Showing papers on "Nonlinear system published in 1998"
TL;DR: In this paper, a new method for analysing nonlinear and nonstationary data has been developed, which is the key part of the method is the empirical mode decomposition method with which any complicated data set can be decoded.
Abstract: A new method for analysing nonlinear and non-stationary data has been developed. The key part of the method is the empirical mode decomposition method with which any complicated data set can be dec...
18,956 citations
Book•
01 Jan 1998
TL;DR: In this article, the authors present techniques from the numerical analysis and applied mathematics literatures and show how to use them in economic analyses, including linear equations, iterative methods, optimization, nonlinear equations, approximation methods, numerical integration and differentiation, and Monte Carlo methods.
Abstract: To harness the full power of computer technology, economists need to use a broad range of mathematical techniques. In this book, Kenneth Judd presents techniques from the numerical analysis and applied mathematics literatures and shows how to use them in economic analyses. The book is divided into five parts. Part I provides a general introduction. Part II presents basics from numerical analysis on R^n, including linear equations, iterative methods, optimization, nonlinear equations, approximation methods, numerical integration and differentiation, and Monte Carlo methods. Part III covers methods for dynamic problems, including finite difference methods, projection methods, and numerical dynamic programming. Part IV covers perturbation and asymptotic solution methods. Finally, Part V covers applications to dynamic equilibrium analysis, including solution methods for perfect foresight models and rational expectation models. A web site contains supplementary material including programs and answers to exercises.
2,880 citations
TL;DR: It is proven that for scalar equations, the LDG methods are L2-stable in the nonlinear case and in the linear case, it is shown that if polynomials of degree k are used, the methods are kth order accurate for general triangulations.
Abstract: In this paper, we study the local discontinuous Galerkin (LDG) methods for nonlinear, time-dependent convection-diffusion systems. These methods are an extension of the Runge--Kutta discontinuous Galerkin (RKDG) methods for purely hyperbolic systems to convection-diffusion systems and share with those methods their high parallelizability, high-order formal accuracy, and easy handling of complicated geometries for convection-dominated problems. It is proven that for scalar equations, the LDG methods are L2-stable in the nonlinear case. Moreover, in the linear case, it is shown that if polynomials of degree k are used, the methods are kth order accurate for general triangulations; although this order of convergence is suboptimal, it is sharp for the LDG methods. Preliminary numerical examples displaying the performance of the method are shown.
2,265 citations
TL;DR: These results may be viewed as generalizing the classical Krasovskii theorem, and, more loosely, linear eigenvalue analysis, and the approach is illustrated by controller and observer designs for simple physical examples.
1,405 citations
TL;DR: In this article, the authors show that Rademacher's theorem for functions with values in Banach spaces implies that the function m is almost everywhere differentiable on (0, T) with dm dt (t)=vt~(t, x) -m( t ) <<.
Abstract: WAVE BREAKING FOR NONLINEAR NONLOCAL SHALLOW WATER EQUATIONS 233 THEOREM 2.1. Let T>O and vE C 1 ([0, T); H 2(R)). Then for every t~ [0, T) there exists at least one point ~(t)ER with ,~(t) := in~ Ivy(t, x)] = ~ ( t , ~(t)), and the function m is almost everywhere differentiable on (0, T) with dm dt (t)=vt~(t,~(t)) a.e. on (O,T). Proof. Let c>0 stand for a generic constant. Fix te[0, T) and define m(t):=infxcR[v~(t,x)]. If m(t))O we have that v(t , . ) is nondecreasing on R and therefore v(t,. ) 0 (recall v(t,. )cL2(R)) , so that we may assume re(t)<0. Since vx(t,. ) c H I ( R ) we see that limlxl~ ~ Vx(t, x)=0 so that there exists at least a ~(t) e R with re(t) =v~(t, ~(t)). Let now s, tC [0, T) be fixed. If re(t) <~rn(s) we have 0 < re(s) .~(t) = i n f [~x (~, x ) ] ~ ( t , ~(t)) <. ~=(~, ~(t)) -~x(t, ~(t)), and by the Sobolev embedding HI(R) C L ~ ( R ) we conclude that Im(8)-.~(t)l ~< Ivx(t)-v~(s)lL~(~) < c Iv~(t)-v~(8)l.l(R). Hence the mean-value theorem for functions with values in Banach spaces-Hi(R) in the present case--yields (see [12]) jm(t)-m(s)l<~clt-s j m a x [IVt~(T)JHI(R)], t, se[O,T). O~T~max{s,t} Since vt~cC([O,T), Hi(R)) , we see that m is locally Lipschitz on [0, T) and therefore Rademacher's theorem (cf. [14]) implies that m is almost everywhere differentiable on (0,T). Fix tC(0, T). We have that v~(t+h)-vx(t)h vt~(t) Hl(R) ---~0 as h--*O, and therefore vx( t+h ,y ) -vx ( t , y ) sup vtx(t,y) --~0 as h---~O, (2.1) ycl~ h in view of the continuous embedding H 1 ( R ) c L ~ (R). 234 A. C O N S T A N T I N AND J. E S C H E R By the definition of m, m(t+h) = v~(t+h, ((t+h)) <. v~(t+h, ((t)). Consequently, given h>0 , we obtain m( t+h) -m( t ) <<. h Letting h--~O + and using (2.1), we find lim sup m(t+h) -m( t ) h~_~0 + h
1,361 citations
TL;DR: A mean-square error lower bound for the discrete-time nonlinear filtering problem is derived based on the van Trees (1968) (posterior) version of the Cramer-Rao inequality and is applicable to multidimensional nonlinear, possibly non-Gaussian, dynamical systems.
Abstract: A mean-square error lower bound for the discrete-time nonlinear filtering problem is derived based on the van Trees (1968) (posterior) version of the Cramer-Rao inequality. This lower bound is applicable to multidimensional nonlinear, possibly non-Gaussian, dynamical systems and is more general than the previous bounds in the literature. The case of singular conditional distribution of the one-step-ahead state vector given the present state is considered. The bound is evaluated for three important examples: the recursive estimation of slowly varying parameters of an autoregressive process, tracking a slowly varying frequency of a single cisoid in noise, and tracking parameters of a sinusoidal frequency with sinusoidal phase modulation.
1,333 citations
TL;DR: It is proved that feasibility of the open-loop optimal control problem at time t = 0 implies asymptotic stability of the closed-loop system.
1,300 citations
TL;DR: The search for a piecewise quadratic Lyapunov function is formulated as a convex optimization problem in terms of linear matrix inequalities and the relation to frequency domain methods such as the circle and Popov criteria is explained.
Abstract: This paper presents a computational approach to stability analysis of nonlinear and hybrid systems. The search for a piecewise quadratic Lyapunov function is formulated as a convex optimization problem in terms of linear matrix inequalities. The relation to frequency domain methods such as the circle and Popov criteria is explained. Several examples are included to demonstrate the flexibility and power of the approach.
1,186 citations
TL;DR: An alternate method based on Fourier series which avoids meshing and which makes direct use of microstructure images is proposed, based on the exact expression of the Green function of a linear elastic and homogeneous comparison material.
1,170 citations
Book•
04 Jun 1998
TL;DR: In this article, the Navier-Stokes equations and the Euler equations are studied in the context of nonlinear partial differential equations (NPDE) and their applications in applied mathematics.
Abstract: One of the most challenging topics in applied mathematics over the past decades has been the developent of the theory of nonlinear partial differential equations. Many of the problems in mechanics, geometry, probability, etc lead to such equations when formulated in mathematical terms. However, despite a long history of contributions, there exists no central core theory, and the most important advances have come from the study of particular equations and classes of equations arising in specific applications. This two volume work forms a unique and rigorous treatise on various mathematical aspects of fluid mechanics models. These models consist of systems of nonlinear partial differential equations like the incompressible and compressible Navier-Stokes equations. The main emphasis in Volume 1 is on the mathematical analysis of incompressible models. After recalling the fundamental description of Newtonian fluids, an original and self-contained study of both the classical Navier-Stokes equations (including the inhomogenous case) and the Euler equations is given. Known results and many new results about the existence and regularity of solutions are presented with complete proofs. The discussion containts many interesting insights and remarks. The text highlights in particular the use of modern analytical tools and methods and also indicates many open problems.
1,149 citations
TL;DR: The THERMOCALC software as mentioned in this paper allows the calculation of a range of other types of phase diagrams involving solid solutions, such as P-T projections and compatibility diagrams, with the equations used for each equilibrium are the equilibrium relationships for an independent set of reactions between the endmembers of the phases in the equilibrium.
Abstract: Phase diagrams involving solid solutions are calculated by solving sets of non-linear equations. In calculating P–T projections and compatibility diagrams, the equations used for each equilibrium are the equilibrium relationships for an independent set of reactions between the end-members of the phases in the equilibrium. Invariant points and univariant lines in P–T projections can be calculated directly, as can coordinates in compatibility diagrams. In calculating P–T and T–x/P–x pseudosections – diagrams drawn for particular bulk compositions – the equilibrium relationship equations are augmented by mass balance equations. Lines in pseudosections, where the mode of one phase in the lower variance equilibrium is zero, and points, where the modes of two phases are zero, can then be calculated directly. The software, THERMOCALC, allows the calculation of these and a range of other types of phase diagram. Examples of phase diagrams and phase diagram movies, with instructions for their production, along with the THERMOCALC input and output files, and the MathematicaTM functions for assembling them, are presented in this paper, partly in hard copy and partly on the JMG web sites (http://www.gly.bris.ac.uk/www/jmg/jmg.html, or equivalent Australian or USA sites).
TL;DR: Some fundamental insights into observer design for the class of Lipschitz nonlinear systems are presented and a systematic computational algorithm is presented for obtaining the observer gain matrix so as to achieve the objective of asymptotic stability.
Abstract: This paper presents some fundamental insights into observer design for the class of Lipschitz nonlinear systems. The stability of the nonlinear observer for such systems is not determined purely by the eigenvalues of the linear stability matrix. The correct necessary and sufficient conditions on the stability matrix that ensure asymptotic stability of the observer are presented. These conditions are then reformulated to obtain a sufficient condition for stability in terms of the eigenvalues and the eigenvectors of the linear stability matrix. The eigenvalues have to be located sufficiently far out into the left half-plane, and the eigenvectors also have to be sufficiently well-conditioned for ensuring asymptotic stability. Based on these results, a systematic computational algorithm is then presented for obtaining the observer gain matrix so as to achieve the objective of asymptotic stability.
TL;DR: Phase-matched harmonic conversion of visible laser light into soft x-rays was demonstrated and the recently developed technique of guided-wave frequency conversion was used to upshift light from 800 nanometers to the range from 17 to 32 nanometers.
Abstract: Phase-matched harmonic conversion of visible laser light into soft x-rays was demonstrated. The recently developed technique of guided-wave frequency conversion was used to upshift light from 800 nanometers to the range from 17 to 32 nanometers. This process increased the coherent x-ray output by factors of 10(2) to 10(3) compared to the non-phase-matched case. This source uses a small-scale (sub-millijoule) high repetition-rate laser and will enable a wide variety of new experimental investigations in linear and nonlinear x-ray science.
Book•
31 Mar 1998
TL;DR: In this article, the authors present a method for solving linear Equations of the Form y(x) - xa K(x, t)y(t)dt = f(x).
Abstract: EXACT SOLUTIONS OF INTEGRAL EQUATIONS Linear Equations of the First Kind with Variable Limit of Integration Linear Equations of the Second Kind with Variable Limit of Integration Linear Equations of the First Kind with Constant Limits of Integration Linear Equations of the Second Kind with Constant Limits of Integration Nonlinear Equations of the First Kind with Variable Limit of Integration Nonlinear Equations of the Second Kind with Variable Limit of Integration Nonlinear Equations of the First Kind with Constant Limits of Integration Nonlinear Equations of the Second Kind with Constant Limits of Integration METHODS FOR SOLVING INTEGRAL EQUATIONS Main Definitions and Formulas: Integral Transforms Methods for Solving Linear Equations of the Form xa K(x, t)y(t)dt = f(x) Methods for Solving Linear Equations of the Form y(x) - xa K(x, t)y(t)dt = f(x) Methods for Solving Linear Equations of the Form xa K(x, t)y(t)dt = f(x) Methods for Solving Linear Equations of the Form y(x) - xa K(x, t)y(t)dt = f(x) Methods for Solving Singular Integral Equations of the First Kind Methods for Solving Complete Singular Integral Equations Methods for Solving Nonlinear Integral Equations Methods for Solving Multidimensional Mixed Integral Equations Application of Integral Equations for the Investigation of Differential Equations SUPPLEMENTS Elementary Functions and Their Properties Finite Sums and Infinite Series Tables of Indefinite Integrals Tables of Definite Integrals Tables of Laplace Transforms Tables of Inverse Laplace Transforms Tables of Fourier Cosine Transforms Tables of Fourier Sine Transforms Tables of Mellin Transforms Tables of Inverse Mellin Transforms Special Functions and Their Properties Some Notions of Functional Analysis References Index
TL;DR: In this paper, the in-plane phase-matching resonances are given by a nonlinear Bragg law and a related nonlinear Ewald construction, which can be used for multiple-beam second-harmonic generation (SHG), ring cavity SHG, or multiple wavelength frequency conversion.
Abstract: Nonlinear frequency conversion in 2D ${\ensuremath{\chi}}^{(2)}$ photonic crystals is theoretically studied. Such a crystal has a 2D periodic nonlinear susceptibility, and a linear susceptibility which is a function of the frequency, but constant in space. It is an in-plane generalization of 1D quasi-phase-matching structures and can be realized in periodic poled lithium niobate or in GaAs. An interesting property of these structures is that new phase-matching processes appear in the 2D plane as compared to the 1D case. It is shown that these in-plane phase-matching resonances are given by a nonlinear Bragg law, and a related nonlinear Ewald construction. Applications as multiple-beam second-harmonic generation (SHG), ring cavity SHG, or multiple wavelength frequency conversion are envisaged.
TL;DR: In this paper, the collapse of self-focusing waves described by the nonlinear Schrodinger (NLS) equation and the Zakharov equations in nonlinear optics and plasma turbulence is reviewed.
TL;DR: A modified adaptive backstepping design procedure is proposed for a class of nonlinear systems with three types of uncertainty: (i)unknown parameters; (ii)uncertain nonlinearities and (iii)unmodeled dynamics.
TL;DR: In this paper, the performance of magnetorheological dampers for seismic response reduction is examined and the results indicate that the MR damper is quite effective for structural response reduction over a wide class of seismic excitations.
Abstract: In this paper, the efficacy of magnetorheological (MR) dampers for seismic response reduction is examined. To investigate the performance of the MR damper, a series of experiments was conducted in which the MR damper is used in conjunction with a recently developed clipped-optimal control strategy to control a three-story test structure subjected to a one-dimensional ground excitation. The ability of the MR damper to reduce both peak responses, in a series of earthquake tests, and rms responses, in a series of broadband excitation tests, is shown. Additionally, because semi-active control systems are nonlinear, a variety of disturbance amplitudes are considered to investigate the performance of this control system over a variety of loading conditions. For each case, the results for three clipped-optimal control designs are presented and compared to the performance of two passive systems. The results indicate that the MR damper is quite effective for structural response reduction over a wide class of seismic excitations.
01 Jan 1998
TL;DR: In this article, the authors present a guided tour of robust stability and robust stability of time-delay systems, including polynomials, quasipolynomials, and nonlinear delay systems.
Abstract: Stability and robust stability of time-delay systems: A guided tour.- Convex directions for stable polynomials and quasipolynomials: A survey of recent results.- Delay-independent stability of linear neutral systems: A riccati equation approach.- Robust stability and stabilization of time-delay systems via integral quadratic constraint approach.- Graphical test for robust stability with distributed delayed feedback.- Numerics of the stability exponent and eigenvalue abscissas of a matrix delay system.- Moving averages for periodic delay differential and difference equations.- On rational stabilizing controllers for interval delay systems.- Stabilization of linear and nonlinear systems with time delay.- Nonlinear delay systems: Tools for a quantitative approach to stabilization.- Output feedback stabilization of linear time-delay systems.- Robust control of systems with a single input lag.- Robust guaranteed cost control for uncertain linear time-delay systems.- Local stabilization of continuous-time delay systems with bounded inputs.
TL;DR: Estimators are obtained which, using only a measured output, can asymptotically estimate to any desired accuracy the system state and the input.
Book•
01 Sep 1998
TL;DR: In this article, the authors present the fundamentals of global stabilization and optimal control of nonlinear systems with uncertain models, including deterministic disturbance attenuation, stochastic control, and adaptive control.
Abstract: From the Publisher:
This monograph presents the fundamentals of global stabilization and optimal control of nonlinear systems with uncertain models. It offers a unified view of deterministic disturbance attenuation, stochastic control, and adaptive control for nonlinear systems. The book addresses researchers in the areas of robust and adaptive nonlinear control, nonlinear H-infinity stochastic control, and other related areas of control and dynamical systems theory.
TL;DR: Based on the homogeneous balance method, a simple and efficient method for obtaining exact solutions of nonlinear partial differential equations is proposed in this paper, where some equations are investigated by this means and new solitary wave solutions or singular traveling wave solutions are found.
Book•
22 Oct 1998
TL;DR: In this paper, the Belousov-Zhabotinsky reaction was used to measure the rate constants with a Ruler, and the BZ waves were used to simulate the Briggs-Rauscher reaction.
Abstract: Part I: Overview 1 Introduction - A Bit of History 2 Fundamentals 3 Apparatus 4 Chemical Oscillations: Synthesis 5 Chemical Oscillations: Analysis 6 Waves and Patterns 7 Computational Tools Part II: Special Topics 8 Complex Oscillations and Chaos 9 Transport and External Field Effects 10 Delays and Differential Delay Equations 11 Polymer Systems 12 Coupled Oscillators 13 Biological Oscillators 14 Turing Patterns 15 Stirring and Mixing Effects Appendix I Demonstrations A11 The Briggs-Rauscher Reaction A12 The Belousov-Zhabotinsky Reaction A13 BZ Waves A14 A Propagating pH Front Appendix 2 Experiments for the Undergraduate Lab A21 Frontal Polymerization A22 Oscillations in the Homogeneous Belousov-Zhabotinsky Reaction A23 Unstirred BZ System: "Measuring Rate Constants with a Ruler" Bibliography
TL;DR: In this article, an optimal two-stage identification algorithm is presented for Hammerstein-Wiener systems, where two static nonlinear elements surround a linear block, and the algorithm is shown to be convergent in the absence of noise and convergent with probability one in the presence of white noise.
TL;DR: A systematic procedure is developed for designing global adaptive control of a class of nonlinear systems that possesses a triangular structure and can be of arbitrary dynamic order.
Abstract: Without a priori knowledge of the signs of the parameters called control directions (since they represent effectively the direction of motion under any control), a systematic procedure is developed for designing global adaptive control of a class of nonlinear systems. The class of systems possesses a triangular structure and can be of arbitrary dynamic order. No growth restrictions are imposed.
TL;DR: The approach is based on conceptual tools of predictive control and consists of adding to a primal compensated nonlinear system a reference governor which online handles the reference to be tracked, taking into account the current value of the state in order to satisfy the prescribed constraints.
Abstract: This paper addresses the problem of satisfying pointwise-in-time input and/or state hard constraints in nonlinear control systems. The approach is based on conceptual tools of predictive control and consists of adding to a primal compensated nonlinear system a reference governor. This is a discrete-time device which online handles the reference to be tracked, taking into account the current value of the state in order to satisfy the prescribed constraints. The resulting hybrid system is proved to fulfil the constraints as well as stability and tracking requirements.
TL;DR: In this article, a new iteration method is proposed to solve nonlinear problems with convolution product nonlinearities and the results reveal that the approximations obtained by the proposed method are uniformly valid for both small and large parameters in non-linear problems.
TL;DR: In this article, the authors studied the behavior of the nonlinear criticalp-heat equation and the related stationaryp-laplacian equation and showed that the behaviour depends on p. The results depend in general on the relation betweenλ and the best constant in Hardy's inequality.
TL;DR: In this article, a numerical solution for the 2 + 1 (long-shore and onshore propagation directions and time) nonlinear shallow-water wave equations, without friction factors or artificial viscosity is presented.
Abstract: A numerical solution for the 2 + 1 (long-shore and onshore propagation directions and time) nonlinear shallow-water wave equations, without friction factors or artificial viscosity is presented. The models use a splitting method to generate two 1 + 1 propagation problems, one in the onshore and the other in long-shore direction. Both are solved in characteristic form using the method of characteristics. A shoreline algorithm is implemented, which is the generalization of the earlier 1 + 1 algorithm used in the code VTCS-2. The model is validated using large-scale laboratory data from solitary wave experiments attacking a conical island. The method is applied then to model the 1993 Okushiri, Japan, the 1994 Kuril Island, Russia, and the 1996 Chimbote, Peru tsunamis. It is found that the model can reproduce correctly overland flow and even extreme events such as the 30-m runup and the 20-m/s inundation velocities inferred during field surveys. The results suggest that bathymetric and topographic resolution ...
TL;DR: In this paper, a meshless Galerkin finite element method (GFEM) based on Local Boundary Integral Equation (LBIE) has been proposed, which is quite general and easily applicable to non-homogeneous problems.
Abstract: The Galerkin finite element method (GFEM) owes its popularity to the local nature of nodal basis functions, i.e., the nodal basis function, when viewed globally, is non-zero only over a patch of elements connecting the node in question to its immediately neighboring nodes. The boundary element method (BEM), on the other hand, reduces the dimensionality of the problem by one, through involving the trial functions and their derivatives, only in the integrals over the global boundary of the domain; whereas, the GFEM involves the integration of the “energy” corresponding to the trial function over a patch of elements immediately surrounding the node. The GFEM leads to banded, sparse and symmetric matrices; the BEM based on the global boundary integral equation (GBIE) leads to full and unsymmetrical matrices. Because of the seemingly insurmountable difficulties associated with the automatic generation of element-meshes in GFEM, especially for 3-D problems, there has been a considerable interest in element free Galerkin methods (EFGM) in recent literature. However, the EFGMs still involve domain integrals over shadow elements and lead to difficulties in enforcing essential boundary conditions and in treating nonlinear problems. The object of the present paper is to present a new method that combines the advantageous features of all the three methods: GFEM, BEM and EFGM. It is a meshless method. It involves only boundary integration, however, over a local boundary centered at the node in question; it poses no difficulties in satisfying essential boundary conditions; it leads to banded and sparse system matrices; it uses the moving least squares (MLS) approximations. The method is based on a Local Boundary Integral Equation (LBIE) approach, which is quite general and easily applicable to nonlinear problems, and non-homogeneous domains. The concept of a “companion solution” is introduced so that the LBIE for the value of trial solution at the source point, inside the domain Ω of the given problem, involves only the trial function in the integral over the local boundary Ω
s
of a sub-domain Ω
s
centered at the node in question. This is in contrast to the traditional GBIE which involves the trial function as well as its gradient over the global boundary Γ of Ω. For source points that lie on Γ, the integrals over Ω
s
involve, on the other hand, both the trial function and its gradient. It is shown that the satisfaction of the essential as well as natural boundary conditions is quite simple and algorithmically very efficient in the present LBIE approach. In the example problems dealing with Laplace and Poisson's equations, high rates of convergence for the Sobolev norms ||·||0 and ||·||1 have been found. In essence, the present EF-LBIE (Element Free-Local Boundary Integral Equation) approach is found to be a simple, efficient, and attractive alternative to the EFG methods that have been extensively popularized in recent literature.