International Journal of Computers, Communications & Control
Vol. III (2008), No. 1, pp. 103-109

State Analysis of Time-Varying Singular Bilinear Systems by RK-Butcher
Algorithms

V. Murugesh, K. Batri

Abstract: The Runge-Kutta (RK)-Butcher algorithm is used to study the time-
varying singular bilinear systems with the exact solutions. The results (discrete so-
lutions) obtained using the Haar wavelets, Single-Term Walsh series (STWS) and
RK-Butcher algorithms are compared with the exact solutions of the time-varying
singular bilinear systems. It is found that the solution obtained using the RK-Butcher
algorithm is closer to the exact solutions of the time-varying singular bilinear sys-
tems. The RK-Butcher algorithm can easily be implemented using a digital computer
and the solution can be obtained for any length of time, which is an added advantage
of this algorithm.
Keywords: Time-varying singular bilinear systems, Haar wavelets, Runge-Kutta
Butcher algorithm, STWS algorithm.

1 Introduction

The development of singular bilinear systems has been studied by some researchers. Campbell [1]
had done some preliminary work, but there was no available closed-form solution in that paper. In some
analysis of neural networks, both singular systems [2] and bilinear systems [3] have been used. The
multipliers and algebraic interconnections between singular systems and bilinear systems are allowed in
dynamical systems. For singular bilinear systems, Lewis et al. [4] have been discussed extensively in the
literature. However, the solution due to Lewis et al. only applies for the time-invariant case. Hsiao and
Wang [5] applied the Haar wavelets for the solution of time -varying singular bilinear systems. Sepehrian
and Razzaghi [6] applied the STWS approach for finding the numerical solution of time-varying bilinear
systems.

Runge-Kutta (RK) methods have become very popular both as computational techniques and as a
topic for research [7-12]. Butcher [8] derived the best RK pair, together with an error estimate, and in all
statistical measures this approach is known as the RK-Butcher algorithm. This algorithm appears to be
of sixth order because it requires six function evaluations, but in practice the ’working order’ is closer to
five (fifth order). However, the accuracy of the results obtained is better than that of all other algorithms
examined including the RK- Fehlberg, RK-Merson, RK-centroidal mean (RKCeM) and RK-arithmetic
mean (RKAM) algorithms.

Murugesh and Murugesan [13-15] introduced the RK-Butcher algorithm in Raster and Time-multiplexing
CNN simulations. Recently, [16, 18] the RK-Butcher algorithm is used to find the numerical solution of
an industrial application problem. In this article, we present a new approach for solving the time-varying
singular bilinear systems using the RK-Butcher algorithm with more accuracy.

2 The RK-Butcher Algorithm

The normal order of an RK algorithm is the approximate number of leading terms of an infinite Taylor
series which calculates the trajectory of a moving point [17]. The remainder of the infinite sum, which
is excluded, is referred to as the local truncation error (LTE). These RK algorithms are forward-looking
predictors, i.e. they do not use any information from preceding steps to predict the future position of a
point. For this reason, they require a minimum of input data and consequently are very simple to program
and use.

Copyright © 2006-2008 by CCC Publications



104 V. Murugesh, K. Batri

The general p-stage Runge-Kutta method for solving an IVP is

y′ = f (x, y) (1)

with the initial condition y(x0) = y0 is defined by

yn+1 = yn + h
p

∑
i=1

biki

where

ki = f

(
xn + cih, yn + h

p

∑
j=1

ai jk j

)
, i = 1, 2, 3, ......, p

and

ci =
p

∑
j=1

ai j, i = i, 2, 3, ....., p

In the preceding equations c and b are p-dimensional vectors and A(ai j) is the p × p matrix. Then
the Butcher array takes the form

c1 a11
c2 a21 a22
c3 a31 a32 a33
· · · ·
· · · ·
· · · ·
· · · · ·
cp ap1 ap2 ap3 app

b1 b2 bp−1 bp

The RK-Butcher algorithm of equation (1) is of the form

k1 = h f (xn, yn)

k2 = h f
(

xn +
h
4
, yn +

k1
4

)

k3 = h f
(

xn +
h
4
, yn +

k1
8

+
k2
8

)

k4 = h f
(

xn +
h
2
, yn −

k2
2

+ k3

)
(2)

k5 = h f
(

xn +
3h
4

, yn +
3k1
16

+
9k4
16

)

k6 = h f
(

xn + h, yn −
3k1
7

+
2k2
7

+
12k3

7
− 12k4

7
+

8k5
7

)

5th order predictor

yn+1 = yn +
1

90
(7k1 + 32k3 + 12k4 + 32k5 + 7k6)

4th order predictor

y∗n+1 = yn +
1
6

(k1 + 4k4 + k6)

The local truncation error estimate (EE) is

EE = yn+1 −y∗n+1
Then the formation of the Butcher array of the above equation (2) takes the following form



State Analysis of Time-Varying Singular Bilinear Systems by RK-Butcher Algorithms 105

0
1
4

1
4

1
4

1
8

1
8

1
2 0 -

1
2 1

3
4

3
16 0 0

9
16

1 37
2
7

12
7

12
7

8
7

7
90 0

32
90

12
90

32
90

7
90

1
6 0 0

4
6 0

1
6

3 Analysis of Time-Varying Singular Bilinear Systems

Consider the linear first order time-varying singular system

K
.
x(t) = Ax(t) + B(t)u(t) (3)

with x0 = x(0), where K is an n×n singular matrix, A is an n×n matrix, B is an n×r matrix. x(t) is
n-state vectors and u(t) is an r-input vector.

The time-varying singular bilinear system is of the form

E(t)
.
x(t) = A(t)x(t) +

q

∑
i=1

Ni(t)x(t)ui(t) + B(t)u(t) (4)

equation (4)is written in the form (3) as

E(t)
.
x(t) =

(
A(t) +

q

∑
i=1

Ni(t)ui(t)

)
x(t) + B(t)u(t) (5)

where the singular matrix E(t) ∈ Rn×n, the state x(t) ∈ Rn , the control u(t) ∈ Rq, A(t) ∈ Rn×n and
B(t) ∈ Rn×q. Ni(t) ∈ Rn×n and ui(t), i = 1, 2, 3, ...., q, are the components of u(t). The response x(t), 0 ≤
t ≤ ti, is required to be found. The time-varying singular bilinear systems are much more difficult to solve
than the time invariant singular bilinear systems. Therefore, many authors have tried various transform
methods to over come these difficulties. In this article, we introduce RK-Butcher algorithms with more
accuracy to solve these time-varying singular bilinear systems.

4 Numerical Example

Consider the time-varying singular bilinear system of the following form (Hsiao and Wang [5]) and
Sepehrian and Razzaghi[6]).

E(t) =




0 −t 0
1 0 t
0 1 0


 A(t) =



−2 t 1
0 −4 2
−2t 0 1


 N1(t) =




1 −t 1
0 3 −2
2t 0 −2


 (6)

B(t) =
[
2 1 3

]T
, u(t) = 1, with initial condition x(0) =

[
12 2 5

]T



106 V. Murugesh, K. Batri

when we solve (5), the analytic solution for x(t) can be shown as

x(t) =




(2−t)
(
ex p

(−t
2

)
+ ex p (t)

)
+ 8

2ex p
(−t

2

)
−ex p (t) + 1

ex p
(−t

2

)
+ ex p (t) + 3


 (7)

The discrete solutions of equation (5) are evaluated using the RK-Butcher algorithms (with step size
t = 0.25 ) represented in equation (2) and the results are compared with the solutions obtained by the
Haar wavelets method by Hsiao and Wang [5] and the STWS method by Sepehrian and Razzaghi [6].
The results are shown in tables 1 - 3 along with the exact solutions calculated using equation (7). Errors
between the exact and discrete solutions are also given in Tables 1 - 3.

S.No Time

Discrete solution x1Values

Exact
Solutions

Haar
Solutions

Haar
Error

STWS
Solutions

STWS
Error

RK-
Butcher
Solutions

RK-
Butcher
Error

1 0 12.000000 12.0000 0.000000 12.00000 0.000000 12.000000 0.000000
2 0.25 11.886053 11.8861 0.000047 11.88605 0.000003 11.886053 0.000000
3 0.5 11.791414 11.7914 0.000014 11.79142 0.000006 11.791414 0.000000
4 0.75 11.711533 11.7115 0.000033 11.71154 0.000007 11.711533 0.000000
5 1 11.641283 11.6413 0.000017 11.64128 0.000003 11.641283 0.000000
6 1.25 11.574810 11.5748 0.000010 11.57481 0.000000 11.574810 0.000000
7 1.5 11.505362 11.5054 0.000038 11.50537 0.000008 11.505362 0.000000
8 1.75 11.425089 11.4251 0.000011 11.42510 0.000011 11.425089 0.000000
9 2 11.324812 11.3248 0.000012 11.32481 0.000002 11.324812 0.000000

Table 1: Solutions for the problem at various values of x1.

S.No Time

Discrete solution x2Values

Exact
Solutions

Haar
Solutions

Haar
Error

STWS
Solutions

STWS
Error

RK-
Butcher
Solutions

RK-
Butcher
Error

1 0 2.000000 2.0000 0.000000 2.00000 0.000000 2.000000 0.000000
2 0.25 1.745678 1.7457 0.000022 1.74568 0.000002 1.745678 0.000000
3 0.5 1.480968 1.4810 0.000032 1.48097 0.000002 1.480968 0.000000
4 0.75 1.203067 1.2031 0.000033 1.20307 0.000003 1.203067 0.000000
5 1 0.908880 0.9089 0.000020 0.90888 0.000000 0.908880 0.000000
6 1.25 0.594985 0.5950 0.000015 0.59498 0.000005 0.594985 0.000000
7 1.5 0.257579 0.2576 0.000021 0.25758 0.000001 0.257579 0.000000
8 1.75 -0.107578 -0.1076 0.000022 -0.10758 0.000002 -0.107578 0.000000
9 2 -0.505221 -0.5052 0.000021 -0.50522 0.000001 -0.505221 0.000000

Table 2: Solutions for the problem at various values of x2.

5 Conclusions

The discrete solutions obtained using the RK-Butcher algorithm gives more accurate values when
compared to the Haar wavelets method discussed by Hsiao and Wang [5] and the STWS method by



State Analysis of Time-Varying Singular Bilinear Systems by RK-Butcher Algorithms 107

S.No Time

Discrete solution x3Values

Exact
Solutions

Haar
Solutions

Haar
Error

STWS
Solutions

STWS
Error

RK-
Butcher
Solutions

RK-
Butcher
Error

1 0 5.000000 5.0000 0.000000 5.00000 0.000000 5.000000 0.000000
2 0.25 5.072562 5.0726 0.000038 5.07256 0.000002 5.072562 0.000000
3 0.5 5.166522 5.1665 0.000022 5.16652 0.000002 5.166522 0.000000
4 0.75 5.284021 5.2840 0.000021 5.28402 0.000001 5.284021 0.000000
5 1 5.427522 5.4275 0.000022 5.42752 0.000002 5.427522 0.000000
6 1.25 5.599862 5.5999 0.000038 5.59986 0.000002 5.599862 0.000000
7 1.5 5.804289 5.8043 0.000011 5.80429 0.000001 5.804289 0.000000
8 1.75 6.044524 6.0445 0.000024 6.04451 0.000014 6.044524 0.000000
9 2 6.324812 6.3248 0.000012 6.32480 0.000012 6.324812 0.000000

Table 3: Solutions for the problem at various values of x3.

Sepehrian and Razzaghi [6]. From the tables 1-3, one can observe that the solutions obtained by the
RK-Butcher algorithm match well with the exact solutions of the time-varying singular bilinear systems,
but the Haar wavelets and STWS methods yields a little error. Hence the RK-Butcher algorithm is more
suitable for studying the time-varying bilinear systems.

Bibliography

[1] Campell, S.L., 1987, Bilinear nonlinear descriptor control systems, CRSC Technical Report
102386-01, Department of Mathematics, N.C. State University, Raleigh, NC 27695.

[2] Declaris, N. and Rindos, A., 1984, Semistate analysis of neural networks in Apysia Californica,
Proceedings of the 27th MSCS, 686-689.

[3] Wiener, N., 1948, Cybernetics (Cambridge, MIT Press).

[4] Lewis, F. L., Mertzios, B. C., and Marszalek, W., 1991, Analysis of singular bilinear systems using
Walsh functions, IEE Proceedings Part-D,138, 89-92.

[5] Hsiao, C. H., and Wang, W. J., 2000, State Analysis of time-varying Singular Bilinear Systems via
Haar wavelets, Mathematics and Computers in Simulation, 52, 11-20.

[6] Sepehrian, B., and Razzaghi, M., 2003, State Analysis of time-varying Singular Bilinear Systmes
by Single-Term Walsh Series, International Journal of Computer Mathematics, 80, 413-418.

[7] Alexander, R.K. and Coyle, J.J.,1990, Runge-Kutta methods for differential-algebraic systems.
SIAM Journal of Numerical Analysis, 27, 736-752.

[8] Butcher, J.C., 1987, The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and
General Linear Methods (Chichester: JohnWiley).

[9] Butcher, J.C., 2003, Numerical Methods for Ordinary Differential Equations (Chichester: JohnWi-
ley).

[10] Shampine, L.F., 1994, Numerical Solution of Ordinary Differential Equations (NewYork: Chapman
& Hall).



108 V. Murugesh, K. Batri

[11] Yaakub, A.R. and Evans, D.J., 1999, A fourth order Runge-Kutta RK(4,4) method with error con-
trol. International Journal of Computer Mathematics, 71, 383-411.

[12] Yaakub, A.R. and Evans, D.J., 1999, New Runge-Kutta starters of multi-step methods. International
Journal of Computer Mathematics, 71, 99-104.

[13] Murugesh, V., and Murugesan, K., 2004, Comparison of Numerical Integration Algorithms in
Raster CNN Simulation, Lecture Notes in Computer Science, 3285, 115-122.

[14] Murugesh, V. and Murugesan, K., 2005, Simulation of Cellular Neural Networks using the RK-
Butcher algorithm, International Journal of Management and Systems, 21, 65-78.

[15] Murugesh, V., and Murugesan, K., 2006, Simulation of Time-Multiplexing Cellular Neural Net-
works with Numerical Integration Algorithms, Lecture Notes in Computer Science, 3991, 115-122.

[16] Devarajan Gopal, Murugesh, V., and Murugesan, K., 2006, Numerical Solution of Second-order
Robot Arm Control Problem using Runge-Kutta Butcher Algorithm, International Journal of Com-
puter Mathematics, 83, 345-356.

[17] Shampine, L.F. and Gordon, M.K., 1975, Computer Solutions of Ordinary Differential Equations:
The Initial Value Problem (San Francisco, CA:W.H. Freeman).

[18] V. Murugesh and K. Batri, “An Efficient Numerical Integration Algorithm for Cellular Neural Net-
work Based Hole-Filler Template Design”, International Journal of Computers, Communications
& Control, Vol. II (2007), No. 4, pp. 367-374.

V. Murugesh
Department of Information and Communication Engineering

Hannam University
133 Ojung-dong Daeduk-gu, Daejeon 306-791, Republic of Korea

E-mail: murugesh72@gmail.com

K. Batri
Department of Computer Science and Engineering

Muthayammal Engineering College
Rasipuram 637 408

India
E-mail: krishnan.batri@gmail.com

Received: November 26, 2007

Dr. V. Murugesh obtained his Bachelor of Science in Com-
puter Science and Master of Computer Applications degree from
Bharathiar University, Coimbatore, India during 1992 and 1995
respectively. Completed his PhD in Computer Science from
Bharathidasan University, Tiruchirappalli, India during 2006. He
has held various positions at National Institute of Technology,
Tiruchirappalli, India and Sona College of Technology, Salem,
India. Currently, he is working as Assistant Professor in the
Department of Information and Communication Engineering at
Hannam University, Daejeon, Republic of Korea. His fields of
interest are in Neural Network based Image Processing and Sci-
entific computing. He has published more than 30 technical pa-
pers in International, National journals and conferences.



State Analysis of Time-Varying Singular Bilinear Systems by RK-Butcher Algorithms 109

Krishnan Batri received the M.E. from Madurai Kamarj Uni-
versity in 2003. He is a Research Scholar with the Depart-
ment of Computer Science and Engineering in the National Insti-
tute of Technology Tiruchirapalli, Tamil Nadu, India. Currently
he is working as a Assistant Proffessor with the department of
Computer Science and Engineering at Muthayammal Engineer-
ing College,Rasipuram, Tamilnadu, India . His research inter-
ests include Information Retrieval, Data fusion and Genetic algo-
rithms.