Showing papers on "Linearization published in 1995"

Nonlinear Models for Repeated Measurement Data

Abstract: Introduction. Nonlinear regression models for individual data. Hierarchical linear models. Hierarchical nonlinear models. Inference based on individual estimates. Inference based on linearization. Nonperametric and semiparametric inference. Bayesian inference. Pharamcokinetic and pharamacodynamic analysis. Analysis of assay data. Further applications. Open problems and discussion. References. Indices.

Non-LTE line-blanketed model atmospheres of hot stars. 1: Hybrid complete linearization/accelerated lambda iteration method

Abstract: A new munerical method for computing non-Local Thermodynamic Equilibrium (non-LTE) model stellar atmospheres is presented. The method, called the hybird complete linearization/accelerated lambda iretation (CL/ALI) method, combines advantages of both its constituents. Its rate of convergence is virtually as high as for the standard CL method, while the computer time per iteration is almost as low as for the standard ALI method. The method is formulated as the standard complete lineariation, the only difference being that the radiation intensity at selected frequency points is not explicity linearized; instead, it is treated by means of the ALI approach. The scheme offers a wide spectrum of options, ranging from the full CL to the full ALI method. We deonstrate that the method works optimally if the majority of frequency points are treated in the ALI mode, while the radiation intensity at a few (typically two to 30) frequency points is explicity linearized. We show how this method can be applied to calculate metal line-blanketed non-LTE model atmospheres, by using the idea of 'superlevels' and 'superlines' introduced originally by Anderson (1989). We calculate several illustrative models taking into accont several tens of thosands of lines of Fe III to Fe IV and show that the hybrid CL/ALI method provides a robust method for calculating non-LTE line-blanketed model atmospheres for a wide range of stellar parameters. The results for individual stellar types will be presented in subsequent papers in this series.

Arbitrary path tracking control of articulated vehicles using nonlinear control theory

Abstract: Designs a path tracking controller for an articulated vehicle (a semitrailer-like vehicle) using time scale transformation and exact linearization. The proposed controller allows articulated vehicles to follow arbitrary paths consisting of arcs and lines, while they are moving forward and/or backward. The experimental result of the 8-shaped path tracking control of the articulated vehicle moving backward is also presented. >

Robust Dynamic Inversion for Control of Highly Maneuverable Aircraft

Abstract: This paper presents a methodology for the design of flight controllers for aircraft operating over large ranges of angle of attack. The methodology is a combination of dynamic inversion and structured singular value (p) synthesis. An inner-loop controller, designed by dynamic inversion, is used to linearize the aircraft dynamics. This inner-loop controller lacks guaranteed robustness to uncertainties in the system model and the measurements; therefore, a robust, linear outer-loop controller is designed using /i synthesis. This controller minimizes the weighted HQO norm of the error between the aircraft response and the specified handling quality model while maximizing robustness to model uncertainties and sensor noise. The methodology is applied to the design of a pitch rate command system for longitudinal control of a high-performance aircraft. Nonlinear simulations demonstrate that the controller satisfies handling quality requirements, provides good tracking of pilot inputs, and exhibits excellent robustness over a wide range of angles of attack and Mach number. The linear controller requires no scheduling with flight conditions. HE objective of this paper is to present a method for design of flight controllers that provides desired handling qualities over a wide range of flight conditions with minimal scheduling. Acceptable stability and performance robustness must be maintained in the presence of unmodeled dynamics, uncertainties in the aircraft design model, and noisy sensor measurements. The aircraft considered in this paper is the NASA high angle-ofattack research vehicle (HARV), which is typical of future fighter aircraft. It is capable of flight at very high angles of attack and has thrust vectoring as well as conventional aerodynamic control surfaces.1 The unaugmented aircraft does not meet handling quality requirements and some type of augmentation is necessary. This paper considers only the longitudinal control. The controller relates pilot longitudinal stick input to the symmetric deflection of the stabilizer and the longitudinal deflection of the thrust vectoring vanes. The control design philosophy is to use an inner-loop, dynamic inversion controller and an outer-loop, linear \JL controller. The dynamic inversion controller linearizes the pitch rate dynamics of the aircraft; however, since model uncertainties prevent exact linearization, there will always be errors associated with this controller. A simple linear fractional transformation model of these errors is developed for use in design of the outer-loop /^ controller. This controller provides pitch rate following by minimizing the weighted //oo-norm of the difference between the actual aircraft pitch rate response to pilot stick inputs and the desired response to these inputs as given by a transfer function model based on standard handling quality specifications. Thus the outer-loop \Ji controller is an implicit model following design, which provides robustness to errors due to the lack of exact cancellation of the pitch rate dynamics by the dynamic inversion controller. Recently a number of papers have appeared that describe controllers for a highly maneuverable aircraft. In Refs. 2-5, application of linear multi-input/multi-output (MIMO) control design techniques to this problem were presented. In every case, excellent

An approximate Kaiman filter for ocean data assimilation: An example with an idealized Gulf Stream model

Abstract: A practical method of data assimilation for use with large, nonlinear, ocean general circulation models is explored. A Kalman filter based on approximation of the state error covariance matrix is presented, employing a reduction of the effective model dimension, the error's asymptotic steady state limit, and a time-invariant linearization of the dynamic model for the error integration. The approximations lead to dramatic computational savings in applying estimation theory to large complex systems. We examine the utility of the approximate filter in assimilating different measurement types using a twin experiment of an idealized Gulf Stream. A nonlinear primitive equation model of an unstable east-west jet is studied with a state dimension exceeding 170,000 elements. Assimilation of various pseudomeasurements are examined, including velocity, density, and volume transport at localized arrays and realistic distributions of satellite altimetry and acoustic tomography observations. Results are compared in terms of their effects on the accuracies of the estimation. The approximate filter is shown to outperform an empirical nudging scheme used in a previous study. The examples demonstrate that useful approximate estimation errors can be computed in a practical manner for general circulation models.

An accelerated learning algorithm for multilayer perceptrons: optimization layer by layer

Abstract: Multilayer perceptrons are successfully used in an increasing number of nonlinear signal processing applications. The backpropagation learning algorithm, or variations hereof, is the standard method applied to the nonlinear optimization problem of adjusting the weights in the network in order to minimize a given cost function. However, backpropagation as a steepest descent approach is too slow for many applications. In this paper a new learning procedure is presented which is based on a linearization of the nonlinear processing elements and the optimization of the multilayer perceptron layer by layer. In order to limit the introduced linearization error a penalty term is added to the cost function. The new learning algorithm is applied to the problem of nonlinear prediction of chaotic time series. The proposed algorithm yields results in both accuracy and convergence rates which are orders of magnitude superior compared to conventional backpropagation learning. >

A linear algebraic framework for dynamic feedback linearization

Abstract: To any accessible nonlinear system we associate a so-called infinitesimal Brunovsky form. This gives an algebraic criterion for strong accessibility as well as a generalization of Kronecker controllability indices. An output function which defines a right-invertible system without zero dynamics is shown to exist if and only if the basis of the Brunovsky form can be transformed into a system of exact differential forms. This is equivalent to the system being differentially flat and hence constitutes a necessary and sufficient condition for dynamic feedback linearizability. >

Dynamic inversion of nonlinear maps with applications to nonlinear control and robotics

Abstract: This dissertation introduces the notion of a dynamic inverse of a nonlinear map. The dynamic inverse is used in the construction of nonlinear dynamical system, called a dynamic inverter, that asymptotically solves inverse problems with time-varying vector-valued solutions. Dynamic inversion generalizes and extends many previous results on the inversion of maps using continuous-time dynamic systems. By posing the dynamic inverse itself as the solution to an inverse problem, we show how one may solve for a dynamic inverse dynamically while simultaneously using the dynamic inverse solution to solve for the time-varying root of interest. Dynamic inversion is a continuous-time dynamic computational paradigm that may be incorporated into controllers in order to continuously provide estimates of time-varying parameters necessary for control. This allows nonlinear control systems to be posed entirely in continuous-time, replacing discrete root-finding algorithms as well as discrete algorithms for matrix inversion with integration. Example applications include solving for the intersection of time-varying polynomials, inversion of nonlinear control systems, regular and generalized inversion of fixed and time-varying matrices, polar decomposition of fixed and time-varying matrices, output tracking of implicitly defined reference trajectories, end-effector tracking control for robotic manipulators, and causal approximate output tracking for nonlinear nonminimum-phase systems. For the problem of output tracking for nonminimum-phase systems, an internal equilibrium manifold is introduced. This manifold is intrinsic to the class of nonlinear nonminimum-phase systems studied. Approximate output tracking is achieved by constructing a controller that makes a neighborhood of the internal equilibrium manifold attractive and invariant. Dynamic inversion is incorporated into the controller to provide a continuous estimate of the manifold location. This estimate is incorporated into the tracking control law. We demonstrate, by application to the tracking problem for the inverted pendulum on a cart, that the resulting internal equilibrium controller significantly outperforms a linear quadratic regulator, where the linearization of the internal equilibrium controller is made identical to the linear quadratic regulator. We also apply internal equilibrium control to the problem of causing a nonlinear, nonholonomic model of a bicycle to track a time-parameterized trajectory in the ground plane while retaining balance.

Adaptation behavior of a feedforward amplifier linearizer

Abstract: Feedforward linearization has advantages in bandwidth and generality over other linearization methods. However, it is based on the subtraction of nearly equal quantities, so its major parameters must adapt to changes in environmental or operating conditions. This paper is the first published analysis of adaptation behavior in feedforward amplifier linearizers. As such, it presents an analytical framework and several new results, including convergence time and coefficient jitter, a bias effect that leads to extreme accuracy requirements in one coefficient, the effect of delay mismatch, and the mitigating effects of a filter inserted in one adaptation path.

A differential geometric setting for dynamic equivalence and dynamic linearization

Abstract: This paper presents an (infinite dimensional) geometric framework for control system, based on infinite jet bundles, where a system is represented by a single vector field and dynamic equivalence (to be precise : equivalence by endogenous dynamic feedback) is conjugation by diffeomorphisms These diffeomorphisms are very much related to Lie-Backlund transformations It is proved in this framework that dynamic equivalence of single-input systems is the same as static equivalence NB: this paper is followed by "Infinitesimal Brunovsky form for nonlinear systems with applications to dynamic linearization", by E Aranda-Bricaire, C H Moog and J-B Pomet, published in the same 1995 volume, which is its natural follow-up This is a corrected version of the reports http://halinriafr/inria-00074360 and http://halinriafr/inria-00074361

A relaxation method for nonconvex quadratically constrained quadratic programs

Abstract: We present an algorithm for finding approximate global solutions to quadratically constrained quadratic programming problems. The method is based on outer approximation (linearization) and branch and bound with linear programming subproblems. When the feasible set is non-convex, the infinite process can be terminated with an approximate (possibly infeasible) optimal solution. We provide error bounds that can be used to ensure stopping within a prespecified feasibility tolerance. A numerical example illustrates the procedure. Computational experiments with an implementation of the procedure are reported on bilinearly constrained test problems with up to sixteen decision variables and eight constraints.

H/sub /spl infin// nonlinear filtering of discrete-time processes

Abstract: This correspondence investigates the problem of H/sub /spl infin// estimation of a discrete-time nonlinear process. An estimator, which may be nonlinear, is introduced so that an H/sub /spl infin//-norm-like of what we call a generalized estimation error is guaranteed to be bounded by a prescribed level. Conditions for the existence of such an estimator, and formulae for its derivation, are obtained utilizing a discrete-time analog of the Hamilton-Jacobi inequality. An approximate filter based on linearization is developed. This filter relates to the extended Kalman filter in the same way that the linear H/sub /spl infin// filter relates to the Kalman filter. >

Output tracking for a non-minimum phase dynamic CTOL aircraft model

Abstract: A dynamic model for the longitudinal axis of a conventional takeoff and landing (CTOL) aircraft is presented. Non-minimum phase characteristics in this model result from the fact that the process of generating an upward pitch moment produces a small downward force, causing the aircraft to lose altitude. The model is not full state linearizable and the internal dynamics which remain after input-output linearization using the coordinates of the center of mass as outputs are unstable. The CTOL model is not flat with respect to fixed points on the aircraft body. The nonlinear inversion technique produces stable trajectories for the states of the internal dynamics, but the corresponding feedforward force inputs required to track these trajectories are large. Approximate linearization techniques which ignore the coupling between the pitch moment and the vertical and horizontal aircraft dynamics, may be used to calculate inputs of smaller magnitude.

A Two-Step Iterative Algorithm For Estimation In Nonlinear Mixed-Effect Models With An Evaluation In Population Pharmacokinetics

Abstract: This article proposes an EM-like algorithm for estimating, by maximum likelihood, the population parameters of a nonlinear mixed-effect model given sparse individual data The first step involves Bayesian estimation of the individual parameters During the second step, population parameters are estimated using a linearization about those Bayesian estimates This algorithm (implemented in P-PHARM) is evaluated on simulated data, mimicking pharmacokinetic analyses and compared to the First-Order method and the First-Order Conditional Estimates method (both implemented in NONMEM) The accuracy of the results, within few iterations, shows the estimation capabilities of the proposed approach

Fault tolerance of magnetic bearings by generalized bias current linearization

Abstract: The mathematical basis for bias linearization of quadratic magnetic actuators, as typified by magnetic bearings, is developed. The approach generalizes prior ad hoc methods of linearizing the relationship between actuator force and electromagnet current, obviating the earlier assumptions of stator symmetry. This relationship is fundamentally quadratic in the regime where the magnetic material is unsaturated and flux is essentially proportional to magnet current. Growing from the properties of a fundamental representation for the current-force relationships in magnetic bearings, conditions are determined under which linearization may be possible. A numerical optimization problem is posed whose solution provides a linearization scheme which maximizes the available force capacity of the actuator. A corollary result is a method for obtaining coil-fault tolerance in magnetic bearings without adding coils to existing actuators. Several paper examples are presented to illustrate linearization of asymmetric actuators, actuators with failed coils, and force/moment actuators. >

Four‐dimensional data assimilation with a wide range of scales

Abstract: A number of experiments investigating four-dimensional variational data assimilation using the adjoint method are presented. It has been proposed that the method will be able to produce improved initial conditions in data-sparse regions. In order to describe the flow in a region where there is little data, it is necessary for observational information either to be advected into the region, or to cascade downscale from larger scales characterizing the separation between observations. We focus on the latter and examine the method's ability to “fill in” small-scale detail determined dynamically from large-scale data. We choose to examine barotropic β-plane flow since it is one of the simplest geophysical settings involving a wide range of scales. In the limit of small error, predictability studies have shown that exponential error growth occurs along the gradients of two highly-correlated realizations. When the realizations have decorrelated, error statistics saturate at climatological levels. By appealing to the adjoint of the linearized equations, the adjoint method accounts for the former behaviour, but not the latter. When the assimilation period exceeds the validity timescale of the linearization, the assimilated fields show spectra which are spuriously shallow in the small scales, following the basic-state gradients. Moreover, it is essential to note that the validity timescale of the linearization is a function of lengthscale. Therefore, for a given assimilation period there is a scale below which useful initial conditions cannot be obtained. Equivalently, for a given model resolution, there is an assimilation period beyond which the exact initial conditions cannot be recovered. Some speculation on the optimal resolution at which to. perform 4D data assimilation as a function of the assimilation period is offered. DOI: 10.1034/j.1600-0870.1995.00204.x

Switched reluctance motor control via fuzzy adaptive systems

Abstract: This article presents the application of fuzzy adaptive systems to the problem of torque ripple reduction in a switched reluctance motor. Conventional methods for torque linearization and decoupling are reviewed briefly, as is the previous application, by the authors, of neural network based techniques. A solution based on the use of fuzzy adaptive systems is then described. Experimental measurements of the static torque production characteristics of a 4 kW, four-phase switched reluctance motor form the basis of simulation studies of this novel approach. The simulation results demonstrate the capability of fuzzy adaptive systems to learn nonlinear current profiles that minimize torque ripple. The use of fuzzy systems in this application has potential advantages where the incorporation of a priori information, expressed linguistically, is concerned. Experimental results illustrate the effectiveness of the approach. >

Comparison of Assembly Tolerance Analysis by the Direct Linearization and Modified Monte Carlo Simulation Methods

Abstract: Two methods for performing statistical tolerance analysis of mechanical assemblies are compared: the Direct Linearization Method (DLM), and Monte Carlo simulation. A selection of 2-D and 3-D vector models of assemblies were analyzed, including problems with closed loop assembly constraints. Closed vector loops describe the small kinematic adjustments that occur at assembly time. Open loops describe critical clearances or other assembly features. The DLM uses linearized assembly constraints and matrix algebra to estimate the variations of the assembly or kinematic variables, and to predict assembly rejects. A modified Monte Carlo simulation, employing an iterative technique for closed loop assemblies, was applied to the same problem set. The results of the comparison show that the DLM is accurate if the tolerances are relatively small compared to the nominal dimensions of the components, and the assembly functions are not highly nonlinear. Sample size is shown to have great influence on the accuracy of Monte Carlo simulation.

Linearization of DAEs along trajectories

Abstract: Over the last decade there has been a considerable amount of research on numerical and analytic aspects of linear and nonlinear differential algebraic equations (DAEs)F(x', x, t, u)=0. Many of these papers have either considered linear equations or based their analysis on linear equations. However, until very recently there has been little rigorous investigation of the relationship between the linearization of a DAE and the original equations. In this paper we carefully examine several aspects of this relationship. Positive results for time varying linearization and counter examples for time invariant linearization are given.