Showing papers on "Linearization published in 2000"

Neural Networks for Modelling and Control of Dynamic Systems: A Practitioner’s Handbook

Abstract: 1. Introduction.- 1.1 Background.- 1.1.1 Inferring Models and Controllers from Data.- 1.1.2 Why Use Neural Networks?.- 1.2 Introduction to Multilayer Perceptron Networks.- 1.2.1 The Neuron.- 1.2.2 The Multilayer Perceptron.- 1.2.3 Choice of Neural Network Architecture.- 1.2.4 Models of Dynamic Systems.- 1.2.5 Recurrent Networks.- 1.2.6 Other Neural Network Architectures.- 2. System Identification with Neural Networks.- 2.1 Introduction to System Identification.- 2.1.1 The Procedure.- 2.2 Model Structure Selection.- 2.2.1 Some Linear Model Structures.- 2.2.2 Nonlinear Model Structures Based on Neural Networks.- 2.2.3 A Few Remarks on Stability.- 2.2.4 Terminology.- 2.2.5 Selecting the Lag Space.- 2.2.6 Section Summary.- 2.3 Experiment.- 2.3.1 When is a Linear Model Insufficient?.- 2.3.2 Issues in Experiment Design.- 2.3.3 Preparing the Data for Modelling.- 2.3.4 Section Summary.- 2.4 Determination of the Weights.- 2.4.1 The Prediction Error Method.- 2.4.2 Regularization and the Concept of Generalization.- 2.4.3 Remarks on Implementation.- 2.4.4 Section Summary.- 2.5 Validation.- 2.5.1 Looking for Correlations.- 2.5.2 Estimation of the Average Generalization Error.- 2.5.3 Visualization of the Predictions.- 2.5.4 Section Summary.- 2.6 Going Backwards in the Procedure.- 2.6.1 Training the Network Again.- 2.6.2 Finding the Optimal Network Architecture.- 2.6.3 Redoing the Experiment.- 2.6.4 Section Summary.- 2.7 Recapitulation of System Identification.- 3. Control with Neural Networks.- 3.1 Introduction to Neural-Network-based Control.- 3.1.1 The Benchmark System.- 3.2 Direct Inverse Control.- 3.2.1 General Training.- 3.2.2 Direct Inverse Control of the Benchmark System.- 3.2.3 Specialized Training.- 3.2.4 Specialized Training and Direct Inverse Control of the Benchmark System.- 3.2.5 Section Summary.- 3.3 Internal Model Control (IMC).- 3.3.1 Internal Model Control with Neural Networks.- 3.3.2 Section Summary.- 3.4 Feedback Linearization.- 3.4.1 The Basic Principle of Feedback Linearization.- 3.4.2 Feedback Linearization Using Neural Network Models..- 3.4.3 Feedback Linearization of the Benchmark System.- 3.4.4 Section Summary.- 3.5 Feedforward Control.- 3.5.1 Feedforward for Optimizing an Existing Control System.- 3.5.2 Feedforward Control of the Benchmark System.- 3.5.3 Section Summary.- 3.6 Optimal Control.- 3.6.1 Training of an Optimal Controller.- 3.6.2 Optimal Control of the Benchmark System.- 3.6.3 Section Summary.- 3.7 Controllers Based on Instantaneous Linearization.- 3.7.1 Instantaneous Linearization.- 3.7.2 Applying Instantaneous Linearization to Control.- 3.7.3 Approximate Pole Placement Design.- 3.7.4 Pole Placement Control of the Benchmark System.- 3.7.5 Approximate Minimum Variance Design.- 3.7.6 Section Summary.- 3.8 Predictive Control.- 3.8.1 Nonlinear Predictive Control (NPC).- 3.8.2 NPC Applied to the Benchmark System.- 3.8.3 Approximate Predictive Control (APC).- 3.8.4 APC applied to the Benchmark System.- 3.8.5 Extensions to the Predictive Controller.- 3.8.6 Section Summary.- 3.9 Recapitulation of Control Design Methods.- 4. Case Studies.- 4.1 The Sunspot Benchmark.- 4.1.1 Modelling with a Fully Connected Network.- 4.1.2 Pruning of the Network Architecture.- 4.1.3 Section Summary.- 4.2 Modelling of a Hydraulic Actuator.- 4.2.1 Estimation of a Linear Model.- 4.2.2 Neural Network Modelling of the Actuator.- 4.2.3 Section Summary.- 4.3 Pneumatic Servomechanism.- 4.3.1 Identification of the Pneumatic Servomechanism.- 4.3.2 Nonlinear Predictive Control of the Servo.- 4.3.3 Approximate Predictive Control of the Servo.- 4.3.4 Section Summary.- 4.4 Control of Water Level in a Conic Tank.- 4.4.1 Linear Analysis and Control.- 4.4.2 Direct Inverse Control of the Water Level.- 4.4.3 Section Summary.- References.

On the interpretation and identification of dynamic Takagi-Sugeno fuzzy models

Abstract: Dynamic Takagi-Sugeno fuzzy models are not always easy to interpret, in particular when they are identified from experimental data. It is shown that there exists a close relationship between dynamic Takagi-Sugeno fuzzy models and dynamic linearization when using affine local model structures, which suggests that a solution to the multiobjective identification problem exists. However, it is also shown that the affine local model structure is a highly sensitive parametrization when applied in transient operating regimes. Due to the multiobjective nature of the identification problem studied here, special considerations must be made during model structure selection, experiment design, and identification in order to meet both objectives. Some guidelines for experiment design are suggested and some robust nonlinear identification algorithms are studied. These include constrained and regularized identification and locally weighted identification. Their usefulness in the present context is illustrated by examples.

Constructive Lyapunov control design for turbocharged diesel engines

Abstract: Presents a control design method for diesel engines equipped with a variable geometry turbocharger and an exhaust gas recirculation valve. Our control objective is to regulate the air-fuel ratio and the fraction of recirculated exhaust gas to their respective set points that depend on engine operating conditions. Interactions between the two actuators and nonlinear behavior of the system make the problem difficult to handle using classical control design methods. Instead, we employ a control Lyapunov function (CLF) based nonlinear control design method because it possesses a guaranteed robustness property equivalent to gain and phase margins. The CLF is constructed using input-output linearization of a reduced order diesel engine model. The controller has been tested in simulations on the full order model as well as experimentally in the dynamometer test cell.

Spurious Welfare Reversals in International Business Cycle Models

Abstract: Papers on international business cycles have documented spurious welfare reversals: incomplete markets produce a higher level of welfare than the complete market. This paper first demonstrates how conventional linearization, as used in King, Plosser, and Rebelo (1988), can generate approximation errors that can result in welfare reversals. Using a two-country production economy, we argue that spurious welfare reversals are not only possible but also plausible under reasonable values for model parameters including labor supply elasticity. As a constructive alternative, this paper then proposes an approximation method that modifies the conventional linearization by a bias correction - the linear approximation around a 'stochastic' steady state. We show that this method can be easily implemented and very well approximates the exact solution. The accuracy of the proposed method is by far better than that of the conventional linearization method and as good as that of a perturbation method involving a second-order expansion.

Heteroscedastic Regression in Computer Vision: Problems with Bilinear Constraint

Abstract: We present an algorithm to estimate the parameters of a linear model in the presence of heteroscedastic noise, i.e., each data point having a different covariance matrix. The algorithm is motivated by the recovery of bilinear forms, one of the fundamental problems in computer vision which appears whenever the epipolar constraint is imposed, or a conic is fit to noisy data points. We employ the errors-in-variables (EIV) model and show why already at moderate noise levels most available methods fail to provide a satisfactory solution. The improved behavior of the new algorithm is due to two factors: taking into account the heteroscedastic nature of the errors arising from the linearization of the bilinear form, and the use of generalized singular value decomposition (GSVD) in the computations. The performance of the algorithm is compared with several methods proposed in the literature for ellipse fitting and estimation of the fundamental matrix. It is shown that the algorithm achieves the accuracy of nonlinear optimization techniques at much less computational cost.

Stochastic effects in physical systems

Abstract: The study of the effects of noise and fluctuations is a well established subject in several different disciplines ranging from pure mathematics (stochastic processes) to physics (fluctuations) and electrical engineering (noise and radiophysics). In traditional statistical physics, fluctuations are of thermal origin giving rise to small departures from a mean value. They tend to zero as one approaches the thermodynamic limit in which different statistical descriptions (different ensembles) become equivalent. Likewise, in more applied contexts fluctuations or noise are usually regarded as small corrections to a deterministic (noise free) behavior that degrades a signal-to-noise ratio or can cause transmission errors. In such framework fluctuations are a correction that can be usually dealt with through some sort of linearization of dynamics around a mean or noise free dynamics. A different point of view about fluctuations emerges, for example, in the study of critical phenomena in the 1970’s. The statistical physics description of these phenomena requires a formulation appropriate for a system dominated by fluctuations and nonlinear it ies. A linear theory only identifies the existence of a critical point by a divergence of fluctuations.

Disturbance Observer Based Tracking Control

Abstract: A disturbance observer based tracking control algorithm is presented in this paper. The key idea of the proposed method is that the plant nonlinearities and parameter variations can be lumped into a disturbance term. The lumped disturbance signal is estimated based on a plant dynamic observer. A state observer then corrects the disturbance estimation in a two-step design. First, a Lyapunov-based feedback estimation law is used. The estimation is then improved by using a feedforward correction term. The control of a telescopic robot arm is used as an example system for the proposed algorithm. Simulation results comparing the proposed algorithm against a standard adaptive control scheme and a sliding mode control algorithm show that the proposed scheme achieves superior performance, especially when large external disturbances are present. @S0022-0434~00!00802-9# Tracking control for uncertain nonlinear systems with unknown disturbances is a challenging problem. To achieve good tracking under uncertainties, one usually needs to combine several or all of the following three mechanisms in the control design: adaptation, feedforward ~plant-inversion!, and high-gain, this paper is no exception. The tracking control of nonlinear systems under plant uncertainties and exogenous disturbances is studied in this paper. However, we will focus on the robotic examples for both literature review and numerical simulations. Many adaptive control schemes for robotic manipulators assume that the structure of the manipulator dynamics is known and/or the unknown parameters influence the system dynamics in an affine manner @1‐5#. There are several inherent difficulties associated with these approaches. First of all, the plant dynamic structure may not be known exactly. Second, it was demonstrated @6,7# that some of these designs may lack robustness against uncertainties. Recently, adaptive control algorithms requiring less model information were proposed @8‐11#. These algorithms adjust the control gains based on the system performance and thus are commonly referred to as performance-based adaptive control. These algorithms require little knowledge of system structures and parameter values. However, the control signal might become quite large. Plant-inversion based methods ~e.g., I/O linearization, backstepping!, roughly speaking, focus on the canceling of unwanted nonlinear dynamics. High-gain approaches such as sliding model controls could guarantee stability but, again, sometimes require very large control signals. While in some cases this may be a viable approach, in many other applications it may not be the best solution. In this paper, a disturbance-estimation based tracking control method is presented. Disturbance observer based control algorithms first appeared in the late 1980s @12#. Since then, they have been applied to many applications @13‐15#. Recently, the H‘ technique has been applied for the design of an optimal disturbance observer @16#. In this paper, we focus on the design for nonlinear systems. The magnitude of the disturbance is estimated based on the state estimation error in a two-step design. The estimated disturbance can then be used to improve the performance of literally any control algorithms. In this paper, a simple computed torque method is selected. The performance of the disturbanceobserver-enhanced method is then compared against those of a simple adaptive control and a simple robust control algorithm.

Robust Autonomous Robot Localization Using Interval Analysis

Abstract: This paper deals with the determination of the position and orientation of a mobile robot from distance measurements provided by a belt of onboard ultrasonic sensors. The environment is assumed to be two-dimensional, and a map of its landmarks is available to the robot. In this context, classical localization methods have three main limitations. First, each data point provided by a sensor must be associated with a given landmark. This data-association step turns out to be extremely complex and time-consuming, and its results can usually not be guaranteed. The second limitation is that these methods are based on linearization, which makes them inherently local. The third limitation is their lack of robustness to outliers due, e.g., to sensor malfunctions or outdated maps. By contrast, the method proposed here, based on interval analysis, bypasses the data-association step, handles the problem as nonlinear and in a global way and is (extraordinarily) robust to outliers.

Reconstructing the Shape of a Deformable Membrane from Image Data

Abstract: In this paper, we study the problem of determining a mathematical description of the surface defined by the shape of a membrane based on an image of it and present an algorithm for reconstructing the surface when the membrane is deformed by unknown external elements. The given data are the projection on an image plane of markings on the surface of the membrane, the undeformed configuration of the membrane, and a model for the membrane mechanics. The method of re construction is based on the principle that the shape assumed by the membrane will minimize the elastic energy stored in the membrane subject to the constraints implied by the measurements. Energy minimization leads to a set of nonlinear partial differential equations. An approximate solution is found using linearization. The initial motivation, and our first application of these ideas, comes from tactile sensing. Experimental results affirm that this approach can be very effective in this context.

Contact with friction between beams in 3‐D space

Abstract: In this paper a formulation to deal with friction between straight beams undergoing large displacements in 3-D space is proposed. The detection of the contact point and the computation of the amount of sliding are carried out using a completely symmetric treatment between the two contacting beams. Starting from the virtual work equation the consistent linearization of the frictional contact contribution is computed and the complete equation set is arranged in matrix form suitable for FE implementation. Some numerical examples are added to show the effectiveness of the method. Copyright © 2000 John Wiley & Sons, Ltd.

Adaptive nonlinear control of agile antiair missiles using neural networks

Abstract: Research has shown that neural networks can be used to improve upon approximate dynamic inversion controllers in the case of uncertain nonlinear systems. In one possible architecture, the neural network adaptively cancels linearization errors through online learning. Learning may be accomplished by a simple weight update rule derived from Lyapunov theory, thus assuring the stability of the closed-loop system. In the paper, the authors discuss the evolution of this methodology and its application in a bank-to-turn autopilot design for an agile antiair missile. First, a control scheme based on approximate inversion of the vehicle dynamics is presented. This nonlinear control system is then augmented by the addition of a feedforward neural network with online learning. Finally, the resulting control law is demonstrated in a nonlinear simulation and its performance is evaluated relative to a conventional gain-scheduled linear autopilot.

Linearization of the activated sludge model ASM1 for fast and reliable predictions

Abstract: In this paper a strategy is proposed to reduce the complexity of the activated sludge model no. 1 (ASM1) which describes the biotransformation processes in a common activated sludge process with N-removal. The key feature of the obtained reduced model is that it combines high predictive value (all state variables keep their biological interpretation) with very low computation time. Therefore, this model is a valuable tool in a risk assessment environment (designed for the evaluation of wastewater treatment plants facing stricter effluent norms) as well as in on-line (MPC) control strategies. The complexity reduction procedure consists of four steps. In the first step representative input/output data sets are generated by simulating the full ASM1 model. In the second step the ASM1 model is rewritten in state space format with linear approximations of the nonlinear (kinetic) terms. In the third step the unknown parameters in the linear terms are identified based on the generated input/output data. To reduce the amount of parameter sets that have to be identified (to cover the full operation range of the plant), a Multi-Model interpolation procedure is introduced as a last step.

The Acoustic Limit for the Boltzmann Equation

Abstract: The acoustic equations are the linearization of the compressible Euler equations about a spatially homogeneous fluid state. We first derive them directly from the Boltzmann equation as the formal limit of moment equations for an appropriately scaled family of Boltzmann solutions. We then establish this limit for the Boltzmann equation considered over a periodic spatial domain for bounded collision kernels. Appropriately scaled families of DiPerna-Lions renormalized solutions are shown to have fluctuations that converge entropically (and hence strongly in L 1 ) to a unique limit governed by a solution of the acoustic equations for all time, provided that its initial fluctuations converge entropically to an appropriate limit associated to any given L 2 initial data of the acoustic equations. The associated local conservation laws are recovered in the limit.

A nonlinear filtering algorithm based on an approximation of the conditional distribution

Abstract: An effective form of the Gaussian or moment approximation method for approximating optimal nonlinear filters with a diffusion signal process and discrete-time observations is presented. Various computational simplifications reduce the dimensionality of the numerical integrations that need to be done. This process, combined with an iterative Gaussian quadrature method, makes the filter effective for real-time use. The advantages are illustrated by a model that captures the general flavour of modeling the highly uncertain behavior of a ship near obstacles such as a shore line into which it cannot go, and must manoeuvre away in some unknown fashion. The observations are of very poor quality, yet the filter behaves well and is quite stable. The procedure does not rely on linearization, but attempts to compute the conditional moments directly by approximating the integrations used by the optimal filter.

Path control of a surface ship in restricted waters using sliding mode

Abstract: This paper addresses the path control problem for a ship steering in restricted waters using sliding mode techniques. The ship's dynamic equations and numerical computation of the bank disturbance forces are briefly described. The ship's path control system is presented, which leads to a nonminimum phase system. Therefore, the traditional input-output linearization strategy in geometric control theory is modified to convert the controller design of a fourth-order nonlinear system to that of a second-order linear system. A continuous sliding mode controller is designed to ensure system's robustness and better performance. Simulation and experimental studies validate the controller design. The simulation results demonstrate the effectiveness of the proposed controller and its practicality in the steering control and navigation of the ship.

Floquet theory and non-linear perturbation analysis for oscillators with differential-algebraic equations

Abstract: Oscillators are key components of electronic systems. In RF communication systems, they are used for frequency translation of information signals and for channel selection, and in digital electronic systems, they are used as a time reference, i.e. a clock signal, in order to synchronize operations. Undesired perturbations in practical electronic systems adversely affect the spectral and timing properties of oscillators, which is a key performance limiting factor, being a major contributor to bit-error-rate (BER) of RF communication systems, and creating synchronization problems in clocked and sampled-data systems. Characterizing how perturbations affect oscillators is therefore crucial for practical applications. The traditional approach to analysing perturbed nonlinear systems (i.e. linearization) is not valid for oscillators. In this paper, we present a theory and efficient numerical methods, for non-linear perturbation and noise analysis of oscillators described by a system of differential-algebraic equations (DAEs). Our techniques can be used in characterizing phase noise and timing jitter due to intrinsic noise in IC devices, and evaluating the effect of substrate and supply noise on the timing properties of practical oscillators. In this paper, we also establish novel results for periodically time-varying systems of linear DAEs, which we rely on in developing the above theory and the numerical methods. Copyright © 2000 John Wiley & Sons, Ltd.

Efficient Computational Model for Inductive Plasma Flows

Abstract: A new approach to the numerical modeling of inductive plasmas is presented. The governing magnetohydrodynamic equations are discretized in a second-order accurate finite volume manner. A straightforward pressure-stabilized flow field solver is introduced as an alternative to the staggered-mesh solvers used in traditional algorithms. It is argued that the widely used integral boundary formulation for the electric field is computationally expensive and cannot be incorporated into an efficient iterative solution procedure. This approach is abandoned in favor of a 'far field' formulation of the electric field, such that powerful iterative methods can be applied to speed up the calculation. The discretized equations are solved through a damped Picard method and an approximate and full Newton method. Efficient linear algebra methods are used to solve the linear systems arising from the iterative methods. An appropriate linearization of the strongly positive Joule heating source term is important for the convergence at the linear level. The new model is tested on an LTE argon inductive plasma computation. The proposed approximate Newton method is found to converge substantially better than the Picard method. The full Newton method appears promising, although its successful use on fine meshes still requires improvements to the linear solver used.

Intelligent Observer and Control Design for Nonlinear Systems

Abstract: 1 Introduction - Control Aspects- 2 Motion Control- 3 Learning in Control Engineering- 4 Nonlinear Function Approximators- 5 Systematic Intelligent Observer Design- 6 Identification of Separable Nonlinearities- 7 Identification and Compensation of Friction- 8 Detection and Identification of Backlash- 9 Identification of Isolated Nonlinearities in Rolling Mills- 10 Input-Output Linearization: an Introduction- 11 Stable Model Reference Neurocontrol- 12 Dynamic Neural Network Compositions- 13 Further Strategies for Nonlinear Control with Neural Networks- List of Figures

A new method for the control of discrete nonlinear dynamic systems using neural networks

Abstract: A new controller design method for nonaffine nonlinear dynamic systems is presented in this paper. An identified neural network model of the nonlinear plant is used in the proposed method. The method is based on a new control law that is developed for any discrete deterministic time-invariant nonlinear dynamic system in a subregion /spl Psi//sub x/, of an asymptotically stable equilibrium point of the plant. The performance of the control law is not necessarily dependent on the distance between the current state of the plant and the equilibrium state if the nonlinear dynamic system satisfies some mild requirements in /spl Psi//sub x/. The control law is simple to implement and is based on a novel linearization of the input-output model of the plant at each instant in time. It can be used to control both minimum phase and nonminimum phase nonaffine nonlinear plants. Extensive empirical studies have confirmed that the control law can be used to control a relatively general class of highly nonlinear multiinput-multioutput (MIMO) plants.

Dynamic Analysis of Harmonics in Electrical Systems

Abstract: Frequency domain analysis and design of power systems is complicated in the presence of harmonics, switching dynamics, nonlinearities, unbalances, and for systems with mixed ac/dc dynamics. The reason is that linearization of the system does not lead to a time invariant system, but a system with periodically time varying dynamics, which implies that there is coupling between different frequencies. Often one has to rely on simplifying assumptions and simulation. The thesis uses linear time periodic (LTP) models to analyze power systems. The harmonic transfer function (HTF) for LTP systems is introduced. Using the HTF, the system can be treated as an infinitely dimensional linear time invariant system, which means that the system, under certain convergence conditions, can be analyzed using the well developed theory for LTI systems. The thesis contains four papers with power system applications. Paper I describes the modeling and analysis of networks including components with switching dynamics, such as diodes and thyristors. An algorithm for parameter estimation from experimental data is presented. Papers II and III treats modeling and analysis of single-phase railway systems. The modeling of the locomotives is performed in collaboration with industry. Paper IV treats analysis and control aspects of a converter for grid connection of a micro-turbine used for distributed power generation. This is a three-phase application done in collaboration with the industry. (Less)