Showing papers on "Nonlinear system published in 1999"
TL;DR: In this paper, the homotopy perturbation technique does not depend upon a small parameter in the equation and can be obtained uniformly valid not only for small parameters, but also for very large parameters.
3,058 citations
TL;DR: A predictive control scheme is proposed which is able to stabilize MLD systems on desired reference trajectories while fulfilling operating constraints, and possibly take into account previous qualitative knowledge in the form of heuristic rules.
2,980 citations
TL;DR: In this paper, Hilbert spectral analysis is proposed as an alternative to wavelet analysis, which provides not only a more precise definition of particular events in time-frequency space, but also more physically meaningful interpretations of the underlying dynamic processes.
Abstract: We survey the newly developed Hilbert spectral analysis method and its applications to Stokes waves, nonlinear wave evolution processes, the spectral form of the random wave field, and turbulence. Our emphasis is on the inadequacy of presently available methods in nonlinear and nonstationary data analysis. Hilbert spectral analysis is here proposed as an alternative. This new method provides not only a more precise definition of particular events in time-frequency space than wavelet analysis, but also more physically meaningful interpretations of the underlying dynamic processes.
1,945 citations
Book•
22 Jun 1999
TL;DR: In this article, the authors compare Linear vs. Nonlinear Control of Differential Geometry with Linearization by State Feedback (LSF) by using Linearization and Geometric Non-linear Control (GNC).
Abstract: 1 Linear vs. Nonlinear.- 2 Planar Dynamical Systems.- 3 Mathematical Background.- 4 Input-Output Analysis.- 5 Lyapunov Stability Theory.- 6 Applications of Lyapunov Theory.- 7 Dynamical Systems and Bifurcations.- 8 Basics of Differential Geometry.- 9 Linearization by State Feedback.- 10 Design Examples Using Linearization.- 11 Geometric Nonlinear Control.- 12 Exterior Differential Systems in Control.- 13 New Vistas: Multi-Agent Hybrid Systems.- References.
1,925 citations
TL;DR: In this paper, the authors investigate the possibility that time series generated by certain physiological control systems may be members of a special class of complex processes, termed multifractal, which require a large number of exponents to characterize their scaling properties.
Abstract: There is evidence that physiological signals under healthy conditions may have a fractal temporal structure. Here we investigate the possibility that time series generated by certain physiological control systems may be members of a special class of complex processes, termed multifractal, which require a large number of exponents to characterize their scaling properties. We report on evidence for multifractality in a biological dynamical system, the healthy human heartbeat, and show that the multifractal character and nonlinear properties of the healthy heart rate are encoded in the Fourier phases. We uncover a loss of multifractality for a life-threatening condition, congestive heart failure.
1,448 citations
TL;DR: In this paper, small group velocities of order 90 m/s and large group delays of greater than 0.26 ms were observed in an optically dense hot rubidium gas ( $\ensuremath{\approx}360\mathrm{K}$).
Abstract: We report the observation of small group velocities of order 90 m/s and large group delays of greater than 0.26 ms, in an optically dense hot rubidium gas ( $\ensuremath{\approx}360\mathrm{K}$). Media of this kind yield strong nonlinear interactions between very weak optical fields and very sharp spectral features. The result is in agreement with previous studies on nonlinear spectroscopy of dense coherent media.
1,042 citations
TL;DR: In this paper, two different families of numerical methods are considered to solve the problem of a homogeneous linear reference material undergoing a nonhomogeneous periodic eigenstrain, and the relative merits of the two methods are compared and several examples are discussed.
1,028 citations
TL;DR: The state of knowledge on spatial soliton interactions is reviewed and effects such as fusion, fission, annihilation, and stable orbiting in three dimensions are reviewed.
Abstract: Spatial solitons, beams that do not spread owing to diffraction when they propagate, have been demonstrated to exist by virtue of a variety of nonlinear self-trapping mechanisms. Despite the diversity of these mechanisms, many of the features of soliton interactions and collisions are universal. Spatial solitons exhibit a richness of phenomena not found with temporal solitons in fibers, including effects such as fusion, fission, annihilation, and stable orbiting in three dimensions. Here the current state of knowledge on spatial soliton interactions is reviewed.
965 citations
TL;DR: In this paper, a review of literature on optimal power flow tracing progress in this area over from 1962-93 is presented. Part I deals with the application of nonlinear and quadratic programming.
Abstract: The paper presents a review of literature on optimal power flow tracing progress in this area over from 1962-93. Part I deals with the application of nonlinear and quadratic programming.
832 citations
Book•
18 Oct 1999
TL;DR: In this article, the authors present a continuation of the first volume by Alberto Isidori on nonlinear control systems, which includes stability analysis of interconnected nonlinear systems, the notion of input-to-state stability and its role in analysing stability of cascade-connected or feedback-connected systems.
Abstract: From the Publisher:
"This book incorporates recent advances in the design of feedback laws to the purpose of globally stabilizing nonlinear systems via state or output feedback. It is a continuation of the first volume by Alberto Isidori on Nonlinear Control Systems. Specifically this second volume will cover: Stability analysis of interconnected nonlinear systems; the notion of input-to-state stability and its role in analysing stability of cascade-connected or feedback-connected systems; the notion of dissipativity and its consequences (passivity and "gain"); robust stabilization in the case of parametric uncertainties; the case of state feedback (global or semi-global stabilization); the case of output feedback (semi-global stabilization); robust stabilization in the case of unstructured perturbations; feedback design via the small-gain approach; robust semi-global stabilization via output feedback; methods for asymptotic tracking, disturbance rejection and model following; global and semi-global analysis; normal forms for multi-input multi-output nonlinear systems form a global point of view; and their role in feedback design."--BOOK JACKET.
811 citations
Book•
23 Mar 1999
TL;DR: In this paper, the authors present a Lie Group Analysis of Ordinary Differential Equations (ODE) for the first order and second order differential equations, respectively, and integrate them into Third Order Equations.
Abstract: Introduction to Differential Equations. Transformation Groups. Lie Group Analysis of Ordinary Differential Equations. Brief on Lie Algebras. First Order Differential Equations. Integration of Second Order Equations. Basic Theory of Linear Equations. Nonlinear Second Order Equations. Integration of Third Order Equations. Nonlinear Superposition Principle. Index.
TL;DR: The authors prove that, although the state dimension is not preserved, the number of input channels is kept fixed and it is proved that a Lie-Backlund isomorphism can be realized by an endogenous feedback.
Abstract: A new system equivalence relation, using the framework of differential geometry of jets and prolongations of infinite order, is studied. In this setting, two systems are said to be equivalent if any variable of one system may be expressed as a function of the variables of the other system and of a finite number of their time derivatives. This is a Lie-Backlund isomorphism. The authors prove that, although the state dimension is not preserved, the number of input channels is kept fixed. They also prove that a Lie-Backlund isomorphism can be realized by an endogenous feedback. The differentially flat nonlinear systems introduced by the authors (1992) via differential algebraic techniques, are generalized and the new notion of orbitally flat systems is defined. They correspond to systems which are equivalent to a trivial one, with time preservation or not. The endogenous linearizing feedback is explicitly computed in the case of the VTOL aircraft to track given reference trajectories with stability.
TL;DR: In this article, the combination of electromagnetically induced transparency based nonlinear optics and cold atom technology, under conditions of ultra-low light propagation, allows nonlinear processes at energies of a few photons per atomic cross section.
Abstract: We show how the combination of electromagnetically induced transparency based nonlinear optics and cold atom technology, under conditions of ultraslow light propagation, allows nonlinear processes at energies of a few photons per atomic cross section.
TL;DR: In this article, a sucient condition for asymptotic stability of a switched linear system in terms of the Lie algebra generated by the individual matrices is presented, and it is shown that any family of linear systems satisfying this condition possesses a quadratic common Lyapunov function.
TL;DR: Jensen's inequality is a mathematical proof that is seldom mentioned in the ecological literature but which provides a powerful tool for predicting some direct effects of environmental variance in biological systems.
Abstract: Many biologists now recognize that environmental variance can exert important effects on patterns and processes in nature that are independent of average conditions. Jensen’s inequality is a mathematical proof that is seldom mentioned in the ecological literature but which provides a powerful tool for predicting some direct effects of environmental variance in biological systems. Qualitative predictions can be derived from the form of the relevant response functions (accelerating versus decelerating). Knowledge of the frequency distribution (especially the variance) of the driving variables allows quantitative estimates of the effects. Jensen’s inequality has relevance in every field of biology that includes nonlinear processes.
TL;DR: It is shown that the performance of a globally bounded partial state feedback control of an input-output linearizable system can be recovered by a sufficiently fast high-gain observer.
Abstract: It is shown that the performance of a globally bounded partial state feedback control of a certain class of nonlinear systems can be recovered by a sufficiently fast high-gain observer. The performance recovery includes recovery of asymptotic stability of the origin, the region of attraction, and trajectories.
TL;DR: Two suboptimal MPC schemes are presented and analyzed that are guaranteed to be stabilizing, provided an initial feasible solution is available and for which the computational requirements are more reasonable.
Abstract: Practical difficulties involved in implementing stabilizing model predictive control laws for nonlinear systems are well known. Stabilizing formulations of the method normally rely on the assumption that global and exact solutions of nonconvex, nonlinear optimization problems are possible in limited computational time. In the paper, we first establish conditions under which suboptimal model predictive control (MPC) controllers are stabilizing; the conditions are mild holding out the hope that many existing controllers remain stabilizing even if optimality is lost. Second, we present and analyze two suboptimal MPC schemes that are guaranteed to be stabilizing, provided an initial feasible solution is available and for which the computational requirements are more reasonable.
TL;DR: In this paper, the same family of controllers semi-globally practically stabilizes the exact discrete-time model of the plant for sufficiently small sampling periods when the controllers are locally bounded, uniformly in the sampling period.
TL;DR: Recent advances in developing nonlinear optical techniques for processing serial digital information at high speed are reviewed and expected to become important in future high-capacity communications networks.
Abstract: Recent advances in developing nonlinear optical techniques for processing serial digital information at high speed are reviewed. The field has been transformed by the advent of semiconductor nonlinear devices capable of operation at 100 gigabits per second and higher, well beyond the current speed limits of commercial electronics. These devices are expected to become important in future high-capacity communications networks by allowing digital regeneration and other processing functions to be performed on data signals “on the fly” in the optical domain.
Book•
01 Jan 1999TL;DR: The purpose of this monograph is to discuss models for optimization and search, as well as some of the principles used in computer programming, which have been used in the design of search engines and mobile devices.
Abstract: Preface 1. Introduction Part I. Analytical Models: 2. Ordinary differential and difference equations 3. Partial differential equations 4. Variational principles 5. Random systems Part II. Numerical Models: 6. Finite differences: ordinary difference equations 7. Finite differences: partial differential equations 8. Finite elements 9. Cellular automata and lattice gases Part III. Observational Models: 10. Function fitting 11. Transforms 12. Architectures 13. Optimization and search 14. Clustering and density estimation 15. Filtering and state estimation 16. Linear and nonlinear time series Appendix 1. Graphical and mathematical software Appendix 2. Network programming Appendix 3. Benchmarking Appendix 4. Problem solutions Bibliography.
Book•
29 Apr 1999
TL;DR: Inverse scattering methods as discussed by the authors have been used to solve the classical soliton equation problem in linear wave theory and nonlinear lattice solitons, as well as in Peturbation Theory.
Abstract: 1. The Birth of a Paradigm 2. Linear Wave Theory 3. The Classical Soliton Equations 4. Reaction-Diffusion Systems 5. Nonlinear Lattices 6. Inverse Scattering Methods 7. Peturbation Theory 8. Quantum Lattice Solitons 9. Looking Ahead Bibliography Index
TL;DR: A Lyapunov-based stabilizing control design method for uncertain nonlinear dynamical systems using fuzzy models is proposed, finding sufficient conditions for stability and stabilizability of fuzzy models using fuzzy state feedback controllers.
Abstract: A Lyapunov-based stabilizing control design method for uncertain nonlinear dynamical systems using fuzzy models is proposed. The controller is constructed using a design model of the dynamical process to be controlled. The design model is obtained from the truth model using a fuzzy modeling approach. The truth model represents a detailed description of the process dynamics. The truth model is used in a simulation experiment to evaluate the performance of the controller design. A method for generating local models that constitute the design model is proposed. Sufficient conditions for stability and stabilizability of fuzzy models using fuzzy state feedback controllers are given. The results obtained are illustrated with a numerical example involving a four-dimensional nonlinear model of a stick balancer.
TL;DR: In this paper, the authors derived stable and accurate interface conditions based on the SAT penalty method for the linear advection?diffusion equation, which are functionally independent of the spatial order of accuracy and rely only on the form of the discrete operator.
Proceedings Article•
29 Nov 1999TL;DR: In this paper, the authors examined properties of the class of Gaussian scale mixtures, and showed that these densities can accurately characterize both the marginal and joint distributions of natural image wavelet coefficients.
Abstract: The statistics of photographic images, when represented using multiscale (wavelet) bases, exhibit two striking types of non-Gaussian behavior. First, the marginal densities of the coefficients have extended heavy tails. Second, the joint densities exhibit variance dependencies not captured by second-order models. We examine properties of the class of Gaussian scale mixtures, and show that these densities can accurately characterize both the marginal and joint distributions of natural image wavelet coefficients. This class of model suggests a Markov structure, in which wavelet coefficients are linked by hidden scaling variables corresponding to local image structure. We derive an estimator for these hidden variables, and show that a nonlinear "normalization" procedure can be used to Gaussianize the coefficients.
01 Jan 1999
TL;DR: This work states that nonlinear model predictive control, i.e. MPC based on a nonlinear plant description, has only emerged in the past decade and the number of reported industrial applications is still fairly low.
Abstract: In the past decade model predictive control (MPC) has become a preferred control strategy for a large number of processes. The main reasons for this preference include the ability to handle constraints in an optimal way and the flexible formulation in the time domain. Linear MPC schemes, i.e. MPC schemes for which the prediction is based on a linear description of the plant, are by now routinely used in a number of industrial sectors and the underlying control theoretic problems, like stability, are well studied. Nonlinear model predictive control (NMPC), i.e. MPC based on a nonlinear plant description, has only emerged in the past decade and the number of reported industrial applications is still fairly low. Because of its additional ability to take process nonlinearities into account, expectations on this control methodology are high.
01 Jan 1999
TL;DR: In this article, the authors present analytical models for linear and nonlinear time series, as well as an analytical model for cellular automata and lattice gases, and a model for partial differential equations.
Abstract: Preface 1. Introduction Part I. Analytical Models: 2. Ordinary differential and difference equations 3. Partial differential equations 4. Variational principles 5. Random systems Part II. Numerical Models: 6. Finite differences: ordinary difference equations 7. Finite differences: partial differential equations 8. Finite elements 9. Cellular automata and lattice gases Part III. Observational Models: 10. Function fitting 11. Transforms 12. Architectures 13. Optimization and search 14. Clustering and density estimation 15. Filtering and state estimation 16. Linear and nonlinear time series Appendix 1. Graphical and mathematical software Appendix 2. Network programming Appendix 3. Benchmarking Appendix 4. Problem solutions Bibliography.
TL;DR: It is proved that in the general case, it is not possible to separate the sources without nonlinear distortion, and this work focuses on specific nonlinear mixtures known as post-nonlinear mixture, which are realistic models in many situations and emphasize interesting properties i.e., in such nonlinearmixtures, sources can be estimated with the same indeterminacies as in instantaneous linear mixtures.
Abstract: We address the problem of separation of mutually independent sources in nonlinear mixtures. First, we propose theoretical results and prove that in the general case, it is not possible to separate the sources without nonlinear distortion. Therefore, we focus our work on specific nonlinear mixtures known as post-nonlinear mixtures. These mixtures constituted by a linear instantaneous mixture (linear memoryless channel) followed by an unknown and invertible memoryless nonlinear distortion, are realistic models in many situations and emphasize interesting properties i.e., in such nonlinear mixtures, sources can be estimated with the same indeterminacies as in instantaneous linear mixtures. The separation structure of nonlinear mixtures is a two-stage system, namely, a nonlinear stage followed by a linear stage, the parameters of which are updated to minimize an output independence criterion expressed as a mutual information criterion. The minimization of this criterion requires knowledge or estimation of source densities or of their log-derivatives. A first algorithm based on a Gram-Charlier expansion of densities is proposed. Unfortunately, it fails for hard nonlinear mixtures. A second algorithm based on an adaptive estimation of the log-derivative of densities leads to very good performance, even with hard nonlinearities. Experiments are proposed to illustrate these results.
01 Jan 1999
TL;DR: The method of lines for partial differential equations consists in replacing spatial derivatives by difference expressions, and then the partial equation is transformed into a system of ordinary differential equations as discussed by the authors, which is used for approximation of solutions of nonlinear differential problems of parabolic type by solutions of ordinary equations ([91, 153, 219, 220, 222, 225, 238]).
Abstract: The method of lines for partial differential equations consists in replacing spatial derivatives by difference expressions. Then the partial equation is transformed into a system of ordinary differential equations. The method is used for approximation of solutions of nonlinear differential problems of parabolic type by solutions of ordinary equations ([91, 153, 219, 220, 222, 225, 238]). The method is also treated as a tool for proving of existence theorems for differential problems corresponding to parabolic equations [223, 224, 227] or first-order hyperbolic systems [101, 157]. Simple examples of the method of lines for nonlinear functional differential equations were considered in [29, 108, 128]. The method for equations of higher orders is studied in [91]. The book [189] demonstrates lots of examples of the use of the numerical method of lines. Convergence analysis of one step difference methods generated by the numerical method of lines was investigated in [186].
Book•
31 Jul 1999
TL;DR: The Faces of Complexity presents a model of nonlinear control of Multivariable Systems that combines Adaptive and Robus Control Design, and nonlinear Systems: Analysis and Design Tools, which describes the design tools used for this model.
Abstract: Preface. Notations and Definitions. 1. Faces of Complexity. 2. Nonlinear Systems: Analysis and Design Tools. 3. Speed-Gradient Method and Partial Stabilization. 4. Nonlinear control of Multivariable Systems. 5. Nonlinear Control of Mimo Systems. 6. Adaptive and Robus Control Design. 7. Decomposition of Adaptive Systems. 8. Control of Mechanical Systems. 9. Physics and Control. A. Appendix. References. Index.