Showing papers on "Linearization published in 2006"

Adaptive approximation based control : unifying neural, fuzzy and traditional adaptive approximation approaches

Abstract: Preface. 1. INTRODUCTION. 1.1 Systems and Control Terminology. 1.2 Nonlinear Systems. 1.3 Feedback Control Approaches. 1.3.1 Linear Design. 1.3.2 Adaptive Linear Design. 1.3.3 Nonlinear Design. 1.3.4 Adaptive Approximation Based Design. 1.3.5 Example Summary. 1.4 Components of Approximation Based Control. 1.4.1 Control Architecture. 1.4.2 Function Approximator. 1.4.3 Stable Training Algorithm. 1.5 Discussion and Philosophical Comments. 1.6 Exercises and Design Problems. 2. APPROXIMATION THEORY. 2.1 Motivating Example. 2.2 Interpolation. 2.3 Function Approximation. 2.3.1 Off-line (Batch) Function Approximation. 2.3.2 Adaptive Function Approximation. 2.4 Approximator Properties. 2.4.1 Parameter (Non)Linearity. 2.4.2 Classical Approximation Results. 2.4.3 Network Approximators. 2.4.4 Nodal Processors. 2.4.5 Universal Approximator. 2.4.6 Best Approximator Property. 2.4.7 Generalization. 2.4.8 Extent of Influence Function Support. 2.4.9 Approximator Transparency. 2.4.10 Haar Conditions. 2.4.11 Multivariable Approximation by Tensor Products. 2.5 Summary. 2.6 Exercises and Design Problems. 3. APPROXIMATION STRUCTURES. 3.1 Model Types. 3.1.1 Physically Based Models. 3.1.2 Structure (Model) Free Approximation. 3.1.3 Function Approximation Structures. 3.2 Polynomials. 3.2.1 Description. 3.2.2 Properties. 3.3 Splines. 3.3.1 Description. 3.3.2 Properties. 3.4 Radial Basis Functions. 3.4.1 Description. 3.4.2 Properties. 3.5 Cerebellar Model Articulation Controller. 3.5.1 Description. 3.5.2 Properties. 3.6 Multilayer Perceptron. 3.6.1 Description. 3.6.2 Properties. 3.7 Fuzzy Approximation. 3.7.1 Description. 3.7.2 Takagi-Sugeno Fuzzy Systems. 3.7.3 Properties. 3.8 Wavelets. 3.8.1 Multiresolution Analysis (MRA). 3.8.2 MRA Properties. 3.9 Further Reading. 3.10 Exercises and Design Problems. 4. PARAMETER ESTIMATION METHODS. 4.1 Formulation for Adaptive Approximation. 4.1.1 Illustrative Example. 4.1.2 Motivating Simulation Examples. 4.1.3 Problem Statement. 4.1.4 Discussion of Issues in Parametric Estimation. 4.2 Derivation of Parametric Models. 4.2.1 Problem Formulation for Full-State Measurement. 4.2.2 Filtering Techniques. 4.2.3 SPR Filtering. 4.2.4 Linearly Parameterized Approximators. 4.2.5 Parametric Models in State Space Form. 4.2.6 Parametric Models of Discrete-Time Systems. 4.2.7 Parametric Models of Input-Output Systems. 4.3 Design of On-Line Learning Schemes. 4.3.1 Error Filtering On-Line Learning (EFOL) Scheme. 4.3.2 Regressor Filtering On-Line Learning (RFOL) Scheme. 4.4 Continuous-Time Parameter Estimation. 4.4.1 Lyapunov Based Algorithms. 4.4.2 Optimization Methods. 4.4.3 Summary. 4.5 On-Line Learning: Analysis. 4.5.1 Analysis of LIP EFOL scheme with Lyapunov Synthesis Method. 4.5.2 Analysis of LIP RFOL scheme with the Gradient Algorithm. 4.5.3 Analysis of LIP RFOL scheme with RLS Algorithm. 4.5.4 Persistency of Excitation and Parameter Convergence. 4.6 Robust Learning Algorithms. 4.6.1 Projection modification. 4.6.2 &sigma -modification. 4.6.3 &epsis -modification. 4.6.4 Dead-zone modification. 4.6.5 Discussion and Comparison. 4.7 Concluding Summary. 4.8 Exercises and Design Problems. 5. NONLINEAR CONTROL ARCHITECTURES. 5.1 Small-Signal Linearization. 5.1.1 Linearizing Around an Equilibrium Point. 5.1.2 Linearizing Around a Trajectory. 5.1.3 Gain Scheduling. 5.2 Feedback Linearization. 5.2.1 Scalar Input-State Linearization. 5.2.2 Higher-Order Input-State Linearization. 5.2.3 Coordinate Transformations and Diffeomorphisms. 5.2.4 Input-Output Feedback Linearization. 5.3 Backstepping. 5.3.1 Second order system. 5.3.2 Higher Order Systems. 5.3.3 Command Filtering Formulation. 5.4 Robust Nonlinear Control Design Methods. 5.4.1 Bounding Control. 5.4.2 Sliding Mode Control. 5.4.3 Lyapunov Redesign Method. 5.4.4 Nonlinear Damping. 5.4.5 Adaptive Bounding Control. 5.5 Adaptive Nonlinear Control. 5.6 Concluding Summary. 5.7 Exercises and Design Problems. 6. ADAPTIVE APPROXIMATION: MOTIVATION AND ISSUES. 6.1 Perspective for Adaptive Approximation Based Control. 6.2 Stabilization of a Scalar System. 6.2.1 Feedback Linearization. 6.2.2 Small-Signal Linearization. 6.2.3 Unknown Nonlinearity with Known Bounds. 6.2.4 Adaptive Bounding Methods. 6.2.5 Approximating the Unknown Nonlinearity. 6.2.6 Combining Approximation with Bounding Methods. 6.2.7 Combining Approximation with Adaptive Bounding Methods. 6.2.8 Summary. 6.3 Adaptive Approximation Based Tracking. 6.3.1 Feedback Linearization. 6.3.2 Tracking via Small-Signal Linearization. 6.3.3 Unknown Nonlinearities with Known Bounds. 6.3.4 Adaptive Bounding Design. 6.3.5 Adaptive Approximation of the Unknown Nonlinearities. 6.3.6 Robust Adaptive Approximation. 6.3.7 Combining Adaptive Approximation with Adaptive Bounding. 6.3.8 Some Adaptive Approximation Issues. 6.4 Nonlinear Parameterized Adaptive Approximation. 6.5 Concluding Summary. 6.6 Exercises and Design Problems. 7. ADAPTIVE APPROXIMATION BASED CONTROL: GENERAL THEORY. 7.1 Problem Formulation. 7.1.1 Trajectory Tracking. 7.1.2 System. 7.1.3 Approximator. 7.1.4 Control Design. 7.2 Approximation Based Feedback Linearization. 7.2.1 Scalar System. 7.2.2 Input-State. 7.2.3 Input-Output. 7.2.4 Control Design Outside the Approximation Region D. 7.3 Approximation Based Backstepping. 7.3.1 Second Order Systems. 7.3.2 Higher Order Systems. 7.3.3 Command Filtering Approach. 7.3.4 Robustness Considerations. 7.4 Concluding Summary. 7.5 Exercises and Design Problems. 8. ADAPTIVE APPROXIMATION BASED CONTROL FOR FIXED-WING AIRCRAFT. 8.1 Aircraft Model Introduction. 8.1.1 Aircraft Dynamics. 8.1.2 Non-dimensional Coefficients. 8.2 Angular Rate Control for Piloted Vehicles. 8.2.1 Model Representation. 8.2.2 Baseline Controller. 8.2.3 Approximation Based Controller. 8.2.4 Simulation Results. 8.3 Full Control for Autonomous Aircraft. 8.3.1 Airspeed and Flight Path Angle Control. 8.3.2 Wind-axes Angle Control. 8.3.3 Body Axis Angular Rate Control. 8.3.4 Control Law and Stability Properties. 8.3.5 Approximator Definition. 8.3.6 Simulation Analysis. 8.4 Conclusions. 8.5 Aircraft Notation. Appendix A: Systems and Stability Concepts. A.1 Systems Concepts. A.2 Stability Concepts. A.2.1 Stability Definitions. A.2.2 Stability Analysis Tools. A.3 General Results. A.4 Prefiltering. A.5 Other Useful Results. A.5.1 Smooth Approximation of the Signum function. A.6 Problems. Appendix B: Recommended Implementation and Debugging Approach. References. Index.

Control Design Techniques in Power Electronics Devices

Abstract: Modelling.- Modelling of DC-to-DC Power Converters.- Controller Design Methods.- Sliding Mode Control.- Approximate Linearization in the Control of Power Electronics Devices.- Nonlinear Methods in the Control of Power Electronics Devices.- Applications.- DC-to-AC Power Conversion.- AC Rectifiers.

VLIDORT: A linearized pseudo-spherical vector discrete ordinate radiative transfer code for forward model and retrieval studies in multilayer multiple scattering media

Abstract: We describe a new vector discrete ordinate radiative transfer model with a full linearization facility. The VLIDORT model is designed to generate simultaneous output of Stokes vector light fields and their derivatives with respect to any atmospheric or surface property. We develop new implementations for the linearization of the vector radiative transfer solutions, and go on to show that the complete vector discrete ordinate solution is analytically differentiable for a stratified multilayer multiply scattering atmospheric medium. VLIDORT will generate all output at arbitrary viewing geometry and optical depth. The model has the ability to deal with attenuation of solar and line-of-sight paths in a curved atmosphere, and includes an exact treatment of the single scatter computation. VLIDORT also contains a linearized treatment for non-Lambertian surfaces. A number of performance enhancements have been implemented, including a facility for multiple solar zenith angle output. The model has been benchmarked against established results in the literature.

Generalized Bouc-Wen model for highly asymmetric hysteresis

Abstract: Bouc-Wen class models have been widely used to efficiently describe smooth hysteretic behavior in time history and random vibration analyses. This paper proposes a generalized Bouc-Wen model with sufficient flexibility in shape control to describe highly asymmetric hysteresis loops. Also introduced is a mathematical relation between the shape-control parameters and the slopes of the hysteresis loops, so that the model parameters can be identified systematically in conjunction with available parameter identification methods. For use in nonlinear random vibration analysis by the equivalent linearization method, closed-form expressions are derived for the coefficients of the equivalent linear system in terms of the second moments of the response quantities. As an example application, the proposed model is successfully fitted to the highly asymmetric hysteresis loops obtained in laboratory experiments for flexible connectors used in electrical substations. The model is then employed to investigate the effect of dynamic interaction between interconnected electrical substation equipment by nonlinear time-history and random vibration analyses.

A two-scale approach for fluid flow in fractured porous media

Abstract: A two-scale numerical model is developed for fluid flow in fractured, deforming porous media. At the microscale the flow in the cavity of a fracture is modelled as a viscous fluid. From the micromechanics of the flow in the cavity, coupling equations are derived for the momentum and the mass couplings to the equations for a fluid-saturated porous medium, which are assumed to hold on the macroscopic scale. The finite element equations are derived for this two-scale approach and integrated over time. By exploiting the partition-of-unity property of the finite element shape functions, the position and direction of the fractures is independent from the underlying discretization. The resulting discrete equations are non-linear due to the non-linearity of the coupling terms. A consistent linearization is given for use within a Newton–Raphson iterative procedure. Finally, examples are given to show the versatility and the efficiency of the approach, and show that faults in a deforming porous medium can have a significant effect on the local as well as on the overall flow and deformation patterns. Copyright © 2006 John Wiley & Sons, Ltd.

An analytical approximate technique for a class of strongly non-linear oscillators

Abstract: An analytical approximate technique for large amplitude oscillations of a class of conservative single degree-of-freedom systems with odd non-linearity is proposed. The method incorporates salient features of both Newton's method and the harmonic balance method. Unlike the classical harmonic balance method, accurate analytical approximate solutions are possible because linearization of the governing differential equation by Newton's method is conducted prior to harmonic balancing. The approach yields simple linear algebraic equations instead of non-linear algebraic equations without analytical solution. With carefully constructed iterations, only a few iterations can provide very accurate analytical approximate solutions for the whole range of oscillation amplitude beyond the domain of possible solution by the conventional perturbation methods or harmonic balance method. Three examples including cubic-quintic Duffing oscillators are presented to illustrate the usefulness and effectiveness of the proposed technique.

Discrete Dynamical Systems

Abstract: One-Dimensional, First-Order Systems.- Multi-Dimensional, First-Order, Linear Systems: Solution.- Multi-Dimensional, First-Order, Linear Systems: Characterization.- Multi-Dimensional, First-Order, Nonlinear Systems.- Higher-Order and Non-Autonomous Systems.- Examples of Two-Dimensional Systems.

Biomimetic motion and balance controllers for use in prosthetics, orthotics and robotics

Abstract: Systems for controlling the motion of multiple articulated elements connected by one or more joints in an artificial appendage system. Four different embodiments includes a controller that reduces the dimension of joint state space by utilizing biomechanically inspired motion primitives; a quadratic proportional-derivative (PD) controller which employs a two-stage linearization method, applies constraints to variables for dynamic stability, and employs a corrective “sliding control” mechanism to account for errors in the linear model used; a non-prioritized balance control approach that employs enforced linear dynamics in which all control variables are truncated to linear terms in joint jerks; and a biomimetic motion and balance controller based on center of mass (CM) energetic and biomimetic zero moment conditions.

Algebraic Methods for Nonlinear Control Systems

Abstract: Methodology.- Preliminaries.- Modeling.- Accessibility.- Observability.- Systems Structure and Inversion.- System Transformations.- Applications to Control Problems.- Input-output Linearization.- Noninteracting Control.- Input-state Linearization.- Disturbance Decoupling.- Model Matching.- Measured Output Feedback Control Problems.

Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials

Abstract: For the spatially homogeneous Boltzmann equation with hard potentials and Grad's cutoff (e.g. hard spheres), we give quantitative estimates of exponential convergence to equilibrium, and we show that the rate of exponential decay is governed by the spectral gap for the linearized equation, on which we provide a lower bound. Our approach is based on establishing spectral gap-like estimates valid near the equilibrium, and then connecting the latter to the quantitative nonlinear theory. This leads us to an explicit study of the linearized Boltzmann collision operator in functional spaces larger than the usual linearization setting.

Absolute value equation solution via concave minimization

Abstract: The NP-hard absolute value equation (AVE) Ax − |x| = b where \(A\in R^{n\times n}\) and \(b\in R^n\) is solved by a succession of linear programs The linear programs arise from a reformulation of the AVE as the minimization of a piecewise-linear concave function on a polyhedral set and solving the latter by successive linearization A simple MATLAB implementation of the successive linearization algorithm solved 100 consecutively generated 1,000-dimensional random instances of the AVE with only five violated equations out of a total of 100,000 equations

Transmission loss reduction based on FACTS and bacteria foraging algorithm

Abstract: An optimal location and parameters of an UPFC along with values of OLTC taps are tuned with a view to minimize the real power losses of a mesh power network. This issue is formulated as a non-linear equality and inequality constrained optimization problem with an objective function incorporating power loss. A new evolutionary algorithm known as Bacteria Foraging is applied for solving, the optimum location and the amount of series injected voltage for the UPFC, and the best values of the taps present in the system. The same problem is also solved with Interior Point Successive Linearization technique using the LINPROG command of MATLAB. A comparison between the two suggests the superiority of the proposed algorithm.

State-dependent delay in regenerative turning processes

Abstract: Stability of a two degrees of freedom model of the turning process is considered. An accurate modeling of the surface regeneration shows that the regenerative delay, determined by the combination of the workpiece rotation and the tool vibrations, is in fact state-dependent. For that reason, the mathematical model considered in this paper is a delay-differential equation with state-dependent time delay. In order to study linearized stability of stationary cutting processes, an associated linear system, corresponding to the state-dependent delay equation, is derived. Stability analysis of this linear system is performed analytically.

On non-linear methods of bogie stability assessment using computer simulations

Abstract: Stability assessment of rail vehicles is probably the most widespread form of dynamic analysis in railway vehicle engineering. The computer simulations using fully non-linear three-dimensional vehicle models constructed in a modern multi-body simulation tool allow detailed non-linear stability analysis for the specified conditions. However, high sensitivity to the wheel/ rail contact conditions and different definitions of stability in mechanics and in railway practise can lead to significant differences between prediction and the measurement. Different methods of non-linear stability analysis, which may be used in industrial applications, are introduced and compared on selected examples of contact geometry wheel set/track with high equivalent conicity. The comparisons show that the linearization of the contact geometry wheel set/track can enable a better assessment of the non-linear stability analyses. A decreasing equivalent conicity function in the range of amplitudes below ∼ 3 mm leads to supe...

An approximation technique for robust nonlinear optimization

Abstract: Nonlinear equality and inequality constrained optimization problems with uncertain parameters can be addressed by a robust worst-case formulation that is, however, difficult to treat computationally. In this paper we propose and investigate an approximate robust formulation that employs a linearization of the uncertainty set. In case of any norm bounded parameter uncertainty, this formulation leads to penalty terms employing the respective dual norm of first order derivatives of the constraints. The main advance of the paper is to present two sparsity preserving ways for efficient computation of these derivatives in the case of large scale problems, one similar to the forward mode, the other similar to the reverse mode of automatic differentiation. We show how to generalize the techniques to optimal control problems, and discuss how even infinite dimensional uncertainties can be treated efficiently. Finally, we present optimization results for an example from process engineering, a batch distillation.

Effect of Approximations of the Discrete Adjoint on Gradient-Based Optimization

Abstract: An exact discrete adjoint of an unstructured nite-volume solver for the RANS equations has been developed. The adjoint is exact in the sense of being based on the full linearization of all terms in the solver, including all turbulence model contributions. From this starting point various approximations to the adjoint are derived with the intention of simplifying the development and memory requirements of the method; considered are many approximations already seen in the literature. The eect of these approximations on the accuracy of the resulting design gradients, and the convergence and nal solution of optimizations is studied, as it applies to a two-dimensional high-lift conguration.

Generalized Hopf Bifurcation for Planar Filippov Systems Continuous at the Origin

Abstract: In this paper, we study the existence of periodic orbits bifurcating from stationary solutions of a planar dynamical system of Filippov type. This phenomenon is interpreted as a generalized Hopf bifurcation. In the case of smoothness, Hopf bifurcation is characterized by a pair of complex conjugate eigenvalues crossing through the imaginary axis. This method does not carry over to nonsmooth systems, due to the lack of linearization at the origin which is located on the line of discontinuity. In fact, generalized Hopf bifurcation is determined by interactions between the discontinuity of the system and the eigen-structures of all subsystems. With the help of geometrical observations for a corresponding piecewise linear system, we derive an analytical method to investigate the existence of periodic orbits that are obtained by searching for the fixed points of return maps.

Symbolic methods to enhance the precision of numerical abstract domains

Abstract: We present lightweight and generic symbolic methods to improve the precision of numerical static analyses based on Abstract Interpretation. The main idea is to simplify numerical expressions before they are fed to abstract transfer functions. An important novelty is that these simplifications are performed on-the-fly, using information gathered dynamically by the analyzer. A first method, called “linearization,” allows abstracting arbitrary expressions into affine forms with interval coefficients while simplifying them. A second method, called “symbolic constant propagation,” enhances the simplification feature of the linearization by propagating assigned expressions in a symbolic way. Combined together, these methods increase the relationality level of numerical abstract domains and make them more robust against program transformations. We show how they can be integrated within the classical interval, octagon and polyhedron domains. These methods have been incorporated within the Astree static analyzer that checks for the absence of run-time errors in embedded critical avionics software. We present an experimental proof of their usefulness.

Model Reference Adaptive Control Design for a Shunt Active Power Filter System

Abstract: In this paper, model reference adaptive control (MRAC) is proposed for a single-phase shunt active power filter (APF) to improve line power factor and to reduce line current harmonics The proposed active power filter controller forces the supply current to be sinusoidal with low current harmonics and in phase with the line voltage The advantages of using MRAC over conventional proportional-integral (PI) control are more flexibility, adaptive and robustness; moreover, MRAC can self-tune the controller gains to assure the system stability Since the APF is a bilinear system, it is hard to design the controller This paper will solve the stability problem when linearization method is used to approach the nonlinearity of the system Besides, using Lyapunov's stability theory and Barbalat's Lemma, an adaptive law is designed to guarantee asymptotic output tracking for the system To verify the proposed APF system, a digital signal controller is adopted to implement the algorithm of MRAC and an 1-kVA laboratory prototype is built to test the feasibility

Neural network-aided adaptive unscented Kalman filter for nonlinear state estimation

Abstract: The extended Kalman filter (EKF) is well known as a state estimation method for a nonlinear system and has been used to train a multilayered neural network (MNN) by augmenting the state with unknown connecting weights. However, EKF has the inherent drawbacks such as instability due to linearization and costly calculation of Jacobian matrices, and its performance degrades greatly, especially when the nonlinearity is severe. In this letter, first a more robust learning algorithm for an MNN-based on unscented Kalman filter (UKF) is derived. Since it gives a more accurate estimate of the linkweights, the convergence performance is improved. The algorithm is then extended further to develop a NN-aided UKF for nonlinear state estimation. The NN in this algorithm is used to approximate the uncertainty of the system model due to mismodeling, extreme nonlinearities, etc. The UKF is used for both NN online training and state estimation simultaneously. Simulation results show that the new algorithm is very effective and is closer to optimal fashion in nonlinear filtering compared with traditional methods

Efficiency Optimizing Control of Induction Motor Using Natural Variables

Abstract: This paper presents a new approach of optimizing the efficiency of induction-motor drives through minimizing the copper and core losses. The induction-machine model, which accounts for the varying core-loss resistance and saturation dependent magnetizing inductance, uses natural and reference frame independent quantities as state variables. Utilization of the nonlinear geometric control methodology of input-output linearization with decoupling permits the implementation of the control in the stationary reference frame. This approach eliminates the need of synchronous reference transformation and flux alignment required in classical vector control schemes. The new efficiency optimizing formulation yields a reference rotor flux, which ensures a minimum loss and yields an improved efficiency of the drive system especially when driving part load. The proposed scheme and its advantages are demonstrated both by computer simulations and some experimental results for motor speed control

Efficiency Improvements of RANS-Based Analysis and Optimization using Implicit and Adjoint Methods on Unstructured Grids

Abstract: The efficiency of an unstructured grid finite volume RANS solver is signicantly improved using two implicit methods based on diering philosophies. The LU-SGS multigrid method aims to improve performance, while maintaining the low memory requirements and robustness of an explicit scheme. The First-Order Krylov Implicit (FOKI) method sacrices these to some extent, in order to achieve high convergence rates and also avoid the use of a multigrid method, whilst care is taken that the method remains practical for large 3d cases. The speeds of the two schemes are compared with that of an existing, highly-tuned Runge-Kutta multigrid method, and it is seen that a factor of two speed-up can be obtained with no additional memory overhead using LU-SGS, and a factor of ten with FOKI. Attention is then turned to the efficiency of aerodynamic design optimization using gradient-based methods. Use of the Jacobian from the implicit methods allows construction of the adjoint of the flow solver. This adjoint is exact in the sense of being based on the full linearization of all terms in the solver, including all turbulence model contributions. From this starting point various approximations to the adjoint are derived with the intention of simplifying the development and reducing the memory requirements of the method. The effect of these approximations on the accuracy of the resulting design gradients, and the convergence and final solution of optimization problems is studied. The result is a tool for extremely rapid sensitivity evaluations.

A least-squares/Newton method for digital predistortion of wideband signals

Abstract: Power amplifiers (PAs) are essential in communication systems, but are inherently nonlinear. To achieve linearity with good efficiency, PA linearization is necessary. Digital baseband predistortion is a highly cost-effective way to linearize PAs, but most existing architectures assume that the PA has a memoryless nonlinearity. For wider bandwidth applications, such as wideband code-division multiple access, PA memory effects can no longer be ignored. Therefore, in order to achieve good linearization performance, the predistorter needs to also have memory structure. In this paper, we propose a new model for the wideband predistorter and a least-squares(LS)/Newton algorithm to estimate the model parameters. Performance of the LS/Newton algorithm is studied through computer simulations. Good linearization performance is achieved by using the new model in an experimental testbed.