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General introduction
Published in Adedeji B. Badiru, Handbook of Industrial and Systems Engineering, 2013
The Maclaurin series expansion is a special case of the Taylor series expansion for a=0. f(x)=f(0)+f'(0)x1!+f''(0)x22!+f'''(0)x33!+⋯+f(n-1)(0)xn-1(n-1)!+Rn
Bending of Thick Plates on Elastic Foundations
Published in Wen L. Li, Weiming Sun, Fourier Methods in Science and Engineering, 2023
In numerical calculations, all the involved series expansions will have to be truncated to contain only a pre-determined number of terms in each direction. Thus, (11.74) actually represents a finite system of linear algebraic equations about the expansion coefficients of the multiscale Fourier series solutions.
Robust design optimisation of adaptive cruise controller considering uncertainties of vehicle parameters and occupants
Published in Vehicle System Dynamics, 2020
Hansu Kim, Tae Hee Lee, Yuho Song, Kunsoo Huh
The Taylor series expansion method involves the approximation of performance function through Taylor series expansion as follows [22,23]: where is mean of random variable , is the first derivative of the performance function, and is the second derivative of the performance function. Performance functions are approximated to their first derivative term owing to the weak nonlinearity of these functions. Statistical moments, such as mean and variance of the performance functions, can be calculated using the approximated performance functions, as follows [24]: where is the number of random variables, is the mean of the performance functions, is the variance of performance functions, and is the standard deviation of the i-th random variable. The derivative term of performance functions is evaluated using the central difference method, which is a numerical differentiation.
Variance-constrained filtering for discrete-time genetic regulatory networks with state delay and random measurement delay
Published in International Journal of Systems Science, 2019
Dongyan Chen, Weilu Chen, Jun Hu, Hongjian Liu
Motivated by the above discussions, in this paper, we aim to investigate the variance-constrained filtering problem for GRNs with state delay and random one-step measurement delay. A Bernoulli distributed random variable with known occurrence probability is introduced to describe the phenomenon of the random one-step measurement delay, which may occur during the data transmission. We employ the Taylor series expansion to deal with the regulatory function in the GRNs, namely, the first order approximation in the Taylor series expansion is utilised during the linearisation. Then, a variance-constrained filtering algorithm is developed to estimate the concentrations of the mRNAs and proteins. The main contributions of this paper lie in the following three aspects: (1) the variance-constrained filtering problem is, for the first time, discussed for discrete-time GRNs with the random one-step measurement delay; (2) a new variance-constrained filtering algorithm is proposed to compensate the impact from state delay and random one-step measurement delay on the filtering algorithm performance; and (3) the effects of different occurrence probabilities of random one-step measurement delay on filtering performance are analysed. In addition, the main difficulties lie in that: (1) how to select appropriate approach to deal with the regulatory function in the GRNs when designing filtering scheme; and (2) how to make full use of the information of the state delay and random one-step measurement delay to improve the estimation accuracy.
Trajectory reconstruction using locally weighted regression: a new methodology to identify the optimum window size and polynomial order
Published in Transportmetrica A: Transport Science, 2018
Suvin P. Venthuruthiyil, Mallikarjuna Chunchu
Let the order of the estimated polynomial function be p. The Taylor series expansion of this function around is The following minimization problem was used to estimate the coefficients of the polynomial function centered at : And finally to produce a fit where is the observed trajectory data centered at within the window size, N; is the estimated polynomial functions corresponding to different polynomial orders; is the weights to the observations used for local estimation; is the estimated polynomial function within the window; is the coefficients of the estimated trajectory; is the vector of polynomial terms.