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.
Citations
More filters
TL;DR: An efficient numerical method is proposed for solving nonlinear mixed type Volterra–Fredholm integral equations, using two-dimensional orthogonal triangular functions (2D-TFs) in a direct approach that has several advantages in reducing computational burden.
35 citations
TL;DR: 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 is established.
34 citations
Cites background from "A new set of orthogonal functions a..."
...The authors of [9] called T1ðtÞ, the left-handed triangular functions (LHTF) and T2ðtÞ, the right-handed triangular functions (RHTF)....
[...]
...The main body of the Section 2, is based on [9]....
[...]
...in [9], presented a new set of orthogonal functions which they called it triangular orthogonal functions, and then, they calculated the operational matrix for integration....
[...]
...For more details about triangular orthogonal functions see [9]....
[...]
Journal Article•
TL;DR: In this article, Babolian and Hatamzadeh-Varmazyar proposed a direct method to compute numerical solutions of the linear Volterra and Fredholm integral equations system.
Abstract: E. Babolian a, Z. Masouri b , S. Hatamzadeh-Varmazyar c (a) Department of Mathematics, Teacher Training University, Tehran, Iran (b) Department of Mathematics, Khorramabad Branch, Islamic Azad University, Khorramabad, Iran (c) Department of Electrical Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran |||||||||||||||||||||||||||||||Abstract A practical direct method to compute numerical solutions of the linear Volterra and Fredholm integral equations system is proposed. This approach is based on vector forms of triangular functions and its operational matrices and without any integration reduces an integral equations system to a system of algebraic equations. Numerical results of some examples show that the method is practical and has high accuracy.
31 citations
Cites background or methods from "A new set of orthogonal functions a..."
...[4] and studied and used by Babolian et al....
[...]
...1 De nition and expansion Two m-sets of triangular functions (TFs) are de ned over the interval [0; T ) as [4]...
[...]
...where P1 and P2 are the operational matrices of integration of TFs as follows [4]...
[...]
TL;DR: This paper presented an approximate method for solving optimal control problem of Volterra integral equations based upon orthogonal triangular functions that has been proved for optimal control and cost functionals.
30 citations
TL;DR: In this article, an effective numerical method for determining the scattered electromagnetic fields from thin wires is presented and discussed, which is modeled by the integral equations of the first kind, and illustrative computations are given for several cases.
Abstract: In this paper an effective numerical method for determining the scattered electromagnetic fields from thin wires is presented and discussed. This problem is modeled by the integral equations of the first kind. The basic mathematical concept is the method of moments. The problem of determining these scattered fields is treated in detail, and illustrative computations are given for several cases.
30 citations
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 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