A new set of orthogonal functions and its application to the analysis of dynamic systems
01 Jan 2006-Journal of The Franklin Institute-engineering and Applied Mathematics (JOURNAL OF THE FRANKLIN INSTITUTE)-Vol. 343, Iss: 1, pp 1-26
TL;DR: 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.
Abstract: 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.
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TL;DR: In this article, a new set of orthogonal basis functions is used to solve these integral equations via collocation method and numerical solutions of these equations are given for some cases of resistance distributions.
Abstract: In this paper the problem of electromagnetic scattering from the resistive surfaces is carefully surveyed. We model this problem by the integral equations of the second kind. A new set of orthogonal basis functions is used to solve these integral equations via collocation method. Numerical solutions of these equations are given for some cases of resistance distributions. Presented method in this paper can be easily generalized to apply to other cases.
30 citations
Cites background from "A new set of orthogonal functions a..."
...But the optimal representation of f(t) can be obtained if the coefficients ci and di are determined from the following two equations [30]: ∫ (i+1)h...
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...These functions are orthogonal [30], so: ∫ 1...
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...[30] as a new set of orthogonal functions....
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TL;DR: In this paper, a numerical method for solving nonlinear two-dimensional Volterra-Fredholm integro-differential equations of the second kind is presented, where the two dimensional triangular operational matrix of integration and differentiation has been presented.
Abstract: This article presents a numerical method for solving nonlinear two-dimensional Volterra–Fredholm integro-differential equations of the second kind. Here, we use the so-called two-dimensional triangular function, First, the two-dimensional triangular operational matrix of integration and differentiation has been presented, then by using this matrices, the nonlinear two-dimensional Volterra–Fredholm integro-differential equation has been reduced to an algebraic system. Finally, some numerical examples are given to clarify the efficiency and accuracy of the presented method.
29 citations
Cites background from "A new set of orthogonal functions a..."
...Orthogonality of 1D-TFs is shown in [15], that is, ∫ 1...
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...More details of 1D-TFs may be found in [15]....
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TL;DR: A numerical method for solving nonlinear Stochastic Ito-Volterra equations is proposed based on delta function approximations and the properties of DFs and their operational matrix of integration together with the Newton-Cotes nodes are presented.
28 citations
TL;DR: In this paper, the authors developed two-dimensional triangular orthogonal functions (2D-TFs) for numerical solution of the linear 2D Fredholm integral equations of the second kind.
27 citations
01 Jan 2013
TL;DR: In this paper, an efficient method for solving numerically stochastic Volterra integral equa- tion is proposed, which has several advantages in reducing computa- tional burden and is more accurate than the BPFs.
Abstract: In this paper, an efficient method for solving numerically stochastic Volterra integral equa- tion is proposed. Here, we consider triangular func- tions and their operational matrix of integration. This method has several advantages in reducing computa- tional burden and is more accurate than the BPFs. An error analysis is valid under fairly restrictive con- ditions. The method is applied to examples to illus- trate the accuracy and implementation of the method.
24 citations
Cites background or methods from "A new set of orthogonal functions a..."
...(2006) in [8], and TF approximation were successfully applied for analysis of dynamic systems [8], variational problems [6], integral equations [3,5], integrodifferential equations [4], Nonlinear Constrained Optimal Control Problems [9], and Volterra-Fredholm integral equations [2,12,13]....
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...From the definition of TFs, it is clear that TFs are disjoint, orthogonal, and complete [8]....
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References
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Book•
01 Jan 1970
TL;DR: This comprehensive treatment of the analysis and design of continuous-time control systems provides a gradual development of control theory and shows how to solve all computational problems with MATLAB.
Abstract: From the Publisher:
This comprehensive treatment of the analysis and design of continuous-time control systems provides a gradual development of control theoryand shows how to solve all computational problems with MATLAB. It avoids highly mathematical arguments, and features an abundance of examples and worked problems throughout the book. Chapter topics include the Laplace transform; mathematical modeling of mechanical systems, electrical systems, fluid systems, and thermal systems; transient and steady-state-response analyses, root-locus analysis and control systems design by the root-locus method; frequency-response analysis and control systems design by the frequency-response; two-degrees-of-freedom control; state space analysis of control systems and design of control systems in state space.
6,634 citations
TL;DR: In der Theorie der Reihenentwicklung der reellen Funktionen spielen die sog. orthogonalen Funktionensysteme eine fuhrende Rolle.
Abstract: In der Theorie der Reihenentwicklung der reellen Funktionen spielen die sog. orthogonalen Funktionensysteme eine fuhrende Rolle. Man versteht darunter ein System von unendlichvielen Funktionen $\phi_1 (s), \phi_2 (s),\ldots$, die in bezug auf die beliebige mesbare Punktmenge $M$ die Orthogonalitatseigenschaft $\int_{(M)}\phi_p(s)\phi_q(s)ds=0$ ($p
eq q, p, q=1,2,\ldots$), $\int_{(M)}(\phi_p(s))^2ds=1$ ($p=1,2,\ldots$) besitzen, wobei die Integrale im Lebesgueschen Sinne genommen sind. acces pdf
1,877 citations
01 Jan 1991
TL;DR: Continuous-time model-based system identification as mentioned in this paper is a well-established field in the field of control systems and is concerned with the determination of particular models for systems that are intended for a certain purpose such as control.
Abstract: System identification is a well-established field. It is concerned with the determination of particular models for systems that are intended for a certain purpose such as control. Although dynamical systems encountered in the physical world are native to the continuous-time domain, system identification has been based largely on discrete-time models for a long time in the past, ignoring certain merits of the native continuous-time models. Continuous-time-model-based system identification techniques were initiated in the middle of the last century, but were overshadowed by the overwhelming developments in discrete-time methods for some time. This was due mainly to the 'go completely digital' trend that was spurred by parallel developments in digital computers. The field of identification has now matured and several of the methods are now incorporated in the continuous time system identification (CONTSID) toolbox for use with Matlab. The paper presents a perspective of these techniques in a unified framework.
373 citations
TL;DR: In this paper, the Walsh operational matrix for performing integration and solving state equations is generalized to fractional calculus for investigating distributed systems and a new set of orthogonal functions is derived from Walsh functions.
Abstract: The Walsh operational matrix for performing integration and solving state equations is generalized to fractional calculus for investigating distributed systems. A new set of orthogonal functions is derived from Walsh functions. By using the new functions, the generalized Walsh operational matrices corresponding to √s, √(s2 + 1), e-s and e-√s etc. are established. Several distributed parameter problems are solved by the new approach.
207 citations