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Model reference adaptive control with fractional derivative
Samir LADACI a,*, Abdelfatah CHAREF b
aUniversité de Skikda,Département d’Informatique,BP:26 Elhadaiek 21000 Skikda, Algeria.
Fax: 00213.38.70.10.04 E-mail: SLADACI@caramail.com
bLaboratoire de Traitement de Signal,Départ Electronique, Université Mentouri,Constantine 25000 Algeria.
E-mail: AFCHAREF@yahoo.com
Abstract- Over the few last years the idea of introducing fractional calculus and systems in adaptive control has
found a great interest, for the benefit one can win in the performances given by such systems. The main idea is to
impose a fractional structure that can improve the behavior of the closed loop. By the mean of introducing a
fractional order derivation at the plant output, and an adapted fractional model reference, we can achieve this
aim, because of the best dynamical stability and robustness against perturbations.
Key words: fractional derivative, fractional order system, Model Reference Adaptive Control.
1. Introduction
There has been considerable research in
the area of fractional order systems and
their application in control (Sun, 1990)
(Oustaloup, 1983-1991) (Loiseau,1998),
and since few years many researches have
focused on the introduction of fractional
order operators in adaptive control, and
especially on MRAC (Model Reference
Adaptive Control)(Vinagre, 2002). In this
approach adaptive algorithms allow the
control of systems on which few
information are known.
First, the use of fractional model reference
in the adaptive scheme has shown an
improvement in system dynamics, due to
the best model reference dynamical
properties(Ladaci, 2002). Then the
introduction of fractional integration has
proven the ability of fractional algorithms
to guarantee stability with a highest level
of performance then the integer order
algorithms (it depends on the choice of the
Integration fractional order).
The originality of this contribution is the
use of both of the too approaches discussed
bellow, by introducing a fractional model
reference and a fractional derivation
(which from an algebraic point of view,
can be assimilated to a advanced
fractional integration in the adaptation
loop).
The use of the derivative action has always
been done with a lot of care in automatics,
because, in the presence of noise it can
damage the desired performance or worst
the system stability. However in the
fractional derivative case, which is a long
memory process (Hotzel,1998), we show
that the perturbation rejection is achieved
at a satisfactory level, joining several
other research conclusions on the
robustness of fractional systems versus
perturbation.
This paper is structured as follows:
Section 2 introduces the fractional order
systems, with both integration and
derivation definitions. Section 3 then
introduces the model reference adaptive
control (MRAC) problem and the use of
fractional operators in the adaptation
algorithm. An example is presented in
section 4, with comments on simulation
results. Finally some concluding remarks
are presented in section 5.
2
2. Fractional order systems
The analysis in Bode plot of many
natural processes, like transmission lines,
dielectric polarisation impedance,
interfaces, cardiac rhythm, spectral density
of physical wave, some types of noise
(Van Der Ziel, 1950) (Duta, 1981) , has
allowed to observe a fractional slope. This
type of process is known as 1/f process or
fractional order system. The used
description equation into frequency
domain of these processes is given as
follows :
m
)
pt
s
(1
k
X(s)
(1)
with m : fractional exponent.
pt : fractional pole which is the cut
frequency.
s : Laplace operator.
For complex systems where puissance is
varying from a real number to another,
they are represented by a multiple poles
function with fractional power :
n
i
m
ii
pts
sX
1)/1(
1
)( 1m0 idd (2)
where (1/pti) : relaxation times.
Performances:
Many precedent works have shown
that fractional systems present best
qualities, in response time and in transition
dynamic stability (Sun, 1990). In fact, for a
second order system, represented by the
following transfer function :
m
n
nw
s
w
s
sG
¸
¸
¹
·
¨
¨
©
§
12
1
)(
2
2
[
(3)
with : wn=10 rd/s, 95.0
[
The step responses for the integer
case (m=1) and the fractional order one
(m=0.55) are given in figure1, and show
the gain in fastness.
Figure 1. Comparative step response
(fractional / integer)
2.1. Fractional integration and
derivative
Fractional derivation and
integration are classical tools in
engineering, they have been used in
mechanics since at least the 1930’s and in
electrochemistry since the 1960’s. In
control field, interesting works have been
achieved in the soviet union (Brin, 1962),
and an increasing interest began mainly
after the contributions done in France
(Oustaloup, 1995).
2.1.1 Definition of fractional integration
(Fliess, 1997-1998)
Let C
D
,0)(
!
D
,Rc
and f a
locally integrable function defined on
[c,+f[. The Dorder integral of f , of
lower bound c is defined as :
³*
t
c
cdf
t
tI WW
D
W)(
)( )(
)(f 1(4)
with
c
t
t
, and
*
is the Euler function.
The formula (4) is called Riemann-
Liouville Integral.
Usually, the control loop is
discreet, and we use a sampled
approximation of (4) given by :
¦'''
*
'
' 1
0
1)()(
)(
)( k
cfkkfI WW
D(5)
with,
'
: Sampling Period.
TIME (sec.)
AMPLITUDE
00.1 0.2 0.3 0.4 0.5 0.6
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Yf
Ye
FRACTIONAL
INTEGER
3
In practice we have found it necessary to
add a constant real c, to obtain a best
dynamical behaviour, so :
¦'''
*
'
' 1
0
1)()(
)(
)( k
cfkckfI WW
D(6)
From a theoretical point of view, we
know that fractional systems are infinite
dimensional systems (Hotzel, 1998). The
)( 'kfIcdepends on all the past values of
f(.) since the initial instant. With the
sampling process the is a certain loss of
information due to the sampling period and
the impossibility to compute )( 'kfIcuntil
the last period k in (6) because we will
have a divide by zero.
2.1.2. Definition of Fractional Order
derivative:
It is defined as follows, by consideration of
the equality: )()( tfItfD c
We can write:
¦
¸
¸
¹
·
¨
¨
©
§
n
k
k
hkhtf
k
n
h
tf
dt
d
tfD
0
0)()1(
1
lim
)()( (7)
And assuming that: )()( tfDtfD h
|,
we have:
¦
¸
¸
¹
·
¨
¨
©
§
k
j
j
hjhkhf
j
htfD 0)()1()( D(8)
With:
)1()1( )1(
)!(! !
**
*
¸
¸
¹
·
¨
¨
©
§
jjjj
jD
D
D
D
D
Where
1
0
¸
¸
¹
·
¨
¨
©
§Dand
*
: Euler function.
Computation of coefficients:
The z-Transform of fractional derivation
can be obtained as follows:
¦¦
¸
¸
¹
·
¨
¨
©
§
00
)(
)1()1( kk
k
k
kk zz
k
zZ
D(9)
with 1
)(
0 Z;
...3,2,1
1
1)( 1
)(
¸
¹
·
¨
©
§
k
kkk Z
D
Z
2.2 Linear approximation of fractional
order Transfer function:
For the purpose of our approach we
need to use an integer order model
approximation of the fractional order
model reference in order to implement the
adaptation algorithm, for this aim we have
used the so-called singularity function
method (Charef, 1991) , and precisely for
the case interesting our approach that is a
fractional second order system of the form
(3) with m a positive real number such that
0<m<0.5 We can approximate
m
ss
sH
¸
¹
·
¨
©
§
12
²
²
1
)(
Z
[
Z
by the function:
¸
¹
·
¨
©
§
¸
¹
·
¨
©
§
¸
¹
·
¨
©
§
12
²
²1
1
)(
Z
D
Z
ZZ
ss
ss
sH (10)
With m
[D And m21
E
,which
also can be represented by the function,
¸
¹
·
¨
©
§
¸
¹
·
¨
©
§
N
ii
N
ii
p
s
z
s
ss
s
sH
1
1
1
)1(
)1(
12
²
²
1
)(
Z
D
Z
Z
(11)
The singularities are given by: Njzaabp j
j,...3,2,1.)( 1
1
1,...3,2)( 1
1 Nizabz i
i
with, bwz
1,)1(10
10
p
a,10
10 p
b ,
)log( )log(
ab
a
E
and p
H: Tolerated error in dB
The order of approximation N is computed
by fixing the frequency band of work,
specified by max
Z, so that:
NN pp max1 ZWhich leads to:
4
11
)log(
log
int 1
max
»
»
»
»
»
¼
º
«
«
«
«
«
¬
ª
¸
¸
¹
·
¨
¨
©
§
ab
p
ofpartegerN
Z
(12)
H(s) can be then be written under a
parametric shape function of order N+2:
2
1
1
2
1
10 .......
....
)(
mN
N
m
NmN
N
m
N
masas
bsbsb
sH (13)
mi
aand mi
bare calculated from the
singularities i
p,i
zand
D
,
Z
.
3. Applying in adaptation algorithm
3.1. Model Reference Adaptive Control
(Astrom, 1995), (Landau, 1979)
It is one of the more used
approaches of adaptive control, in which
the desired performance is specified by the
choice of a reference model. Adjustment of
parameters is achieved by the mean of the
error between the output of the plant and
the model reference output. Which can be
represented like in figure 2.
Figure 2. Direct Model Reference Adaptive Control
3.2. M.I.T. Rule:
We consider a closed loop system
where the controller has an adjustable
parameter vector T. A model which output
is ymspecifies the desired closed loop
response. Let ebe the error between the
closed loop system output yand the model
one ym, one possibility is to adjust the
parameters such that the cost function:
2
e
2
1
)(J T
(14)
be minimised. In order to make Jsmall it
is reasonable to change parameters in the
direction of negative gradient J,so:
Tw
w
J
Tw
w
J
T
e
e
J
dt
d
(15)
3.3. Introducing Fractional Operators:
Let the Reference model Gmod be a
fractional order transfer function of second
order like in (3). We introduce also a
fractional derivation bloc at the process
output, as it is represented in figure 3. We
assume that the relative degree of the siso
process transfer function n is known. It
means that )( )(
)( sden sNum
sG with
Deg(Den(s))-
Deg(Num(s))=n (16)
Figure 3. Bloc diagram of Adaptation algorithm
So when sis of high value, we can write:
mm
n
n
c
ms
w
s
w
s
u
y
sG 2
2
2
mod 1
12
1
)( |
¸
¸
¹
·
¨
¨
©
§
[
and also, n
s
u
y
sG 1
)( | (17)
So in order to compare ym(t) with
dt
tyd )(
we must assure this equality,
mn
s
s
s2
1
|in other words:
mn 2
D
(18)
uc
Gmod
s
J
G
6
3
3
ym
-
u
T
+
Process
Fractional Model
y
s
Reference
Model
Adjustment
mechanism
Contr
Proces
y
mod
(t)
U(t)
y(t)
U
c
(t)
5
4. Example
Taking:
5
10
s
G,
With a model transfer function
m
ss
G18.1²
1
mod
Where 1
Z
;9.0
[
This transfer
function is sampled and approximated to
an integer order model by the singularity
function method (13).
For m = 0.4, 0.35, we obtain the simulation
results given in figures 4. And 5.
Comments
xThe stability of the closed loop is
maintained, with a good level of
performances.
xWhen m<<0.5 which means that
n
|
D
(for example m= 0.1 18.0
|
D
)
the rejection of perturbation is not
achieved and the response is very bad,
as the order of derivation is quite
integer.
xHowever for higher values of m there
is an effective rejection of perturbation
as expected because as we can see in
(7), the calculus of the fractional
derivative is dependent of all the
history of the signal, which will
moderate the effect of last variations.
5. Conclusion
A fractional Model adaptive control
algorithm which includes the use of
fractional derivation was presented that
can guarantee the closed loop stability with
a good level of performances and high
ability to reject perturbation. The
simulations showed the improvement of
performances of the adaptive control
algorithm even when there are
perturbations. This approach is interesting
also in the case of fractional order
processes. The stability and robustness
conditions of such systems are under
study.
6. References
Astrom, K.J., & Wittenmark, B. (1995)
Adaptive Control, Addison-Wesley,
USA.
Brin, I. A. (1962) On the stability of
certain systems with distributed and
lumped parameters. Automat. Remote
Control, vol. 23, 798-807.
Charef A., (1991) Analysis and Synthesis
of fractal systems, Ph.D. Thesis, Drexel
University.
Duta P. & P.M. Horn, (1981) Low
frequency fluctuations in solids : 1/f
noise. Review of modern physics, Vol.
53, No 3.
Fliess M. & Hotzel R. (1998) On linear
systems with a fractional derivation:
Introductory theory with examples” in
Mathematics and computers in
Simulation 45, 385-395
Fliess M. & Hotzel, R. (1997) Sur les
systèmes linéaires à dérivation non
entière. C.R.Acad. Sci. Paris, série II b,
(signal, informatique), vol. 324, 99-105.
Hotzel R., (1998) “Contribution à la
théorie structurelle de la commande des
systèmes linéaires fractionnaires”,
Thèse de Doctorat, Université de Paris-
Sud, Centre d’Orsay.
Ladaci S. & Charef A. (2002) Commande
Adaptative à modèle de référence
d’ordre Fractionnaire d’un bras
artificiel. Revue Sciences et
Communication, ENSET Oran Algeria,
N°1, October, 53-55.
Landau, Y.D., (1979) Adaptive Control :
The model reference Approach, Marcel
Dekker, New York.
Loiseau, J.J., & Mounier, H. (1998)
Stabilisation de l’équation de la chaleur
commandée en flux, ESAIM, Proc.,
131-144.
Oustaloup A., (1983) Systèmes asservis
d’ordre fractionnaire. Masson, Paris.
Oustaloup A., (1991) La commande
CRONE. Hermès, Paris.
Oustaloup A., (1995) La dérivation non
entière, Hermès, Paris.
6
Sun, H., & Charef, A. (1990) Fractal
System-A time domain Approach ,
Annals of Biomedical Ing.Vol :18, 597-
621.
Van Der Ziel, A. (1950) On the noise
spectra of semiconductor noise and of
flikker effects. Physica. 16, 359-372.
Vinagre, B. M., & Petras, I., & Podlubny,
I., & Chen, Y.Q. (2002) Using
Fractional Order Adjustment Rules and
Fractional Order Reference Models in
Model-Reference Adaptive Control. In
Nonlinear Dynamics 29, 269-279.
Figure 4. Output of the process and its fractional derivation for m = 0.4
a-without perturbations
b-with perturbation of 10% (Amplitude).
Figure 5. Output of the process and its fractional derivation for m = 0.35
a-without perturbations
b-with perturbation of 10% reference signal amplitude.
0 5 10 15 20 25 30
-1.5
-1
-0.5
0
0.5
1
1.5 Output of MRAC with fractional derivation m=0. 35
time in sec 0 5 10 15 20 25 30
-15
-10
-5
0
5
10
15 Fractional derived output m=0.35
time in sec
0 5 10 15 20 25 30
-15
-10
-5
0
5
10
15 Fractional derived output witn perturbati on (+/-10%) m=0.35
time in sec
0 5 10 15 20 25 30
-1.5
-1
-0.5
0
0.5
1
1.5 Output of Perturbed MRAC with fractional deriv ation m=0.35
time in sec
b
a
0 5 10 15 20 25 30
-20
-15
-10
-5
0
5
10
15
20 Fractional derived output m=0.4
time in sec
0 5 10 15 20 25 30
-1.5
-1
-0.5
0
0.5
1
1.5 Output of MRAC with fractional deriv ation m=0.4
time in sec
0 5 10 15 20 25 30
-20
-15
-10
-5
0
5
10
15 Fractional derived output witn perturbati on (+/-10%) m=0.4
time in sec
0 5 10 15 20 25 30
-2
-1.5
-1
-0.5
0
0.5
1
1.5 Output of Perturbed MRAC with fractional deriv ation m=0.4
time in sec
a
b
