1
On the Sliding Mode Control of a Ball on a
Beam System
Naif B. Almutairi and Mohamed Zribi
Department of Electrical Engineering, College of Engineering and Petroleum, Kuwait University.
P. O. Box 5969, Safat 13060, KUWAIT
Tel.: (+965) 498-5845, Fax.: (+965) 481-7451, E-mail: [email protected]
Abstract-- This paper investigates the sliding mode control of
the Ball on a Beam system. A static and a dynamic sliding mode
controllers are designed using a simplified model of the system;
the simplified model renders the system feedback linearizable.
Then, a static and a dynamic sliding mode controllers are
designed using the complete model of the Ball on a Beam system.
Simulation results indicate that the proposed controllers work
well. The four proposed controllers are implemented using an
experimental setup. Implementation results indicate that the
proposed control schemes work well. As expected, it is found that
the proposed two controllers which are designed using the
complete model of the system gave better performances than the
ones designed using the simplified model of the system. In
addition, the experimental results indicate the two dynamic
controllers greatly reduce the chattering usually associated with
sliding mode controllers.
I. DYNAMIC MODEL OF THE BALL ON A BEAM SYSTEM
y
L
R
Ball
constant; K g : gear ratio; d : lever arm offset; J 1 : moment
of inertia of the beam; Kb : back EMF constant. The
parameters k1 , k 2 , k3 , and k 4 are functions of the system
parameters as follows: k1 =
k2 =
Rm J m L
Km Kg d
+ J1 ;
L ⎛⎜ K m Kb
K
R B ⎞⎟
+ Kb + m m ⎟⎟ ; k3 = 1 + m ; k 4 = 7 5 ;
⎜
Rm
d ⎜⎝ Rm
K m K g ⎠⎟
vin (t ) : input voltage to the motor ; u (t ) = k 3vin (t ) is the
control input to the Ball on a Beam system.
II. SOME RESULTS
A. Design of a dynamic Sliding Mode Controller Using
the Complete Model of the Ball on a Beam System
Motivated by the work done in [2, 3], we propose a dynamic
sliding mode controllers for the Ball on a Beam system using
the complete model of the system.
r
mg sin α
Beam
α
Define the sliding surface,
mg
z
Mg
s 2 = eα + λ1eα + λ2eα + λ3er + λ4er
d
θ
+ λ1α + λ2 α + λ3r + λ4 (r − rd )
=α
where λ1 , λ2 , λ3 , and λ4 are scalars such that λ1 > 0 ,
Motor
Figure 1. Schematic diagram of the Ball on a Beam System.
The equations of motion describing the Ball on a Beam
system can be written as [1]:
(mr 2
(
+ (2mrr + k 2 ) α + mgr +
+ k1 ) α
L
2
)
Mg cos α = u
2
k 4r − r α + g sin α = 0
where,
α (t ) : beam
angle;
r (t ) : ball
position;
θ (t ) : servo gear angle; g : the gravitational constant;
λ2 > 0 ,
2
λ1 λ4 > λ1λ2λ3 +
beam;
J m : effective
Rm : armature
resistance
moment of inertia;
of
the
K m : motor
motor;
g
λ4 < 0 ,
g
k4
λ3 > 0 ,
2
k4
λ3 .
Proposition:
The following discontinuous dynamic control scheme,
⎧
⎪
⎪
⎪
⎩
2
u = (mr + k1 ) ⎪
⎨−f +
2mrr
(mr 2
2
+ k1 )
(
u − (2mrr + k 2 ) α − mgr +
−λ1
(mr
torque
2
−λ2 α − λ3
Manuscript submitted on February 15, 2008.
λ1λ2 +
Let Γ4 be a positive scalar.
m : mass of the ball; M : mass of the beam; L : length of
the
λ3 < 0 ,
2
r α − g sin α
k4
+ k1 )
u
L
2
)
Mg cos α
⎫
⎪
⎪
⎪
⎭
− λ4r − Γ 4 sgn ( s2 )⎪
⎬
and
2
when applied to the Ball on a Beam system, asymptotically
stabilizes the states of the system to their desired values.
Proof of this proposition is in the full version of the paper.
B. Simulation results
The controller parameters used are λ1 = 72 , λ2 = 1342 ,
λ3 = −362.5 , λ4 = −342.5 and Γ 4 = 6 . Figure 2 shows the
simulation results when the dynamic sliding mode controller
is used. It can be seen from Figure 2 (a) that the output
y = r (t ) converges to its desired signal rd in about 15 sec.
The control input vin (t ) is shown in Figure 2 (b); note that
the chattering is greatly reduced.
C. Experimental results
The parameters of the controller are the same as the
parameters used in the simulation results section. Figure 3 (a)
shows the ball position while Figure 3 (b) shows the applied
voltage to the DC motor. It is noticed that the best result is
obtained when using the complete model in designing the
dynamic SMC. This confirms the most important finding of
this paper. That is, when using the dynamic SMC designed
using the complete model of the system, better results in terms
of system performance and bigger reduction in the chattering
of the control signal are obtained.
Experimental Results : Dynamic SMC (Complete Model)
40
r(t)
rd(t)
35
30
Simulation Results : Dynamic SMC (Complete Model)
40
Ball Position r(t) in (cm)
rd(t)
r(t)
35
Ball Position r(t) in (cm)
30
25
25
20
15
10
20
5
15
0
10
0
20
40
60
80
120
140
160
180
200
160
180
200
(a)
5
0
100
Time (sec)
Experimental Results : Dynamic SMC (Complete Model)
0
20
40
60
80
100
Time (sec)
120
140
160
180
5
200
4
(a)
3
Simulation Results : Dynamic SMC (Complete Model)
5
Applied Voltage vin(t) (V)
2
4
3
−2
0
−3
−1
−4
−2
−5
−3
−5
0
20
40
60
80
100
Time (sec)
120
140
(b)
−4
0
20
40
60
80
100
Time (sec)
120
140
160
180
200
(b)
Figure 2
0
−1
1
in
Applied Voltage v (t) (V)
2
1
System response when the dynamic SMC is used
(using the complete model).
It can be concluded that although the responses of the ball
position in all cases are very good, the chattering is greatly
reduced for the two cases when the dynamic SMC are used.
This is an expected result and actually it is one of the
properties of the dynamic SMC. Also, it is worth mentioning
that the chattering is reduced further when using the complete
model in the design of the controller.
Figure 3
Experimental results when the dynamic SMC is used
(using complete model)
REFERENCES
[1] F. O. Rodríguez, W. Yu, R. L. Feregrino, and J. d. J. M. Serrano,
"Stable PD Control for ball and beam system," in Proc. International
Symposium on Robotics and Automation, Querétaro, México, 2004, pp. 333338.
[2] H. Ashrafiuon and R. S. Erwin, "Sliding control approach to
underactuated multibody systems," in Proc. American control conference,
Boston, Massachusetts, USA, 2004, pp. 1283-1288.
[3] M. Nikkhah and H. Ashrafiuon, "Optimal sliding mode control for
underactuated systems," in Proc. American Control Conference, Minneapolis,
Minnesota, USA, 2006, pp. 4688-4693.