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SIMULATION OF SYSTEMS WITH VARIOUS TIME DELAYS
USING PADE'S APPROXIMATION
Mujo Hebibovic, Bakir Lacevic, Jasmin Velagic
Faculty of Electrical Engineering, University of Sarajevo, Zmaja od Bosne bb, 71000 Sarajevo, B&H
Keywords: Time delay, Pade's function, Simulation, Time constant, Taylor series
Abstract: In this paper, Pade's rational functions have been simulated for approximating several characteristic values
of time delay regarding the plant time constant. Several representative plants were tested in order to show in
which cases Pade’s function approximates time-delay block well. Only if the ratio of time delay versus time
constant of the plant is rather great, or the plant contains emphasized numerator dynamics; approximation
capabilities get poorer. The convergence rate of n-order Pade’s function has been also analyzed by using
Taylor series and phase-frequency characteristics.
1 INTRODUCTION
Many of industrial processes and process control
systems, along with their structural presentations,
contain one or more time delay components
(Dugard, Verriest, 1998; Chen, et al., 2003). These
make an inherent part of the mathematical models
used to describe the systems' dynamics of
management and biological systems as well. Padé
approximations are widely used to approximate a
dead-time in continuous control systems (Vajta,
2000). It provides a finite-dimensional rational
approximation of a dead-time. The accuracy of
applied time delay blocks is particularly important in
computer simulation of complex dynamic systems,
described by high-order equations, then in
computation of convolution integrals etc (Beek, et
al., 1999). The principal problem in their realization
is that their transfer function appears in transcendent
form, what is not quite appropriate for simulation
(Hebibovic, 1991; Vajta, 2000). In order to avoid the
problem, it has long been the practice to
approximate the time delay transfer function with a
rational function (Hebibovic, 1998).
In this paper MATLAB/Simulink features were
exploited and comparison of the first four Pade’s
functions has been done regarding several typical
plants and typical ratios of time-delay versus plant
time constant. Convergence that can be seen well
from simulations is supported by theoretical analysis
using Taylor series.
2 SYSTEM DESCRIPTION
Let's consider a time function u(t) as an input of the
time delay block. The output of this system is the
same function, but with time delay τ, which can be
described by Eq.1.
)t(u)t(x
(1)
Time delay block can be described in Laplace
form which can be derived from Eq.1.
s
s
e
)s(U
)s(Ue
)s(U
)s(X
)s(G
τ−
τ−
=== (2)
Pade's approximation of time delay block is
very favorable in practice because of good
convergence rate of this approximation. It is also
very interesting theoretical case when Pade's
approximation order reaches infinity.
Pade's function is a rational function determined
by Eqs 3-5 (Hebibovic, 1998).
)s(D
)s(N
)s(W
τ−
τ−
=τ−
µν
µν
µν
(3)
i
1i
)s(
)!(!i
)!i(
)!i(
!
)s(N τ−
ν+µ
−ν+µ
−ν
ν
=τ−
∑
ν
=
µν
(4)
j
1j
)s(
)!(!j
)!j(
)!j(
!
)s(D τ
ν+µ
−ν+µ
−µ
µ
=τ−
∑
ν
=
µν
(5)
289
Hebibovic M., Lacevic B. and Velagic J. (2004).
SIMULATION OF SYSTEMS WITH VARIOUS TIME DELAYS USING PADE’S APPROXIMATION.
In Proceedings of the First International Conference on Informatics in Control, Automation and Robotics, pages 291-295
DOI: 10.5220/0001127402910295
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