Block pulse functions, the most fundamental of all piecewise constant basis functions
01 Feb 1994-International Journal of Systems Science (Taylor & Francis Group)-Vol. 25, Iss: 2, pp 351-363
TL;DR: In this article, it is shown that block pulse functions (BPFs) are superior to the delayed unit step function (DUSF) proposed by Hwang (1983) due to the most elemental nature of BPFs in comparison to any other PCBF function.
Abstract: It is established that block pulse functions (BPFs) are superior to the delayed unit step function (DUSF) proposed by Hwang (1983). The superiority is mainly due to the most elemental nature of BPFs in comparison to any other PCBF function. It is also proved that the operational matrix for integration in the BPF domain is connected to the integration operational matrix in the DUSF domain by simple linear transformation involving invertible Toeplitz matrices. The transformation appears to be transparent because the integration operational matrices are found to match exactly. The reason for such transparency is explained mathematically. Finally, Hwang claimed superiority of DUSFs compared to Walsh functions in obtaining the solution of functional differential equations using a stretch matrix in the DUSF domain. It is shown that the stretch matrices of Walsh and DUSF domains are also related by linear transformation and use of any of these two matrices leads to exactly the same result. This is supported by a...
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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.
152 citations
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.
90 citations
TL;DR: In this paper, the authors used piecewise constant two-dimensional block-pulse functions (2D-BPFs) and their operational matrices for solving mixed nonlinear Volterra-Fredholm integral equations of the first kind.
54 citations
TL;DR: In this paper, a set of piecewise constant orthogonal functions, termed sample-and-hold functions (SHF), is introduced for the analysis of control systems with SISO.
Abstract: The present work searches for a suitable set of orthogonal functions for the analysis of control systems with sample-and-hold ( S/H ). The search starts with the applicability of the well known block pulse function (BPF) set and uses an operational technique by defining a block pulse operational transfer function ( BPOTF ) to analyse a few control systems. The results obtained are found to be fairly accurate. But this method failed to distinguish between an input sampled system and an error sampled system. To remove these limitations, another improved approach was followed using a sample-and-hold operational matrix, but it also failed to come up with accurate results. Further, the method needed a large number of component block pulse functions leading to a much larger amount of storage as well as computational time. To search for a more efficient technique, a new set of piecewise constant orthogonal functions, termed sample-and-hold functions (SHF), is introduced. The analysis, based upon a similar operational technique, in the SHF domain results in the same accuracy as the conventional z -transform analysis. Here, the input signal is expressed as a linear combination of sample-and-hold functions; the plant having a Laplace transfer function G(s) is represented by an equivalent sample-and-hold operational transfer function ( SHOTF ), and the output in the SHF domain is obtained by means of simple matrix multiplication. This technique is able to do away with the laborious algebraic manipulations associated with the z -transform technique without sacrificing accuracy. Also, the accuracy does not depend upon m and the presented method does not need any kind of inverse transformation. A few linear sample-and-hold SISO control systems, open loop as well as closed loop, are analysed as illustrative examples. The results are found to match exactly with the z -transform solutions. Finally, an error analysis has been carried out to estimate the upper bound of the mean integral squared error (m.i.s.e.) of the SHF approximation of a function f(t) of Lebesgue measure.
26 citations
References
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Book•
01 Jan 1969
TL;DR: When you read more every page of this transmission of information by orthogonal functions, what you will obtain is something great.
Abstract: Read more and get great! That's what the book enPDFd transmission of information by orthogonal functions will give for every reader to read this book. This is an on-line book provided in this website. Even this book becomes a choice of someone to read, many in the world also loves it so much. As what we talk, when you read more every page of this transmission of information by orthogonal functions, what you will obtain is something great.
366 citations
271 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
Book•
01 Jun 1983
TL;DR: In this article, the authors proposed piecewise constant orthogonal basis functions (PCF) for linear and non-linear linear systems, and the optimal control of linear lag-free and time-lag systems.
Abstract: I Piecewise constant orthogonal basis functions.- II Operations on square integrable functions in terms of PCBF spectra.- III Analysis of lumped continuous linear systems.- IV Analysis of time delay systems.- V Solution of functional differential equations.- VI Analysis of non-linear and time-varying systems.- VII Optimal control of linear lag-free systems.- VIII Optimal control of time-lag systems.- IX Solution of partial differential equations (PDE) [W55].- X Identification of continuous lumped parameter systems.- XI Parameter identification in distributed systems.
188 citations
01 Jun 1977
TL;DR: The paper presents a method of numerically integrating a system of differential equations based on an idea of orthogonal approximation of functions that gives piecewise constant solutions with minimal mean-square error and is computationally similar to the familiar trapezoidal rule of integration.
Abstract: The paper presents a method of numerically integrating a system of differential equations based on an idea of orthogonal approximation of functions. Here, block-pulse functions are chosen as the orthogonal set. The method gives piecewise constant solutions with minimal mean-square error and is computationally similar to the familiar trapezoidal rule of integration. Design of piecewise constant controls or feedback gains for dynamic systems can be simplified following this approach.
160 citations