Numerical instability of deconvolution operation via block pulse functions

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Abstract

This paper characterizes oscillations found in block pulse function (BPF) domain identification of open loop first-order systems with step input. A useful condition for occurrence of such oscillations is presented mathematically. For any positive value of ‘ah’, oscillations are observed to occur, where h is the width of BPF domain sub-interval and 1/a is the time constant of the first-order system under consideration.

Introduction

Piecewise constant basis functions [1] were introduced by Alfred Haar in 1910. Its square wave nature attracted many researchers and other such functions, e.g., Walsh functions, slant functions, block pulse functions (BPF) [1], [2] followed. Among all these functions, the BPF set proved to be the most fundamental [3] and it enjoyed immense popularity in different applications in the area of control systems [4], including analysis and identification [5] problems.

However, it has been observed that BPF domain identification of a control system, in absence of measurement noise, yields oscillatory results when ‘deconvolution’ [5] based operational technique is used. It is implied that presence of measurement noise will aggravate the situation further.

In the present work, the reason for such ‘instability’ has been explored and a mathematical explanation has been provided with suitable numerical examples.

The main objectives of the present work are:

  • (i)

    To identify an open loop single-input–single-output (SISO) system in the BPF domain, in absence of measurement noise, using established knowledge and draw attention to undesired oscillations.

  • (ii)

    To find mathematical reasons for such undesired oscillations.

Section snippets

Brief review of BPF domain convolution and deconvolution process [4,5]

A set of BPF Ψ(m)(t) containing m-component functions in the semi-open interval [0,T) is given byΨ(m)(t)[ψ0(t)ψ1(t)ψ2(t)ψi(t)ψ(m-1)(t)]T,where []T denotes transpose.

The ith component ψi(t) of the BPF vector Ψ(m)(t) is defined as ψi(t)=1iht<(i+1)h,0elsewhere,where h=T/m and i=0,1,2,,(m-1).

A square integrable time function f(t) of Lebesgue measure may be expanded into an m-term BPF series in t[0,T) asf(t)=[f0f1f2fif(m-1)]Ψ(m)(t)=FTΨ(m)(t).The constant coefficients fi's in Eq. (2) are

Identification of open loop first-order systems in BPF domain

Now we proceed to identify a simple plant in the BPF domain using Eq. (10).

Consider the system with input r(t)=u(t) and the output c(t)=14[1-exp(-4t)]. The plant is g(t)=exp(-4t). For T=1s and m=8, the BPF domain result for g(t) is given in Table 1.

Comparing first and second columns of the result, it is noticed that there is predominant oscillation of the result via deconvolution compared to the result via direct expansion. Also, BPF domain identification gives highly erroneous result with a

Conclusion

This paper has established the oscillatory nature of results for solving identification problems, in absence of measurement noise, for a SISO first-order system via deconvolution in BPF domain as presented in Ref. [5]. The reason for such oscillation has also been established mathematically with supporting interpretation. The investigation implies that, system identification via deconvolution in the BPF domain is always error prone and the numerical example in Ref. [5] is no exception. In [5],

References (5)

  • K.G. Beauchamp

    Applications of Walsh and Related Functions with an Introduction to Sequency theory

    (1984)
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    (1983)
There are more references available in the full text version of this article.

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