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Showing papers on "Nonlinear system published in 2001"


Book ChapterDOI
01 Jan 2001
TL;DR: In this paper, the clssical filleting and prediclion problem is re-examined using the Bode-Shannon representation of random processes and the?stat-tran-sition? method of analysis of dynamic systems.
Abstract: The clssical filleting and prediclion problem is re-examined using the Bode-Shannon representation of random processes and the ?stat-tran-sition? method of analysis of dynamic systems. New result are: (1) The formulation and Methods of solution of the problm apply, without modification to stationary and nonstationary stalistics end to growing-memory and infinile -memory filters. (2) A nonlinear difference (or differential) equalion is dericed for the covariance matrix of the optimal estimalion error. From the solution of this equation the coefficients of the difference, (or differential) equation of the optimal linear filter are obtained without further caleulations. (3) Tke fillering problem is shoum to be the dual of the nois-free regulator problem. The new method developed here, is applied to do well-known problems, confirming and extending, earlier results. The discussion is largely, self-contatained, and proceeds from first principles; basic concepts of the theory of random processes are reviewed in the Appendix.

15,391 citations


Book
15 Jun 2001
TL;DR: The Time Scales Calculus as discussed by the authors is a generalization of the time-scales calculus with linear systems and higher-order linear equations, and it can be expressed in terms of linear Symplectic Dynamic Systems.
Abstract: Preface * The Time Scales Calculus * First Order Linear Equations * Second Order Linear Equations * Self-Adjoint Equations * Linear Systems and Higher Order Equations * Dynamic Inequalities * Linear Symplectic Dynamic Systems * Extensions * Solutions to Selected Problems * Bibliography * Index

2,581 citations


Book
01 Jan 2001
TL;DR: The paradoxes of irreversibility as mentioned in this paper is a well-known problem in nonlinear problems, and it has been studied extensively in the literature for a long time, e.g. in the context of projection operators.
Abstract: 1. Brownian Motion and Langevin equations 2. Fokker-Planck equations 3. Master equations 4. Reaction rates 5. Kinetic models 6. Quantum dynamics 7. Linear response theory 8. Projection operators 9. Nonlinear problems 10. The paradoxes of irreversibility Appendices

2,050 citations


Journal ArticleDOI
TL;DR: In this article, a Jacobi elliptic function expansion method was proposed to construct the exact periodic solutions of nonlinear wave equations, which includes some shock wave solutions and solitary wave solutions.

1,231 citations


Reference EntryDOI
25 Apr 2001
TL;DR: The most exotic form of nonlinear dynamics is Chaos as mentioned in this paper, in which deterministic interactions produce apparently irregular fluctuations, and small changes in the initial state of the system are magnified through time.
Abstract: Nonlinear dynamics deals with more-or-less regular fluctuations in system variables caused by feedback intrinsic to the system (as opposed to external forces). Chaos is the most exotic form of nonlinear dynamics, in which deterministic interactions produce apparently irregular fluctuations, and small changes in the initial state of the system are magnified through time.7 Keywords: chaos; population cycles; population dynamics; nonlinear time-series analysis

1,190 citations


Journal ArticleDOI
TL;DR: The input-to-state stability property and small-gain theorems are introduced as the cornerstone of new stability criteria for discrete-time nonlinear systems.

1,179 citations


Journal ArticleDOI
13 Dec 2001
TL;DR: In this article, the authors systematically developed the procedure of holographic renormalization for RG flows dual to asymptotically AdS domain walls by adding covariant local boundary counterterms determined by the near-boundary behavior of bulk fields.
Abstract: We systematically develop the procedure of holographic renormalization for RG flows dual to asymptotically AdS domain walls All divergences of the on-shell bulk action can be cancelled by adding covariant local boundary counterterms determined by the near-boundary behavior of bulk fields This procedure defines a renormalized action from which correlation functions are obtained by functional differentiation The correlators are finite and well behaved at coincident points Ward identities, corrected for anomalies, are satisfied The correlators depend on parts of the solution of the bulk field equations which are not determined by near-boundary analysis In principle a full nonlinear solution is required, but one can solve linearized fluctuation equations to define a bulk-to-boundary propagator from which 2-point correlation functions are easily obtained We carry out the procedure explicitly for two known RG flows obtained from the maximal gauged D=5 supergravity theory, obtaining new results on correlators of vector currents and related scalar operators and giving further details on a recent analysis of the stress tensor sector

816 citations


Journal ArticleDOI
TL;DR: A new version of the DelPhi program, which provides numerical solutions to the nonlinear Poisson−Boltzmann (PB) equation, is reported in this paper, which can divide space into multiple regions containing...
Abstract: A new version of the DelPhi program, which provides numerical solutions to the nonlinear Poisson−Boltzmann (PB) equation, is reported. The program can divide space into multiple regions containing ...

806 citations


Journal ArticleDOI
TL;DR: A differential geometric approach to the problem of fault detection and isolation for nonlinear systems derived in terms of an unobservability distribution, which is computable by means of suitable algorithms.
Abstract: We present a differential geometric approach to the problem of fault detection and isolation for nonlinear systems. A necessary condition for the problem to be solvable is derived in terms of an unobservability distribution, which is computable by means of suitable algorithms. The existence and regularity of such a distribution implies the existence of changes of coordinates in the state and in the output space which induce an "observable" quotient subsystem unaffected by all fault signals but one. For this subsystem, a fault detection filter is designed.

802 citations


Journal ArticleDOI
TL;DR: This survey describes the 'activation' of stability, optimality and uncertainty concepts into design tools and constructive procedures in nonlinear control theory and concludes with four representative applications.

720 citations


Book
30 Nov 2001
TL;DR: In this article, the Stable Adaptive Neural Network Control offers an in-depth study of stable adaptive control designs using approximation-based techniques, and presents rigorous analysis for system stability and control performance.
Abstract: While neural network control has been successfully applied in various practical applications, many important issues, such as stability, robustness, and performance, have not been extensively researched for neural adaptive systems. Motivated by the need for systematic neural control strategies for nonlinear systems, Stable Adaptive Neural Network Control offers an in-depth study of stable adaptive control designs using approximation-based techniques, and presents rigorous analysis for system stability and control performance. Both linearly parameterized and multi-layer neural networks (NN) are discussed and employed in the design of adaptive NN control systems for completeness. Stable adaptive NN control has been thoroughly investigated for several classes of nonlinear systems, including nonlinear systems in Brunovsky form, nonlinear systems in strict-feedback and pure-feedback forms, nonaffine nonlinear systems, and a class of MIMO nonlinear systems. In addition, the developed design methodologies are not only applied to typical example systems, but also to real application-oriented systems, such as the variable length pendulum system, the underactuated inverted pendulum system and nonaffine nonlinear chemical processes (CSTR).

Journal ArticleDOI
TL;DR: It is shown that the closed-loop system resulting from the control law can maintain its local finite-time stability regardless of some nonlinear perturbations, indicating that the law actually applies to a large class of nonlinear second order systems.
Abstract: Studies the problem of finite-time output feedback stabilization for the double integrator system. A class of output feedback controllers that can achieve global finite-time stability for the double integrator system are constructed based on a "finite-time separation principle." Furthermore, it is shown that the closed-loop system resulting from our control law can maintain its local finite-time stability regardless of some nonlinear perturbations. Thus, our control law actually applies to a large class of nonlinear second order systems.

Journal ArticleDOI
28 Jun 2001-Nature
TL;DR: This work uses a key simplification to investigate the theoretical limits to the information capacity of an optical fibre arising from these nonlinearities and relates the nonlinear channel to a linear channel with multiplicative noise, for which it is able to obtain analytical results.
Abstract: The exponential growth in the rate at which information can be communicated through an optical fibre is a key element in the 'information revolution' However, as for all exponential growth laws, physical limits must be considered The nonlinear nature of the propagation of light in optical fibre has made these limits difficult to elucidate Here we use a key simplification to investigate the theoretical limits to the information capacity of an optical fibre arising from these nonlinearities The success of our approach lies in relating the nonlinear channel to a linear channel with multiplicative noise, for which we are able to obtain analytical results In fundamental distinction to linear channels with additive noise, the capacity of a nonlinear channel does not grow indefinitely with increasing signal power, but has a maximal value The ideas presented here may have broader implications for other nonlinear information channels, such as those involved in sensory transduction in neurobiology These have been often examined using additive noise linear channel models but, as we show here, nonlinearities can change the picture qualitatively

Journal ArticleDOI
TL;DR: This work develops a maximum a posteriori probability (MAP) estimation approach for interferometric radar techniques, and derives an algorithm that approximately maximizes the conditional probability of its phase-unwrapped solution given observable quantities such as wrapped phase, image intensity, and interferogram coherence.
Abstract: Interferometric radar techniques often necessitate two-dimensional (2-D) phase unwrapping, defined here as the estimation of unambiguous phase data from a 2-D array known only modulo 2pi rad. We develop a maximum a posteriori probability (MAP) estimation approach for this problem, and we derive an algorithm that approximately maximizes the conditional probability of its phase-unwrapped solution given observable quantities such as wrapped phase, image intensity, and interferogram coherence. Examining topographic and differential interferometry separately, we derive simple, working models for the joint statistics of the estimated and the observed signals. We use generalized, nonlinear cost functions to reflect these probability relationships, and we employ nonlinear network-flow techniques to approximate MAP solutions. We apply our algorithm both to a topographic interferogram exhibiting rough terrain and layover and to a differential interferogram measuring the deformation from a large earthquake. The MAP solutions are complete and are more accurate than those of other tested algorithms.

Journal ArticleDOI
TL;DR: In this article, the authors propose to decompose the problem into a fluid and a structural part through an additive decomposition of the space of kinematically admissible test functions, which can be discretised in time by implicit, stable, energy conserving time integration schemes and solved by simple, iterative uncoupled algorithms.

Dissertation
01 Jan 2001
TL;DR: This main contribution is to find explicit change of coordinates and control that transform several classes of underactuated systems into cascade nonlinear systems with structural properties that are convenient for control design purposes.
Abstract: This thesis is devoted to nonlinear control, reduction, and classification of underactuated mechanical systems. Reduction and nonlinear control of high-order underactuated systems with kinetic symmetry is the main focus of this thesis. Our main contribution is to find explicit change of coordinates and control that transform several classes of underactuated systems, which appear in robotics and aerospace applications, into cascade nonlinear systems with structural properties that are convenient for control design purposes. The obtained cascade normal forms are three classes of nonlinear systems, namely, systems in strict feedback form, feedforward form, and nontriangular linear-quadratic form. The triangular normal forms of underactuated systems can be controlled using existing backstepping and feedforwarding procedures. However, control of the nontriangular normal forms is a major open problem. We address this problem for important classes of nontriangular systems of interest by introducing a new stabilization method based on the solutions of fixed-point equations as stabilizing nonlinear state feedback laws. As a result of the reduction process, one obtains a reduced nonlinear subsystem in cascade with a linear subsystem. For many classes of underactuated systems, this reduced nonlinear subsystem is physically meaningful. In fact, the reduced nonlinear subsystem is itself a Lagrangian system with a well-defined lower-order configuration vector. The key analytical tools that allow reduction of high-order underactuated systems using transformations in explicit forms are “normalized generalized momentums and their integrals” (whenever integrable). Both of them can be obtained from the Lagrangian of the system. Based on some basic properties of underactuated systems as actuation/passivity of shape variables, integrability/non-integrability of appropriate normalized momentums, and presence/lack of input coupling; I managed to classify underactuated systems to 8 classes. Examples of these 8 classes cover almost all major applications in robotics, aerospace systems, and benchmark systems. For underactuated systems with nonholonomic velocity constrains and symmetry, we obtained normal forms as the cascade of the constraint equation and a reduced-order Lagrangian control system which is underactuated or fully-actuated. (Abstract shortened by UMI.) (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.)

Journal ArticleDOI
TL;DR: In this paper, the authors study energy pumping in an impulsively excited, two-degrees-of-freedom damped system with essential (nonlinearizable) nonlinearities by means of two analytical techniques.
Abstract: We study energy pumping in an impulsively excited, two-degrees-of-freedom damped system with essential (nonlinearizable) nonlinearities by means of two analytical techniques. First, we transform the equations of motion using the action-angle variables of the underlying Hamiltonian system and bring them into the form where two-frequency averaging can be applied. We then show that energy pumping is due to resonance capture in the 1:1 resonance manifold of the system, and perform a perturbation analysis in an O (√e) neighborhood of this manifold in order to study the attracting region responsible for the resonance capture. The second method is based on the assumption of 1:1 internal resonance in the fast dynamics of the system, and utilizes complexification and averaging to develop analytical approximations to the nonlinear transient responses of the system in the energy pumping regime. The results compare favorably to numerical simulations. The practical implications of the energy pumping phenomenon are discussed.

BookDOI
01 Jan 2001
TL;DR: This website will be so easy for you to access the internet service, so you can really keep in mind that the book is the best book for you.
Abstract: We present here because it will be so easy for you to access the internet service. As in this new era, much technology is sophistically offered by connecting to the internet. No any problems to face, just for this day, you can really keep in mind that the book is the best book for you. We offer the best here to read. After deciding how your feeling will be, you can enjoy to visit the link and get the book.

Book
09 Mar 2001
TL;DR: The Galerkin Method and its Variants and Finite Element Analysis have been used in this paper to solve the problem of finding the optimal solution of the Fredholm Integral Equations of the Second Kind.
Abstract: Preface 1 Linear Spaces 2 Linear Operators on Normed Spaces 3 Approximation Theory 4 Nonlinear Equations and Their Solution by Iteration 5 Finite Difference Method 6 Sobolev Spaces 7 Variational Formulations of Elliptic Boundary Value Problems 8 The Galerkin Method and Its Variants 9 Finite Element Analysis 10 Elliptic Variational Inequalities and Their Numerical Approximations 11 Numerical Solution of Fredholm Integral Equations of the Second Kind 12 Boundary Integral Equations References Index.

Book
15 Jun 2001
TL;DR: In this paper, the authors present a generalized Abelian Higgs Equations and a Chern-Simons system for the non-Abelian case of the Higgs equation.
Abstract: Preface * Primer of Field Theory * Sigma Models * Multiple Instantons and Characteristic Classes * Generalized Abelian Higgs Equations * Chern-Simons Systems: Abelian Case * Chern-Simons Systems: Non-Abelian Case * Electroweak Vortices * Dyons * Ordinary Differential Equations * Strings in Cosmology * Vortices and Antivortices * Born-Infeld Solutions * References * Bibliography * Index

Journal ArticleDOI
TL;DR: The effectiveness of the proposed controller design methodology is finally demonstrated through numerical simulations on the chaotic Lorenz system, which has complex nonlinearity.
Abstract: Addresses the robust fuzzy control problem for nonlinear systems in the presence of parametric uncertainties. The Takagi-Sugeno (T-S) fuzzy model is adopted for fuzzy modeling of the nonlinear system. Two cases of the T-S fuzzy system with parametric uncertainties, both continuous-time and discrete-time cases are considered. In both continuous-time and discrete-time cases, sufficient conditions are derived for robust stabilization in the sense of Lyapunov asymptotic stability, for the T-S fuzzy system with parametric uncertainties. The sufficient conditions are formulated in the format of linear matrix inequalities. The T-S fuzzy model of the chaotic Lorenz system, which has complex nonlinearity, is developed as a test bed. The effectiveness of the proposed controller design methodology is finally demonstrated through numerical simulations on the chaotic Lorenz system.

Journal ArticleDOI
TL;DR: In this paper, the Jacobi elliptic functions are applied in Jacobi function expansion method to construct the exact periodic solutions of nonlinear wave equations and it is shown that more new periodic solutions can be obtained by this method and more shock wave solutions or solitary wave solution can be got at their limit condition.

Journal ArticleDOI
TL;DR: This work presents a classical solution in terms of the parallel connection of a robust stabilizer and an internal model, where the latter is adaptively tuned to the device that reproduces the steady-state control necessary to maintain the output-zeroing condition.
Abstract: We address the problem of output regulation for nonlinear systems driven by a linear, neutrally stable exosystem whose frequencies are not known a priori. We present a classical solution in terms of the parallel connection of a robust stabilizer and an internal model, where the latter is adaptively tuned to the device that reproduces the steady-state control necessary to maintain the output-zeroing condition. We obtain robust regulation (i.e. in presence of parameter uncertainties) with a semi-global domain of convergence for a significant class of nonlinear minimum-phase system.

Journal ArticleDOI
TL;DR: It is shown that the analysis results provide an efficient technique for the design of fuzzy controllers and a stabilization approach for nonlinear retarded systems through fuzzy state feedback and fuzzy observer-based controller is proposed.

Journal ArticleDOI
TL;DR: In this paper, the authors presented numerical evidence of energy pumping in coupled nonlinear mechanical oscillators, i.e., of one-way (irreversible) channeling of externally imparted energy from the linear to the nonlinear part of the system, provided that the energy is above a critical level.
Abstract: The systems considered in this work are composed of weakly coupled, linear and essentially nonlinear (nonlinearizable) components. In Part I of this work we present numerical evidence of energy pumping in coupled nonlinear mechanical oscillators, i.e., of one-way (irreversible) channeling of externally imparted energy from the linear to the nonlinear part of the system, provided that the energy is above a critical level. Clearly, no such phenomenon is possible in the linear system. To obtain a better understanding of the energy pumping phenomenon we first analyze the dynamics of the underlying Hamiltonian system (corresponding to zero damping). First we reduce the equations of motion on an isoenergetic manifold of the dynamical flow, and then compute subharmonic orbits by employing nonsmooth transformation of coordinates which lead to nonlinear boundary value problems. It is conjectured that a 1:1 stable subharmonic orbit of the underlying Hamiltonian system is mainly responsible for the energy pumping phenomenon. This orbit cannot be excited at sufficiently low energies. In Part II of this work the energy pumping phenomenon is further analyzed, and it is shown that it is caused by transient resonance capture on a 1:1 resonance manifold of the system.

Journal ArticleDOI
TL;DR: This paper considers an emerging family of high dimensional model representation concepts and techniques capable of dealing with large numbers of input variables, typically a nonlinear relationship.
Abstract: In the chemical sciences, many laboratory experiments, environmental and industrial processes, as well as modeling exercises, are characterized by large numbers of input variables. A general objective in such cases is an exploration of the high-dimensional input variable space as thoroughly as possible for its impact on observable system behavior, often with either optimization in mind or simply for achieving a better understanding of the phenomena involved. An important concern when undertaking these explorations is the number of experiments or modeling excursions necessary to effectively learn the system input → output behavior, which is typically a nonlinear relationship. Although simple logic suggests that the number of runs could grow exponentially with the number of input variables, broadscale evidence indicates that the required effort often scales far more comfortably. This paper considers an emerging family of high dimensional model representation concepts and techniques capable of dealing with s...

BookDOI
01 Jan 2001
TL;DR: Holmes et al. as mentioned in this paper proposed a model of low-dimensional models of turbulence, including lattice dynamical systems and extended systems, and three lectures on mathematical fluid mechanics.
Abstract: Preface. Introduction J.C. Robinson, P.A. Glendinning. Spatial correlations and local fluctuations in host-parasite models M.J. Keeling, D.A. Rand. Lattice dynamical systems L.A. Bunimovich (assisted by C. Giberti). Attractors and dynamics in partial differential equations J.K. Hale. Nonlinear dynamics of extended systems P. Collet. Three lectures on mathematical fluid mechanics P. Constantin. Low-dimensional models of turbulence P.J. Holmes, et al.

Journal ArticleDOI
TL;DR: In this paper, the authors present an overview of experimental evidence and present understanding of nonlinear dielectric, elastic and piezoelectric relationships in PEG ceramics.
Abstract: The paper presents an overview of experimental evidence and present understanding of nonlinear dielectric, elastic and piezoelectric relationships in piezoelectric ceramics. This topic has gained an increasing recognition in recent years due to the use of such materials under extreme operating conditions, for example in electromechanical actuators and high power acoustic transducers. Linear behaviour is generally confined to relatively low levels of applied electric field and stress, under which the dielectric, elastic and piezoelectric relationships are described well by the standard piezoelectric constitutive equations. Nonlinear relationships are observed above certain ‘threshold’ values of electric field strength and mechanical stress, giving rise to field and stress-dependent dielectric (e), elastic (s) and piezoelectric (d) coefficients. Eventually, strong hysteresis and saturation become evident above the coercive field/stress due to ferroelectric/ferroelastic domain switching. The thermodynamic method provides one approach to describing nonlinear behaviour in the ‘intermediate’ field region, prior to large scale domain switching, by extending the piezoelectric constitutive equations to include nonlinear terms. However, this method seems to fail in its prediction of the amplitude and phase of high frequency harmonic components in the field-induced polarisation and strain waveforms, which arise directly from the nonlinear dielectric and piezoelectric relationships. A better fit to experimental data is given by the empirical Rayleigh relations, which were first developed to describe nonlinear behaviour in soft magnetic materials. This approach also provides an indication of the origins of nonlinearity in piezoelectric ceramics, in terms of ferroelectric domain wall translation (at intermediate field/stress levels) and domain switching (at high field/stress levels). The analogy with magnetic behaviour is also reflected in the use of Preisach-type models, which have been successfully employed to describe the hysteretic path-dependent strain-field relationships in piezoelectric actuators. The relative merits and limitations of the different modelling methods are compared and possible areas of application are identified.

Journal ArticleDOI
TL;DR: In this paper, an asymptotic theory for nonlinear regression with integrated processes is developed, and sufficient conditions for weak consistency are given and a limit distribution theory is provided, which is mixed normal with mixing variates that depend on the sojourn time of the limiting Brownian motion of the integrated process.
Abstract: An asymptotic theory is developed for nonlinear regression with integrated processes. The models allow for nonlinear effects from unit root time series and therefore deal with the case of parametric nonlinear cointegration. The theory covers integrable and asymptotically homogeneous functions. Sufficient conditions for weak consistency are given and a limit distribution theory is provided. The rates of convergence depend on the properties of the nonlinear regression function, and are shown to be as slow as n 1/4 for integrable functions, and to be generally polynomial in n 1/2 for homogeneous functions. For regressions with integrable functions, the limiting distribution theory is mixed normal with mixing variates that depend on the sojourn time of the limiting Brownian motion of the integrated process.

Journal ArticleDOI
TL;DR: A direct adaptive output feedback control design procedure is developed for highly uncertain nonlinear systems, that does not rely on state estimation, and extends the universal function approximation property of linearly parameterized neural networks to model unknown system dynamics from input/output data.