Sunit K. Sen

Block pulse functions, the most fundamental of all piecewise constant basis functions

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...

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.

A new set of pulse-width modulated generalized block pulse functions (PWM-GBPF) and its application to cross-/auto-correlation of time-varying functions

The present work proposes a significant modification to conventional block pulse functions (BPF) by describing a novel set of pulse-width modulated generalized block pulse functions (PWM-GBPF) which has been utilized to develop generalized correlation matrices of operational nature. These matrices are used to determine cross-/auto-correlation of time-varying functions. Numerical examples are treated to establish the validity of the proposal. Also, a representational error analysis has been carried out for the PWM-GBPF to show that this kind of BPF set introduces a smaller error than the conventional equal width BPF

Walsh Functions and their Applications: A Review

The piecewise constant Walsh function is not new in the field of mathematics because it was proposed in the year 1923 by J L Walsh. But its suitability to qualify as an elegant mathematical tool for the analysis of physical systems became apparent to the researchers quite late. Though other functions of the same piecewise constant orthogonal class attracted attentions of scientists as well as technologists to find frequent applications, the Walsh function has maintained its leadership among its clan having wide application areas ranging from the field of communication to the area of systems and control. The applications encompass die areas of digital signal processing image processing, logic circuit design etc.The present paper gives an overview of the Walsh analysis with special stress upon its application in the fields of communication, systems, control and other areas as well. The paper also reviews other non-sinusoidal orthogonal functions of the same class. Considering the present state of applicatio...

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.

Optical Computing Techniques

A review of the contemporary issues in optical computing is presented. The optical systems and techniques which are used to implement different functional sub units of a digital computer are elaborated. Several architectures that are being explored for digital optical computing are described. Some issues relating to organisational and functional aspects of optical computing, such as cellular automata, symbolic substitution, optical artificial intelligence and optical neural network are also discussed.

Cross-/ auto-correlation of time-varying signals using block pulse functions and determination of power spectral densities

The present work proposes a method based on block pulse function ( BPF) analysis for the determination of cross-/ auto-correlation of time-varying signals. These results are used subsequently to evaluate power spectral densities ( PSD) of a few codes for communication. Two Lebesgue integrable time-varying functions are represented by means of their respective equivalent BPF spectra. The BPF spectral coefficients are used to develop positive and negative correlation matrices of an operational nature. These matrices, when combined, result in a correlation matrix that is used to determine cross-/ auto-correlation of the input functions over the entire time scale. The results thus obtained in the BPF domain are utilized to determine the PSDs of a few communication codes employing the well-known fast Fourier transform algorithm. Numerical experiments are carried out to establish the validity of the proposed method and the results are found to be in close agreement with the exact solutions

Microprocessor based industrial motor control and protection using interrupt

The authors present a microprocessor based control scheme for an industrial motor with interrupt driven cause/effect protection. The scheme requests service from the microprocessor only under unhealthy conditions. The scheme also takes care of the causes as well as the effect of the fault. Under healthy conditions, the microprocessor generates the required control signals based upon the desired logic sequence. This arrangement increases the efficiency of the microprocessor, throughput of the system and also reduces the software complexity. The interrupt driven protection scheme allows the microprocessor to run other programs under normal operating conditions.<<ETX>>

A method for tracking power line frequency with millihertz resolution and high/low cut-off

Abstract A method for the accurate measurement, tracking and displaying of power line frequency with millihertz resolution is presented which has the facility of generating trip signals in the case of violations of presettable high or low frequency limits. A phase locked loop is used in the multiplier configuration to increase the measurement accuracy. Design equations are established and transient response of the system is simulated.