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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 1−p, 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

ℜ

∈

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, ,

ℜ

∈

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.

Podlubny. I. (1999a). Fractional-order systems and

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.

Tseng, C. (2001). Design of fractional order digital

FIR differentiators, IEEE Signal Processing Lett.

8(3), pp. 77–79.

Vinagre, B., Chen, Y., Petras, I. (2003). Two direct

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.