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: In this article, the authors have dealt with seven kinds of non-linear Volterra and Fredholm classes of equations and formulated an algorithm for solving the aforementioned equation types via Hybrid Function (HF) and Triangular Function (TF) piecewise-linear orthogonal approach.
Abstract: In this paper the authors have dealt with seven kinds of non-linear Volterra and Fredholm classes of equations. The authors have formulated an algorithm for solving the aforementioned equation types via Hybrid Function (HF) and Triangular Function (TF) piecewise-linear orthogonal approach. In this approach the authors have reduced integral equation or integro-differential equation into equivalent system of simultaneous non-linear equation and have employed either Newton’s method or Broyden’s method to solve the simultaneous non-linear equations. The authors have calculated the L2-norm error and the max-norm error for both HF and TF method for each kind of equations. Through the illustrated examples, the authors have shown that the HF based algorithm produces stable result, on the contrary TF-computational method yields either stable, anomalous or unstable results.
1 citations
TL;DR: In this article, a method for solving a problem in biologic system that is formulated by stochastic Volterra integral equations is given, where triangular functions, block pulse functions and their operational matrices of integration are considered.
Abstract: The goal of this paper is to give useful method for solving a problem in biologic system that is formulated by stochastic Volterra integral equations. Here, we consider triangular functions, block pulse functions and their operational matrices of integration. Illustrative example is included to demonstrate the validity and applicability of the operational matrices.
1 citations
Cites background or methods from "A new set of orthogonal functions a..."
...2 Triangular functions (TFs) Two m-sets of triangular functions are defined [5] over the interval [0, T ) as T 1 i (t) = { 1 − t−ih h ih ≤ t < (i + 1))h, 0 elsewhere (17)...
[...]
...(19) From the definition of TFs, it is clear that TFs are disjoint, orthogonal, and complete [5]....
[...]
...TF approximation has been successfully used to analysis of dynamic systems [5], variational problems [1], integral equations [2], integro-differential equations [3], and Volterra–Fredholm integral equations [4]....
[...]
01 Jan 2016
TL;DR: This chapter presents the genesis of hybrid functions (HF) mathematically and different elementary properties and operational rules of HF are discussed.
Abstract: Starting with a brief review of block pulse functions (BPF), sample-and-hold functions (SHF) and triangular functions (TF), this chapter presents the genesis of hybrid functions (HF) mathematically. Then different elementary properties and operational rules of HF are discussed. The chapter ends with a qualitative comparison of BPF, SHF, TF and HF.
1 citations
01 Jan 2016
TL;DR: This chapter is devoted to develop the theory of one-shot operational matrices, which are useful for multiple integration and are superior to repeated integration using the first order integration matrices.
Abstract: This chapter is devoted to develop the theory of one-shot operational matrices. These matrices are useful for multiple integration and, in general, are superior to repeated integration using the first order integration matrices. Theory of one-shot operational matrices is presented and the one-shot operational matrices of n-th order integration have been derived. Three examples with nine figures and four tables elucidate the technique.
1 citations
TL;DR: In this article, a numerical method to solve linear two-dimensional Fredholm integral equations system of the second kind was proposed, which converted the given two dimensional integral equation system into a linear system of algebraic equations by using twoodimensional triangular functions.
Abstract: In this paper, we intend to offer a numerical method to solve linear two-dimensional Fredholm integral equations system of the second kind. This method converts the given two-dimensional Fredholm integral equations system into a linear system of algebraic equations by using twodimensional triangular functions. Moreover, we prove the convergence of the method. Finally the proposed method is illustrated by two examples and also results are compared with the exact solution by using computer simulations.
1 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