Read Stability Analysis of PD Regulation for Ball and Beam System text version
Proceedings of the 2005 IEEE Conference on Control Applications Toronto, Canada, August 2831, 2005
MC4.7
Stability analysis of PD regulation for ball and beam system
Wen Yu, Floriberto Ortiz
Abstract Many nonlinear controllers for ball and beam system can achieve in some cases asymptotic stability. Only a few of them apply their nonlinear control theory results in real experiment. On the other hand, many laboratories use PD control for the ball and beam system, but theory analysis is based on a linear approximate model. Little effort has been made to analyze the PD control via the complete nonlinear model. In this paper we propose an asymptotically stabilizing PD controller that ensures that, for a well defined set of initial conditions, the ball remains on any point of the bar. Complete nonlinear model of ball and beam system and stability analysis of PD control are proposed in this paper. Real experiments are applied to test our theory results.
I. I NTRODUCTION Ball and beam system is one of the most enduringly popular and important laboratory models for teaching control systems engineering. It is widely used because many important classical and modern design methods can be studied based on it. The system (shown in Fig.1) is very simple a steel ball rolling on the top of a long beam. One side of the beam is fixed, the other side is mounted on the output shaft of an electric motor and so the beam can be tilted by applying an electrical control signal to the motor amplifier. The position of the ball can be measured using a special sensor. It has a very important property open loop unstable, because the system output (the ball position) increases without limit for a fixed input (beam angle). The control job is to automatically regulate the position of the ball by changing the position of the motor. This is a difficult control task because the ball does not stay in one place on the beam when 6= 0 but moves with an acceleration that is proportional to the tilt of the beam. This standard experiment can be approximated by a linear model, many universities use it for education of classical control theory. Linear feedback control or PID control can be applied, the stability analysis are based on linear statespace model or transfer function [9]. Resent results show that the stabilization problem of the ball and beam can be solved by nonlinear controllers. Approximate inputoutput linearization used state feedback to linearize ball and beam system first, the a tracking controller based on the approximates system can stabilize the ball and beam system [5]. But this controller is very complex for real application.
In order to solve transient performance problem, energy shaping method uses a nonlinear static state feedback that is derived from the interconnection and damping assignment [3]. But it requires the kinetic and potential energies shaping [8]. Sliding mode controller can overcome the problem associated with singular states [6]. But chattering in sliding mode is a big problem in application. Observerbased nonlinear control in [7] uses the same coordinate transformation as in [5] to design a nonlinear observer for the velocities of ball and beam system. The controller is more complex than [5]. Some intelligent controllers for ball and beam can also be found, such as fuzzy control [10], sliding mode fuzzy control, neural control [1], fuzzy neural control [2], etc. These intelligent controllers are derived from some prior information or inputoutput data of ball and beam system. There are two problems for ball and beam control: 1) many laboratories use simple controllers such as PD control, but theory analysis is based on linear models. 2) nonlinear controllers for ball and beam system have good theory results, but they are seldom used in real applications. In this paper we will analyze the stability of the PD control with the complete nonlinear model. To the best of our knowledge, stability analysis of PD control based on nonlinear model of ball and beam system has not yet been established in the literature. Since the dynamic equations of ball and beam system are not suitable for Lyapunov method, some special transformations are applied. Also we propose a modifiedPD controller to guarantee asymptotic stability of the regulation procedure. A real experiment is applied to test our theory results. II. BALL AND BEAM MODEL AND PD CONTROL
For the ball and beam system described schematically in Fig.1, a ball is placed on a beam where it is allowed to roll with 1 degree of freedom along the length of the beam. A lever arm is attached to the beam at one end and a servo gear at the other. As the servo gear turns by an angle , the lever changes the angle of the beam by . When the angle is changed from the horizontal position, gravity causes the ball to roll along the beam. The basic mathematical description of this system consists of DC servomotor dynamic and ball on the beam model. Wen Yu and Floriberto Ortiz are with the Departamento de Control Modelling DC servomotor can be divided into electrical Automatico, CINVESTAVIPN, Av.IPN 2508, México D.F., 07360, México [email protected] and mechanical two subsystems. The electrical system is by 0780393546/05/$20.00 ©2005 IEEE 517
y
ball
N R
r
F
L
L
mg sin mg
beam
d
z
Mg
d
Fig. 2. Relation between motor position and beam angle.
motor
Fig. 1.
Ball and beam system
summarize the two equation in (5) with
··
sin
+
·· ··
cos
based on Kirchhoff´s voltage law =
·
+
+
·
(1)
is armature current, and where is input voltage, are the resistance and inductance of the armature, · is back emf constant, is angular velocity. Compared to and the term is very small. In order to simplify the modeling and as most DC motor modeling methods, we neglected the term is µ ·· + + The mechanical subsystem is µ ·· 1 +
· · · ·
= Use the conditions ³· ´ · cos sin , =
··
= ³·
sin sin sin
··
=0 2 + 5
·
··
+
cos
= +
·2
cos
´
··
=
sin
+ cos
··
=
DC motor model ¶
·
=
(2)
It is the second equation of (4). When the system is near · to stable point, 0 the acceleration of the ball is given by 5 ·· sin = 7 Since is a small angle, sin The approximation linear model for the ball and beam system becomes ( )=
2
where is gear ratio, is the effective moment of is viscous friction coefficient, is the torque inertia, produced at the motor shaft. The electrical and mechanical subsystems are coupled to each other through an algebraic torque equation = where is torque constant of the motor. By Lagrange method we can obtain the mathmatical model of the ball and beam system ¢ ¡ ¢ ¡ · · 2 ·· +2 + + 2 cos = 1+ (4) Remark 1: The second equation of (4) can be derived from force relation directly. In Fig.1 = ·· =
·· 7 ·· 5 ·2
¶
(6) ¸ · ¸
=
(3)
In state space form, it is " · # · ¸· 0 1 1 = · 0 0 2 ¸ · 1 = [1 0]
2 ·
1 2
+
0
+ sin
=0
where 1 = 1 = Remark 2: The model (4) is different with the most popular used ball and beam system as in [5], where the motor is fixed in the body center of the beam, in our case the fixed point is in one side of the beam. So the gravity of the beam cannot be neglected. Also the beam angle and motor position are not same, we use Fig.2 to calculate them. The arc distances in the two circle are equal, i.e., = (7)
where · =
The control problem is to design a controller which computes the applied voltage for the motor to move (5) the ball in such a way that the actual position of the ball reaches desired one. The controllers are constructed by · is friction, is rotational force, = introducing nonlinear compensation terms into the tradi2 = 2 . Multiply with sin and cos and tional PD controller. Two types of PD controllers will be 5 518 + cos + sin + cos sin
r* +
Ball controller C2
*

+

Motor controller C1
+ +
d/L
U
Motor m odel M1
Ball m odel M2
r
(a) Serial control
The whole ball and beam system is (2), (4) and (7), ¡ 2 ¢ ·· ¡ ¢ · · + 1 +2 + + 2 cos · = 2 (12) 3
4 ·· ·2
+ sin
=0 + ´
1 4 2
r*
 +
Ball controller C2
Motor controller C1 1
+
+ +
U
Motor m odel M1
Ball m odel M2
r
(b) Parallel control
d/L
Fig. 3.
PD control for ball and beam system.
designed for this system. The first one is serial PD control which is shown in Fig.3 (a). The beam angle (or motor position ) can be controlled by PD controller 1. This constitutes the innerloop. The outerloop controls the ball position with PD controller 2. is a compensator which can assure asymptotically stable. The serial PD control has the following form ³· ´ · ( )+ + = ³· ´ (8) · = ( )+ and are positive constants, which correwhere spond to proportional and derivative coefficients for motor and are proportional and derivative gains control, for the ball control. The second one is parallel PD control which is shown in Fig.3 (b). Because the final position of the motor must be 0 = 0 The feedback such that the ball does not move, so control of motor position becomes 1 The parallel PD control has the following form ³· ´ h ´i ³ · · + ( )+ + = (9)
For regulation problem the control aim is to stabilize · the ball in a desired position so = 0 The two PD controllers (8) and (9) can be rewritten in a unique form =
1e 2 · 3 ·· 4 5 ·
+
2 5
(10) =
4
where serial PD control 1 = + ) ( 3 = 4 = = parallel PD control 1 = 2 0 ( = 1 · · · 5) 5 =
= for
3
=0
=
] has an isolated solution [ ]=[ Proof: Substitute into the first equation of (13), we have ·· (14) 2 1e 2 3 2 4 =0 From the second equation of (13), we can conclude 1 sin So (14) becomes 4
2 1e + 2 3 4 ··
where is desired variable, =[ ] . For ball and · beam system, in the balance position =0 = 0 So = [0 ] is the desired ball position. It is difficult to apply the dynamic equation of the ball and beam system (12) and PD control (10) for Lyapunov method directly. On the other hand, it is well known that we can prove the stability of robots with PD control by Lyapunov method. In this paper we will transfer (12) and (10) into the form of the robot dynamics, then we will prove that the ball and beam system has similar properties as robots. The closedloop system is obtained by substituting the control voltage from the control law (10) into ball and beam system (12) ³ ·´ · ·· + ( ) = e+ ( ) + · ¸ ³ ·´ 2 1+ 2 3 = where ( ) = 0 # 4 " · ¸ · 2 5+ 3 2 2+2 2 4 2 1 = · 0 0 0 · ¡ ¸ ¢ ¸ · + 2 cos 2 ( )= . Before = 0 sin presenting the stability analysis, we give the following lemma. Lemma 1: The following equations ¢ ¡ cos = 2 + 2 ·· =0 (13) 4 + sin ¡ ¢ ·· + 2 + 12 cos = 1e 3 4
We define system state as = [ is e=
where ³
1
= + +
= 1 +
7 5,
3
=
=
]
the regulation error
0 ( = 1 · · · 4)
=
III. S TABILITY ANALYSIS OF PD REGULATION In this section, PD regulation for ball and beam system is proposed. By (3) we have ¶ µ · = (11)
sin
2 4
=0
It can be rewritten as sin =
4 4 3 4 1 3
519
e
(15)
The only possible solution for is = 0 otherwise the ball has to move. For any 6= 0 e cannot be a constant, so (15) has no solution. When = 0 form (15) we know e = 0 = 0 this allows us to conclude [ ]=[ ] Because is the unique solution for (13). The stability of the closedloop system is stated in the following theorem. Theorem 1: The serial or parallel PD control as in (10) with a compensator as h i · · 3) = 12 { 2 2 + (2 (16) ¡ ¢ + + 2 cos }
Using
·
4
··
·2
+ sin
·
=0
·2
+ [
2
+( 2
so
If we choose the compensator as µ · 1 · · = + + (2 3)
2 ·
= ( 2 5 + 3) ¡ · · + 3) + 2 2
4) ·2
¶
¢
cos
2 2
·
]
cos
+
2 2
·
¸
(
·
3 5
+
(19)
can guarantee asymptotic stability of the ball and beam system (12), i.e., from any defined set of initial condition 0 (0) , the PD control (10) ensures that the ball remains on any point of the bar. Proof: Because ( ) and are positive definite matrices, we choose the following positive definite quadratic form as Lyapunov function candidate ³ ·´ 1 · 1 · 4 ·2 = ( ) + e e+ (17) 2 2 2 Differentiating it with respect to time, and recalling that is constant, yields = Since
· ··
is a negativesemidefinite funcSince ( 3 5 + 4 ) 0 tion. Therefore, by invoking the Lyapunov's direct method, it can be concluded that [ ] = [0 ] ( = 0) is a stable equilibrium. In order to prove asymptotic stability, we use LaSalle´s theorem. In the region ¾ ½ · = [ ]: =0 the invariant set is obtained from the closedloop system · (12) when = 0 that is (13). Furthermore, according to = 0). Lemma 1, (13) is satisfied for [ ] = [0 ] ( Therefore, invoking LaSalle´s theorem, we can assure that the equilibrium [ ] = [0 ] is asymptotically stable. This means that (20) lim e = 0 =0 lim ( )=0 ( )=
· ·
( ) + e+ ·
··
1· 2
·
( ) =
·
=
1· 2
·
( ) ³ ·´ 2 ³
·
· ·
e+
· · ··
4
· ··
( )
( )
+ Because " So
·
·
[ e 4
( )+ ³
·
·
2 2
· ·
0
( ) 2 # · ·
· · ·
2
2 5
´
e] + =
3
´¸ ·
For ball and beam system, in the balance position is the desired ball position, (20) means lim
4
+2 0
2
2 2
0 #
¸
=
1· 2 ·
+2
" 2
· ·
2 0
4 · ··
·
·
( )+ +2 0
· 3
where 2
· · · 1· 2
= " =
·
·
· ·
2
2 5
2 2
0 # +
·
+ ¸ .
·2
( )=
· 2
2 0 ¡ (
=
2 ·2
2 ¢
(
·
1)
· ·
cos
+ sin
· ·
=
2 5
+
3)
2 2
The controllers (8) or (9) with (21) are very simple and easy implement. The control parameters of PD control are independent of system parameters, the compensator uses two motor parameters and the masses of the ball and beam. Although the pure PD controller (with = 0) can also stabilize the system as many laboratories' experimental proofs, the control performance under the pure PD control is very unsatisfactory (especially for the confutation of our type), due to the gravities of ball and beam. Remark 4: To the best of our knowledge, theoretical (18) analysis of PD controller for ball and beam system based on complete nonlinear model has not yet been established in the literature. Many stability analysis are based on complex nonlinear controller, and these controller have to use the 520
Remark 3: Since the velocities and in (16) are very small in regulation case, the main compensation is gravities of the ball and beam ¶ µ 1 cos (21) + 2 2
5 4 3 2 1 0 1 2 3
r(cm )
N ormal PD control for simplified m odel
M odified PD control for com plete m odel
N orm al PD control for complete m odel
4 0
Tim e (m s) 20 40 60 80 100
Fig. 4.
Fig. 5.
Ball and beam control system.
nonlinear model of the ball and beam system. On the other hand, many laboratories use modelfree controllers (e.g. PD controller), the theoretical analysis use the simplified linear model as in (6). Because PD controller is also linear system, traditional control theory can be applied for stability analysis. IV. S IMULATION AND EXPERIMENTAL CASE STUDY First we give some simulation to compare our controller with the other existing methods. For the simulation we = 0 01176 + = 0 58823 [9]. choose If we do not consider energy effect, the whole dynamic equation is
7 ·· 5 ·2 ··
=
sin
·
0 01176 + 0 45823 = 1 = 16 The regulation results of normal PD control ³ ´ h ³· · = 58 0 1 + 2 2( )+0 8
·
(22)
(23) When the kinetic energy of the system is considered, the first equation of (22) becomes (12). We use the parameters as = 0 06 = 9 8 = 0 12 = 0 6 The modified PD control is (23) with compensation = 0 2 + 0 1 cos The simulation results are shown in Fig.4. The normal PD control is suitable for simplified model, but it does not work for the complete nonlinear model. The modified PD control proposed in this paper can work. The response is similar as in [5], but the transient performance is worse than [8]. We note that the nonlinear controllers of [5] and [8] need the complete model of ball and beam system. They only give simulation results. Our modified PD control does not 0 3 (0 588 + 0 353) cos require the nonlinear model of ball and beam system. It is It can be approximated as when the ball and beam system does not move quickly easier for application. The experiment is carried out on the Quanser ball and The response of the parallel PD control for ball and beam beam system [9] (see Fig.5). The beam is 60cm long. The system is shown in Fig.6. The serial PD control with the same compensator as the ball is about 60g. Input to the system is motor control we move the ball for 1cm, voltage , outputs are the positions of motor ( ) and ball parallel one. At time = 200 521
´i
( ). Power module is also Quanser PA0103 with ±12V and 3A output. A/DD/A board is based on a a Xilinx FPGA microprocessor, which is a multifunction analog and digital timing I/O board dedicated to realtime data acquisition and control in the Windows XP environment. The board is mounted in a PC PentiumIII 500MHz host computer. Because Xilinx FPGA chip supports realtime operations without introducing latencies caused by the Windows default timing system, the control program is operated in Windows XP with Matlab 6.5/Simulink. The sampling time is about 10ms. The motor and ball controllers are both of PD type and require direct velocity measurements, but they are unavailable. We use derivative block of Simulink to calculate them. This require position signals are smooth enough, first order lowpass filters are applied. Because the closedloop system appears (from the step input) to exhibit second order behaviour with a natural frequency around 1 rad/s. Faster filters are used (a rule of thumb would suggest at least 5 to 10 times faster than the fasted closedloop modes). For motor position we use the following firstorder filter 7 1 ( ) = +7 For ball position we use the following first8 order filter 2 ( ) = +8 For the serial PD control (8) =2 = 01 = 05 = 0 1 For we use =2 = 05 the parallel PD control (9) we use =04 = 0 1 The parameters for this experiment = 16 = 0 06 = 98 = 03 are + = 0 12 = 0 6 We only use gravity compensation (21) The compensator is h i · · { ( + ) + (2 3) = + + ¡ ¢ + + 2 cos }
2
5
r (cm)
r (cm)
4 3 2 1 0 1 2 3 4 0 500 1000 1500 2000
1 0 1 2 3
Time (ms)
4 0
500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Time (ms)
2500 3000 3500 4000 4500 5000
Fig. 8. Fig. 6. Parallel PD control with gravity compensation.
Serial PD control without compensator.
3
r (cm)
2
1
0
1
2
Time (ms)
3 0 500 1000 1500 2000 2500 Time (ms) 3000 3500 4000 4500
Fig. 7.
Serial PD control with gravity compensation and disturbance.
this likes a external disturbance. The response is shown in Fig.7. When we use pure PD control, the response of the serial PD control without compensator is shown in Fig.8. We can see that PD control with exact compensation is effective for ball and beam system V. C ONCLUSION The main contributions of the paper are, 1) A class of asymptotically stable PD controllers has been presented for regulation of ball and beam system. The new controllers, which have some attractive advantages for practical applications, are constructed by introducing a nonlinear compensator into the traditional PD controller. 2) By using Lyapunov´s direct method, we have shown for a well defined set of initial conditions, the ball remains on any point of the bar. 3) Experimental results are presented to illustrate the control system stability and performance. R EFERENCES
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[2] P.H.Eaton, D.V.Prokhorov, D.C. Wunsch II, Neurocontroller alternatives for "fuzzy" ballandbeam systems with nonuniform nonlinear friction, IEEE Trans. Neural Networks, Vol.11, No.2, 423435, 2000. [3] F.Gordillo, F.GómezEstern, R.Ortega, and J.Aracil, On the ball and beam problem: regulation with guaranteed transient performance and tracking periodic orbits, International Symposium on Mathematical Theory of Networks and Systems, University of Notre Dame, IN, USA. August, 2002. [4] J.S.Glower, J.Munighan, Designing fuzzy controllers from a variable structures standpoint, IEEE Trans. on Fuzzy Systems, Vol.5, No.1, 138 144, 1997. [5] J.Hauser, S.Sastry and P.Kokotovic, Nonlinear control via approximate inputoutput linearization: ball and beam example, IEEE Trans. on Automatic Control, Vol.37, No.3,392398., 1992. [6] R. M. Hirschorn, Incremental sliding mode control of the ball and beam, IEEE Trans. on Automatic Control, Vol.47, No.10, 16961700, 2002. [7] N.H. Jo, J.H.Seo, A state observer for nonlinear systems and its application to ball and beam system, IEEE Trans. on Automatic Control, Vol.45, No.5, 968 973, 2000. [8] R. Ortega, M.W. Spong, F. GómezEstern, G. Blankenstein, Stabilization of a class of underactuated mechanical systems via interconnection and damping assignment, IEEE Trans. on Automatic Control, Vol.47, No.8, 1218 1233, 2002. [9] Ball and BeamExperiment and Solution, Quanser Consulting, 1991. [10] L.X.Wang, Stable and optimal fuzzy control of linear systems, IEEE Trans. on Fuzzy Systems, Vol.6, No.1, 137 143, 1998.
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Stability Analysis of PD Regulation for Ball and Beam System
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