Showing papers on "Linearization published in 1999"

Variational iteration method – a kind of non-linear analytical technique: some examples

Abstract: In this paper, a new kind of analytical technique for a non-linear problem called the variational iteration method is described and used to give approximate solutions for some well-known non-linear problems. In this method, the problems are initially approximated with possible unknowns. Then a correction functional is constructed by a general Lagrange multiplier, which can be identified optimally via the variational theory. Being different from the other non-linear analytical methods, such as perturbation methods, this method does not depend on small parameters, such that it can find wide application in non-linear problems without linearization or small perturbations. Comparison with Adomian’s decomposition method reveals that the approximate solutions obtained by the proposed method converge to its exact solution faster than those of Adomian’s method.

Nonlinear Systems: Analysis, Stability, and Control

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.

A Nonlinear Primal-Dual Method for Total Variation-Based Image Restoration

Abstract: We present a new method for solving total variation (TV) minimization problems in image restoration The main idea is to remove some of the singularity caused by the nondifferentiability of the quantity $| abla u|$ in the definition of the TV-norm before we apply a linearization technique such as Newton's method This is accomplished by introducing an additional variable for the flux quantity appearing in the gradient of the objective function, which can be interpreted as the normal vector to the level sets of the image u Our method can be viewed as a primal-dual method as proposed by Conn and Overton [ A Primal-Dual Interior Point Method for Minimizing a Sum of Euclidean Norms, preprint, 1994] and Andersen [Ph D thesis, Odense University, Denmark, 1995] for the minimization of a sum of Euclidean norms In addition to possessing local quadratic convergence, experimental results show that the new method seems to be globally convergent

A Lie-Backlund approach to equivalence and flatness of nonlinear systems

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.

Robust extended Kalman filtering

Abstract: Linearization errors inherent in the specification of an extended Kalman filter (EKF) can severely degrade its performance. This correspondence presents a new approach to the robust design of a discrete-time EKF by application of the robust linear design methods based on the H/sub /spl infin// norm minimization criterion. The results of simulations are presented to demonstrate an advantage for signal demodulation and nonlinear equalization applications.

Matrix diagonal stability in systems and computation

Abstract: This text provides a reference for methods and analysis to dynamical systems described by linear and nonlinear ordinary differential equations and difference equations

Nonlinear ordinary differential equations : an introduction to dynamical systems

Abstract: 1. Second-order differential equations in the phase plane 2. Plane autonomous systems and linearization 3. Geometrical aspects of plane autonomous systems 4. Periodic solutions averaging methods 5. Perturbation methods 6. Singular perturbation methods 7. Forced oscillations: harmonic and subharmonic response, stability and entrainment 8. Stability 9. Determination of stability by solution perturbation 10. Liapunov methods for determining stability of the zero solution 11. The existence of periodic solutions 12. Bifurcations and manifolds 13. Poincare series, homoclinic bifurcations and chaos HINTS AND ANSWERS TO THE PROBLEMS Appendix A. Existence and uniqueness theorems Appendix B. Topographic systems Appendix C. Norms for vectors and matrices Appendix D. A contour integral References and further reading Index

A strong tracking extended Kalman observer for nonlinear discrete-time systems

Abstract: The authors show how the extended Kalman filter, used as an observer for nonlinear discrete-time systems or extended Kalman observer (EKO), becomes a useful state estimator when the arbitrary matrices, namely R/sub k/ and Q/sub k/, are adequately chosen. As a first step, we use the linearization technique given by Boutayed et al. (1997), which consists of introducing unknown diagonal matrices to take the approximation errors into account. It is shown that the decreasing Lyapunov function condition leads to a linear matrix inequality (LMI) problem, which points out the connection between a good convergence behavior of the EKO and the instrumental matrices R/sub k/ and Q/sub k/. In order to satisfy the obtained LMI, a particular design of Q/sub k/ is given. High performances of the proposed technique are shown through numerical examples under the worst conditions.

Lazy learning for local modelling and control design

Abstract: This paper presents local methods for modelling and control of discrete-time unknown non-linear dynamical systems, when only input-output data are available. We propose the adoption of lazy learning, a memory-based technique for local modelling. The modelling procedure uses a query-based approach to select the best model configuration by assessing and comparing different alternatives. A new recursive technique for local model identification and validation is presented, together with an enhanced statistical method for model selection. A lso, three methods to design controllers based on the local linearization provided by the lazy learning algorithm are described. In the first method the lazy technique returns the forward and inverse models of the system which are used to compute the control action to take. The second is an indirect method inspired by self-tuning regulators where recursive least squares estimation is replaced by a local approximator. The third method combines the linearization provided by t...

Feedforward Linear Power Amplifiers

Abstract: Radio System Introduction. Linearity and Signal Description. Power Amplifier Operation. System Design. Linearization Techniques. Feedforward Analysis. Feedforward Components. Index.

Nonlinear EEG analysis based on a neural mass model.

Abstract: The well-known neural mass model described by Lopes da Silva et al. (1976) and Zetterberg et al. (1978) is fitted to actual EEG data. This is achieved by reformulating the original set of integral equations as a continuous-discrete state space model. The local linearization approach is then used to discretize the state equation and to construct a nonlinear Kalman filter. On this basis, a maximum likelihood procedure is used for estimating the model parameters for several EEG recordings. The analysis of the noise-free differential equations of the estimated models suggests that there are two different types of alpha rhythms: those with a point attractor and others with a limit cycle attractor. These attractors are also found by means of a nonlinear time series analysis of the EEG recordings. We conclude that the Hopf bifurcation described by Zetterberg et al. (1978) is present in actual brain dynamics.

Broad-band linearization of a Mach-Zehnder electrooptic modulator

Abstract: Analog optical-link dynamic range in excess of 75 dB in a 1-MHz band has been achieved using specially designed electrooptic modulators that minimize one or more orders of harmonic and intermodulation distortion. To date, however, such "linearized" modulators have only enabled improved link dynamic ranges at frequencies below 1 GHz. Additionally, linearization across more than an octave bandwidth has required precise balancing of the signal voltage levels on multiple electrodes in a custom modulator, which represents a significant implementation challenge. In this paper, a link linearization technique that uses a standard Mach-Zehnder lithium-niobate modulator with only one RF and one dc-bias electrode to achieve broad-band linearization is discussed, resulting in a dynamic range of 74 dB in 1 MHz across greater than an octave bandwidth (800-2500 MHz). Instead of balancing the voltages on two RF electrodes, the modulator in this new link architecture simultaneously modulates optical carriers at two wavelengths, and it is the ratio of these optical carrier powers that is adjusted for optimum distortion canceling. The paper concludes by describing a second analogous link architecture in which it is the ratio of optical power at two modulated polarizations that is adjusted in order to achieve broad-band linearization.

Nash moser methods for the solution of quasilinear schrödinger equations

Abstract: Using new Nash Moser techniques for Frechet spaces by M. Poppenberg the local existence, uniqueness and continuous dependence of smooth solutions of a special quasilinear evolutionary Schrodinger equation is proved; as basic function space H∞ (IRn), the intersection of all Sobolev spaces Hk(IRn), is used. The method consists in finding an appropriate linearization of the given nonlinear Schrodinger equation, and proving that this linear Schrodinger equation admits a strongly continuous evolution operator which provides the necessary a priori estimates for any derivative; this is shown by a transformation procedure using a time dependent metric which overcomes the difficulty arising from nondissipativity of the linearized Schrodinger equation

Sliding mode input-output linearization and field orientation for real-time control of induction motors

Abstract: This paper deals with real-time control of an induction motor based on a digital signal processor (DSP) TMS320C31/40-MHz-based system. A sliding mode controller (SMC) is presented and compared with the well-known field orientation and input-output linearization techniques. To estimate the rotor flux, a sliding mode observer is used. Experimental results are given to highlight the performances and disadvantages of these methods with respect to rotor resistance variations.

An examination of the accuracy of the linearization of a mesoscale model with moist physics

Abstract: The accuracy of tangent linear and adjoint versions of a primitive-equation model with moist physics is examined with respect to growing perturbations having significant initial magnitudes. The Jacobians for the convective parametrizations are approximated using a perturbation method. These Jacobians are then quality controlled to ensure that the approximations are suitable. Results show that: (I) linearization of the diabatic moist physics can have a significant impact; (2) even where such impacts are large, the linearized versions of the model can yield good approximations to the nonlinear behaviour for significant perturbations, especially if there is sufficient dynamical influence: (3) poor approximations can be obtained when convection dominates the results; and (4) a straightforward linearization of some parametrization schemes may be inadequate. The results are encouraging for quantitative applications of some moist adjoint models to extratropical cyclones in the winter, but suggest some tangent linear approximations may be unsuitable in the tropics or over continents in the summer, except if qualitative agreements with nonlinear results are sufficient. Detailed comparisons of linear and nonlinear results should be made. particularly using optimal perturbations, prior to any applications of tangent linear or adjoint models.

Variable-Speed Wind Turbine Controller Systematic Design Methodology: A Comparison of Non-Linear and Linear Model-Based Designs

Abstract: Variable-speed, horizontal axis wind turbines use blade-pitch control to meet specified objectives for three regions of operation. This paper focuses on controller design for the constant power production regime. A simple, rigid, non-linear turbine model was used to systematically perform trade-off studies between two performance metrics. Minimization of both the deviation of the rotor speed from the desired speed and the motion of the actuator is desired. The robust nature of the proportional-integral-derivative (PID) controller is illustrated, and optimal operating conditions are determined. Because numerous simulation runs may be completed in a short time, the relationship of the two opposing metrics is easily visualized. Traditional controller design generally consists of linearizing a model about an operating point. This step was taken for two different operating points, and the systematic design approach was used. A comparison of the optimal regions selected using the n on-linear model and the two linear models shows similarities. The linearization point selection does, however, affect the turbine performance slightly. Exploitation of the simplicity of the model allows surfaces consisting of operation under a wide range of gain values to be created. This methodology provides a means of visually observing turbine performance based upon the two metrics chosen for this study. Design of a PID controller is simplified, and it is possible to ascertain the best possible combination of controller parameters. The wide, flat surfaces indicate that a PID controller is very robust in this variable-speed wind turbine application.

Controllable viscous damping: an experimental study of an electrorheological long-stroke damper under proportional feedback control

Abstract: It is now well known that smart fluids (electrorheological (ER) and magnetorheological) can form the basis of controllable vibration damping devices. With both types of fluid, however, the force/velocity characteristic of the resulting damper is significantly nonlinear, possessing the general form associated with a Bingham plastic. In a previous paper the authors suggested that by using a linear feedback control strategy it should be possible to produce the equivalent of a viscous damper with a continuously variable damping coefficient. In the present paper the authors describe a comprehensive investigation into the implementation of this linearization strategy on an industrial scale ER long-stroke vibration damper. Using mechanical excitation frequencies up to 5 Hz it is shown that linear behaviour can be obtained between well defined limits and that the slope of the linearized force/velocity characteristic can be specified through the choice of a controller gain term.

Tutorial on nonlinear backstepping: Applications to ship control

Abstract: The theoretical foundation of nonlinear backstepping designs is presented in a tutorial setting. This includes a brief review of integral backstepping, extensions to SISO and MIMO systems in strict feedback form and physical motivated case studies. Parallels and differences to feedback linearization where it is shown how so-called good nonlinearities can be exploited in the design are also made. Nonlinear, optimal and robust backstepping are discussed in a separate section where parallels to linear quadratic optimal control and H∞-control are drawn. In addition, inverse optimality is discussed as a nonlinear design tool. Physics is put into control by using mechanical systems like mass-damper-springs and ship models in the case studies. Lyapunov theory is used to prove convergence and stability for all control laws where energy dissipation is obtained by exploiting physical model properties.

X-33 Ascent Flight Controller Design by Trajectory Linearization: A Singular Perturbational Approach

Abstract: The flight control of X-33 poses a challenge to conventional gain-scheduled flight controllers due to its large attitude maneuvers from liftoff to orbit and reentry. In addition, a wide range of uncertainties in vehicle handling qualities and disturbances must be accommodated by the attitude control system. Nonlinear tracking and decoupling control by trajectory linearization can be viewed as the ideal gain-scheduling controller designed at every point on the flight trajectory. Therefore it provides robust stability and performance at all stages of flight without interpolation of controller gains, and eliminates costly controller redesigns due to minor airframe alteration or mission reconfiguration. A prototype trajectory linearization design for X-33 ascent flight controller was designed and tested with 3-DOF and 6-DOF simulations during the 10 weeks SFFP. It is noted that the 6-DOF results were obtained from the 3-DOF design with only a few hours of tuning, which demonstrates the inherent robustness of the design technique. It is this "plug-and-play" feature that is much needed by NASA for the development, test and routine operations of the RLVs. Plans for further research are also presented.

In-line predistorter for linearization of electronic and optical signals

Abstract: An electronic circuit provides a substantially linear output from a nonlinear transmission device such as a laser. The input signal (22) to the nonlinear device (20) is applied to an in-line electrical path coupled to the nonlinear device. In the in-line predistorter of the present invention, the desired real and imaginary distortion terms may be synthesized by summing the distortion contributions from several different distorter elements. In the simplest case, one distorter produces a constant real distortion (16), another produces distortion proportional to frequency, f, and so on. However, it is not essential to have the simplest set of distorters. Distorters with more complex distortion characteristics can be used so long as they provide an independent set. A number of circuits are disclosed that can be combined to provide the building blocks of an in-line circuit.

Analytic framework for blended multiple model systems using linear local models

Abstract: In this paper it is shown that the dynamics of a conventional type of blended multiple model system are only weakly related to the local models from which it is formed. A novel class of velocity-based blended multiple model systems is proposed for which the dynamics are directly related to the local models. Indeed, the solution to the blended multiple model system, locally to a specific operating point, is approximated by the weighted linear combination of the solutions to the local models. Moreover, in contrast to conventional blended multiple model systems, the velocity-based blended multiple model systems employs linear local models, thereby providing a degree of continuity with established linear methods and, consequently, facilitating analysis and design.

A Scale-Space Approach to Nonlocal Optical Flow Calculations

Abstract: This paper presents an interpretation of a classic optical flow method by Nagel and Enkelmann as a tensor-driven anisotropic diffusion approach in digital image analysis. We introduce an improvement into the model formulation, and we establish well-posedness results for the resulting system of parabolic partial Differential equations. Our method avoids linearizations in the optical flow constraint, and it can recover displacement fields which are far beyond the typical one-pixel limits that are characteristic for many Differential methods for optical flow recovery. A robust numerical scheme is presented in detail. We avoid convergence to irrelevant local minima by embedding our method into a linear scale-space framework and using a focusing strategy from coarse to fine scales. The high accuracy of the proposed method is demonstrated by means of a synthetic and a real-world image sequence.

An observer-based anti-windup scheme for non-linear systems with input constraints

Abstract: This article addresses the problem of anti-windup design for feedback linearizable non-linear systems with input constraints, subject to dynamic controllers with linear dynamics and non-linear gains. A method for designing the controller gains and a non-linear 'observer-based' anti-windup modification is proposed, which allows reducing the behaviour of the closed-loop system under the dynamic feedback controller to that under a given static feedback linearizing controller, arbitrarily fast. The proposed anti-windup method allows, thus, to attenuate the effect of windup arbitrarily fast and guarantee that the region of closed-loop asymptotic stability under the dynamic controller is limited only by the corresponding region under the static feedback linearizing controller. The application and advantages of the method are illustrated through a simulation study on a methyl-methacrylate polymerization reactor.

Calculation of timing and amplitude jitter in dispersion-managed optical fiber communications using linearization

Abstract: An approach based on linearization that allows us to calculate the timing and amplitude jitter for arbitrary pulse shapes in dispersion-managed fibers is developed. We apply this approach to calculate the jitter for dispersion-managed soliton, return-to-zero (RZ), and nonreturn-to-zero (NRZ) transmission formats. We then estimate the bit error rates. The approach described here yields more precise results than Monte Carlo simulations at a fraction of the computational cost.