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A note on block pulse function operational matrix for

integration
a
K. R. PALANISAMY
a
Department of Electronics and Communication Engineering , Government College of
Technology , Coimbatore, 641013, India.
Published online: 10 May 2007.

To cite this article: K. R. PALANISAMY (1983) A note on block pulse function operational matrix for integration, International
Journal of Systems Science, 14:11, 1287-1290, DOI: 10.1080/00207728308926530

To link to this article: http://dx.doi.org/10.1080/00207728308926530

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 INT. J. SYSTEMS SCI., 1983, VOL. 14, NO. 11, 1287-1290

A note on block pulse function operational matrix

for integration
K. R. PALANISAMYt

This paper clearly brings out the relationship between the block pulse functions
(b.p.f.) operational matrix for integration and the single step method of b.p.I. It
is also shown that the operational matrix for integration may be obtained by using
b.p.f, delay matrices.

1. Introduction
Corrington (1973) proposed a method of solving non-linear differential
and integral equations using a set of Walsh functions (WF) as the basis. The
Downloaded by [New York University] at 13:08 31 May 2015

method required previously prepared tables of coefficients for integrating

WFs. Chen and Hsiao (1975 a, b) and Rao et al. (1975) introduced the
'operational matrix' to perform the integration. This operational matrix
approach has been applied to various problems such as identification (Rao
et al. 1975, Tzafestas 1978 b, 'I'zafestas and Chrysochoides 1977) time domain
analysis and synthesis (Chen and Hsiao 1975 b), piecewise constant feedback
gain determination for optimal control of linear systems (Chen and Hsiao
1975 c, Tzafestas and Chrysochoides 1977), design of observers and filters in
optimal control systems (Tzafestas and Chrysochoides 1978 a, Stavroulakis
and 'I'zafestas 1977, 1978) and for inverting irrational Laplace transforms
(Chen et al. 1977). Gopalsami and Deekshatulu (1976) introduced the block
pulse function operational matrix for numerical integration. Sannuti (1977)
presented a recursive formula for the analysis of dynamical systems via b.p.f.
Rao et al, (1978) employed b.p.f. techniques to the analysis of linear and non-
linear delay systems. Shieh et al, (1978) presented a method to obtain the
solution of state space equation via b.p.f. Palanisamy (1982) and Palanisamy
and Bhattacharya (1982) applied a single step method to the analysis of
non-linear and stiff systems.
The present paper brings out the relationship between the single step
method of b.p.f. and the operational matrix for integration in b.p.f. The
b.p.f. delay matrices are also closely related to b.p.f. operational matrix for
integration.

2. Operational matrix for integration and the single step method

A set of b. p.f. is defined as
i-I i
for --<t<-
m m i=1,2, .. ,,1n (1)
otherwise

Received 17 January 1983.

t Department of Electronics and Communication Engineering, Government
College of Technology, Coimbatore-641, 013, India.
 1288 K. R. Palanisamy

Represented concisely as m-vector <1>(t), the set may be used to approximate

an integrable function x(t) in the interval (0, 1) as

x(t) ~ BT<1>(t) (2)

where the elements B i of the coefficient vector B are chosen such that
1
€ = J [{(t) - BT<1>(t)]2 dt (3)
o

is minimized. For such a least-squares fit

ilm i-I i
Bi=m J {(t) dt,
(i-1)/,.
-<t<-,
m m
i=l, 2, 3, ... , m (4)

The optimal m-segment piecewise constant approximation is unique

Downloaded by [New York University] at 13:08 31 May 2015

whatever the set of piecewise constant basis functions chosen for approxima-
tioo. 1
Now we would like to expand the integral J <1>(t) dt, into the set of basic
o
functions <1>(t). The coefficients of expansion can be calculated using eqn. (4).
Arranging the coefficients in a matrix form we have
1
J <1>(t) dt ~ P<1>(t) (5)
o

where the P is m x m matrix given by

1 1

Let us expand x(t), x(t) into b.p.f. as

m
1/2

0
1

............

m
J (6)

x(t) = L Ciq,i(t) = CT<1>(t), x(t) ~ L Biq,i(t) = BT<1>(t) (7)

i=l i=l

The following identity is considered

t
J x(t) dt = x(t) - x(O) (8)
o
Inserting the b.p.f. expansion (7) and replacing the integral by the operational
matrix P, we have
CTP<1>(t)=BT<1>(t)_[X(O) '" x(O)] (9)
~

m terms
We can write
BT=CTP+[X(O) ... x(O)] . (10)
~

In terms
 Block pulse function operational matrix for integration 1289

By expanding eqn. (10) and using (6)

1
B I = - C I +x(O)
2m

(11)
IIi-I
B.=- C.+-
2m m
:E
;~I
Cj+x(O)

1 1 .. - I
B m = - C.+- :E C;+x(O)
2m m ;~I
The terms inside square brackets are actually the discrete values of x(t)
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1
x(l) = - C I +x(O)
m
1
x(2) = - C 2+x(l) (12)
m

1
x(i)=- C.+x(i-l)
m •
The above equations clearly indicate how the block pulse values of rate
variable are added together to get the new discrete values x(I), x(2), ... etc.
These values are extracted as in eqn. (12), where x(i) represents the discrete
time values at the junctions. If we let m = 1 in (11) and (12) we get a general
expression B.=1/2C.+X(i-l)}
(13)
x(i)=C i+x(i-l)
This result is the same as that obtained by Rao et al. (1980) via the single
term Walsh series approach. The above expression provides an insight into
the working of operational matrix for integration in b.p.f.

3. B.p.f. delay matrix and operational matrix for integration

The b.p.f. delay matrices were first introduced by Rao et al. (1978). The
operational matrix for integration

p mXm =~ [~\ DI+ ... + Dm-I] (14)

where D is the delay matrix and the superscript indicates the delay in number
of sub-intervals. For example DI attains the form with m = 4 as

n,-[ ~ ~ !] (15)
 1290 Block pulse function operational matrix for integration

Using eqn. (14)

P 4 X 4 =!
[10 ~1+1[
o I
o 0
0
0
I
I
0
0
0
I
0 ]+1 [~
0
0
0
I
0
0
]
".
o 0 0 0 0 0 0

[~ ]
0 0
0 0
(16)
0 0
+!
0 0
The P matrix may be easily generated using the delay matrices while analysing
delay systems.
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4. Conclusion
A clear insight into the working of the b.p.f. operational matrix for integra-
tion is brought out. This results in the single step method and discrete time
values in addition to b.p.f. values of the solution. The new method is shown
to generate the operational matrix for integration by using b.p.f. delay
matrices.
ACKNOWLEDGMENTS
The author sincerely thanks Professor Ganti Prasada Rao, Department of
Electrical Engineering, Indian Institute of Technology, Kharagpur for his
kind help and discussions.

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