Ball and Beam
Content
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
Dept of Electrical Eng. Yildiz Technical University Istanbul, Turkey [email protected]
Haluk Gorgun
Dept of Electrical Eng. Yildiz Technical University Istanbul, Turkey [email protected]
Erkan Zergeroglu
Dept of Computer Eng. Gebze Institute of Tech. Izmit, Turkey [email protected]
Galip Cansever
Dept of Electrical Eng. Yildiz Technical University Istanbul, Turkey cansever @ yildiz.edu.tr
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.
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.
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.
II.
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
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
+
(
+ +
2)
*2
(mgx
+ Mlg sin
2
(r
(1)
where m,R and Jb
are
the
mass
(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
(m + Jb/R)
(r) (2)
m x()
2
+
mgsin (I)
0
(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
=
e2 +
k2e2
sides of (11) we obtain
o ( j5 + vy ( ZmgxV+ 33 Mgl) COS Z o 33.5
T
1K
T
(3)
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
e1
(01 + 03X2) %2 = (01 + 03X2) ((3d + k2e2) +203X±y( + (04X + 02) COS ( so) -T ± 03X±Z2- (12)
We can linearly parameterized (12) to obtain open-loop dynamics of the angle of the motor shaft as
(01
vector,
+ 03X2) Z2 = YO -T-03XZ2
error
(13)
=Xd-X
(4)
(5)
where Y is the regression are defined as
row
vector and 0 is the parameter
z1
=
ej + kle,
where k, is a positive constant. By taking derivative of (5) and putting (2) into it, we obtain,
¾l =
Y-=
2 d + k2 _(I
+ 2)COS
((d ++k2C2 (
I- W)
X±
IT
(14)
('d + kje1) +
sin (1675).
(6)
0
=
COS
( I ])
03
04
((9d+ ( + k2e2)
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
=
[
01
02
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.
sin
(175
(7)
error
After putting that into (2) we obtain the open-loop dynamics of the position of the ball as
l =
a nonnegative function as
(-~d + kj,) +
some
5
-7
(8)
We
can
rewrite (3) by defining
12
2
constant system param-
(16)
eters as
(o1 + 03X2) ( + 203X±y( + (04X + 02) COS =T.
can
Taking time derivative of (16) and substituting (8) into it gives
Y)
(9)
be
T1h s=hZl e(conrojsd
nd
a:+k )+
as:
(17)
Likewise, error and filtered tracking defined for beam angle as
e2 =
error
signals
Thus, the control law is defined
7
U =-
d-O
(10)
where k
is
a
5g
[(-~d +kj~,) +k1z1z
(18)
positive constant. After plugging (18) in (17),
V1
=-kzl Z12
(19)
Fig. 1 Ball and Beam System
2
Kz1 2 is obtained. Taking norms of (16) and (19) gives V1 < V1 < kz 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
16.75 arccos {-[d + kl l) + kzl1 }
(20)
V3
=
Z2 [YO -T2] 0
10
(28)
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].
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.
is obtained. Designing applied torque and parameter estimation update law respectively will be
T2= Y + kz2 Z2
(29)
(30)
0
I
=
ryT Z2Y
and substituting them into (28) will result in
(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, ~, P, T2, 0 C ,. Also one can show that from (3) and (11) ( L Ce 4, by taking derivative of (11) we see Z2 which indicates Z2 is uniformly continuous. By integrating (31) from zero to t and taking square root we obtain
1>3 =
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
V2 =
(01 + 03X) Z22
(21) (22)
12
=
(01 + 03X2) Z2%2 + 03X±z22
1>2
=
respectively. Substituting (13) into (22) gives
Z2
X|V3(t) V3(0)
Z dt k Z2 Iz22
(32)
[YO-
Tj]
(23) which shows that z2(t)
Defining applied torque as following,
(24) Tj= YO + kz2 Z2 where k,2 is a positive constant and substituting it into (23) gives
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
V2 = k z22
(25)
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.
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
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
0
[7.56 x 10-5 2.281 x 10-4 0.0374 0.0234 ]
0
0
0
(26)
where 0 denotes dynamic parameter estimation vector. We can define a nonnegative function here as,
while in direct adaptive controller k2=3, k Z2=5 and varying parameter vector which was defined in (30) is as
(33)
V3
2(o +o3x2) z2
1+oTF 10
O(t)
=
O(0) + XZ2(t)Fy (t)dt
0t
(34)
(27)
where F is a 4x4, invertible, positive definite matrix. By taking derivative of (27) and plugging (13) into it
3
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
F
=
diag {5 x10-3, 0.03, 0.005, 0.1} .
0.435
l l
(35)
E =n
0.4 0.35 0.3 0.25
0.2
desired -real
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.
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-
0.15 0.1 0.05
O0 _
8
10
(a)
60
40
desired -real
20
E
20
-40
-60
0
2
4
6
8
10
(b)
40
35
30
E
LD
,
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
20
io
15
i0
0
2
4
6
8
10
(c)
Fig. 3 Constant Reference Response of EMK Controller. (a)Ball Position, (b)Beam Angle, (c)Control Signal
0.41
0.35 0.3
E 0.25
desired -real
02
0.15 0.1
U.Ub
6
8
10
Fig. 2 Experimental Setup
4
(a)
60 F
40i
- desired
20
E~
LD
*n
0
10
(b)
40
(c)
0.03, 0.025 0.02
E~
25
ID
Fz 20 LD 15
C'
*-E0.015
*n
1u0
12 0
0.01 0.005
u
-5
0
10
° °° 51 0
4
A 6
A
10
(c)
(d)
Fig. 4 Sinusoidal Reference Response of EMK Controller. (a)Ball Position, (b)Beam Angle, (c)Control Signal
0.40.35 0.3 - desired real
Fig. 5 Constant Reference Response of Direct Adaptive Controller. (a)Ball Position, (b)Beam Angle, (c)Control Signal, (d)Parameter Estimates
0.45
|
~~~~~~~~~~~real
E0.2
-<
0.20.15.^
035K
0 03
0.15 10 0.1
I1
0.05_
0
0
10
15
(a)
60 F
40
40 ds ' '
dS
(a)
- desired
60
-----desired|
real
E 0
61 ID g
ID
20
-20 -40
0 0
.
0
E
ID
-20
-60, -I.u 0
-40 2 4 6 8 0 -60'_ 0
5
10
(b)
(b)
5
E~
(c)
0.03,
I0.025l
E E
0.015
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
REFERENCES
[1] J. Hauser, S. Sastry, P. Kokotovic, "Nonlinear control via approximate input-output linearization: the ball and beam example," IEEE Trans. Automatic Control,, Vol. 37, Issue 3, pp. 392-398, March 1992. [2] Y Guo, D.J. Hill, Z.P. Jiang, "Global nonlinear control of the ball and beam system,"IEEE Int. Con. on Decision and Control, vol. 3., pp. 28182823, Dec. 1996. [3] Y.L. Gu, "A direct adaptive control scheme for under-actuated dynamic systems," IEEE Int. Con. on Decision and Control, vol. 2, pp. 1625-1627, Dec. 1993. [4] H.K. Kim, D.H. Lee, T.Y Kuc, T.C. Yi, "A backstepping design of adaptive robust learning controller for fast trajectory tracking of ball-beam dynamic systems," IEEE Int. Con. on Systems, Man, and Cybernetics, vol.3, pp. 2311 -2314, Oct. 1996. [5] W. Yu, F. Ortiz, "Stability analysis of PD regulation for ball and beam system," IEEE Int. Con. On Control Applications, pp. 517-522, Aug. 2005. [6] YC. Chu, J. Huang, "A neural-network method for the nonlinear servomechanism problem," IEEE Trans. Neural Networks, Vol. 10, Issue. 6, pp. 1412-1423, Nov. 1999. [7] P.H. Eaton, D.V. Prokhorov, D.C. Wunsch, "Neurocontroller alternatives for "fuzzy" ball-and-beam systems with nonuniform nonlinear friction," IEEE Trans. Neural Networks, Vol. 11, Issue 2, pp. 423-435, March 2000. [8] H. Sira-Ramirez, "On the control of the "ball and beam" system: a trajectory planning approach," IEEE Int. Con. on Decision and Control, vol.4, pp. 4042-4047, Dec. 2000. [9] J.J.E.Slotine, W. Li, Applied Nonlinear Control, NJ. Prentice Hall, Englewood Cliff, 1991. [10] S. Uran, K. Jezernik, "Control of a Ball and Beam Like Mechanism," IEEE Int. Workshop Advanced Motion Control, pp. 376-380, July. 2002.
6
TuAl -A-1
Exact Model Knowledge and Direct Adaptive Controllers on Ball and Beam
Turker Turker
Dept of Electrical Eng. Yildiz Technical University Istanbul, Turkey [email protected]
Haluk Gorgun
Dept of Electrical Eng. Yildiz Technical University Istanbul, Turkey [email protected]
Erkan Zergeroglu
Dept of Computer Eng. Gebze Institute of Tech. Izmit, Turkey [email protected]
Galip Cansever
Dept of Electrical Eng. Yildiz Technical University Istanbul, Turkey cansever @ yildiz.edu.tr
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.
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.
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.
II.
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
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
+
(
+ +
2)
*2
(mgx
+ Mlg sin
2
(r
(1)
where m,R and Jb
are
the
mass
(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
(m + Jb/R)
(r) (2)
m x()
2
+
mgsin (I)
0
(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
=
e2 +
k2e2
sides of (11) we obtain
o ( j5 + vy ( ZmgxV+ 33 Mgl) COS Z o 33.5
T
1K
T
(3)
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
e1
(01 + 03X2) %2 = (01 + 03X2) ((3d + k2e2) +203X±y( + (04X + 02) COS ( so) -T ± 03X±Z2- (12)
We can linearly parameterized (12) to obtain open-loop dynamics of the angle of the motor shaft as
(01
vector,
+ 03X2) Z2 = YO -T-03XZ2
error
(13)
=Xd-X
(4)
(5)
where Y is the regression are defined as
row
vector and 0 is the parameter
z1
=
ej + kle,
where k, is a positive constant. By taking derivative of (5) and putting (2) into it, we obtain,
¾l =
Y-=
2 d + k2 _(I
+ 2)COS
((d ++k2C2 (
I- W)
X±
IT
(14)
('d + kje1) +
sin (1675).
(6)
0
=
COS
( I ])
03
04
((9d+ ( + k2e2)
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
=
[
01
02
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.
sin
(175
(7)
error
After putting that into (2) we obtain the open-loop dynamics of the position of the ball as
l =
a nonnegative function as
(-~d + kj,) +
some
5
-7
(8)
We
can
rewrite (3) by defining
12
2
constant system param-
(16)
eters as
(o1 + 03X2) ( + 203X±y( + (04X + 02) COS =T.
can
Taking time derivative of (16) and substituting (8) into it gives
Y)
(9)
be
T1h s=hZl e(conrojsd
nd
a:+k )+
as:
(17)
Likewise, error and filtered tracking defined for beam angle as
e2 =
error
signals
Thus, the control law is defined
7
U =-
d-O
(10)
where k
is
a
5g
[(-~d +kj~,) +k1z1z
(18)
positive constant. After plugging (18) in (17),
V1
=-kzl Z12
(19)
Fig. 1 Ball and Beam System
2
Kz1 2 is obtained. Taking norms of (16) and (19) gives V1 < V1 < kz 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
16.75 arccos {-[d + kl l) + kzl1 }
(20)
V3
=
Z2 [YO -T2] 0
10
(28)
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].
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.
is obtained. Designing applied torque and parameter estimation update law respectively will be
T2= Y + kz2 Z2
(29)
(30)
0
I
=
ryT Z2Y
and substituting them into (28) will result in
(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, ~, P, T2, 0 C ,. Also one can show that from (3) and (11) ( L Ce 4, by taking derivative of (11) we see Z2 which indicates Z2 is uniformly continuous. By integrating (31) from zero to t and taking square root we obtain
1>3 =
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
V2 =
(01 + 03X) Z22
(21) (22)
12
=
(01 + 03X2) Z2%2 + 03X±z22
1>2
=
respectively. Substituting (13) into (22) gives
Z2
X|V3(t) V3(0)
Z dt k Z2 Iz22
(32)
[YO-
Tj]
(23) which shows that z2(t)
Defining applied torque as following,
(24) Tj= YO + kz2 Z2 where k,2 is a positive constant and substituting it into (23) gives
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
V2 = k z22
(25)
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.
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
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
0
[7.56 x 10-5 2.281 x 10-4 0.0374 0.0234 ]
0
0
0
(26)
where 0 denotes dynamic parameter estimation vector. We can define a nonnegative function here as,
while in direct adaptive controller k2=3, k Z2=5 and varying parameter vector which was defined in (30) is as
(33)
V3
2(o +o3x2) z2
1+oTF 10
O(t)
=
O(0) + XZ2(t)Fy (t)dt
0t
(34)
(27)
where F is a 4x4, invertible, positive definite matrix. By taking derivative of (27) and plugging (13) into it
3
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
F
=
diag {5 x10-3, 0.03, 0.005, 0.1} .
0.435
l l
(35)
E =n
0.4 0.35 0.3 0.25
0.2
desired -real
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.
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-
0.15 0.1 0.05
O0 _
8
10
(a)
60
40
desired -real
20
E
20
-40
-60
0
2
4
6
8
10
(b)
40
35
30
E
LD
,
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
20
io
15
i0
0
2
4
6
8
10
(c)
Fig. 3 Constant Reference Response of EMK Controller. (a)Ball Position, (b)Beam Angle, (c)Control Signal
0.41
0.35 0.3
E 0.25
desired -real
02
0.15 0.1
U.Ub
6
8
10
Fig. 2 Experimental Setup
4
(a)
60 F
40i
- desired
20
E~
LD
*n
0
10
(b)
40
(c)
0.03, 0.025 0.02
E~
25
ID
Fz 20 LD 15
C'
*-E0.015
*n
1u0
12 0
0.01 0.005
u
-5
0
10
° °° 51 0
4
A 6
A
10
(c)
(d)
Fig. 4 Sinusoidal Reference Response of EMK Controller. (a)Ball Position, (b)Beam Angle, (c)Control Signal
0.40.35 0.3 - desired real
Fig. 5 Constant Reference Response of Direct Adaptive Controller. (a)Ball Position, (b)Beam Angle, (c)Control Signal, (d)Parameter Estimates
0.45
|
~~~~~~~~~~~real
E0.2
-<
0.20.15.^
035K
0 03
0.15 10 0.1
I1
0.05_
0
0
10
15
(a)
60 F
40
40 ds ' '
dS
(a)
- desired
60
-----desired|
real
E 0
61 ID g
ID
20
-20 -40
0 0
.
0
E
ID
-20
-60, -I.u 0
-40 2 4 6 8 0 -60'_ 0
5
10
(b)
(b)
5
E~
(c)
0.03,
I0.025l
E E
0.015
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
REFERENCES
[1] J. Hauser, S. Sastry, P. Kokotovic, "Nonlinear control via approximate input-output linearization: the ball and beam example," IEEE Trans. Automatic Control,, Vol. 37, Issue 3, pp. 392-398, March 1992. [2] Y Guo, D.J. Hill, Z.P. Jiang, "Global nonlinear control of the ball and beam system,"IEEE Int. Con. on Decision and Control, vol. 3., pp. 28182823, Dec. 1996. [3] Y.L. Gu, "A direct adaptive control scheme for under-actuated dynamic systems," IEEE Int. Con. on Decision and Control, vol. 2, pp. 1625-1627, Dec. 1993. [4] H.K. Kim, D.H. Lee, T.Y Kuc, T.C. Yi, "A backstepping design of adaptive robust learning controller for fast trajectory tracking of ball-beam dynamic systems," IEEE Int. Con. on Systems, Man, and Cybernetics, vol.3, pp. 2311 -2314, Oct. 1996. [5] W. Yu, F. Ortiz, "Stability analysis of PD regulation for ball and beam system," IEEE Int. Con. On Control Applications, pp. 517-522, Aug. 2005. [6] YC. Chu, J. Huang, "A neural-network method for the nonlinear servomechanism problem," IEEE Trans. Neural Networks, Vol. 10, Issue. 6, pp. 1412-1423, Nov. 1999. [7] P.H. Eaton, D.V. Prokhorov, D.C. Wunsch, "Neurocontroller alternatives for "fuzzy" ball-and-beam systems with nonuniform nonlinear friction," IEEE Trans. Neural Networks, Vol. 11, Issue 2, pp. 423-435, March 2000. [8] H. Sira-Ramirez, "On the control of the "ball and beam" system: a trajectory planning approach," IEEE Int. Con. on Decision and Control, vol.4, pp. 4042-4047, Dec. 2000. [9] J.J.E.Slotine, W. Li, Applied Nonlinear Control, NJ. Prentice Hall, Englewood Cliff, 1991. [10] S. Uran, K. Jezernik, "Control of a Ball and Beam Like Mechanism," IEEE Int. Workshop Advanced Motion Control, pp. 376-380, July. 2002.
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