Showing papers on "Nonlinear system published in 1994"

Linear Matrix Inequalities in System and Control Theory

Abstract: Preface 1. Introduction Overview A Brief History of LMIs in Control Theory Notes on the Style of the Book Origin of the Book 2. Some Standard Problems Involving LMIs. Linear Matrix Inequalities Some Standard Problems Ellipsoid Algorithm Interior-Point Methods Strict and Nonstrict LMIs Miscellaneous Results on Matrix Inequalities Some LMI Problems with Analytic Solutions 3. Some Matrix Problems. Minimizing Condition Number by Scaling Minimizing Condition Number of a Positive-Definite Matrix Minimizing Norm by Scaling Rescaling a Matrix Positive-Definite Matrix Completion Problems Quadratic Approximation of a Polytopic Norm Ellipsoidal Approximation 4. Linear Differential Inclusions. Differential Inclusions Some Specific LDIs Nonlinear System Analysis via LDIs 5. Analysis of LDIs: State Properties. Quadratic Stability Invariant Ellipsoids 6. Analysis of LDIs: Input/Output Properties. Input-to-State Properties State-to-Output Properties Input-to-Output Properties 7. State-Feedback Synthesis for LDIs. Static State-Feedback Controllers State Properties Input-to-State Properties State-to-Output Properties Input-to-Output Properties Observer-Based Controllers for Nonlinear Systems 8. Lure and Multiplier Methods. Analysis of Lure Systems Integral Quadratic Constraints Multipliers for Systems with Unknown Parameters 9. Systems with Multiplicative Noise. Analysis of Systems with Multiplicative Noise State-Feedback Synthesis 10. Miscellaneous Problems. Optimization over an Affine Family of Linear Systems Analysis of Systems with LTI Perturbations Positive Orthant Stabilizability Linear Systems with Delays Interpolation Problems The Inverse Problem of Optimal Control System Realization Problems Multi-Criterion LQG Nonconvex Multi-Criterion Quadratic Problems Notation List of Acronyms Bibliography Index.

Nonlinear dynamics and Chaos

Abstract: We explore several basic aspects of chaos, chaotic systems, and non-linear dynamics through three different setups. The logistic map, a canonical one-dimensional system exhibiting surprisingly complex and aperiodic behavior, is modeled as a function of its chaotic parameter. We consider maps of its phase space, and the progression through period-doubling bifurcations to the onset of chaos. The Feigenbaum ratio of successive bifurcation periods is estimated at 4.674, in good agreement with the accepted value. The Liapunov exponent, governing the exponential growth of small perturbations in chaotic systems, is calculated and its fractal structure compared to the corresponding bifurcation diagram for the logistic map. Using a non-linear p-n junction circuit we analyze the return maps and power spectrums of the resulting time series at various types of system behavior. Similarly, an electronic analog to a ball bouncing on a vertically driven table provides insight into real-world applications of chaotic motion. For both systems we calculate the fractal information dimension and compare with theoretical behavior for dissipative versus Hamiltonian systems. Subject headings: non-linear dynamics; non-linear dynamical systems; fractal dimension; chaos; strange attractors; logistic map

Some simple chaotic flows.

Abstract: A systematic examination of general three-dimensional autonomous ordinary differential equations with quadratic nonlinearities has uncovered 19 distinct simple examples of chaotic flows with either five terms and two nonlinearities or six terms and one nonlinearity. The properties of these systems are described, including their critical points, Lyapunov exponents, and fractal dimensions.

Practical Bifurcation and Stability Analysis

Abstract: and Prerequisites.- Basic Nonlinear Phenomena.- Applications and Extensions.- Principles of Continuation.- Calculation of the Branching Behavior of Nonlinear Equations.- Calculating Branching Behavior of Boundary-Value Problems.- Stability of Periodic Solutions.- Qualitative Instruments.- Chaos.

A general result on the stabilization of linear systems using bounded controls

Abstract: We present two constructions of controllers that globally stabilize linear systems subject to control saturation. We allow essentially arbitrary saturation functions. The only conditions imposed on the system are the obvious necessary ones, namely that no eigenvalues of the uncontrolled system have positive real part and that the standard stabilizability rank condition hold. One of the constructions is in terms of a "neural-network type" one-hidden layer architecture, while the other one is in terms of cascades of linear maps and saturations. >

Algebro-geometric approach to nonlinear integrable equations

Abstract: A brief but self-contained exposition of the basics of Riemann surfaces and theta functions prepares the reader for the main subject of this text, namely the application of these theories to solving non-linear integrable equations for various physical systems. Physicists and engineers involved in studying solitons, phase transitions or dynamical (gyroscopic) systems, and mathematicians with some background in algebraic geometry and Abelian and automorphic functions, are the targeted audience. This book is suitable for use as a supplementary text to a course in mathematical physics. The authors introduce Riemann surfaces and theta functions as mathematical methods used to analyzse solitons, dynamical systems, phase transitions, etc, and to obtain the solutions of the related non-linear integrable equations.

Hyperbolic conservation laws with stiff relaxation terms and entropy

Abstract: We study the limiting behavior of systems of hyperbolic conservation laws with stiff relaxation terms. Reduced systems, inviscid and viscous local conservation laws, and weakly nonlinear limits are derived through asymptotic expansions. An entropy condition is introduced for N × N systems that ensures the hyperbolicity of the reduced inviscid system. The resulting characteristic speeds are shown to be interlaced with those of the original system. Moreover, the first correction to the reduced system is shown to be dissipative. A partial converse is proved for 2 × 2 systems. This structure is then applied to study the convergence to the reduced dynamics for the 2 × 2 case. © 1994 John Wiley & Sons, Inc.

Chaos and physiology: deterministic chaos in excitable cell assemblies

Abstract: In this review we examined the emerging science of deterministic chaos (nonlinear systems theory) and its application to selected physiological systems. Although many of the popular images of fractals represent fascination and beauty that by analogy corresponds to nature as we see it, the question remains as to its ultimate meaning for physiological processes. It was our intent to help clarify this somewhat popular, somewhat obscure area of nonlinear dynamics in the context of an ever-changing procedural base. We examined not only the basic concepts of chaos, but also its applications ranging from observations in single cells to the complexity of the EEG. We have not suggested that nonlinear dynamics will answer all of our questions; however, we did attempt to illustrate ways in which this approach may help us to answer new questions and to rearticulate old ones. Chaos is revolutionary in that the overall approach requires us to adopt a different frame of reference which, at times, may move us away from previous concerns and methods of data analysis. In sections I-IV, we summarized the nonlinear dynamics approach and described its application to physiology and neural systems. First, we presented a general overview of the application of nonlinear dynamical techniques to neural systems. We discussed the manner in which even apparently simple deterministic systems can behave in an unpredictable manner. Second, we described the principles of nonlinear dynamical systems including the derived analytical techniques. We now see a variety of procedures for delineating whether frenetic chaotic behavior results from a nonlinear dynamical system with a few degrees of freedom, or whether it is caused by an infinite number of variables, i.e., noise. Third, we approached the applications of nonlinear procedures to the cardiovascular systems and to the neurosciences. In terms of time series, we described initial studies which applied the now "traditional" measures of dimensionality (e.g., based on the algorithm by Grassberger and Procaccia) and information change (e.g., Lyapunov exponents). Examples include our own work and that of Pritchard et al., demonstrating that the dynamics of neural mass activity reflect psychopathological states. Today, however, the trend has expanded to include the use of surrogate data and statistical null hypotheses testing to examine whether a given time series can be considered different from that of white or colored noise (cf. Ref. 262). One of the most important potential applications is that of quantifying changes in nonlinear dynamics to predict future states of the system.(ABSTRACT TRUNCATED AT 400 WORDS)

Oscillation Theory for Functional Differential Equations

Abstract: Preliminaries oscillations of first order delay differential equations oscillation of first order neutral differential equations oscillation and nonoscillation of second order differential equations with deviating arguments oscillation of higher order neutral differential equations oscillation of systems of neutral differential equations boundary value problems for second order functional differential equations.

Practical bifurcation and stability analysis : from equilibrium to chaos

Abstract: Basic Nonlinear Phenomena Practical Problems Principles of Continuation Calculation of the Branching Behaviour of Nonlinear Equations Calculating Branching Behaviour of Boundary-value Problems Stability of Periodic Solutions Qualitative Instruments Chaos.

Exponential Stability of Stochastic Differential Equations

Abstract: This unique, self-contained reference presents a systematic study of current developments in stochastic differential delay equations driven by nonlinear integrators - detailing various exponential stabilities for stochastic differential equations and large-scale systems.

Observer design for a class of nonlinear systems

Abstract: A viable design methodology to construct observers for a class of nonlinear systems is developed. The proposed technique is based on the off-line solution of a Riccati equation, and can be solved using commercially available software packages. For globally valid results, the class of systems considered is characterized by globally Lipschitz nonlinearities. Local results relax this assumption to only a local requirement. For a more general description of nonlinear systems, the methodology yields approximate observers, locally. The proposed theory is used to design an observer for a single-link flexible joint robot. This observer estimates the robot link variables based on the joint measurements.

A novel two-grid method for semilinear elliptic equations

Abstract: A new finite element discretization technique based on two (coarse and fine) subspaces is presented for a semilinear elliptic boundary value problem The solution of a nonlinear system on the fine space is reduced to the solution of two small (one linear and one nonlinear) systems on the coarse space and a linear system on the fine space It is shown, both theoretically and numerically, that the coarse space can be extremely coarse and still achieve asymptotically optimal approximation As a result, the numerical solution of such a nonlinear equation is not significantly more expensive than the solution of one single linearized equation

Observability and Observers for Nonlinear Systems

Abstract: This paper deals with observability and observers for general nonlinear systems. In the non-control-affine case, we characterize systems that are observable independently of the inputs. An exponential observer for these systems is also exhibited.

Modelling and identification of nonlinear dynamic loads in power systems

Abstract: This paper describes an approach for experimental determination of aggregate dynamic loads in power systems. The work is motivated by the importance of accurate load modeling in voltage stability analysis. The models can be expressed in general as nonlinear differential equations or equivalently realised in block diagram form as interconnections of nonlinear (memoryless) functions and linear dynamic blocks. These components are parameterized by load indexes and time constants. Experimental results from tests in Southern Sweden on the identification of these parameters are described. >

A fast numerical method for computing the linear and nonlinear mechanical properties of composites

Abstract: This Note is devoted to a new iterative algorithm to compute the local and overall response of a composite from images of its (complex) microstructure. The elastic problem for a heterogeneous material is formulated with the help of a homogeneous reference medium and written under the form of a periodic Lippman-Schwinger equation. Using the fact that the Green's function of the pertinent operator is known explicitely in Fourier space, this equation is solved iteratively.The method is extended to the case where the individual constituents are elastic-plastic Von Mises materials with isotropic hardening

Advanced data assimilation in strongly nonlinear dynamical systems

Abstract: Advanced data assimilation methods are applied to simple but highly nonlinear problems. The dynamical systems studied here are the stochastically forced double well and the Lorenz model. In both systems, linear approximation of the dynamics about the critical points near which regime transitions occur is not always sufficient to track their occurrence or nonoccurrence. Straightforward application of the extended Kalman filter yields mixed results. The ability of the extended Kalman filter to track transitions of the double-well system from one stable critical point to the other depends on the frequency and accuracy of the observations relative to the mean-square amplitude of the stochastic forcing. The ability of the filter to track the chaotic trajectories of the Lorenz model is limited to short times, as is the ability of strong-constraint variational methods. Examples are given to illustrate the difficulties involved, and qualitative explanations for these difficulties are provided. Three generalizations of the extended Kalman filter are described. The first is based on inspection of the innovation sequence, that is, the successive differences between observations and forecasts; it works very well for the double-well problem. The second, an extension to fourth-order moments, yields excellent results for the Lorenz model but will be unwieldy when applied to models with high-dimensional state spaces. A third, more practical method--based on an empirical statistical model derived from a Monte Carlo simulation--is formulated, and shown to work very well. Weak-constraint methods can be made to perform satisfactorily in the context of these simple models, but such methods do not seem to generalize easily to practical models of the atmosphere and ocean. In particular, it is shown that the equations derived in the weak variational formulation are difficult to solve conveniently for large systems.

Radial basis function neural network for approximation and estimation of nonlinear stochastic dynamic systems

Abstract: This paper presents a means to approximate the dynamic and static equations of stochastic nonlinear systems and to estimate state variables based on radial basis function neural network (RBFNN). After a nonparametric approximate model of the system is constructed from a priori experiments or simulations, a suboptimal filter is designed based on the upper bound error in approximating the original unknown plant with nonlinear state and output equations. The procedures for both training and state estimation are described along with discussions on approximation error. Nonlinear systems with linear output equations are considered as a special case of the general formulation. Finally, applications of the proposed RBFNN to the state estimation of highly nonlinear systems are presented to demonstrate the performance and effectiveness of the method. >

Solving systems of polynomial equations

Abstract: Geometric and solid modelling deal with the representation and manipulation of physical objects. Currently most geometric objects are formulated in terms of polynomial equations, thereby reducing many application problems to manipulating polynomial systems. Solving systems of polynomial equations is a fundamental problem in these geometric computations. The author presents an algorithm for solving polynomial equations. The combination of multipolynomial resultants and matrix computations underlies this efficient, robust and accurate algorithm. >

Determination of size distribution from small‐angle scattering data for systems with effective hard‐sphere interactions

Abstract: Methods for the free-form determination of size distributions for systems with hard-sphere interactions are described. An approximation, called the local monodisperse approximation, is introduced. Model calculations show that this approximation gives relatively small errors even at relatively high polydispersities and large volume fractions. The size distributions are determined by least-squares methods with smoothness and non-negativity constraints. The local monodisperse approximation leads to normal equations that are linear in the amplitude of the size distribution. This is used when solving the least-squares problem: only the two effective parameters describing the interference effects are treated as nonlinear parameters in an external optimization routine. The parameters describing the size distribution are determined by a linear least-squares method. The size distribution is also determined using the nonlinear equations from the calculation of the scattering intensity in the Percus–Yevick approximation. For this, a nonlinear least-squares routine with a smoothness constraint and a non-negativity constraint is used. Both approaches are tested by analysis of simulated examples calculated by the analytical expressions in the Percus–Yevick approximation. Finally, the methods are applied to two sets of experimental data from silica particles and from δ′ precipitates in an Al–Li alloy. For the simulated examples, good agreement is found with the input distributions. For the experimental examples, the results agree with the expected and known properties of the samples.

Signal and image restoration using shock filters and anisotropic diffusion

Abstract: The authors define a new class of filters for noise elimination and edge enhancement by using shock filters and anisotropic diffusion. Some nonlinear partial differential equations used as models f...

Chaos in the Colpitts oscillator

Abstract: In this work, we present experimental results and SPICE simulations of chaos in a Colpitts oscillator. We show that the nonlinear dynamics of this oscillator may be modeled by a third-order autonomous continuous-time circuit consisting of a linear inductor, two linear capacitors, two linear resistors, two independent voltage sources, a linear current-controlled current source, and a single voltage-controlled nonlinear resistor. The nonlinear resistor has a two-segment piecewise-linear DP characteristic. With the appropriate choice of parameters, the piecewise-linear circuit model has a positive Lyapunov exponent. >

Memory neuron networks for identification and control of dynamical systems

Abstract: This paper discusses memory neuron networks as models for identification and adaptive control of nonlinear dynamical systems. These are a class of recurrent networks obtained by adding trainable temporal elements to feedforward networks that makes the output history-sensitive. By virtue of this capability, these networks can identify dynamical systems without having to be explicitly fed with past inputs and outputs. Thus, they can identify systems whose order is unknown or systems with unknown delay. It is argued that for satisfactory modeling of dynamical systems, neural networks should be endowed with such internal memory. The paper presents a preliminary analysis of the learning algorithm, providing theoretical justification for the identification method. Methods for adaptive control of nonlinear systems using these networks are presented. Through extensive simulations, these models are shown to be effective both for identification and model reference adaptive control of nonlinear systems. >

Adaptive control of a class of nonlinear systems with fuzzy logic

Abstract: An adaptive tracking control architecture is proposed for a class of continuous-time nonlinear dynamic systems, for which an explicit linear parameterization of the uncertainty in the dynamics is either unknown or impossible. The architecture employs fuzzy systems, which are expressed as a series expansion of basis functions, to adaptively compensate for the plant nonlinearities. Global asymptotic stability of the algorithm is established in the Lyapunov sense, with tracking errors converging to a neighborhood of zero. Simulation results for an unstable nonlinear plant are included to demonstrate that incorporating the linguistic fuzzy information from human experts results in superior tracking performance. >

An Introduction to Nonlinear Partial Differential Equations

Abstract: Preface. 1. Partial Differential Equations. 1.1 Partial Differential Equations. 1.1.1 PDEs and Solutions. 1.1.2 Classification. 1.1.3 Linear vs. Nonlinear. 1.1.4 Linear Equations. 1.2 Conservation Laws. 1.2.1 One Dimension. 1.2.2 Higher Dimensions. 1.3 Constitutive Relations. 1.4 Initial and Boundary Value Problems. 1.5 Waves. 1.5.1 Traveling Waves. 1.5.2 Plane Waves. 1.5.3 Plane Waves and Transforms. 1.5.4 Nonlinear Dispersion. 2. First-Order Equations and Characteristics. 2.1 Linear First-Order Equations. 2.1.1 Advection Equation. 2.1.2 Variable Coefficients. 2.2 Nonlinear Equations. 2.3 Quasi-linear Equations. 2.3.1 The general solution. 2.4 Propagation of Singularities. 2.5 General First-Order Equation. 2.5.1 Complete Integral. 2.6 Uniqueness Result. 2.7 Models in Biology. 2.7.1 Age-Structure. 2.7.2 Structured predator-prey model. 2.7.3 Chemotherapy. 2.7.4 Mass structure. 2.7.5 Size-dependent predation. 3. Weak Solutions To Hyperbolic Equations. 3.1 Discontinuous Solutions. 3.2 Jump Conditions. 3.2.1 Rarefaction Waves. 3.2.2 Shock Propagation. 3.3 Shock Formation. 3.4 Applications. 3.4.1 Traffic Flow. 3.4.2 Plug Flow Chemical Reactors. 3.5 Weak Solutions: A Formal Approach. 3.6 Asymptotic Behavior of Shocks. 3.6.1 Equal-Area Principle. 3.6.2 Shock Fitting. 3.6.3 Asymptotic Behavior. 4. Hyperbolic Systems. 4.1 Shallow Water Waves Gas Dynamics. 4.1.1 Shallow Water Waves. 4.1.2 Small-Amplitude Approximation. 4.1.3 Gas Dynamics. 4.2 Hyperbolic Systems and Characteristics. 4.2.1 Classification. 4.3 The Riemann Method. 4.3.1 Jump Conditions for Systems. 4.3.2 Breaking Dam Problem. 4.3.3 Receding Wall Problem. 4.3.4 Formation of a Bore. 4.3.5 Gas Dynamics. 4.4 Hodographs and Wavefronts. 4.4.1 Hodograph Transformation. 4.4.2 Wavefront Expansions. 4.5 Weakly Nonlinear Approximations. 4.5.1 Derivation of Burgers' Equation. 5. Diffusion Processes. 5.1 Diffusion and Random Motion. 5.2 Similarity Methods. 5.3 Nonlinear Diffusion Models. 5.4 Reaction-Diffusion Fisher's Equation. 5.4.1 Traveling Wave Solutions. 5.4.2 Perturbation Solution. 5.4.3 Stability of Traveling Waves. 5.4.4 Nagumo's Equation. 5.5 Advection-Diffusion Burgers' Equation. 5.5.1 Traveling Wave Solution. 5.5.2 Initial Value Problem. 5.6 Asymptotic Solution to Burgers' Equation. 5.6.1 Evolution of a Point Source. 6. Reaction-Diffusion Systems. 6.1 Reaction-Diffusion Models. 6.1.1 Predator-Prey Model. 6.1.2 Combustion. 6.1.3 Chemotaxis. 6.2 Traveling Wave Solutions. 6.2.1 Model for the Spread of a Disease. 6.2.2 Contaminant transport in groundwater. 6.3 Existence of Solutions. 6.3.1 Fixed-Point Iteration. 6.3.2 Semi-Linear Equations. 6.3.3 Normed Linear Spaces. 6.3.4 General Existence Theorem. 6.4 Maximum Principles. 6.4.1 Maximum Principles. 6.4.2 Comparison Theorems. 6.5 Energy Estimates and Asymptotic Behavior. 6.5.1 Calculus Inequalities. 6.5.2 Energy Estimates. 6.5.3 Invariant Sets. 6.6 Pattern Formation. 7. Equilibrium Models. 7.1 Elliptic Models. 7.2 Theoretical Results. 7.2.1 Maximum Principle. 7.2.2 Existence Theorem. 7.3 Eigenvalue Problems. 7.3.1 Linear Eigenvalue Problems. 7.3.2 Nonlinear Eigenvalue Problems. 7.4 Stability and Bifurcation. 7.4.1 Ordinary Differential Equations. 7.4.2 Partial Differential Equations. References. Index.