems. The approach that can be used to analyze incremental stability properties of so-called Filippov systems (or variable structure systems) is based on the use of regularization, a procedure to make the vector field of interest differentiable before analyzing its properties. We show that by using this extension of contraction theory to nondifferentiable vector fields, it is possible to design observers for a large class of Piecewise Smooth systems using not only Euclidean norms, as also done in previous literature, but also non-Euclidean norms. This allows greater flexibility in the design and encompasses the case of both Piecewise-linear and Piecewise-Smooth (nonlinear) systems. The theoretical methodology is illustrated via a set of representative examples.
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Discontinuity-induced bifurcations of Piecewise Smooth dynamical systems
Philosophical transactions. Series A Mathematical physical and engineering sciences, 2010Co-Authors: Mario Di Bernardo, Stephen John HoganAbstract:This paper presents an overview of the current state of the art in the analysis of discontinuity-induced bifurcations (DIBs) of Piecewise Smooth dynamical systems, a particularly relevant class of hybrid dynamical systems. Firstly, we present a classification of the most common types of DIBs involving non-trivial interactions of fixed points and equilibria of maps and flows with the manifolds in phase space where the system is non-Smooth. We then analyse the case of limit cycles interacting with such manifolds, presenting grazing and sliding bifurcations. A description of possible classification strategies to predict and analyse the scenarios following such bifurcations is also discussed, with particular attention to those methodologies that can be applied to generic n-dimensional systems.
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discontinuity induced bifurcations in a Piecewise Smooth satellite power subsystem model
IFAC Proceedings Volumes, 2009Co-Authors: Luiz Felipe Ramos Turci, Elbert E N Macau, Mario Di Bernardo, Takashi YoneyamaAbstract:Abstract In this work, a simple Piecewise-Smooth model for satellite power subsystem is presented and analyzed. The system presents a multitude of nonlinear phenomena like coexistence of attractors, chaos, and bifurcations induced by discontinuity; analyzed here by applying numeric and analytical approaches.
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discontinuity induced bifurcations of equilibria in Piecewise Smooth and impacting dynamical systems
Physica D: Nonlinear Phenomena, 2008Co-Authors: Mario Di Bernardo, Arne Nordmark, Gerard OlivarAbstract:A rich variety of dynamical scenarios has been shown to occur when a fixed point of a non-Smooth map undergoes a border-collision. This paper concerns a closely related class of discontinuity-induced bifurcations, those involving equilibria of n-dimensional Piecewise-Smooth flows. Specifically, transitions are studied which occur when a boundary equilibrium, that is one lying within a discontinuity manifold, is perturbed. It is shown that such equilibria can either persist under parameter variations or can disappear giving rise to different bifurcation scenarios. Conditions to classify among the possible simplest scenarios are given for Piecewise-Smooth continuous, Filippov and impacting systems. Also, we investigate the possible birth of other attractors (e.g. limit cycles) at a boundary-equilibrium bifurcation.
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Piecewise-Smooth Dynamical Systems: Theory and Applications
2007Co-Authors: Mario Di Bernardo, Chris Budd, Alan R Champneys, Piotr KowalczykAbstract:Qualitative theory of non-Smooth dynamical systems.- Border-collision in Piecewise-linear continuous maps.- Bifurcations in general Piecewise-Smooth maps.- Boundary equilibrium bifurcations in flows.- Limit cycle bifurcations in impacting systems.- Limit cycle bifurcations in Piecewise-Smooth flows.- Sliding bifurcations in Filippov systems.- Further applications and extensions.
Helena E Nusse - One of the best experts on this subject based on the ideXlab platform.
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robust dangerous border collision bifurcations in Piecewise Smooth systems
Physical Review Letters, 2004Co-Authors: Munther A Hassouneh, Eyad H Abed, Helena E NusseAbstract:Physical and computer experiments involving systems describable by Piecewise Smooth continuous maps that are nondifferentiable on some surface in phase space exhibit novel types of bifurcations in which an attracting fixed point exists before and after the bifurcation. The striking feature of these bifurcations is that they typically lead to "unbounded behavior" of orbits as a system parameter is slowly varied through its bifurcation value. This new type of border-collision bifurcation is fundamental and robust. A method that prevents such "dangerous border-collision bifurcations" is given. These bifurcations may be found in a variety of experiments including circuits.
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border collision bifurcations for Piecewise Smooth one dimensional maps
International Journal of Bifurcation and Chaos, 1995Co-Authors: Helena E Nusse, James A YorkeAbstract:We examine bifurcation phenomena for continuous one-dimensional maps that are Piecewise Smooth and depend on a parameter μ. In the simplest case, there is a point c at which the map has no derivative (it has two one-sided derivatives). The point c is the border of two intervals in which the map is Smooth. As the parameter μ is varied, a fixed point (or periodic point) Eμ may cross the point c, and we may assume that this crossing occurs at μ=0. The investigation of what bifurcations occur at μ=0 reduces to a study of a map fμ depending linearly on μ and two other parameters a and b. A variety of bifurcations occur frequently in such situations. In particular, Eμ may cross the point c, and for μ 0 there may be a period-3 attractor or even a three-piece chaotic attractor which shrinks to E0 as μ→0. More generally, for every integer m≥2, bifurcations from a fixed point attractor to a period-m attractor, a 2m-piece chaotic attractor, an m-piece chaotic attractor, or a one-piece chaotic attractor can occur for Piecewise Smooth one-dimensional maps. These bifurcations are called border-collision bifurcations. For almost every point in the region of interest in the (a, b)-space, we state explicitly which border-collision bifurcation actually does occur. We believe this phenomenon will be seen in many applications.
Daniel Cremers - One of the best experts on this subject based on the ideXlab platform.
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real time minimization of the Piecewise Smooth mumford shah functional
European Conference on Computer Vision, 2014Co-Authors: Evgeny Strekalovskiy, Daniel CremersAbstract:We propose an algorithm for efficiently minimizing the Piecewise Smooth Mumford-Shah functional. The algorithm is based on an extension of a recent primal-dual algorithm from convex to non-convex optimization problems. The key idea is to rewrite the proximal operator in the primal-dual algorithm using Moreau’s identity. The resulting algorithm computes Piecewise Smooth approximations of color images at 15-20 frames per second at VGA resolution using GPU acceleration. Compared to convex relaxation approaches [18], it is orders of magnitude faster and does not require a discretization of color values. In contrast to the popular Ambrosio-Tortorelli approach [2], it naturally combines Piecewise Smooth and Piecewise constant approximations, it does not require an epsilon-approximation and it is not based on an alternation scheme. The achieved energies are in practice at most 5% off the optimal value for one-dimensional problems. Numerous experiments demonstrate that the proposed algorithm is well-suited to perform discontinuity-preserving Smoothing and real-time video cartooning.
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on the statistical interpretation of the Piecewise Smooth mumford shah functional
International Conference on Scale Space and Variational Methods in computer vision, 2007Co-Authors: Thomas Brox, Daniel CremersAbstract:In region-based image segmentation, two models dominate the field: the Mumford-Shah functional and statistical approaches based on Bayesian inference. Whereas the latter allow for numerous ways to describe the statistics of intensities in regions, the first includes spatially Smooth approximations. In this paper, we show that the Piecewise Smooth Mumford-Shah functional is a first order approximation of Bayesian a-posteriori maximization where region statistics are computed in local windows. This equivalence not only allows for a statistical interpretation of the full Mumford-Shah functional. Inspired by the Bayesian model, it also offers to formulate an extended Mumford-Shah functional that takes the variance of the data into account.
Jerrold E Marsden - One of the best experts on this subject based on the ideXlab platform.
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the complex geometry of weak Piecewise Smooth solutions of integrable nonlinear pde s of shallow water and dym type
Communications in Mathematical Physics, 2001Co-Authors: Mark Alber, Roberto Camassa, Yuri N Fedorov, Darryl D Holm, Jerrold E MarsdenAbstract:An extension of the algebraic-geometric method for nonlinear integrable PDE's is shown to lead to new Piecewise Smooth weak solutions of a class of N-component systems of nonlinear evolution equations. This class includes, among others, equations from the Dym and shallow water equation hierarchies. The main goal of the paper is to give explicit theta-functional expressions for Piecewise Smooth weak solutions of these nonlinear PDE's, which are associated to nonlinear subvarieties of hyperelliptic Jacobians. The main results of the present paper are twofold. First, we exhibit some of the special features of integrable PDE's that admit Piecewise Smooth weak solutions, which make them different from equations whose solutions are globally meromorphic, such as the KdV equation. Second, we blend the techniques of algebraic geometry and weak solutions of PDE's to gain further insight into, and explicit formulas for, Piecewise-Smooth finite-gap solutions. The basic technique used to achieve these aims is rather different from earlier papers dealing with peaked solutions. First, profiles of the finite-gap Piecewise Smooth solutions are linked to certain finite dimensional billiard dynamical systems and ellipsoidal billiards. Second, after reducing the solution of certain finite dimensional Hamiltonian systems on Riemann surfaces to the solution of a nonstandard Jacobi inversion problem, this is resolved by introducing new parametrizations. Amongst other natural consequences of the algebraic-geometric approach, we find finite dimensional integrable Hamiltonian dynamical systems describing the motion of peaks in the finite-gap as well as the limiting (soliton) cases, and solve them exactly. The dynamics of the peaks is also obtained by using Jacobi inversion problems. Finally, we relate our method to the shock wave approach for weak solutions of wave equations by determining jump conditions at the peak location.
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the complex geometry of weak Piecewise Smooth solutions of integrable nonlinear pde s of shallow water and dym type
arXiv: Chaotic Dynamics, 2001Co-Authors: Mark Alber, Roberto Camassa, Yuri N Fedorov, Darryl D Holm, Jerrold E MarsdenAbstract:An extension of the algebraic-geometric method for nonlinear integrable PDE's is shown to lead to new Piecewise Smooth weak solutions of a class of $N$-component systems of nonlinear evolution equations. This class includes, among others, equations from the Dym and shallow water equation hierarchies. The main goal of the paper is to give explicit theta-functional solutions of these nonlinear PDE's, which are associated to nonlinear subvarieties of hyperelliptic Jacobians. The main results of the present paper are twofold. First, we exhibit some of the special features of integrable PDE's that admit Piecewise Smooth weak solutions, which make them different from equations whose solutions are globally meromorphic, such as the KdV equation. Second, we blend the techniques of algebraic geometry and weak solutions of PDE's to gain further insight into, and explicit formulas for, Piecewise-Smooth finite-gap solutions.
Celso Grebogi - One of the best experts on this subject based on the ideXlab platform.
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border collision bifurcations in two dimensional Piecewise Smooth maps
Physical Review E, 1999Co-Authors: Soumitro Banerjee, Celso GrebogiAbstract:Recent investigations on the bifurcations in switching circuits have shown that many atypical bifurcations can occur in Piecewise Smooth maps that cannot be classified among the generic cases like saddle-node, pitchfork, or Hopf bifurcations occurring in Smooth maps. In this paper we first present experimental results to establish the need for the development of a theoretical framework and classification of the bifurcations resulting from border collision. We then present a systematic analysis of such bifurcations by deriving a normal form - the Piecewise linear approximation in the neighborhood of the border. We show that there can be eleven qualitatively different types of border collision bifurcations depending on the parameters of the normal form, and these are classified under six cases. We present a partitioning of the parameter space of the normal form showing the regions where different types of bifurcations occur. This theoretical framework will help in explaining bifurcations in all systems, which can be represented by two-dimensional Piecewise Smooth maps.