Journal ArticleDOI

# 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.

AbstractThe 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.

##### Citations
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Journal ArticleDOI
, Ning Sun1
TL;DR: A way to solve the fractional differential equations using the Riemann-Liouville fractional integral for repeated fractional integration and the generalized block pulse operational matrices of differentiation are proposed.
Abstract: The Riemann-Liouville fractional integral for repeated fractional integration is expanded in block pulse functions to yield the block pulse operational matrices for the fractional order integration. Also, the generalized block pulse operational matrices of differentiation are derived. Based on the above results we propose a way to solve the fractional differential equations. The method is computationally attractive and applications are demonstrated through illustrative examples.

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Journal ArticleDOI
TL;DR: An effective direct method to determine the numerical solution of the specific nonlinear Volterra-Fredholm integro-differential equations is proposed, based on new vector forms for representation of triangular functions and its operational matrix.
Abstract: An effective direct method to determine the numerical solution of the specific nonlinear Volterra-Fredholm integro-differential equations is proposed. The method is based on new vector forms for representation of triangular functions and its operational matrix. This approach needs no integration, so all calculations can be easily implemented. Some numerical examples are provided to illustrate the accuracy and computational efficiency of the method.

81 citations

### Cites background or methods from "A new set of orthogonal functions a..."

• ...From the definition of TFs, it is clear that triangular functions are disjoint, orthogonal, and complete [9]....

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• ...[9] and studied and used by Babolian et al....

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• ...Operational matrix of integration Expressing ∫ s 0 T1(τ )dτ and ∫ s 0 T2(τ )dτ in terms of the triangular functions follows [9] ∫ s...

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Journal ArticleDOI
TL;DR: A numerical method based on an m-set of general, orthogonal triangular functions (TF) is proposed to approximate the solution of nonlinear Volterra-Fredholm integral equations.
Abstract: A numerical method based on an m-set of general, orthogonal triangular functions (TF) is proposed to approximate the solution of nonlinear Volterra–Fredholm integral equations. The orthogonal triangular functions are utilized as a basis in collocation method to reduce the solution of nonlinear Volterra–Fredholm integral equations to the solution of algebraic equations. Also a theorem is proved for convergence analysis. Some numerical examples illustrate the proposed method.

64 citations

Journal ArticleDOI
TL;DR: Two-dimensional orthogonal triangular functions are presented as a new set of basis functions for expanding 2D functions and used to approximate solutions of nonlinear two-dimensional integral equations by a direct method.
Abstract: Two-dimensional orthogonal triangular functions (2D-TFs) are presented as a new set of basis functions for expanding 2D functions. Their properties are determined and an operational matrix for integration obtained. Furthermore, 2D-TFs are used to approximate solutions of nonlinear two-dimensional integral equations by a direct method. Since this approach does not need integration, all calculations can be easily implemented, and several advantages in reducing computational burdens arise. Finally, the efficiency of this method will be shown by comparison with some numerical results.

50 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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• ...[15] integrate each element of T1(s) and T2(s), and express the results in terms of 1D-TF vector, ∫ s...

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• ...A review of one-dimensional triangular functions One-dimensional triangular functions were introduced by Deb et al. (2006) in [15]....

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Journal ArticleDOI
Esmail Babolian
TL;DR: The present work proposes a method for solving Fredholm integral equations using a complementary pair of orthogonal triangular functions set derived from the well-known block pulse functions set, and demonstrates validity and applicability of the method.
Abstract: 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.

41 citations

### Cites background from "A new set of orthogonal functions a..."

• ...For more details about triangular orthogonal functions, see [1]....

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• ...The main body of Section 2 is based on [1]....

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• ...The authors of [1] called T1ðtÞ the left-handed triangular functions (LHTF) and T2ðtÞ the right-handed triangular functions (RHTF)....

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##### References
More filters
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 theoryand 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.

6,629 citations

Journal ArticleDOI
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

1,751 citations

01 Apr 2005

475 citations

Journal ArticleDOI
G.P. Rao
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

Journal ArticleDOI
, Tao Wu2
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.

203 citations