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Showing papers on "Nonlinear system published in 2005"


Journal ArticleDOI
TL;DR: The current capabilities of the codes, along with some of the algorithms and heuristics used to achieve efficiency and robustness, are described.
Abstract: SUNDIALS is a suite of advanced computational codes for solving large-scale problems that can be modeled as a system of nonlinear algebraic equations, or as initial-value problems in ordinary differential or differential-algebraic equations. The basic versions of these codes are called KINSOL, CVODE, and IDA, respectively. The codes are written in ANSI standard C and are suitable for either serial or parallel machine environments. Common and notable features of these codes include inexact Newton-Krylov methods for solving large-scale nonlinear systems; linear multistep methods for time-dependent problems; a highly modular structure to allow incorporation of different preconditioning and/or linear solver methods; and clear interfaces allowing for users to provide their own data structures underneath the solvers. We describe the current capabilities of the codes, along with some of the algorithms and heuristics used to achieve efficiency and robustness. We also describe how the codes stem from previous and widely used Fortran 77 solvers, and how the codes have been augmented with forward and adjoint methods for carrying out first-order sensitivity analysis with respect to model parameters or initial conditions.

2,124 citations


MonographDOI
01 Feb 2005
TL;DR: The theory of linear difference equations applied to population growth and the applications of nonlinear difference equations to population biology are explained.
Abstract: Part I. Discrete Process in Biology: 1. The theory of linear difference equations applied to population growth 2. Nonlinear difference equations 3. Applications of nonlinear difference equations to population biology Part II. Continuous Processes and Ordinary Differential Equations: 4. An introduction to continuous models 5. Phase-plane methods and qualitative solutions 6. Applications of continuous models to population dynamics 7. Models for molecular events 8. Limit cycles, oscillations, and excitable systems Part III. Spatially Distributed Systems and Partial Differential Equation Models: 9. An introduction to partial differential equations and diffusion in biological settings 10. Partial differential equation models in biology 11. Models for development and pattern formation in biological systems Selected answers Author index Subject index.

1,925 citations


Journal ArticleDOI
TL;DR: An algorithm for computing the set of reachable states of a continuous dynamic game based on a proof that the reachable set is the zero sublevel set of the viscosity solution of a particular time-dependent Hamilton-Jacobi-Isaacs partial differential equation.
Abstract: We describe and implement an algorithm for computing the set of reachable states of a continuous dynamic game. The algorithm is based on a proof that the reachable set is the zero sublevel set of the viscosity solution of a particular time-dependent Hamilton-Jacobi-Isaacs partial differential equation. While alternative techniques for computing the reachable set have been proposed, the differential game formulation allows treatment of nonlinear systems with inputs and uncertain parameters. Because the time-dependent equation's solution is continuous and defined throughout the state space, methods from the level set literature can be used to generate more accurate approximations than are possible for formulations with potentially discontinuous solutions. A numerical implementation of our formulation is described and has been released on the web. Its correctness is verified through a two vehicle, three dimensional collision avoidance example for which an analytic solution is available.

1,107 citations


Journal ArticleDOI
TL;DR: A backstepping based control design for a class of nonlinear systems in strict-feedback form with arbitrary uncertainty is developed and is able to eliminate the problem of "explosion of complexity" inherent in the existing method.
Abstract: The dynamic surface control (DSC) technique was developed recently by Swaroop et al. This technique simplified the backstepping design for the control of nonlinear systems in strict-feedback form by overcoming the problem of "explosion of complexity." It was later extended to adaptive backstepping design for nonlinear systems with linearly parameterized uncertainty. In this paper, by incorporating this design technique into a neural network based adaptive control design framework, we have developed a backstepping based control design for a class of nonlinear systems in strict-feedback form with arbitrary uncertainty. Our development is able to eliminate the problem of "explosion of complexity" inherent in the existing method. In addition, a stability analysis is given which shows that our control law can guarantee the uniformly ultimate boundedness of the solution of the closed-loop system, and make the tracking error arbitrarily small.

1,079 citations


Journal ArticleDOI
TL;DR: It is shown that the constrained optimal control law has the largest region of asymptotic stability (RAS) and the result is a nearly optimal constrained state feedback controller that has been tuned a priori off-line.

1,045 citations


Journal ArticleDOI
TL;DR: It is proved that global finite-time stabilizability of uncertain nonlinear systems that are dominated by a lower-triangular system can be achieved by Holder continuous state feedback.

982 citations


Journal ArticleDOI
TL;DR: A modification of the exponential time-differencing fourth-order Runge--Kutta method for solving stiff nonlinear PDEs is presented that solves the problem of numerical instability in the scheme as proposed by Cox and Matthews and generalizes the method to nondiagonal operators.
Abstract: A modification of the exponential time-differencing fourth-order Runge--Kutta method for solving stiff nonlinear PDEs is presented that solves the problem of numerical instability in the scheme as proposed by Cox and Matthews and generalizes the method to nondiagonal operators. A comparison is made of the performance of this modified exponential time-differencing (ETD) scheme against the competing methods of implicit-explicit differencing, integrating factors, time-splitting, and Fornberg and Driscoll's "sliders" for the KdV, Kuramoto--Sivashinsky, Burgers, and Allen--Cahn equations in one space dimension. Implementation of the method is illustrated by short MATLAB programs for two of the equations. It is found that for these applications with fixed time steps, the modified ETD scheme is the best.

921 citations


Journal ArticleDOI
TL;DR: In this article, a different approach is adopted, and proper orthogonal decomposition is considered, and modes extracted from the decomposition may serve two purposes, namely order reduction by projecting high-dimensional data into a lower-dimensional space and feature extraction by revealing relevant but unexpected structure hidden in the data.
Abstract: Modal analysis is used extensively for understanding the dynamic behavior of structures. However, a major concern for structural dynamicists is that its validity is limited to linear structures. New developments have been proposed in order to examine nonlinear systems, among which the theory based on nonlinear normal modes is indubitably the most appealing. In this paper, a different approach is adopted, and proper orthogonal decomposition is considered. The modes extracted from the decomposition may serve two purposes, namely order reduction by projecting high-dimensional data into a lower-dimensional space and feature extraction by revealing relevant but unexpected structure hidden in the data. The utility of the method for dynamic characterization and order reduction of linear and nonlinear mechanical systems is demonstrated in this study.

838 citations


Book
01 Jan 2005
TL;DR: In this paper, the authors present a system of equations for evolving pseudomonotone or weakly continuous mappings with set-valued mappings, and a set of auxiliary tools.
Abstract: Preface.- Preface to the 2nd edition.- Notational conventions.- 1 Preliminary general material.- I Steady-state problems.- 2 Pseudomonotone or weakly continuous mappings.- 3 Accretive mappings.- 4 Potential problems: smooth case.- 5 Nonsmooth problems variational inequalities.- 6. Systems of equations: particular examples.- II Evolution problems.- 7 Special auxiliary tools.- 8 Evolution by pseudomonotone or weakly continuous mappings.- 9 Evolution governed by accretive mappings.- 10 Evolution governed by certain set-valued mappings.- 11 Doubly-nonlinear problems.- 12 Systems of equations: particular examples.- References.- Index.

740 citations


Proceedings ArticleDOI
08 Jun 2005
TL;DR: Todorov et al. as discussed by the authors presented an iterative linear-quadratic-Gaussian method for locally-optimal feedback control of nonlinear stochastic systems subject to control constraints.
Abstract: We present an iterative linear-quadratic-Gaussian method for locally-optimal feedback control of nonlinear stochastic systems subject to control constraints. Previously, similar methods have been restricted to deterministic unconstrained problems with quadratic costs. The new method constructs an affine feedback control law, obtained by minimizing a novel quadratic approximation to the optimal cost-to-go function. Global convergence is guaranteed through a Levenberg-Marquardt method; convergence in the vicinity of a local minimum is quadratic. Performance is illustrated on a limited-torque inverted pendulum problem, as well as a complex biomechanical control problem involving a stochastic model of the human arm, with 10 state dimensions and 6 muscle actuators. A Matlab implementation of the new algorithm is availabe at www.cogsci.ucsd.edu//spl sim/todorov.

730 citations


Journal ArticleDOI
TL;DR: In this paper, the dynamics of open flows are considered as a superposition of linear or nonlinear instability waves that behave at each streamwise station as if the flow were homogeneous in the streamwise direction.
Abstract: The objective of this review is to critically assess the different approaches developed in recent years to understand the dynamics of open flows such as mixing layers, jets, wakes, separation bubbles, boundary layers, and so on. These complex flows develop in extended domains in which fluid particles are continuously advected downstream. They behave either as noise amplifiers or as oscillators, both of which exhibit strong nonlinearities (Huerre & Monkewitz 1990). The local approach is inherently weakly nonparallel and it assumes that the basic flow varies on a long length scale compared to the wavelength of the instability waves. The dynamics of the flow is then considered as a superposition of linear or nonlinear instability waves that, at leading order, behave at each streamwise station as if the flow were homogeneous in the streamwise direction. In the fully global context, the basic flow and the instabilities do not have to be characterized by widely separated length scales, and the dynamics is then viewed as the result of the interactions between Global modes living in the entire physical domain with the streamwise direction as an eigendirection. This second approach is more and more resorted to as a result of increased computational capability. The earlier review of Huerre & Monkewitz (1990) emphasized how local linear theory can account for the noise amplifier behavior as well as for the onset of a Global mode. The present survey first adopts the opposite point of view by demonstrating how fully global theory accounts for the noise amplifier behavior of open flows. From such a perspective, there is strong emphasis on the very peculiar nonorthogonality of linear Global modes, which in turn allows a novel interpretation of recent numerical simulations and experimental observations. The nonorthogonality of linear Global modes also imposes severe constraints on the extension of linear global theory to the fully nonlinear regime. When the flow is weakly nonparallel, this limitation is so severe that the linear Global mode theory is of little help. It is then much more appropriate to develop a fully nonlinear formulation involving the presence of a front separating the base state region from the bifurcated state region.

Posted Content
TL;DR: In this article, an introduction to the nonlinear equations for completely symmetric bosonic higher spin gauge fields in anti de Sitter space of any dimension is provided, and some related issues such as the MacDowell-Mansouri-Stelle-West formulation of gravity, unfolded formulation of dynamical systems in terms of free differential algebras and Young tableaux symmetry properties in terms with Howe dual algesbras are discussed.
Abstract: In this article, an introduction to the nonlinear equations for completely symmetric bosonic higher spin gauge fields in anti de Sitter space of any dimension is provided. To make the presentation self-contained we explain in detail some related issues such as the MacDowell-Mansouri-Stelle-West formulation of gravity, unfolded formulation of dynamical systems in terms of free differential algebras and Young tableaux symmetry properties in terms of Howe dual algebras.

Journal ArticleDOI
TL;DR: In this paper, the mathematical setting of stationary systems modelled by elliptic partial differential equations with stochastic coefficients (random fields) is investigated and stability with respect to stability.

Journal ArticleDOI
TL;DR: In this paper, a new method is presented to look for exact solutions of nonlinear differential equations by using the general solutions of the simplest nonlinear equations and taking into consideration all possible singularities of equation studied.
Abstract: New method is presented to look for exact solutions of nonlinear differential equations. Two basic ideas are at the heart of our approach. One of them is to use the general solutions of the simplest nonlinear differential equations. Another idea is to take into consideration all possible singularities of equation studied. Application of our approach to search exact solutions of nonlinear differential equations is discussed in detail. The method is used to look for exact solutions of the Kuramoto–Sivashinsky equation and the equation for description of nonlinear waves in a convective fluid. New exact solitary and periodic waves of these equations are given.

Journal ArticleDOI
TL;DR: A method is proposed for designing an antiwindup gain that maximizes an estimate of the basin of attraction of the closed-loop system that can be modeled by a linear system with a deadzone nonlinearity.
Abstract: This note addresses the design of antiwindup gains for obtaining larger regions of stability for linear systems with saturating inputs. Considering that a linear dynamic output feedback has been designed to stabilize the linear system (without saturation), a method is proposed for designing an antiwindup gain that maximizes an estimate of the basin of attraction of the closed-loop system. It is shown that the closed-loop system obtained from the controller plus the antiwindup gain can be modeled by a linear system with a deadzone nonlinearity. A modified sector condition is then used to obtain stability conditions based on quadratic Lyapunov functions. Differently from previous works these conditions are directly in linear matrix inequality form. Some numerical examples illustrate the effectiveness of the proposed design technique when compared with the previous ones.

Journal ArticleDOI
TL;DR: The proposed dependence measures provide a natural framework for a limit theory for stationary processes and present limit theorems for partial sums, empirical processes, and kernel density estimates under conditions with quite simple forms.
Abstract: Based on the nonlinear system theory, we introduce previously undescribed dependence measures for stationary causal processes. Our physical and predictive dependence measures quantify the degree of dependence of outputs on inputs in physical systems. The proposed dependence measures provide a natural framework for a limit theory for stationary processes. In particular, under conditions with quite simple forms, we present limit theorems for partial sums, empirical processes, and kernel density estimates. The conditions are mild and easily verifiable because they are directly related to the data-generating mechanisms.

Journal ArticleDOI
TL;DR: The dynamic model of a wheeled inverted pendulum (e.g., Segway, Quasimoro, and Joe) is analyzed from a controllability and feedback linearizability point of view and two novel controllers are designed.
Abstract: In this paper, the dynamic model of a wheeled inverted pendulum (eg, Segway, Quasimoro, and Joe) is analyzed from a controllability and feedback linearizability point of view First, a dynamic model of this underactuated system is derived with respect to the wheel motor torques as inputs while taking the nonholonomic no-slip constraints into considerations This model is compared with the previous models derived for similar systems The strong accessibility condition is checked and the maximum relative degree of the system is found Based on this result, a partial feedback linearization of the system is obtained and the internal dynamics equations are isolated The resulting equations are then used to design two novel controllers The first one is a two-level velocity controller for tracking vehicle orientation and heading speed set-points, while controlling the vehicle pitch (pendulum angle from the vertical) within a specified range The second controller is also a two-level controller which stabilizes the vehicle's position to the desired point, while again keeping the pitch bounded between specified limits Simulation results are provided to show the efficacy of the controllers using realistic data

Journal ArticleDOI
TL;DR: This paper is concerned with sliding mode control for uncertain stochastic systems with time-varying delay, and an integral sliding surface is first constructed, and a sufficient condition is derived to guarantee the global Stochastic stability of the stoChastic dynamics in the specified switching surface for all admissible uncertainties.

Book
01 Jan 2005
TL;DR: In this article, a sampling of design methodologies Linear and nonlinear potential shaping for stabilization and tracking for fully actuated systems Stabilization and tracking using oscillatory controls Motion planning for underactuated systems Appendices Timedependent vector fields Some proofs.
Abstract: Part I: Modeling of mechanical systems Introductory examples and problems Linear and multilinear algebra Differential geometry Simple mechanical control systems Lie groups, systems on groups, and symmetries.- Part II: Analysis of mechanical control systems Stability Controllability Low-order controllability and kinematic reduction Perturbation analysis.- Part III: A sampling of design methodologies Linear and nonlinear potential shaping for stabilization Stabilization and tracking for fully actuated systems Stabilization and tracking using oscillatory controls Motion planning for underactuated systems Appendices Time-dependent vector fields Some proofs.

Journal ArticleDOI
TL;DR: In this paper, the authors present an experimental study on wave propagation in highly nonlocal optically nonlinear media, for which far-away boundary conditions significantly affect the evolution of localized beams.
Abstract: We present an experimental study on wave propagation in highly nonlocal optically nonlinear media, for which far-away boundary conditions significantly affect the evolution of localized beams. As an example, we set the boundary conditions to be anisotropic and demonstrate the first experimental observation of coherent elliptic solitons. Furthermore, exploiting the natural ability of such nonlinearities to eliminate azimuthal instabilities, we perform the first observation of stable vortex-ring solitons. These features of highly nonlocal nonlinearities affected by far-away boundary conditions open new directions in nonlinear science by facilitating remote control over soliton propagation.

Journal ArticleDOI
TL;DR: In this article, the authors studied the linearized Navier-stokes (LNS) equations in channel flows from an input-output point of view by analysing their spatio-temporal frequency responses.
Abstract: We study the linearized Navier–Stokes (LNS) equations in channel flows from an input–output point of view by analysing their spatio-temporal frequency responses. Spatially distributed and temporally varying body force fields are considered as inputs, and components of the resulting velocity fields are considered as outputs into these equations. We show how the roles of Tollmien–Schlichting (TS) waves, oblique waves, and streamwise vortices and streaks in subcritical transition can be explained as input–output resonances of the spatio-temporal frequency responses. On the one hand, we demonstrate the effectiveness of input field components, and on the other, the energy content of velocity perturbation components. We establish that wall-normal and spanwise forces have much stronger influence on the velocity field than streamwise force, and that the impact of these forces is most powerful on the streamwise velocity component. We show this using the relative scaling of the different input–output system components with the Reynolds number. We further demonstrate that for the streamwise constant perturbations, the spanwise force localized near the lower wall has, by far, the strongest effect on the evolution of the velocity field. In this paper, we analyse the dynamical properties of the Navier–Stokes (NS) equations with spatially distributed and temporally varying body force fields. These fields are considered as inputs, and different combinations of the resulting velocity fields are considered as outputs. This input–output analysis can in principle be done in any geometry and for the full nonlinear NS equations. In such generality, however, it is difficult to obtain useful results. We therefore concentrate on the geometry of channel flows, and the input–output dynamics of the linearized Navier–Stokes (LNS)

Book
01 Jan 2005
TL;DR: This paper presents a meta-modelling framework called Bayesian Modeling for Generalized Linear Models and Its Extensions (GLS) and some examples of this model include GANs, Bayesian models, and many others.
Abstract: The Art of Modeling.- Linear Models and Regression.- Nonlinear Models and Regression.- Variance Models, Weighting, and Transformations.- Case Studies in Linear and Nonlinear Modeling.- Linear Mixed Effects Models.- Nonlinear Mixed Effects Models: Theory.- Nonlinear Mixed Effects Models: Practical Issues.- Nonlinear Mixed Effects Models: Case Studies.- Appendix.- References.- Index.

Journal ArticleDOI
TL;DR: An approach for the efficient solution of motion-planning problems for time-invariant dynamical control systems with symmetries, such as mobile robots and autonomous vehicles, under a variety of differential and algebraic constraints on the state and on the control input.
Abstract: In this paper, we introduce an approach for the efficient solution of motion-planning problems for time-invariant dynamical control systems with symmetries, such as mobile robots and autonomous vehicles, under a variety of differential and algebraic constraints on the state and on the control inputs. Motion plans are described as the concatenation of a number of well-defined motion primitives, selected from a finite library. Rules for the concatenation of primitives are given in the form of a regular language, defined through a finite-state machine called a Maneuver Automaton. We analyze the reachability properties of the language, and present algorithms for the solution of a class of motion-planning problems. In particular, it is shown that the solution of steering problems for nonlinear dynamical systems with symmetries and invariant constraints can be reduced to the solution of a sequence of kinematic inversion problems. A detailed example of the application of the proposed approach to motion planning for a small aerobatic helicopter is presented.

Journal ArticleDOI
TL;DR: This work presents a new algorithm based on the foraging behavior of E. coli bacteria in the authors' intestine to estimate the harmonic components present in power system voltage/current waveforms, presenting the hybrid method.
Abstract: Harmonic estimation for a signal distorted with additive noise has been an area of interest for researchers in many disciplines of science and engineering. This work presents a new algorithm based on the foraging behavior of E. coli bacteria in our intestine to estimate the harmonic components present in power system voltage/current waveforms. The basic foraging strategy is made adaptive, through a Takagi-Sugeno fuzzy scheme, depending on the operating condition to make the convergence faster. Besides, the harmonic estimation is linear in amplitude and nonlinear in phase. As the proposed algorithm does not rely on Newton-like gradient descent methods, this is used for phase estimation whereas the linear least square scheme estimates the amplitude, thereby presenting the hybrid method. The improvement in %error, as well as the processing time compared with the conventional discrete Fourier transform and genetic algorithm method is demonstrated in this paper. Besides, the performance is quite acceptable even in the presence of decaying dc component as well as to change in amplitude and phase angle of harmonic components.

Journal ArticleDOI
TL;DR: A new fault detection and identification method based on kernel principal component analysis (PCA) that uses kernel functions, which is a challenging problem in nonlinear PCA, is formulated based on a robust reconstruction error calculation.

Journal ArticleDOI
TL;DR: In this article, the authors present a modern approach to the theoretical and experimental study of complex nonlinear behavior of a semiconductor laser with optical injection-an example of a widely applied and technologically relevant forced nonlinear oscillator, and show that careful bifurcation analysis of a rate equation model yields a deeper understanding of already studied physical phenomena, and discovery of new dynamical effects, such as multipulse excitability.

Journal ArticleDOI
TL;DR: Using switched Lyapunov functions, some new general criteria for exponential stability and asymptotic stability with arbitrary and conditioned impulsive switching are established and a new hybrid impulsive and switching control strategy for nonlinear systems is developed.
Abstract: In this note, a new class of hybrid impulsive and switching models is introduced and their asymptotic stability properties are investigated. Using switched Lyapunov functions, some new general criteria for exponential stability and asymptotic stability with arbitrary and conditioned impulsive switching are established. In addition, a new hybrid impulsive and switching control strategy for nonlinear systems is developed. A typical example, the unified chaotic system, is given to illustrate the theoretical results.

Journal ArticleDOI
TL;DR: Two identification algorithms are developed for Hammerstein nonlinear systems with memoryless nonlinear blocks and linear dynamical blocks described by ARMAX/CARMA models to replace unmeasurable noise terms in the information vectors by their estimates, and to compute the noise estimates based on the obtained parameter estimates.

Book
01 Jan 2005
TL;DR: In this paper, the authors provide an introduction to the theory of nonlinear Fokker-planck equations and highlight what systems described by these equations have in common with synergetic many-body systems.
Abstract: Nonlinear Fokker–Planck equations have found applications in various fields such as plasma physics, surface physics, astrophysics, the physics of polymer fluids and particle beams, nonlinear hydrodynamics, theory of electronic circuitry and laser arrays, engineering, biophysics, population dynamics, human movement sciences, neurophysics, psychology and marketing. In spite of the diversity of these research fields, many phenomena addressed therein have a fundamental physical mechanism in common. They arise due to cooperative interactions between the subsystems of many-body systems. These cooperative interactions result in a reduction of the large number of degrees of freedom of many-body systems and, in doing so, bind the subunits of manybody systems by means of self-organization into synergetic entities. These synergetic many-body systems admit low dimensional descriptions in terms of nonlinear Fokker–Planck equations that capture and uncover the essential dynamics underlying the observed phenomena. The phenomena that will be addressed in this book range from equilibrium and nonequilibrium phase transitions and the multistability of systems to the emergence of power law and cut-off distributions and the distortion of Boltzmann distributions. We will study possible asymptotic behaviors of systems such as the approach to stationary distributions and the emergence of nonstationary traveling wave distributions. We will be concerned with normal and anomalous diffusion and we will examine how correlation functions evolve with time in these kinds of synergetic systems. We will discuss a Fokker–Planck approach to quantum statistics, linear nonequilibrium thermodynamics, and generalized extensive and nonextensive thermostatistics. The aim of this book is to provide an introduction to the theory of nonlinear Fokker–Planck equations and to highlight what systems described by nonlinear Fokker–Planck equations have in common. Theoretical considerations and concepts will be illustrated by various examples and applications. Due to the ramifications of the theory of nonlinear Fokker–Planck equations in various scientific fields, this book is designed for graduate students and researchers in physics and related fields such as biology, neurophysics, human movement sciences, and psychology. I hope that this book will make graduate students interested in the topic of nonlinear Fokker–Planck equa tions and will make researchers aware of the connections between the different areas in which nonlinear Fokker–Planck equations have been applied so far.

Journal ArticleDOI
TL;DR: It is demonstrated that global asymptotic stabilization is possible if a suitable relationship holds between the number of values taken by the encoder, the sampling period, and a system parameter, provided that a feedback law achieving input-to-state stability with respect to measurement errors can be found.
Abstract: This note is concerned with the problem of stabilizing a nonlinear continuous-time system by using sampled encoded measurements of the state. We demonstrate that global asymptotic stabilization is possible if a suitable relationship holds between the number of values taken by the encoder, the sampling period, and a system parameter, provided that a feedback law achieving input-to-state stability with respect to measurement errors can be found. The issue of relaxing the latter condition is also discussed.