# On improvement of the integral operational matrix in block pulse function analysis

01 Jul 1995-Journal of The Franklin Institute-engineering and Applied Mathematics (Pergamon)-Vol. 332, Iss: 4, pp 469-478

TL;DR: In this paper, Chen and Chung (1987) showed that the trapezoidal rule is equivalent to evaluating the BPF coefficients of the integrated function by the well known trapezoid rule.

Abstract: It has been shown by Chen and Chung (1987) that the use of the conventional kintegral operational matrix P in block pulse function (BPF) analysis is equivalent to evaluating the BPF coefficients of the integrated function by the well known trapezoidal rule. They have improved upon P by employing a three-point interpolation polynomial in the Lagrange form to develop a new integral operational matrix P 1 (say). In the present paper, it has been established that once a function f(t) is represented by a BPF series, application of P to integrate f(t) in the staircase form is exact. Also, the method proposed by Chen and Chung (1987) is merely an extension of the trapezoidal rule wherein only one term of the remainder has been considered. Consideration of two terms from the remainder improves upon the integral operational matrix P 1 further and this improved operational matrix P 2 (say) has been employed to illustrate its superiority. Inclusion of still further terms from the remainder will improve upon P 2 further, but the rate of improvement will diminish gradually as evident from the illustrative examples.

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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.

24 citations

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TL;DR: In this article, a modified block Pulse Operational Transfer Function (MBPOTF) is proposed for linear SISO control system analysis in the block pulse function domain. But the results are not so accurate when compared with the direct expansion of the exact solution in the BPF domain.

Abstract: The present work makes use of the block pulse domain operational matrix for differentiation D1(m) to find out an operational transfer function. Analysis of simple control systems using this Block Pulse Operational Transfer Function (BPOTF) shows that the results are not so accurate when compared with the direct expansion of the exact solution in the BPF domain. To remove this defect, one shot operational matrices for repeated integration (OSOMRI) are obtained and these matrices are used to develop a Modified Block Pulse Operational Transfer Function (MBPOTF) for linear SISO control system analysis in the block pulse function domain. A few linear SISO control systems are analysed using the developed MBPOTF s as illustrative examples. The results are found to match exactly with the direct BPF expansions of the exact solutions.

9 citations

##### References

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01 Jan 1978

TL;DR: In this article, the authors present a solution to the Matrix Eigenvalue Problem for linear systems of linear equations, based on linear algebra and linear algebra with nonlinear functions, which they call linear algebraic integration.

Abstract: Error: Its Sources, Propagation, and Analysis. Rootfinding for Nonlinear Equations. Interpolation Theory. Approximation of Functions. Numerical Integration. Numerical Methods for Ordinary Differential Equations. Linear Algebra. Numerical Solution of Systems of Linear Equations. The Matrix Eigenvalue Problem. Appendix. Answers to Selected Problems. Index.

2,603 citations

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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.

361 citations

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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

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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

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01 Jan 1987

TL;DR: The theory and application of numerical methods for the solution of engineering problems using personal computers, which encompass linear and nonlinear algebraic equations, finite difference methods, ordinary and partial differential equations, and linear andNonlinear regression analysis are presented.

Abstract: From the Publisher:
This book is designed for advanced undergraduate and graduate students of engineering,science,and mathematics. Its purpose is to present the theory and application of numerical methods for the solution of engineering problems using personal computers. These algorithms encompass linear and nonlinear algebraic equations,finite difference methods,ordinary and partial differential equations,and linear and nonlinear regression analysis. Each method in the book is demonstrated with worked examples which require programs for their solution. These programs have been written in advanced BASIC language,the most widely used computer language today. The programs are interactive and user-friendly. They have been written in a general manner so that they may be applied to the solution of other problems,and are given in the source code so that they can be modified easily to suit the need of the user. The programs are described in detail to provide the reader with thorough background and understanding of the BASIC programming language.

162 citations

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