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Derive
Published in Paul W. Ross, The Handbook of Software for Engineers and Scientists, 2018
The Taylor series is a technique used to approximate a function by a polynomial expression. The Taylor series of a function f(x) at the expansion point x = a is given by f(x)=f(a)+f′(a)(x−a)+f″(a)2(x−a)2+⋯+fn(a)n!(z−a)n
Systems Integration Management
Published in Gary O. Langford, Engineering Systems Integration, 2016
A Taylor series expansion is often used to approximate a function as a polynomial of terms whose first terms turn out to be reasonably close approximations to that which would otherwise seems mathematically complicated (Mason et al. 2003). The first derivative of a Taylor series expansion taken about the target value is a quadratic curve when the target value is set to zero. The curve’s minimum (or nominal position) is centered on the target value, which (Taguchi et al. 1989) has shown to provide the best performance in the eyes of the customer. However, identifying the appropriate performance measures as well as selecting the best target value can be challenging. Designers sometimes offer their best guess. The quadratic form was chosen by Taguchi because it was both simple, and as it turned out, useful. Further, after the Taylor expansion, higher powers in the series change the loss at the target value by a very small margin, and for practical purposes can be ignored within experimental error. Symmetric formulations of loss functions are assumed to be approximate and accurate to a first order. This assumption is shown to be accurate since the result of development is indeed what is placed into service by users, that is, that which is equivalent to a Taguchi validated quadratic form of loss function. The general loss function discussed in this book, provides the quantitative means to evaluate integration from conceptualization through disposal by adapting the order of the loss function to the desired phase in the product’s lifecycle. Asymmetric loss functions are most useful when integrating systems into a system of systems. The reason for this situation (as distinct and different from that of integrating a system) is the requirement for reversibility of actions to allow a system to remain a system when it is no longer a part of a system of systems.
Introduction
Published in Vlad P. Shmerko, Svetlana N. Yanushkevich, Sergey Edward Lyshevski, Computer Arithmetics for Nanoelectronics, 2018
Vlad P. Shmerko, Svetlana N. Yanushkevich, Sergey Edward Lyshevski
In the classic Taylor series, the coefficients are calculated as derivatives of the initial function at a certain point. By analogy, the coefficients of the logic Taylor series are Boolean differences (derivatives) with respect to a variable, and multiple Boolean differences at certain points (assignments of Boolean variables). A certain point in logic Taylor expansion is called the polarity. The polarity specifies which variables in Taylor expansion are complemented.
Data-Driven Forecasting of Nonlinear System with Herding via Multi-Dimensional Taylor Network
Published in Cybernetics and Systems, 2022
Hong-Sen Yan, Guo-Biao Wang, Bo Zhou, Xiao-Qin Wan, Jiao-Jun Zhang
In addition, the function approximated by the Taylor series is known, and its coefficients are just the derivatives of its order. In the case of multiple variables, it can be taken as the MISO (multiple input single output) system. However, the function approximated by MTN is unknown, and its coefficients are also unknown and need to be determined through training or learning. In the case of multiple variables, it can be taken as the MIMO (multiple input multiple output) system with complex coupling relationships among variables, that is, a recurrent network. From the viewpoint of regression, MTN contains many equations. There exist complicated coupling relationships known as multi-dimensional multi-element polynomial regression, which is the extension of the existing multi-element polynomial regression containing only one single equation. From the perspective of mechanism and non-mechanism, MTN is a semi-mechanism model between the mechanism model and non-mechanism one (e.g., NN) because its linear terms are generally of clear physical sense.
Recursive approximate solution to time-varying matrix differential Riccati equation: linear and nonlinear systems
Published in International Journal of Systems Science, 2018
Saeed Rafee Nekoo, Mohsen Irani Rahaghi
The intention of presenting different numerical methods always was (and is) to provide a faster, simpler and more applicable solution. Taylor series expansion is a prominent method for representing a function around a point as a polynomial. The motivation for using Taylor method to generate a solution is simplicity along with fast convergence. In this present research, Taylor series is employed in a recursive manner to give a precise and simple solution to a class of systems with fewer sampling time rather than common numerical solutions. The application of this approach is presented in finite-horizon optimal control: linear and nonlinear systems. This work is concerned with a solution to matrix Riccati equation (in finite horizon) which the LTV case of that results in linear quadratic regulator (LQR) problem, and nonlinear extension of that is so-called ‘state-dependent Riccati equation (SDRE).’
New LSTM Deep Learning Algorithm for Driving Behavior Classification
Published in Cybernetics and Systems, 2023
Nesrine Kadri, Ameni Ellouze, Mohamed Ksantini, Sameh Hbaieb Turki
Taylor series are an approximation of functions by a series of powers or sum of whole powers of polynomials. Researchers resort to this approximation when computations are time consuming or too complex or also when this function is unknown and we just need to get a rough idea of its main properties. The only thing that is required is to be able to calculate this function at one or more points and to make an estimate for all the other values. The idea then is to obtain a good approximation of the function in a neighborhood of the reference point.