Fractional Order Controller Strategy for Nonlinear

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Fractional Order Controller Strategy for Nonlinear Dynamic Systems
H. Delavari*, R. Ghaderi*,
A. Ranjbar N. 1**, S. Momani***

*Noushirvani University of Technology, Faculty of Electrical and Computer Engineering,
P.O. Box 47135-484, Babol, Iran, ([email protected] ),
** Golestan University, Gorgan P.O. Box 386, Iran ([email protected] )
*** Department of Mathematics, Mutah University, P.O. Box: 7, Al-Karak, Jordan
Abstract: In this paper, a fractional order controller is proposed for nonlinear system. In this method the
fractional controller converts the system with integer derivatives into a system with desired fractional
derivatives in order to increases the degree of freedom of the stability. Preliminary fractional order
controller is designed for general form of the nonlinear systems, and then the performance of the
controller is investigated for two nonlinear dynamic case studies. The stability of the closed loop system
is also verified in presence of the proposed controller. At first this controller is designed for a robot
system and thereafter for a twin-tank model. Then numerical simulations are presented to show the
effectiveness of the proposed controller.
Keywords: Fractional Derivative, Fractional Control, Stability, Robotic System, Twin-Tank.

1. INTRODUCTION
Fractional calculus is an old mathematical topic since 17th
century. Although it has a long history, its applications to
physics and engineering are just a recent focus of interest. A
fractional-order controller for stabilizing the unstable fixed
points of an unstable open-loop system is proposed in
(Tavazoei et al., 2008). In (Calderón et al., 2006) several
alternative methods for the control of power electronic buck
converters applying fractional order control (FOC) are
presented. Many systems are known to display fractional
order dynamics, such as chaotic equations in control
engineering (Ge and Ou, 2008), earthquake oscillation (He,
1998), wave equation (Jafari and
Momani, 2007), and
Riccati differential equation (Odibat, and Momani, 2008;
Cang et. al,. 2007). An algorithm to determine parameters of
an active sliding mode controller in synchronizing to
different chaotic systems has been studied in (Tavazoei and
Haeri, 2007). In (Ahmad et al., 2004) the problem of chaos
control of three types of fractional order systems using simple
state feedback gains is studied. The control of a special class
of Single Input Single Output (SISO) switched fractional
order systems (SFOS) from the viewpoints of the Generalized
Proportional Integral (GPI) feedback control approach and a
sliding mode based Σ − Δ modulation implementation of an
average model based designed feedback controller is
addressed in (Sira-Ramirez et al., 2006). In the robotics
context, feedback linearization has accomplished (Park et al.,
2000). Using this method, all the coupling nonlinearities in
the dynamical model of the nonlinear system are first
compensated, transforming the nonlinear system into a linear
one. Then, the controller is designed on the basis of the linear
and decoupled plant. In (Ferreira et al., 2003), the
implementation of fractional order algorithms in the
position/force hybrid control of robotic manipulators is
1
studied. The signal propagation and the fractional-order
dynamics during, the evolution of a genetic algorithm, for
generating a robot manipulator trajectory are addressed in
(Pires et al., 2003).
In this paper, we propose a fractional-order controller to
control nonlinear dynamic systems. Also, we show that the
proposed controller can reduce the reaching time and error in
tracking control. This paper is organized as follows. Section 2
includes basic concepts in fractional calculus. In Section 3, a
fractional-order controller is proposed. The implementation
of proposed controller for robotic system and twin-tank
model are studied in section 3.1 and 3.2 respectively. Finally,
concluding remarks are drawn in Section 4.
2. FRACTIONAL CALCULUS
Fractional calculus is a mathematical topic with more than
300-year history, but the application in physics and
engineering has been recently attracted lots of attention.
During past three centuries, this subject was with
mathematicians, and only in last few years, this was pulled to
several (applied) fields of engineering, science and
economics. Perhaps the fractional calculus will be the
calculus of twenty-first century. The fractional-order
differentiator can be denoted by a general fundamental
operator a Dtq as a generalization of the differential and
integral operators, which is defined as follows (Calderón et
al., 2006):
 q
d
 dt q

q
a Dt  1
t
 ( d )  q

a

, R(q)  0
, R (q)  0
, R(q)  0
Corresponding Author: [email protected] , Tel: +98 911 112 0971, Fax: +98 111 32 34 201
(1)
where q is the fractional order which can be a complex
number, the constant a is related to the initial conditions.
There are two commonly used definitions for the general
fractional differentiation and integration, i.e., the Grünwald–
Letnikov (GL) and the Riemann Liouville (RL). The GL
definition is as:
(2)
 (t  q ) h 
1
q
j q
D
f
(
t
)

lim
(

1)
f
(
t

jh
)
 
a t
h 0 h q
 j

j 0
Where . is a flooring-operator while the RL definition is
given by:
t
(3)
1
dn
f ( )
q
d
a Dt f (t ) 
(n  q) dt n (t   )q n 1

a
For (n  1  q  n) and ( x) is the well known Euler’s
Gamma function. Also, there is another definition of
fractional differintegral introduced by (Caputo, 1967).
Caputo’s definition can be written as:
t
(4)
1
f ( n) ( )
q
D
f
(
t
)

d

,
n

1

q

n
a t
(q  n) (t   )q  n 1

a
Fractional order differential equations are at least as stable as
their integer orders counterparts, This is because; systems
with memory are typically more stable than their memoryless alternatives (Ahmed et al, 2007), Consider the following
autonomous commensurate fractional order system:
(5)
D q x  f ( x) ,
Whilst q is the commensurate order of the fractional timederivative in the Riemann-Liouville sense, and 0  q  1
and x 2 R . The equilibrium points of system (5) are
calculated when:
(6)
f (x )  0 .
These points are locally and asymptotically stable if all
eigenvalues of Jacobian matrix J  f / x , which are
evaluated at the equilibrium points, satisfy the following
condition (Matignon, 1996):
(7)
arg(eig ( J ))  q / 2 .
n
3. CONTROLLER DESIGN:
Applying the proposed fractional order controller, a system of
integer derivatives will be transformed to a system of
fractional order. Consequently, the degree of achieving the
stability will be increased (Tavazoei et al., 2008). Suppose
there is a nonlinear dynamic system which the governing
dynamical equations can be written as:
(8)
x2 k 1 (t )  x2 k (t )
x2 k (t )  f k ( x, t )  bk ( x)uk (t ),
x0  x(t0 )
Where t  R ,
k  1, 2,..., n
x(t )  [ x1 (t ) x2 (t ) x3 (t )... x2n (t )]T  R2n
(9)
x(t ) is the state vector, u(t )  R is the control action vector,
x0  x(t0 ) is the arbitrary initial conditions given at initial
time t0 , bk ( x) and fk ( x, t ) , k  1,2,..., n are the control gains and
the nonlinear dynamics of the system respectively. The
desired state variables are defined as:
xd (t )  [ xd1 (t ) xd 2 (t ) xd 3 (t )... xd 2n (t )]T  R2n
(10)
The tracking error e(t )  R2n can be defined as:
(11)
e(t )  x(t )  xd (t )
The control objective is to get the states x(t ) to track the
specific states xd (t ) . In other meaning, it is required to drive
the tracking error asymptotically to zero for any arbitrary
initial conditions. The proposed fractional controller in
(Tavazoei and Haeri, 2008) increases the boundary of the
stability.
The general form of the proposed controller is written as
follows:
 x2k (t )   k D q ( x2k (t )) 

(12)




uk (t )  bk1 ( x)  f k ( x, t )   k D1 q ( xd 2k 1 (t )) 



 K D1 q (e2k 1 (t ))   k K0 D q (e2k 1 (t )) 

 k 1
where K0 and K1 are positive constant. Meanwhile the rate of
convergence will be adjusted by other parameter, i.e.  k .
Then replacing (12) in (8) results the tracking error of the
closed loop system as:
e2k 1  K1e2k 1  K0 e2k 1  0
(13)
which represents an exponentially stable error dynamics, and
the proposed fractional order controller guarantees the
convergence of x(t ) to its desired value xd (t ) . In next
section the proposed method is implemented for two case
studies.
3.1. Implementation of Proposed Controller in robot
manipulator
A two-degree of freedom polar robot manipulator has one
rotational and sliding joint in the (x,y) plane. Neglecting the
gravity forces and normalizing unit mass and unit length of
the arm, the mathematical model of the two-degree of
freedom polar robot can be expressed as follows:
x1 (t )  x2 (t )
(14)
  x1 (t )  M ( x1 (t )  a)  x42 (t )  
x2 (t )  
 (   m)
u1 (t )  d1 (t ) 

x3 (t )  x4 (t )
2   x1 (t )  M ( x1 (t )  a)  
x4 (t )  

 x2 (t ) x4 (t )  u2 (t )  d 2 (t ) 
 J1  J 2 

 2


x
(
t
)

 1

 M ( x (t )  a ) 2 
1


where  is the mass of the motional link, M is the payload, J1
and J2 are moments of inertia of the motional link with
respect to the vertical axis through c and o respectively.
The robot manipulator joints are driven according to the
following desired trajectory:
(15)
xd 1  (0.5 )sin( t / 20) m,
xd3 (t),x3 (t),e 3 (t),rad.
3
xd 3  (0.9 )sin( t / 20) rad
x 3(t)
e 3(t)
1
0
-1
-2
Applying the proposed controller yields:
-3
 x2 (t )  1 D q ( x2 (t )) 



2
  x1 (t )  M ( x1 (t )  a )  x4 (t )  
u1 (t )  (   m)  

1 q
1 q
1 D ( xd 1 (t ))  1 K1 D (e1 (t )) 
 K D q (e (t ))  d (t )

1
1
 1 0

x d3(t)
2
0
5
10
15
20
25
30
35
40
30
35
40
Time(second)
(16)
(b)
0.4
 x4 (t )   2 D q ( x4 (t ))



2   x1 (t )  M ( x1 (t )  a )  
2
 J1  J 2   x1 (t )  

u2 (t )  
  x2 (t ) x4 (t )   2 D1 q ( xd 3 (t )) 
2  

M
(
x
(
t
)

a
)


1
 
1 q
 2 K 3 D (e3 (t ))

 K D q (e (t ))  d (t )

3
2
 2 2

u1 (t),N.
0.2
0
-0.2
-0.4
5
10
15
20
25
Time(second)
(c)
For the simulation studies, the following numerical values are
employed for the robot model parameters: M=1.2kg,
  1kg , J1=J2=1 kgm2, a=1m, and the initial conditions
are [ x1 (0), x2 (0), x3 (0), x4 (0)]T  [-0.25, -0.2, 0.6, 0.8]T . and
the controller parameters are chosen from the author
experience, 1  0.5 , k1  1 , k0  1 ,  2  0.5 , k2  1 , k3  1
, q  0.8 . The simulation result for this controller has been
shown in Fig. 1. After reaching time the actual trajectory
response x1(t) is almost identical to the desired command
x1d(t), the same results is noticed for x3(t)and x3d(t). Then
using (16) in (14) results the tracking errors of the closed
loop system:
e1  K1e1  K0 e1  0
1.5
xd1 (t),x1 (t),e 1 (t),m.
-0.5
5
10
15
20
25
30
35
40
Time(second)
(d)
Fig.1. Proposed fractional order controller (a): Tracking
response of joint1 (b): Tracking response of joint2 (c):
Control signal u1(t) (d): Control signal u2(t)
A twin-tank system consists of two small tanks coupled by an
orifice and a pump that supplies water to the first tank. The
pump only increases the liquid level and is not responsible
for pumping the water out of the tank. It is assumed that the
back pressure created by the water-head does not affect the
flow rate of the pump significantly. Tracking control of twintank has been studied already such as (Khan, et al., 2006;
Delavari et al., 2007a, b).
2
1
0.5
0
-0.5
x d1(t)
-1
x 1(t)
-1.5
e 1(t)
5
0
3.2. Implementation of Proposed Controller in Twin-tank
Model
This represents an exponentially stable error dynamics.
0
0.5
(17)
e3  K3e3  K2 e3  0
-2
u2 (t),N.m.
1
10
15
20
25
Time(second)
(a)
30
35
40
A twin-connected tanks system is a nonlinear dynamic
system which the governing dynamic equation can be written
as:
h1  U in  U12  A
h2  (U12  U out ) A
where
(18)
U12  a12 2 g (h1  h2 )
,For h1  h2
(19)
U out  a2 2 gh2 ,For h2  0
where h1 and h2 are the total water heads in Tank 1 and Tank
2, respectively, Uin is the inlet flow rate, U12 is the flow rate
from Tank 1 to Tank 2, A is the cross-section area of Tank 1
and Tank 2, a12 is the area of the coupling orifice, a2 is the
area of the outlet orifice and g is the gravitational constant.
Moreover, Uin  0 means that pump can only force water
into the tank.
Uin
U12
Tank 1
 x2 (t )  1 D q ( x2 (t )) (c12  2c2 2 ) 2   (27)


z1 ) 
(c1c2 2)( z1 z2  z2

U in  (2 A z2 c2 ) 

1 q
1 q
1 D ( H )  1 K1 D (e1 (t ))



q


K
D
(
e
(
t
))
1
 1 0

where e=x1(t)-xd(t)=h2(t)-H, is the tracking error. With k0 and
k1 are positive constants, then using (27) in (24) results the
tracking error of the closed loop system:
(28)
e1  k1e1  k0 e1  0
This represents an exponentially stable error dynamics. The
height h2(t) will asymptotically converge to the desired value
H. Hence the proposed controller guarantees the asymptotic
convergence of the output to the desired amount.
Let 1  0.5 , k1  1 , k0  1 , q  0.8 , simulation results for
this controller have shown in Fig.3.
h1
h2
g= 981 cm/sec.2
H= 7cm
(0 cm3/sec<Uin<50 cm3/sec.
Applying the control strategy for this system one can obtain
the input control signal as:
Uout
Tank 2
Fig.2. Interconnected Twin-Tank schematic
7.5
Let:
z1  h2  0, z2  h1  h2  0
(20)
c1  a2 2 g A , c2  a12 2 g A
Also the output of the coupled tanks is taken h2(t) , hence the
dynamic model in (1) and (2) can be written as:
z1  c1 z1  c2 z2
2
z2  z2
2
100
150
100
150
100
150
3
uin (t),cm /sec.
40
30
20
10
0
0
50
Time(second)
(b)
2
1.5
2
1
0.5
(25)
0
z2  z2
z1 )
(26)
-0.5
0
50
Time(second)
(c)
2A z )
2
Let parameters of system for this model are as:
A=180 cm2
50
2.5
in
f ( x)  (c1  2c2 ) 2  (c1c2 2)( z1
b ( x )  (c
0
50
z1 ) 
y  x1
It can be easily expressed as
x  f ( x)  b( x)u
where
2
5.5
(a)
e(t),cm.
2 A z2 c2 U
x 1 (t)
6
Time(second)
y1  z1
Then the goal is to regulate the system output (h2(t)) to the
desired value (H). Let:
(22)
z1

 x1  



x 
 2   c1 z1  c2 z2 
The inverse mapping from x to z is given by:
(23)
x1

 z1  



z 
2
 2  ((c1 x1  x2 ) c2 ) 
Then the model in (21) can be expressed as:
(24)
x1  x2
2
x d1 (t)
6.5
5
(21)
z2  c1 z1  2c2 z2  U in A
x  (c1  2c2 ) 2  (c1c2 2)( z1
2
xd1 (t),x1 (t),cm.
7
a2= 0.24 cm2
a21=0.5 cm2
Fig.3. (a): water level in Tank2 (b): The inflow rate into
Tank1 (c): Tracking error using proposed fractional order
controller.
Simulation results of employing the proposed controller with
+5% variations in system parameters have been shown in
Table 1, 2 respectively. A fast tracking response is observed
by employing the proposed controller in comparison with the
response obtained by employing the feedback linearization.
From Table 1, 2 it can be seen that the Reaching times (Rt)
and the mean absolute value of errors (E) by employing the
proposed fractional order controller is less than conventional
feedback linearization.
Table 1. Results of controller performances with 5%
variation in parameters of the system
Robot
q
0.8
1
Rt1(sec.)
7.5121
8.1512
Rt2(sec.)
9.1121
10.4182
E1
0.0062
0.0071
E3
0.0152
0.0161
Table 2. Results of controller performances with 5%
variation in parameters of the system
Twin-Tanks
q
0.75
1
Rt1(sec.)
55.12
65.34
E1
0.3404
0.4108
6. CONCLUSIONS
In this paper, a fractional-order controller is proposed to
control nonlinear dynamic systems. This controller converts
the system of integer type to a desired fractional one. The
proposed controller has also increased the degree of freedom
and of course the rate of convergence. It can also reduce the
reaching time and error in tracking control. This kind of
controller will be applied on some other dynamic. It is
therefore concluded that the integrated performance of
proposed fractional order controller is superior to the
traditional Feedback linearization control.
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