The present work proposes a new set of hybrid functions (HF) which evolved from the synthesis of sample-and-hold functions (SHF) and triangular functions (TF). Traditional block pulse functions (BPF) still continue to...
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The present work proposes a new set of hybrid functions (HF) which evolved from the synthesis of sample-and-hold functions (SHF) and triangular functions (TF). Traditional block pulse functions (BPF) still continue to be attractive to many researchers in the arena of control theory. Block pulse functions also gave birth to a few useful variants. Two such variants are SHF and TF. The former is efficient for analyzing sample-and-hold control systems, while triangular functions established their superiority in obtaining piecewise linear solution of various control problems. After developing the basic theory of HF, a few square integrable functions are approximated via this set in a piecewise linear manner. For such approximation, it is shown, the mean integral square error (MISE) is much less than block pulse function based approximation. The operational matrices for integration in HF domain are also derived. Finally, this new set is employed for solving identification problem from impulse response data. The results are compared with the solutions obtained via BPF, SHF, TF, etc. and many illustrations are presented. (C) 2011 Elsevier Inc. All rights reserved.
The present work proposes application of a new set of orthogonal hybrid functions (HF) which evolved from the synthesis of orthogonal sample-and-hold functions (SHF) and orthogonal triangular functions (TF). This HF s...
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ISBN:
(纸本)9781467322706;9781467322713
The present work proposes application of a new set of orthogonal hybrid functions (HF) which evolved from the synthesis of orthogonal sample-and-hold functions (SHF) and orthogonal triangular functions (TF). This HF set is employed for determining the result of convolution of two time functions and the result has been used for solving linear control system analysis and synthesis problems. The theory is supported by examples and the results are compared with the exact solutions.
The present work proposes a method for solving Fredholm integral equations. This is demonstrated by using a complementary pair of orthogonal triangular functions set derived from the well-known block pulse functions s...
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The present work proposes a method for solving Fredholm integral equations. This is demonstrated by using a complementary pair of orthogonal triangular functions set derived from the well-known block pulse functions set. The operational matrices for integration, product of two triangular functions and some formulas for calculating definite integral of them are derived and utilized to reduce the solution of Fredholm integral equation to the solution of algebraic equations. Illustrative examples are included to show the high accuracy of the estimation, and to demonstrate validity and applicability of the method. (c) 2007 Elsevier Inc. All rights reserved.
Most integral equations of the first kind are ill posed, and obtaining their numerical solution often requires solving a linear system of algebraic equations of large condition number. Solving this system may be diffi...
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Most integral equations of the first kind are ill posed, and obtaining their numerical solution often requires solving a linear system of algebraic equations of large condition number. Solving this system may be difficult or impossible. Since many problems in one-dimensional (1D) and 2D scattering from perfectly conducting bodies can be modelled by linear Fredholm integral equations of the first kind, the main focus of this study is to present a fast numerical method for solving them. This method is based on vector forms for representation of triangular functions. By using this approach, solving the first kind integral equation reduces to solving a linear system of algebraic equations. To construct this system, the method uses sampling of functions. Hence, the calculations are performed very quickly. Its other advantages are the low cost of setting up the equations without applying any projection method such as collocation, Galerkin, etc;setting up a linear system of algebraic equations of appropriate condition number and good accuracy. To show the computational efficiency of this approach, some practical 1D and 2D scatterers are analysed by it.
The present work proposes a new set of orthogonal hybrid functions (HF) which evolved from synthesis of orthogonal sample-and-hold functions (SHF) and triangular functions (TF). The HF set is used to approximate a tim...
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ISBN:
(纸本)9781457711091
The present work proposes a new set of orthogonal hybrid functions (HF) which evolved from synthesis of orthogonal sample-and-hold functions (SHF) and triangular functions (TF). The HF set is used to approximate a time function in a piecewise linear manner. The algorithm of one-shot operational matrices for integration of different orders in HF domain are derived and employed for more accurate multiple integration and solution of third order non-homogeneous differential equations. The results are compared with the exact solution and the results obtained via 4th order Runge-Kutta method.
This paper establishes a Ritz direct method for solving variational problems via a set of complementary pair of triangular orthogonal functions, derived from well-known block pulse functions. The properties of triangu...
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This paper establishes a Ritz direct method for solving variational problems via a set of complementary pair of triangular orthogonal functions, derived from well-known block pulse functions. The properties of triangular functions are presented, and the operational matrices for integration, product and some formulas for calculating definite integration of product are derived and utilized to obtain a clear procedure to reduce a variational problem to the solution of algebraic equations. Illustrative examples are included to show the high accuracy of the estimation, and to demonstrate validity and applicability of the method. (c) 2007 Elsevier Inc. All rights reserved.
Aiming at fast and effectively evaluating harmonics in the power system, a triangular neural network is constructed, of which the hidden neurons are activated with triangular functions. Based on gradient descent metho...
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ISBN:
(纸本)9780769537283
Aiming at fast and effectively evaluating harmonics in the power system, a triangular neural network is constructed, of which the hidden neurons are activated with triangular functions. Based on gradient descent method, the learning rules (i.e., weights-iterative-formula) for the constructed neural network are derived. Then global-convergence of the weights-iterative-formula is proved. As the results, a weights-direct-determination method is achieved, which could obtain the optimal weights of such a neural network in one step by using pseudo-inverse. Furthermore, several numerical tests have been conducted to apply this method to some harmonics models. The simulation results substantiate this method can be used to fast and precisely evaluate the harmonic components.
The present work proposes a complementary pair of orthogonal triangular function (TF) sets derived from the well-known block pulse function (BPF) set. The operational matrices for integration in TF domain have been co...
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The present work proposes a complementary pair of orthogonal triangular function (TF) sets derived from the well-known block pulse function (BPF) set. The operational matrices for integration in TF domain have been computed and their relation with the BPF domain integral operational matrix is shown. It has been established with illustration that the TF domain technique is more accurate than the BPF domain technique as far as integration is concerned, and it provides with a piecewise linear solution. As a further study, the newly proposed sets have been applied to the analysis of dynamic systems to prove the fact that it introduces less mean integral squared error (MISE) than the staircase solution obtained from BPF domain analysis, without any extra computational burden. Finally, a detailed study of the representational error has been made to estimate the upper bound of the MISE for the TF approximation of a function f(t) of Lebesgue measure. (c) 2005 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
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