1. Proceedings of International Conference on Mechatronics
Kumamoto Japan, 8-10 May 2007
TuAl -A-1
Exact Model Knowledge and Direct Adaptive
Controllers on Ball and Beam
Turker Turker
Haluk Gorgun
Erkan Zergeroglu
Galip Cansever
Dept of Electrical Eng.
Yildiz Technical University
Istanbul, Turkey
turker@yildiz.edu.tr
Dept of Electrical Eng.
Yildiz Technical University
Istanbul, Turkey
gorgun@yildiz.edu.tr
Dept of Computer Eng.
Gebze Institute of Tech.
Izmit, Turkey
ezerger@bilmuh.gyte.edu.tr
Dept of Electrical Eng.
Yildiz Technical University
Istanbul, Turkey
cansever @ yildiz.edu.tr
asymptotically stable PD controller has been analyzed for nonlinear ball and beam system in [5]. To be able to use Lyapunov
method for analysis they have been use transformations and
applied modified asymptotically stable PD controller requiring
well defined initial conditions afterwards. Since the system is
nonminumum phase nonlinear system and only approximate
solutions could be found, intelligent control algorithms are also
suggested for ball and beam in literature [6,7]. In this paper
we aimed to analyze and to implement exact model knowledge
control and direct adaptive control for cascaded dynamics of
ball and beam system. Second order ball position and second
order beam angle dynamics are cascaded. Following, proposed
controllers for these two dynamics implemented individually.
Abstract-This paper presents analysis and implementation of
Exact Model Knowledge (EMK) and Direct Adaptive control
schemes on the 4th order ball and beam system in which the
dynamics of the ball position and the dynamics of the beam
angle are cascaded. For the controller analysis the error dynamic
equations for ball position and the beam angle are derived
for both cases. Following, experimental studies are conducted
based on the proposed control approaches and it is presented
that constant and sinusoidal references for the ball position are
tracked asymptotically.
I. INTRODUCTION
The position of the ball on the beam is controlled by dc motor
which is pivoted to the one end of the beam. The ball moves
on frictionless beam according to angle change of the rotor.
The main difficulty in such system is it has one passive joint
and since the ball is not actuated directly such systems are
called under-actuated system.
In the following sections,the dynamics of the system is described and the control problem of the system is defined.
Next, the controller designs with two approaches: exact model
knowledge and direct adaptive are proposed. Results on application of two controllers for the ball and beam system are
presented and discussed. Finally concluding remarks are given.
The control of this type of structure has attracted attention
and several studies are conducted and published in literature.
Feedback linearization method is not suitable for ball and
beam system since the relative degree of the system is not
well defined. In [1], by using geometric transformation the
system converted to have well defined relative degree and
input output feedback linearization is applied accordingly.
Linearization around the operating point can be applied to
such system but one should aware that it could be nothing
more than a local solution. Even though it is good enough
for small region of attraction but it fails in larger domain. To
extend the solution region, one way is to have two different
algorithms as in [2]. First to have piecewise linearization
such that the singularity around the origin would be avoided
and then away from the origin another algorithm which will
apply input output linearization is used to have larger solution
domain for the system. Another approach to overcome the
difficulty mentioned earlier has been proposed by [3] in which
extended dynamic model and a normal form augmentation
is presented. The main idea behind in this study is to cover
parameter linearity and then implement direct adaptive control
algorithm. Teel has been used simplified model of the system
and achieved semi-global stabilization for unknown ball mass
and beam inertia. Two stages of adaptive scheme are proposed
and backstepping algorithm is simulated in [4]. More recently,
1-4244-1 184-X/07/$25.00©2007 IEEE
II.
PROBLEM FORMULATION
Fig.1 illustrates the ball and beam system used in this study.
Dynamic model for the system was derived using EulerLagrange equations, and the Lagrangian of the system can
be derived as,
L
(
+
2
2
+
(
(mgx
where m,R and Jb
are
the
+
+
+ Mlg sin
mass
2
2)
(r
*2
(1)
(kg), radius (im) and the
moment of inertia (kg.m2) of the ball, respectively. Likewise,
M,l and J are the mass (kg), length (im) and the moment of
inertia (kg.m2) of the beam, respectively. o is the motor shaft
angle (rad), is the position of the ball (im) and g is the
acceleration of the gravity (m/s2). r is the distance between
the shaft and contact point of the disc and the beam (im). The
ratio between r and I is constant and given as 1/16.75 for
our example. Applying Euler-Lagrange equations, the model
of the system can be found as,
x
1

2. (11)
(2) where k2 is a positive constant. Taking time derivative of (1 1),
combining the result with (9) and adding term 03X±z2 to both
Z2
(m + Jb/R)
m
x()
(r) (2)
+
mgsin (I)
0
2
1K
(3)
T
T
We can linearly parameterized (12) to obtain open-loop
dynamics of the angle of the motor shaft as
(01
ej + kle,
('d + kje1) +
=
sin
(13)
vector and 0 is the parameter
row
(5)
sin (1675).
Y-=
((d ++k2C2
(
+
2)COS
2
_(I d + k2
0
After putting that into (2) we obtain the open-loop
dynamics of the position of the ball as
01
((9d+ ( + k2e2)
X±
(14)
( I ])
02
03
04
1
(15)
respectively.
CONTROL DESIGN
This section includes controllers for derived error dynamics.
For (8) we can derive a control law directly assuming the
dynamics given by (8) known exactly. In that way we define
III.
(7)
(175
[
=
IT
I- W)
COS
(6)
Since we can change the position of the ball by only changing
the angle of the motor shaft, the control input which can be
assumed as desired beam angle can be defined as in
u
error
vector,
where k, is a positive constant. By taking derivative of (5)
and putting (2) into it, we obtain,
¾l =
+ 03X2) Z2 = YO -T-03XZ2
where Y is the regression
are defined as
(4)
=Xd-X
=
k2e2
(01 + 03X2) %2 = (01 + 03X2) ((3d + k2e2)
+203X±y( + (04X + 02) COS ( so) -T ± 03X±Z2- (12)
where is applied torque to the beam. The middle term in (3)
can be omitted since it's value is very close to zero during the
operation of the ball and beam system [8]. Here we have the
whole fourth order ball and beam model which has two parts
consisting of two second order equations. For the position of
the ball we can define error and filtered tracking error signals,
respectively, as follows
z1
e2 +
sides of (11) we obtain
o Z o
+ vy
( ZmgxV+ 33 Mgl) COS ( j5
33.5
e1
=
error
a nonnegative function as
l =
We
can
5
(-~d + kj,) +
rewrite (3) by defining
some
(8)
-7
12
constant system param-
2
Taking time derivative of (16) and substituting (8) into it gives
eters as
(o1 + 03X2) ( + 203X±y( + (04X + 02) COS
Likewise, error and filtered tracking
defined for beam angle as
error
Y)
(9)
=T.
signals
can
T1h s=hZl e(conrojsd
be
Thus, the control law is defined
7
U
e2 =
d-O
(16)
(10)
=-
nd
a:+k )+
as:
[(-~d +kj~,) +k1z1z
5g
where k
is
a
(17)
(18)
positive constant. After plugging (18) in (17),
V1
=-kzl Z12
(19)
Kz1 2
is obtained. Taking norms of (16) and (19) gives V1 <
- kz
V1
<
2
and
respectively. Combining these two
<- kz V means V(t) and also z12 decreasing
results in
exponentially. Based on these facts it can be concluded that
V1 (t) C L£, and zi (t) C L,. Also from (4), (5) and assuming
desired signal and its first and second derivatives are bounded,
so as el1,e,x,x,, C L£X
[9].Thus, all the signals in (8) are
bounded and el approaches to zero exponentially. Note that,
extracting o from (7) gives
Fig. 1
Ball and Beam System
2

3. 16.75 arccos {-[d + kl l) + kzl1 }
V3
(20)
=
Z2 [YO -T2] 0
I
0
=
(01 + 03X2) Z2%2 + 03X±z22
=
Z2
[YO-
Tj]
(21)
(22)
C L2. By using these results, due to
Barbalat's Lemma [9] we can conclude that limt,O Z2 (t) = 0
and therefore, limt,O e2(t) = 0.
IV. EXPERIMENTAL RESULTS
The controllers designed for the system given by (1) and (2)
have been verified experimentally on Quanser Ball and Beam
module in which the beam was actuated with a DC servomotor.
A P4 3.00 GHz computer implemented with Quanser Q4-PCIDAQ was used to process feedback signals and derive the
control input for the system. There is also a power opamp
module between DAQ and DC servo providing the input
signal for the motor. The mechanical system parameters are
m=0.064kg, Jb=1.65xlO5kgm2, R=0.0254m, J=0.0106kgm2,
M=0.2kg, 1=0.4m and the acceleration of the gravity is
g=9.8tm/s2. Constant gain values and adaptation gain matrix
were tuned while the experiment. After tuning, the gains for
both controllers k1=2 and k l=2, in EMK controller k2=15,
k Z2=0.5 and the constant parameter vector given by (15) is as
(25)
B. Direct Adaptive Controller
In this section the parameter vector, 0, is assumed to be
uncertain. We can define a parameter estimation error vector
as follows
0
(33)
where 0 denotes dynamic parameter estimation vector. We can
define a nonnegative function here as,
V3
2(o z2
+o3x2)
1+oTF 10
[7.56 x 10-5 2.281 x 10-4 0.0374 0.0234 ]
while in direct adaptive controller k2=3, k Z2=5 and varying
parameter vector which was defined in (30) is as
(26)
0
(32)
(23) which shows that z2(t)
Similar to analysis performed before, one can show that
V2, and based on that Z22 decreases exponentially. Thus,
V2(t),z2(t) C L£X,. Also from (10) and (11) and assuming
desired signal and its derivatives are bounded, it is clear that
e2, e2, 0, (, Ti e L,o. We can conclude that all the signals in
the closed loop system are bounded and e2 approaches to zero
exponentially.
0
Z
k Z2 Iz22 dt
X|V3(t) V3(0)
(24)
Tj= YO + kz2 Z2
where k,2 is a positive constant and substituting it into (23)
gives
0
(30)
(31)
-2 z22.
Note that the adaptive control algorithm is applied to stabilize
the system against uncertain parameters. From (21) and (25)
one can show that V2(t), z2(t),0 C L,o. Based on these
facts and assuming that desired signal and its derivatives are
bounded, it is clear that e2, e2, ~, , T2, 0 C ,. Also one can
P
show that from (3) and (11) ( Le by taking derivative of
C 4,
(11) we see Z2 which indicates Z2 is uniformly continuous.
By integrating (31) from zero to t and taking square root we
obtain
Defining applied torque as following,
V2 = k z22
ryT
Z2Y
1>3 =
respectively. Substituting (13) into (22) gives
1>2
=
and substituting them into (28) will result in
A. Exact Model Knowledge Controller
In this type of the controller, we assume that we know all the
system parameters exactly. Defining a nonnegative function
with respect to Z2 as follows and taking time derivative of it
we obtain
12
(29)
T2= Y + kz2 Z2
In order to develop controller for the dynamics given by (13),
two approaches have been applied. While in the first approach
the dynamics assumed to be known exactly, in the second one
some parameters in dynamics are assumed uncertain.
(01 + 03X) Z22
(28)
is obtained. Designing applied torque and parameter estimation
update law respectively will be
which is bounded and can be considered as a desired trajectory
for the motor shaft angle. It is need to be assumed that fourth
derivative of desired trajectory for the ball is a smooth function
[10].
V2 =
10
O(t)
(27)
0t
=
O(0) + XZ2(t)Fy (t)dt
(34)
and all initial conditions have been assumed to zero (0(0) =
0). As noted before, the adaptation gain matrix was tuned and
the final value is as
where F is a 4x4, invertible, positive definite matrix. By taking
derivative of (27) and plugging (13) into it
3

4. F
=
diag {5 x10-3, 0.03, 0.005, 0.1} .
0.435
(35)
l l
desired
-real
0.4
0.35
We have performed the experiment to track two references:
constant and sinusoidal for both controllers. Initial position
of the ball is set to 0.4m. We first present experimental
results when EMK controller is implemented for the system.
Figures 3 and 4 illustrate the system response for constant
and sinusoidal references respectively. As it can be clearly
seen from Fig. 3, ball reaches the desired position rapidly.
To show the robustness of the proposed controllers, opposite
disturbance torque is applied immediately after Ssecs. Despite
resulting transients because of the disturbance, EMK controller
achieves very fast convergence to the reference value. For the
case of sinusoidal reference is set to track, the ball follows the
reference trajectory with an acceptable error. The reference
and reel values are indistinguishable. Similar experimental
results are obtained when the direct adaptive controller has
been implemented. The for this case have been demonstrated
in Figures 5 and 6. In short, from the results presented above,
the controllers have achieved successful performance.
0.3
E
0.25
=n
0.2
0.15
0.1
0.05
O0 _
8
10
(a)
60
desired
-real
40
20
E
20
-40
V. CONCLUSION
The control of ball and beam system stands for controlling of
the position of the ball that freely rolls on frictionless beam by
changing beam angle. In exact model knowledge control all
model parameters are assumed to be known whereas in direct
adaptive control the model parameters are assumed to be uncertain. We have analyzed and employed exact model knowl-
-60
0
2
4
6
8
10
6
8
10
(b)
40
35
30
edge and direct adaptive controller for cascaded dynamics of
ball and beam system. The performances of the controllers
have been presented with experimental results. It has been
shown that the reference constant and sinusoidal signals have
been tracked successfully for each case of controllers. Also
robustness of the controllers is shown by applying opposite
torque on ball position. For the future study authors plan on
focusing on designing and implementing nonlinear observers.
25
E
20
LD
,
15
io
i0
0
2
4
(c)
Fig. 3
Constant Reference Response of EMK Controller.
(a)Ball Position, (b)Beam Angle, (c)Control Signal
0.41
desired
-real
0.35
0.3
E 0.25
02
0.15
0.1
U.Ub
6
(a)
Fig. 2
Experimental Setup
4
8
10

5. 60 F
- desired
40i
20
E~
0
LD
*n
10
(b)
(c)
0.03,
40
0.025
0.02
E~
25
ID
*-E0.015
Fz 20
LD
15
*n
0.01
12
0
1u0
C'
0.005
u
-5
0
° °° 51
0
10
A
6
4
A
10
(c)
(d)
Fig. 4
Sinusoidal Reference Response of EMK Controller.
(a)Ball Position, (b)Beam Angle, (c)Control Signal
Fig. 5
Constant Reference Response of Direct Adaptive Controller.
(a)Ball Position, (b)Beam Angle,
(c)Control Signal, (d)Parameter Estimates
0.4-
- desired
real
0.35
0.45
|
~~~~~~~~~~~real
0.3
035K
E0.2
-<
03
0
0.20.15.^
I1
0.15
0.05_
0
10
0.1
0
10
15
(a)
(a)
60 F
- desired
60
40
-----desired|
real
40
61
ID
g
E
0
ds
' '
20
ID
.
0
-20
0
E
0
ID
-20
-40
-60,0
-I.u
-40
2
4
6
8
0
-60'_
0
10
5
(b)
(b)
5
dS

6. E~
(c)
0.03,
I0.025l
E
0.015
E
0.01
0.005
-0.005
0
5
10
15
(d)
Fig. 6
Sinusoidal Reference Response of Direct Adaptive Controller.
(a)Ball Position, (b)Beam Angle,
(c)Control Signal, (d)Parameter Estimates
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