45 results found
Haasdonk B, Hamzi B, Santin G, et al., 2021, Kernel methods for center manifold approximation and a weak data-based version of the Center Manifold Theorem, PHYSICA D-NONLINEAR PHENOMENA, Vol: 427, ISSN: 0167-2789
Hamzi B, Owhadi H, 2021, Learning dynamical systems from data: A simple cross-validation perspective, part I: Parametric kernel flows, PHYSICA D-NONLINEAR PHENOMENA, Vol: 421, ISSN: 0167-2789
Bittracher A, Klus S, Hamzi B, et al., 2021, Dimensionality Reduction of Complex Metastable Systems via Kernel Embeddings of Transition Manifolds, Publisher: SPRINGER
Colonius F, Hamzi B, 2020, Entropy for practical stabilization, Publisher: arXiv
For deterministic continuous-time nonlinear control systems,epsilon-practical stabilization entropy and practical stabilization entropy areintroduced. Here the rate of attraction is specified by a KL-function. Upperand lower bounds for the diverse entropies are proved, with special attentionto exponential KL-functions. Two scalar examples are analyzed in detail.
Klus S, Nueske F, Hamzi B, 2020, Kernel-based approximation of the koopman generator and schrodinger operator, Entropy: international and interdisciplinary journal of entropy and information studies, Vol: 22, Pages: 1-22, ISSN: 1099-4300
Many dimensionality and model reduction techniques rely on estimating dominant eigenfunctions of associated dynamical operators from data. Important examples include the Koopman operator and its generator, but also the Schrödinger operator. We propose a kernel-based method for the approximation of differential operators in reproducing kernel Hilbert spaces and show how eigenfunctions can be estimated by solving auxiliary matrix eigenvalue problems. The resulting algorithms are applied to molecular dynamics and quantum chemistry examples. Furthermore, we exploit that, under certain conditions, the Schrödinger operator can be transformed into a Kolmogorov backward operator corresponding to a drift-diffusion process and vice versa. This allows us to apply methods developed for the analysis of high-dimensional stochastic differential equations to quantum mechanical systems.
Giesl P, Hamzi B, Rasmussen M, et al., 2020, Approximation of Lyapunov functions from noisy data, Journal of Computational Dynamics, Vol: 7, Pages: 57-81, ISSN: 2158-2491
Methods have previously been developed for the approximation of Lyapunovfunctions using radial basis functions. However these methods assume that theevolution equations are known. We consider the problem of approximating a givenLyapunov function using radial basis functions where the evolution equationsare not known, but we instead have sampled data which is contaminated withnoise. We propose an algorithm in which we first approximate the underlyingvector field, and use this approximation to then approximate the Lyapunovfunction. Our approach combines elements of machine learning/statisticallearning theory with the existing theory of Lyapunov function approximation.Error estimates are provided for our algorithm.
Hamzi B, Abed EH, 2020, Local modal participation analysis of nonlinear systems using Poincare linearization, NONLINEAR DYNAMICS, Vol: 99, Pages: 803-811, ISSN: 0924-090X
Haasdonk B, Hamzi B, Santin G, et al., 2020, Greedy Kernel Methods for Center Manifold Approximation, Publisher: Springer International Publishing
<jats:title>Abstract</jats:title><jats:p>For certain dynamical systems it is possible to significantly simplify the study of stability by means of the center manifold theory. This theory allows to isolate the complicated asymptotic behavior of the system close to a non-hyperbolic equilibrium point, and to obtain meaningful predictions of its behavior by analyzing a reduced dimensional problem. Since the manifold is usually not known, approximation methods are of great interest to obtain qualitative estimates. In this work, we use a data-based greedy kernel method to construct a suitable approximation of the manifold close to the equilibrium. The data are collected by repeated numerical simulation of the full system by means of a high-accuracy solver, which generates sets of discrete trajectories that are then used to construct a surrogate model of the manifold. The method is tested on different examples which show promising performance and good accuracy.</jats:p>
Hamzi B, Colonius F, 2019, Kernel methods for the approximation of discrete-time linear autonomous and control systems, SN Applied Sciences, Vol: 1, ISSN: 2523-3963
Methods from learning theory are used in the state space of linear dynamical and control systems in order to estimate relevant matrices and some relevant quantities such as the topological entropy. An application to stabilization via algebraic Riccati equations is included by viewing a control system as an autonomous system in an extended space of states and control inputs. Kernel methods are the main techniques used in this paper and the approach is illustrated via a series of numerical examples. The advantage of using kernel methods is that they allow to perform function approximation from data and, as illustrated in this paper, allow to approximate linear discrete-time autonomous and control systems from data.
Hamzi B, Kuehn C, Mohamed S, 2019, A note on kernel methods for multiscale systems with critical transitions, MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Vol: 42, Pages: 907-917, ISSN: 0170-4214
Hamzi B, Colonius F, 2019, Kernel Methods for Discrete-Time Linear Equations, 19th Annual International Conference on Computational Science (ICCS), Publisher: SPRINGER INTERNATIONAL PUBLISHING AG, Pages: 178-191, ISSN: 0302-9743
Hamzi B, AlOtaiby TN, AlShebeili S, et al., 2018, Kernel Methods and the Maximum Mean Discrepancy for Seizure Detection, 1st International Conference on Computer Applications and Information Security (ICCAIS), Publisher: IEEE
Bouvrie J, Hamzi B, 2017, KERNEL METHODS FOR THE APPROXIMATION OF NONLINEAR SYSTEMS, SIAM JOURNAL ON CONTROL AND OPTIMIZATION, Vol: 55, Pages: 2460-2492, ISSN: 0363-0129
Hamzi B, Lamb JSW, Lewis D, 2015, A Characterization of Normal Forms for Control Systems, JOURNAL OF DYNAMICAL AND CONTROL SYSTEMS, Vol: 21, Pages: 273-284, ISSN: 1079-2724
Hamzi B, Abed EH, 2014, Local Mode-in-State Participation Factors for Nonlinear Systems, 53rd IEEE Annual Conference on Decision and Control (CDC), Publisher: IEEE, Pages: 43-48, ISSN: 0743-1546
Bouvrie J, Hamzi B, 2012, Empirical Estimators for Stochastically Forced Nonlinear Systems: Observability, Controllability and the Invariant Measure, Proc. American Control Conference (ACC)
We introduce a data-based approach to estimating key quantities which arise in the study of nonlinear control systems and random nonlinear dynamical systems. Our approach hinges on the observation that much of the existing linear theory may be readily extended to nonlinear systems - with a reasonable expectation of success - once the nonlinear system has been mapped into a high or infinite dimensional feature space. In particular, we develop computable, non-parametric estimators approximating controllability and observability energy functions for nonlinear systems, and study the ellipsoids they induce. In all cases the relevant quantities are estimated from simulated or observed data. It is then shown that the controllability energy estimator provides a key means for approximating the invariant measure of an ergodic, stochastically forced nonlinear system.
Bouvrie J, Hamzi B, 2011, Model Reduction for Nonlinear Control Systems using Kernel Subspace Methods
We introduce a data-driven order reduction method for nonlinear control systems, drawing on recent progress in machine learning and statistical dimensionality reduction. The method rests on the assumption that the nonlinear system behaves linearly when lifted into a high (or infinite) dimensional feature space where balanced truncation may be carried out implicitly. This leads to a nonlinear reduction map which can be combined with a representation of the system belonging to a reproducing kernel Hilbert space to give a closed, reduced order dynamical system which captures the essential input-output characteristics of the original model.Empirical simulations illustrating the approach are also provided.
Hamzi B, Krener AJ, 2007, The Controlled Center Systems, IEEE TRANSACTIONS ON AUTOMATIC CONTROL, Vol: 52, Pages: 2188-2192, ISSN: 0018-9286
Hamzi B, Krener AJ, Kang W, 2006, The controlled center dynamics of discrete time control bifurcations, SYSTEMS & CONTROL LETTERS, Vol: 55, Pages: 585-596, ISSN: 0167-6911
Hamzi B, Krener AJ, 2006, The controlled center systems, 45th IEEE Conference on Decision and Control, Publisher: IEEE, Pages: 6414-6418, ISSN: 0743-1546
Kang W, Hamzi B, Krener AJ, 2006, On the Convergence and Behavior of Three Dimensional Normal Forms, 1st IFAC Conference on the Analysis and Control of Chaotic Systems
Krener AJ, Kang W, Hamzi B, et al., 2005, Low codimension control singularities for single input nonlinear systems, Conference in Honor of Clyde Martins 60th Birthday on New Directions and Applications in Control Theory, Publisher: SPRINGER-VERLAG BERLIN, Pages: 181-192, ISSN: 0170-8643
Hamzi B, Kang W, Krener AJ, 2005, The controlled center dynamics, MULTISCALE MODELING & SIMULATION, Vol: 3, Pages: 838-852, ISSN: 1540-3459
Krener AJ, Kang W, Hamzi B, et al., 2005, Low codimension control singularities for single input nonlinear systems, Publisher: Springer Verlag, ISBN: 9783540239536
This volume contains a collection of papers in control theory and applications presented at a conference in honor of Clyde Martin on the occasion of his 60th ...
Hamzi B, Kang W, Barbot JP, 2004, Analysis and control of Hopf bifurcations, SIAM JOURNAL ON CONTROL AND OPTIMIZATION, Vol: 42, Pages: 2200-2220, ISSN: 0363-0129
Hamzi B, Kang W, 2003, Resonant terms and bifurcations of nonlinear control systems with one uncontrollable mode, SYSTEMS & CONTROL LETTERS, Vol: 49, Pages: 267-278, ISSN: 0167-6911
Hamzi B, Tall IA, 2003, Normal forms for nonlinear discrete time control systems, 42nd IEEE Conference on Decision and Control, Publisher: IEEE, Pages: 1357-1361
Hamzi B, Krener AJ, 2003, Practical stabilization of systems with a fold control bifurcation, Symposium on New Trends in Nonlinear Dynamics and Control, and Their Applications, Publisher: SPRINGER-VERLAG BERLIN, Pages: 37-48, ISSN: 0170-8643
Hamzi B, Kang W, Krener AJ, 2003, Control of center manifolds, 42nd IEEE Conference on Decision and Control, Publisher: IEEE, Pages: 2065-2070
Hamzi B, 2002, Analysis and stabilization of nonlinear systems with a zero-Hopf control bifurcation, 41st IEEE Conference on Decision and Control, Publisher: IEEE, Pages: 3912-3917, ISSN: 0191-2216
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