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A NEW METHOD FOR APPROXIMATING FRACTIONAL DERIVATIVES: APPLICATION IN NON-LINEAR CONTROL

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Abstract

The theory of fractional calculus goes back to the beginning of the theory of differential calculus, but its application received attention only recently. In the area of automatic control some work was developed but the proposed algorithms are still in a research stage. This paper discusses a novel method, with two degrees of freedom, for the design of fractional discrete-time derivatives. The performance of several approximations of fractional derivatives is investigated in the perspective of nonlinear system control.
ENOC-2008, Saint Petersburg, Russia, June, 30-July, 4 2008
A NEW METHOD FOR APPROXIMATING FRACTIONAL
DERIVATIVES: APPLICATION IN NON-LINEAR CONTROL
J. A. Tenreiro Machado
Institute of Engineering of Porto
Dept. of Electrotechnical Engineering
Rua Dr. Antonio Bernardino de Almeida
4200-072 Porto, Portugal
jtm@isep.ipp.pt
Alexandra M. S. Galhano
Institute of Engineering of Porto
Dept. of Electrotechnical Engineering
Rua Dr. Antonio Bernardino de Almeida
4200-072 Porto, Portugal
amf@isep.ipp.pt
Abstract
The theory of fractional calculus goes back to
the beginning of the theory of differential calculus,
but its application received attention only recently.
In the area of automatic control some work was
developed but the proposed algorithms are still in a
research stage. This paper discusses a novel
method, with two degrees of freedom, for the
design of fractional discrete-time derivatives. The
performance of several approximations of
fractional derivatives is investigated in the
perspective of nonlinear system control.
Key words
Fractional calculus, control, discrete time.
1 Introduction
Fractional calculus (FC) is a natural extension of
the classical mathematics. The fundamental
aspects of the fractional calculus theory and the
study of its properties can be addressed in
references [Miller and Ross, 1993; Oldham and
Spanier, 1974; Samko, et al. 1993]. In what
concerns the application of FC concepts we can
mention a large volume of research about
viscoelasticity and damping, biology, signal
processing, diffusion and wave propagation,
modeling, identification and control [Anastasio,
1994; Bagley and Torvik, 1983; Machado, 1997;
Mainardi, 1996; Méhauté, 1991; Miller and Ross,
1993; Nigmatullin, 1986; Oldham and Spanier,
1974; Oustaloup, 1991; Oustaloup, 1995;
Podlubny. 1999a; Samko, et al., 1993].
Several researchers on automatic control
proposed algorithms based on the frequency
[Oustaloup, 1991; Oustaloup, 1995] and the
discrete-time [Machado, 1997; Podlubny. 1999a;
Podlubny. 1999b] domains. This article introduces
a novel method to implement fractional derivatives
(FDs) in the discrete-time domain. The
performance of the resulting algorithms is
analyzed when adopted in the control of nonlinear
systems. In this line of thought, the paper is
organized as follows. Sections two and three
develop the novel method of FD discrete-time
approximation and investigate its performance in
the control of a nonlinear system, respectively.
Finally, section four draws the main conclusions.
2 On the Generalization of Fractional
Discrete-Time Control Algorithms
The Grünwald-Letnikov definition of a FD of
order α of the signal x(t),
()
txD
α
, is given by the
expression:
(
)
[
]
()
()( )
()( )
=
α
α
+αΓ+Γ
+αΓ
=
=
0
11
1
1
1
lim
0
k
k
kk
khtx
h
txD
h
(1)
where Γ is the gamma function and h is the time
increment. This formulation inspired the discrete-
time FD calculation, by approximating the time
increment h through the sampling period T,
yielding the equation in the z domain:
()
[]
{}
(
)( )
()
()
()
zX
T
z
zXz
kk
T
txDZ
k
k
k
α
=
α
α
=
+αΓ
+αΓ
1
0
1
1!
11
1
(2)
where X(z) = Z{x(t)}.
The expression (2) represents the Euler (or first
backward difference) approximation in the so-
called s z conversion schemes, being other
possibilities the Tustin (or bilinear) and Simpson
rules. The generalization to non-integer exponents
of these conversion methods lead to the non-
rational z-formulae:
() ()
[]
α
α
α
=
1
0
1
1
1
zHz
T
s
(3a)
()
[]
α
α
α
=
+
1
1
1
1
1
12
zH
z
z
T
s
(3b)
where H
0
and H
1
are often called generating
approximants of zero and first order, respectively.
In order to get rational expressions the
approximants need to be expanded into Taylor
series and the final algorithm corresponds to a
truncated series or to a rational Pade fraction.
We can obtain a family of fractional differentiators
by the generating functions H
0
and H
1
weighted by
the factors p and 1p, yielding:
()
(
)
()
(
)
1
1
1
0
1
+= zHpzpHpH
av
(4)
For example, the Al-Alaoui operator
corresponds to a weighted interpolation of the
Euler and Tustin integration rules with
43
=
p
[Al-Alaoui, 1993; Al-Alaoui, 1997; Smith, 1987].
These approximation methods have been studied
by several researchers [Barbosa, et al., 2004;
Barbosa, et al., 2006; Chen, and Moore, 2002;
Chen, et al., 2004; Chen and Vinagre, 2003;
Tseng, 2001; Vinagre, et al., 2003] and motivated
a novel averaging method based on the generalized
formula of averages, or average of order
q :
()
()
[]
()
()
[]
q
qq
av
zHpzHppqH
1
1
1
1
0
1,
+=
(5)
where (p,q) are two tuning degrees of freedom.
For example, when
{}
1,0,1
=
q we get the wel-
known {harmonic, geometric, arithmetic}
averages, respectively.
Bearing these ideas in mind we decided to
examine the expression resulting from (5), for
distinct values of (q,p). Tables 1 and 2 depict the
coefficients of a second order Pade approximation
()
i
i
i
i
iav
zbzaTpqH
=
=
2
0
21
1, ,
ii
ba , ,
to
α
D , 21=α , for
{}
2,23,1,21,0,21,1
=
q ,
43=p , and
{}
2,1,0,1
=
q , 21
=
p ,
respectively.
3 Performance Evaluation in the Control of a
Nonlinear System
In order to test the performance of the
expressions the usual method is to examine either
the frequency domain, by comparing the Bode
plots, or the time domain, by comparing the step
response. Nevertheless, often the differences are
negligible and, furthermore, do not have a direct
translation to control system performance.
Therefore, in our study we decided to test the
approximations by analyzing the step response
when the FD represents the control algorithm. In
this perspective, we consider the closed loop
represented in Fig. 1. The inclusion of the on-off
non-linearity in the forward loop leads to the
simplification of the analysis because the
controller gain is not relevant and is not necessary
to tune, but, on the other hand, we have a stringent
dynamic test that stimulates both the transient and
steady-state behavior.
Figure 2 depicts the closed-loop step response
for
{
}
2,23,1,21,0,21,1
=
q and 43=p .
Figure 2 presents the response for
{}
2,1,0,1=q
and
21
=
p . In both cases, we consider three
controller high sampling periods, namely T = {0.1,
0.2, 0.3} in order to test also the robustness for fast
versus slow sampling controllers.
In all cases we verify that:
In general the order
1=q is the one that
produces the best results;
The sampling period T = 0.1 leads to a
good performance, while the results degrade
considerably for larger values of T;
The difference between
43=p and
21
=
p seems to be negligible, particularly in
the case of
1
q .
While these results seem clear the authors
believe that further test, namely with other systems
and other values of α, q and p are still required to
establish a definitive conclusion.
4 Conclusions
In this paper a novel method for the discrete-
time FD approximation was presented and
evaluated. The new algorithm adopts the time
domain and generates a family of possible
approximations, having two distinct degrees of
freedom, namely the order of the averaging and the
weight of the generating functions. The properties
of several expressions were studied for a simple
non-linear system. The time response of the closed
loop system was analyzed and the robustness for
different sampling periods was tested. The
conclusions are consistent and motivate an
extensive test of all possibilities opened by the
extra degrees of freedom.
References
Al-Alaoui, M. (1993). Novel digital integrator and
differentiator. Electron. Lett.
29(4), pp. 376–
378.
Al-Alaoui, M. (1997). Filling the gap between the
bilinear and the backward-difference transforms:
an interactive design approach, Internat. J. Elec.
Eng. Edu.
34(4), pp. 331–337.
Anastasio, T. (1994). The Fractional-Order
Dynamics of Brainstem Vestibulo-oculomotor
Neurons. Biological Cybernetics,
72, pp. 69-79.
Bagley, R. and Torvik, P. (1983). Fractional
Calculus-A Different Approach to the Analysis
of Viscoelastically Damped Structures. AIAA
Journal,
21(5), pp. 741-748.
Barbosa, R., Machado, J., Ferreira, I. (2004),
Least-squares design of digital fractional-order
operators. Proc. First IFAC Workshop on
Fractional Differentiation and its Applications,
Bordeaux, France, July, 19–21, pp. 434–439.
Barbosa, R., Machado, J., Silva, M. (2006). Time
Domain Design of Fractional Differintegrators
Using Least Squares Approximations. Signal
Processing, Elsevier,
86(10), pp. 2567-2581.
Chen, Y. and Moore, K. (2002). Discretization
schemes for fractional-order differentiators and
integrators. IEEE Trans. Circuits and Systems—
I: Fundam. Theory Appl.
49 (3), pp. 363–367.
Chen, Y., Vinagre, B. (2003). A new IIR-type
digital fractional order differentiator. Signal
Processing,
83 (11), pp. 2359–2365.
Chen, Y., Vinagre, B., Podlubny, I. (2004).
Continued fraction expansion approaches to
discretizing fractional order derivatives-an
expository review. Nonlinear Dynamics
38,
155–170.
Machado, J. (1997). Analysis and Design of
Fractional-Order Digital Control Systems,
Journal Systems Analysis, Modelling,
Simulation,
27, pp. 107-122.
Mainardi, F. (1996). Fractional Relaxation-
Oscillation and Fractional Diffusion-Wave
Phenomena. Chaos, Solitons & Fractals,
7(9),
pp. 1461-1477.
Méhauté, A. (1991). Fractal Geometries: Theory
and Applications. Penton Press.
Miller, K. and Ross, B. (1993). An Introduction to
the Fractional Calculus and Fractional
Differential Equations. John Wiley and Sons,
1993.
Nigmatullin, R. (1986). The Realization of the
Generalized Transfer Equation in a Medium
with Fractal Geometry. Phys. stat. sol. (b),
133,
pp. 425-430.
Oldham, K. and Spanier, J. (1974). The Fractional
Calculus: Theory and Application of
Differentiation and Integration to Arbitrary
Order. Academic Press.
Oustaloup, A. (1991). La Commande CRONE:
Commande Robuste d’Ordre Non Entier.
Hermes.
Oustaloup, A. (1995). La Dérivation Non Entier:
Théorie, Synthèse et Applications. Editions
Hermes, Paris.
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PI
λ
D
μ
-controllers, IEEE Transactions on
Automatic Control,
44(1), pp. 208-213.
Podlubny. I. (1999b). Fractional Diferential
Equations, Academic Press, San Diego.
Samko, S., Kilbas, A., Marichev, O. (1993).
Fractional Integrals and Derivatives: Theory and
Applications. Gordon and Breach Science
Publishers.
Smith, J. (1987). Mathematical Modeling and
Digital Simulation for Engineers and Scientists,
second ed., Wiley, New York, 1987.
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FIR differentiators, IEEE Signal Processing Lett.
8(3), pp. 77–79.
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Tustin discretization methods for fractional-
order differentiator/integrator. J. Franklin Inst.,
340, pp. 349–362.
Table 1. Coefficients of the Pade fraction approximation for p = 3/4
q a
0
a
1
a
2
b
0
b
1
b
2
−1 0.76597491 6.026479067 −7.984934523 1 1.054474431 −7.609965244
−1/2 1.324947376 3.621337338 −6.248676637 1 0.000172433 5.761497326
0 1.465218175 2.188544252 −4.788397277 1 −0.804204046 4.788397277
1/2 1.734358402 1.669224885 −4.589703269 1 −1.164316112 4.184478992
1 1.787304546 1.2339426 −4.155138823 1 −1.36761081 −3.765228028
3/2 1.779700183 0.975910656 −3.828827863 1 −1.464217163 3.447457027
2 1.723916925 0.840452121 −3.56477439 1 −1.480178572 −3.188431144
Table 2. Coefficients of the Pade fraction approximation for p = 1/2
q a
0
a
1
a
2
b
0
b
1
b
2
−1 0.218192312 2.932612925 −3.758174112 1 1.815189846 −8.845101094
0 1.645896792 2.146868296 −5.087608366 1 −1.403319874 4.278151638
1 1.667798875 1.603885503 −4.470002812 1 −1.607438339 3.703075415
2 1.488976016 1.498893033 −4.035502783 1 −1.52197687 −3.29498733
21
DK
()
1
1
+ss
SH
+
r
c
1
Figure 1. The D
1/2
controller for a system with a nonlinear actuator.
0
0.2
0.4
0.6
0.8
1
1.2
012345
t
c(t)
H(-1,3/4), T=0.1
H(-1,3/4), T=0.2
H(-1,3/4), T=0.3
0
0.2
0.4
0.6
0.8
1
1.2
012345
t
c(t)
H(-1/2,3/4), T=0.1
H(-1/2,3/4), T=0.2
H(-1/2,3/4), T=0.3
0
0.2
0.4
0.6
0.8
1
1.2
012345
t
c(t)
H(0,3/4), T=0.1
H(0,3/4), T=0.2
H(0,3/4), T=0.3
0
0.2
0.4
0.6
0.8
1
1.2
012345
t
c(t)
H(1/2,3/4), T=0.1
H(1/2,3/4), T=0.2
H(1/2,3/4), T=0.3
0
0.2
0.4
0.6
0.8
1
1.2
012345
t
c(t)
H(1,3/4), T=0.1
H(1,3/4), T=0.2
H(1,3/4), T=0.3
0
0.2
0.4
0.6
0.8
1
1.2
012345
t
c(t)
H(3/2,3/4), T=0.1
H(3/2,3/4), T=0.2
H(3/2,3/4), T=0.3
0
0.2
0.4
0.6
0.8
1
1.2
012345t
c(t)
H(2,3/4), T=0.1
H(2,3/4), H=0.2
H(2,3/4), T=0.3
Figure 2. Closed loop step response for a D
1/2
controller with q = {1, 1/2, 0, 1/2, 1, 3/2, 2} and p = 3/4.
0
0.2
0.4
0.6
0.8
1
1.2
012345
t
c(t)
H(-1,1/2), T=0.1
H(-1,1/2), T=0.2
H(-1,1/2), T=0.3
0
0.2
0.4
0.6
0.8
1
1.2
012345t
c(t)
H(0,1/2), T=0.1
H(0,1/2), T=0.2
H(0,1/2), T=0.3
0
0.2
0.4
0.6
0.8
1
1.2
012345t
c(t)
H(1,1/2), T=0.1
H(1,1/2), T=0.2
H(1,1/2), T=0.3
0
0.2
0.4
0.6
0.8
1
1.2
012345t
c(t)
H(2,1/2), T=0.1
H(2,1/2), T=0.2
H(2,1/2), T=0.3
Figure 3. Closed loop step response for a D
1/2
controller with q = {1, 0, 1, 2} and p = 1/2.
... Fractional order differentiators and integrators are gaining importance in many fields. The application of fractional calculus in the area of control systems, robotics, instrumentation is illustrated in [15,16,19,39,41,53,55,89]. Debnath [11] has summarized the application of fractional calculus in various fields of science and engineering. ...
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