Piecewise Linear Form Research Articles

Optimization of Vibration System with a Damper Using Electro-Rheological Fluid.

This paper deals with optimum design of a damper using electro-rheological (ER) fluid having apparent viscosity that can be modified by an electric field. The authors propose new characteristics of damping force that are expressed as a function of road profile and horizontal position of the damper on the road in piecewise linear form, using the fact that the damping force of the ER damper and piston speed of the damper are almost independent each other. The maximum acceleration of a vibration system with the optimum ER damper which passes over a versine protuberance is less than that with the optimum existing damper for which the damping force is expressed as a function of piston speed of the damper.

Evolution of reaction-diffusion patterns in infinite and bounded domains

Abstract We introduce a semi-analytical method to study the evolution of spatial structures in reaction-diffusion systems. It consists in writing an integral equation for the relevant densities, from the propagator of the linear part of the evolution operator. In order to test the method, we perform an exhaustive study of a one-dimensional reaction-diffusion model associated to an electrical device - the ballast resistor. We consider the evolution of step and bubble-shaped initial density profiles in free space as well as in a semi-infinite domain with Dirichlet and Neumann boundary conditions. The piecewise-linear form of the reaction term, which preserves the basic ingredients of more complex nonlinear models, makes it possible to obtain exact wave-front solutions in free space and stationary solutions in the bounded domain. Short and long-time behaviour can also be analytically studied, whereas the evolution at intermediate times is analyzed by numerical techniques. We paid particular attention to the features introduced in the evolution by boundary conditions.

Fuzzy controller design by using neural network techniques

This paper investigates the relationship between the piecewise linear fuzzy controller (PLFC), in which the membership functions for fuzzy variables and the associated inference rules are all in piecewise linear forms, and a Gaussian potential function network based controller (GPFNC), in which the network output is a weighted summation of hidden responses from a series of Gaussian potential function units (GPFU's). Systematic procedures are proposed for transformation from a PLFC to its GPFNC counterpart, and vice versa. Based on these transformation principles, a series of systematic and feasible steps is presented for the design of an optimized PLFC (PLFC*) by using neural network techniques. In the design procedures, the simplified PLFC is used as the initial controller structure, then a GPFNC, which gives the approximate control response to the initially given PLFC, is found for further optimization. A neutralization process is used to demonstrate the feasibility and the potential applicability of these intelligent controllers on the regulation of highly nonlinear chemical processes. >

Use of Piecewise Linear Value Functions in Interactive Multicriteria Decision Support: A Monte Carlo Study

This paper describes a Monte Carlo study conducted to evaluate the effects of modelling assumptions and design parameters on the behaviour of interactive methods for the discrete choice MCDM problem, based on explicit value function models. The purpose of the study is to identify those assumptions and parameters which lead to the most efficient use of preference judgements made by the decision maker, and to the greatest robustness to judgmental errors. It is concluded that nonlinearities in the value function need to be modelled, achieved here by use of a piecewise linear form. It was also found that search for indifference points, rather than using simple preference judgements alone, is of great advantage, best realized by expressing judgements in terms of pairwise trade-offs. Methods incorporating these features are highly robust to judgmental errors. Interactive methods of this class are compared with a priori fitting of similar value functions, and found to give a very similar quality of solution.

A Note on Particle-Like Solutions for a Nonlinear Complex-Valued Klein– Equation

Solutions of a nonlinear complex-valued Klein-Gordon equation in one spatial dimension are considered. Localized rotating solutions are examined for finite intervals with periodic boundary conditions. Explicit solutions are given when the nonlinearity is of a simple piecewise-linear form. An example is given where interesting interference effects occur for moving solutions. Half integer values reminiscent of certain quantum-mechanical problems arise in a natural way even though the example is of a purely classical nature.

A network of oscillators

A model of neuronal behaviour capable of accounting for the oscillatory, plateau and rebound properties of biological neurons is derived, discussed and analysed. The model is based on a piecewise linear form of the FitzHugh-Nagumo equations, but reduces to a set of maps very similar to those of the Hopfield model (1982). In particular, the binary description of individual neurons and the well studied form of the synaptic current Sigma JijSj are preserved, although the model is capable of reproducing behaviours on the slow timescales characteristic of plateau and oscillation. By coupling two model cells together a mutually inhibitory or half-centred oscillator and an oscillator, fixed-phase follower pairs are constructed.

Ritz method for dynamic analysis of large discrete linear systems with non‐proportional damping

AbstractReal and complex Ritz vector bases for dynamic analysis of large linear systems with non‐proportional damping are presented and compared. Both vector bases are generated utilizing load dependent vector algorithms that employ recurrence equations analogous to the Lanczos algorithm. The choice of static response to fixed spatial loading distribution, as a starting vector in recurrence equations, is motivated by the static correction concept. Different phases of dynamic response analysis are compared with respect to computational efficiency and accuracy. It is concluded that the real vector basis approach is approximately eight times more efficient than the complex vector basis approach. The complex vector basis has some advantages with respect to accuracy, if the excitation is of piecewise linear form, since the exact solution can be utilized. In addition, it is demonstrated that both Ritz vector bases, real and complex, possess superior accuracy over the adequate eigenvector bases.

Shape matching using curvature processes

Shape matching is a fundamental problem of vision in general and interpretation of deforming shapes in particular. The objective of matching in this instance is to recover the deformation and therefore generalizes the notion of correlation, which aims to only produce a numerical measure of the similarity between two shapes. To address shape matching, we introduce a new representation of a closed 2D shape a cyclic sequence of the extended circular images of the convex and concave segments of its contour. This representation is then used to establish correspondences between segments of the two contours using dynamic programming. Finally, we compute a recovery of the differences between two similar contours in terms of the action of curvature process. Computation of convex and concave segments of the contours, given in piecewise linear form, is accomplished using the analytic representation of a local B-spline fit. We show the result of our deformation recovery scheme appliedto dynamic cloud silhouette analysis using hand-traced input from real satellite images.

Some extensions of fuzzy linear programming

In the formulation of fuzzy linear programming proposed by Zimmermann, it is assumed that fuzzy goals and fuzzy constraints are explicitly given as fuzzy sets on appropriate real lines respectively and are actually characterized by membership functions in linear form. For relaxing those assumptions, two kinds of extensions are derived in this paper. In the first extension, it is shown that fuzzy goals and fuzzy constraints expressed as fuzzy relations with fuzzy parameters can be treated as fuzzy sets on different real lines under some assumptions. In the second extension, it is shown that optimization in the case where the membership functions of the fuzzy goals and the fuzzy constraints are given in piecewise linear form can be achieved by using a standard linear programming technique repetitionally.

A multiplicative model for efficiency analysis

This paper develops theory and algorithms for a “multiplicative” Data Envelopment Analysis (DEA) model employing virtual outputs and inputs as does the CCR ratio method for efficiency analysis. The frontier production function results here are of piecewise log-linear rather than piecewise linear form.

An implementation of estimation techniques to a hydrological model for prediction of runoff to a hydroelectric power station

Parameter and state estimation algorithms have been applied to a hydrological model of a catchment area in southern Norway to yield improved control of the household of water resources and better economy and efficiency in the running of the power station, as experience proves since the system was installed on-line in the summer of 1978. A nonlinear conceptual state-space model of a Nordic hydrological system is presented in this paper. The model is transformed analytically to a piecewise linear time-discrete form in order to obtain straightforward updating of the gain matrix in the estimator through solution of the Riccati equation. Estimator gains can then be expressed approximately as a function of the state. Discharge coefficients (time constants) are estimated by the maximum likelihood method. Results are presented which show estimated and observed runoffs, and estimates of the state variables.

Computer Control of Traffic: Combining Subareas

The paper describes an investigation of the delays incurred by traffic passing between junctions with different cycle times. A range of traffic and road conditions is considered by means of simulation and for each of these conditions the cycle time of the downstream junction is varied between approximately 0.2 and 3.0 times each of the upstream cycle times tested. It is found that the relationship between the uniform component of delay and the ratio of the downstream to upstream cycle times can be represented in a piecewise linear form when the degree of saturation is kept constant. The nature of this relationship is investigated and restated for the case where the degree of saturation is a function of the cycle time. The analysis results in a procedure, which utilizes the TRANSYT program, for determining whether sub-areas which, when considered by themselves, operate best with different cycle times, should or should not be combined into a larger area or sub-area with a common cycle time. The criterion used is the total delay to traffic in the network.

Continuous Parameter Dependence in a Class of Volterra Equations

Conditions are found under which the solution of the Volterra integral equation $x'(t) + \int_0^t {a(t - s,\lambda )x(s)ds} = k,\quad x(0) = x_0 ,$ is continuous in $\lambda $, uniformly in $\{ 0 \leqq t < \infty \} $, when $a(t,\lambda )$ is nonnegative, nonincreasing, and convex as a function of t, for each $\lambda $. The main theorem concerns the case where the kernel has a special piecewise linear form and solutions are asymptotic $(t \to \infty )$ to nondegenerate periodic functions. This is the case excluded in similar earlier results of the author.The significance of these results for certain related Volterra equations in Hilbert space is summarized.

Modeling of three terminal devices: A black box approach

A unified black box approach is presented for synthesizing nonlinear dc circuit models of 3-terminal devices characterized by two families of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\upsilon-i</tex> curves in piecewise-linear form. In order to model arbitrary curves with nonuniform spacings and slopes, three simple but versatile building blocks-a controlled linear resistance, a controlled concave resistor, and a controlled convex resistor-are introduced and shown to be essential ingredients. The circuit parameters and functions characterizing each element in the black box model can be determined easily from the slopes and breakpoints associated with the segments of the prescribed characteristic curves. The paper concludes with the presentation of several nonlinear dc circuit models for four widely used devices; namely, bipolar transistors, field-effect transistors (FET's), unijunction transistors, and triacs.

Two-Stage Least-Squares Estimation with Shifts in the Structural Form

1. IN THIS NOTE we consider the estimation of linear models when the coefficients of the structural form are not the same for all observations for which the model is postulated to be valid. An example of such a model is given in [3], where some structural relations have a piecewise linear form. Another example is the water melon market model of Suits [2] where there are two alternative harvest supply schedules. Also discussed here is the case where for one part of the sample period one or more variables are endogenously determined while for another part they are exogenous, for instance, the wage rate or the rate of exchange. Such a change in the nature of the model can also be interpreted as a change in the coefficients of the structural form. It is assumed throughout that it is known a priori for what observations each specification holds. 2. A shift in the value of the coefficients of a predetermined variable does not cause special problems. If, say, only one shift occurs, one defines two predetermined variables to replace the original one. The vector of observations for the first of these consists of the observations on the original variable with the exception of those observations for which the second value is supposed to hold. These latter observations are replaced by zero's. The vector of observations on the second variable is simply the difference between the vector of observations for the original variable and the one for the first variable. 3. Next, consider the case where there is a shift in the value of one or more structural coefficients associated with an endogenous variable. Let the model in structural form be (1) Yt = y'B + x'C + ur where yt is an M-element vector of endogenous or jointly dependent variables, xt an L-element vector of predetermined variables, while ut is the M-element vector of structural disturbances. The matrix B is the M x M matrix of coefficients associated with the endogenous variables and C is the L x M matrix of coefficients associated with the predetermined variables. It is assumed that one or more elements of B take for some observations a different value than for others. The superscripts a and b are used to distinguish between the two situations. The following partitioning of the sets of T observations in two subsets of T7 and Tb observations, respectively, are introduced:

The Determination of Time Intervals on Phase-Plane Trajectories

An analytical method is given for determining the time intervals between points on the phase-plane trajectory of linear second-order systems. The procedure uses the concept of elapsedtime loci and is particularly useful where nonlinear systems can be reduced to piecewise-linear form or where the forcing function is linearly time dependent.

A mathematical technique for the conversion of radio wave interaction data toD-region electron density profiles

In this paper a method for the conversion of amplitude and phase interaction data to electron density profiles is developed. The method, using an iterative least squares error technique, synthesizes electron density profiles having a piece-wise linear functional form. With model studies the method is evaluated with respect to the errors that may be produced in the synthesized profile arising from uncertain values of collision frequency, disturbing pulse power, energy loss coefficient, and coefficients of interaction. Also, a demonstration of the effectiveness of the combined use of amplitude and phase interaction in producing a more accurate profile than could be obtained with amplitude data alone is presented. The method has been very successful in the synthesis of electron density profiles from actual interaction data and is presently being used for the reduction of data procured at the Ionosphere Research Laboratory during the International Quiet Sun Year. Some of these preliminary D-region profiles are presented.