Anish Deb

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.

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.

Transfer function identification from impulse response via a new set of orthogonal hybrid functions (HF)

The present work proposes a new set of hybrid functions (HF) which evolved from the synthesis of sample-and-hold functions (SHF) and triangular functions (TF). Traditional block pulse functions (BPF) still continue to be attractive to many researchers in the arena of control theory. Block pulse functions also gave birth to a few useful variants. Two such variants are SHF and TF. The former is efficient for analyzing sample-and-hold control systems, while triangular functions established their superiority in obtaining piecewise linear solution of various control problems. After developing the basic theory of HF, a few square integrable functions are approximated via this set in a piecewise linear manner. For such approximation, it is shown, the mean integral square error (MISE) is much less than block pulse function based approximation. The operational matrices for integration in HF domain are also derived. Finally, this new set is employed for solving identification problem from impulse response data. The results are compared with the solutions obtained via BPF, SHF, TF, etc. and many illustrations are presented.

Analysis of pulse-fed power electronic circuits using Walsh function

A new operational method for the analysis of pulse–fed power electronic circuits is suggested, where input waveforms are expressed by a series combination of Walsh functions. The output response is obtained in terms of Walsh functions after operation by Walsh operational transfer function (WOTF). The current waveform of a DC chopper fed R-L load is approximated by piecewise constant solution and various average and r.m.s. currents of the same power electronic circuit are computed as an illustration.

Approximation, integration and differentiation of time functions using a set of orthogonal hybrid functions (HF) and their application to solution of first order differential equations

Abstract Differential equations of different types and orders are of utmost importance for mathematical modeling of control system problems. State variable method uses the concept of expressing n number of first order differential equations in vector matrix form to model and analyze/synthesize control systems. The present work proposes a new set of orthogonal hybrid functions (HF) which evolved from synthesis of sample-and-hold functions (SHF) and triangular functions (TF). This HF set is used to approximate a time function in a piecewise linear manner with the mean integral square error (MISE) much less than block pulse function based approximation which always provides staircase solutions. The operational matrices for integration and differentiation in HF domain are also derived and employed for solving non-homogeneous and homogeneous differential equations of the first order as well as state equations. The results are compared with exact solutions, the 4th order Runge–Kutta method and its further improved versions proposed by Simos [6] . The presented HF domain theory is well supported by a few illustrations.

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.

Sample shifting technique (SST) for estimation of harmonic power in polluted environment

In this paper, an attempt has been made to estimate real and reactive power components for each harmonic in a polluted sinusoidal environment based on a newly proposed sample shifting technique (SST). A special feature of this technique is that each harmonic power component is estimated from the samples of voltage and current signals only, without measuring the power factor angle, or adopting FFT or DFT etc. as used by traditional methods. The present method also determines the DC power component along with the sinusoids. The method of SST is supported by simulation results along with experimental validation.

Analysis of a continuously variable pulse-width modulated system via Walsh functions

A new operational technique for the analysis of a continuously pulse-width modulated system is suggested, where the input pulse wave is expressed by a series combination of Walsh functions. A Walsh operational transfer function is utilized instead of a conventional transfer function to obtain the output response, e.g. device current, freewheeling current, load current, average speed, etc. in the Walsh domain. The method is suitable for computer manipulations and has no restriction on the variation of width, repetition rate and amplitude of the input wave. The results are found to be in good agreement with the exact solutions.

A note on oscillations in Walsh-domain analysis of first-order systems

Oscillations found in Walsh-domain analysis of first-order systems are characterized. A useful condition for occurrence of such oscillations is given, and the expression for the percentage maximum overshoot for different system time constants, scaling constants and number of Walsh functions in the set considered are derived. Several graphs are presented to show the variational behavior of the percentage maximum overshoot. The reasons for this oscillatory phenomenon are discussed. Comparisons are also made between the actual time-domain solution and the Walsh-domain solution which gives rise to oscillations if the derived constraints are violated. >