Manabrata Bhattacharjee

Bio: Manabrata Bhattacharjee is an academic researcher from University of Calcutta. The author has contributed to research in topics: Transfer function & Transformation (function). The author has an hindex of 3, co-authored 4 publications receiving 38 citations.

A new set of piecewise constant orthogonal functions for the analysis of linear siso systems with sample-and-hold

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.

All-integrator approach to linear SISO control system analysis using block pulse functions (BPF)

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.

Analysis of linear discrete SISO control systems via a set of delta functions

Abstract: The paper presents a computational technique through operational matrices using a set of mutually disjoint delta functions (DF) for the analysis of linear discrete control systems. Following a brief review of the well known block pulse functions (BPF), a new set of delta functions is viewed in the same light. This set is used to develop operational transfer functions in the delta function domain (DOTF) and employed for discrete system analysis which results in the same accuracy as the conventional z-transform method. The presented technique uses simple matrix manipulations and is able to do away with laborious and involved algebraic steps, including inverse transformation, associated with the z-transform analysis without losing accuracy. Also, the accuracy of sample values of the output does not depend upon m (or the sampling interval h). A few linear discrete SISO control systems, open loop as well as closed loop, having different typical plant transfer functions, are analysed as illustrative examples.

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

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.

Numerical solution of fractional differential equations using the generalized block pulse operational matrix

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.

A new set of orthogonal functions and its application to the analysis of dynamic systems

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.

A new set of piecewise constant orthogonal functions for the analysis of linear siso systems with sample-and-hold

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.

A numerical method for solving Linear Non-homogenous Fractional Ordinary Differential Equation

Abstract: In this paper, a numerical method for solving LNFODE (Linear Non-homogenous Fractional Ordinary Differential Equa- tion) is presented. The method presented is based on Bernstein polynomials approximation. The operational matrices of integration, differentiation and products are introduced and utilized to reduce the LNFODE problem in order to solve algebraic equations. The method is general, easy to implement, and yields very accurate results. Illustrative examples are included to demonstrate the validity and applicability of the technique.