Static & Dynamic Sliding Mode Control of Ball and Beam System

Static & Dynamic Sliding Mode Control of Ball and Beam System

Abdul Manan Khanl, Amir Iqbal Bhatti 2, Sami-ud-Din 3, Qudrat Khan 4

Department of Electronics Engineering, Muhammad Ali Jinnah University, Islamabad, Pakistan [email protected], [email protected]

Ahstract-This paper presents an application of static

and dynamic sliding mode control of ball and beam

system. Conventional PID controllers use incomplete

simplified models and are mostly designed for linear

systems. Although, some PID controllers are designed

for nonlinear system but they are using simplified

incomplete model which do not cater for matched

disturbances. Our proposed control laws using static & dynamic sliding mode control use complete nonlinear

system without model approximation. Moreover, static

sliding mode control (SSMC) caters for matched

disturbance rejection as well. There is an inherent issue

of chattering with static sliding mode control (SSMC).

However, dynamics sliding mode control (DSMC)

counter it well. DSMC is not only equally effective when

it comes to matched disturbance rejection but also

removes chattering as well. In the end detailed

comparative analysis is presented and experimental

results confirm the superiority of DSMC.

Keywords-static sliding mode control(SSMC), dynamic sliding mode control (DSMC)

1. INTRODUCTION

Ball and beam system is a very interesting system by its nonlinear dynamics and it's under actuated phenomenon. These types of systems, have wide range of industrial applications including passenger's platform balancing for comfort in luxury cars, control of exothermic chemical process where addition of heat accelerate the process, control of rocket and aircraft vertical takeoff and liquid carrying tankers on the roads where liquid behaves like a ball on beam. Together with the interesting application, its dynamics gives a wide range of flexibility to implement classical, modem and advanced control techniques. It is ideal laboratory equipment to test classical, modem and advanced control system theory.

So far, several techniques have been applied for stabilizing ball position on the ball on beam including linear and nonlinear one [1]-[4]. For [1], PID or linear feedback control can be applied based on linear approximate simplified model. However, it does not cater for matched disturbances. In [7] it was concluded that relative degree for the system is ill defined so input-output linearization is not possible. This problem is solved via approximating nonlinear model which

978-1-4577-1929-5/12/$26.00 ©2011 IEEE

simplifies according to the method given in [7]. Using simplified incomplete model, system can be linearized using input-output feedback linearization followed by the controller design [7] but it was noticed that system has chattering in the output. Therefore complete model for system is used for controller design [8]. In [8], sliding mode control techniques are discussed but disturbances effect is not addressed. Moreover, intelligent control techniques including genetic, fuzzy and neural network based advanced adaptive control techniques are also experimented [8]-[11] but the controllers are very complex for application and require training input output data for the controller's development. In this draft, SSMC and DSMC application is developed based on the complete nonlinear model of ball and beam system and matched disturbance rejection is addressed is particularly.

The paper organization is as follows. In Section II system's mathematical model is presented which includes equilibrium point and open loop analysis. In Section III, static SMC design technique is discussed. In Section IV dynamics SMC technique is presented. All the simulation results are elaborated in Section V. In Section VI, conclusion of both the controllers is described.

II. MODELING OF THE BALL AND BEAM SYSTEM

Ball and Beam system is shown in Figure 1. It consists of a ball on the top of a beam. Ball is free to move on the beam. Beam is attached to a fixed support from one end while the other end is mounted on the electric motor shaft. The beam can be moved up and down, by a control signal to the motor amplifier. Ball position can be measured using any suitable position sensor. Open loop system is unstable as the ball position keeps on increasing once a fixed input to the motor is applied. Therefore, a controller is required to automatically regulate the position of the ball by changing the control signal to the motor.

System modeling consists of DC servo motor with attached gear and ball on beam system physics. DC servo motor can be modeled with simple Kirchoff's current law and Kirchoff's voltage law while its mechanical part requires simple statics to balance forces [12]. Ball and Beam can be modeled by using Lagrangian energy method. In its modeling, friction between ball and beam is assumed to be negligible.

Proceedings of 2012 9th International Bhurban Conference on Applied Sciences & Technology (IBCAST) 32 Islamabad, Pakistan, 9th - 12th January, 2012

Figure 1 Ball and Beam System

Dynamic equations of motion for the system can be considered as [8]

((mrZ + C1)P + (2mrr + CZ)/3) = U +(mgr + L/2 Mg) cosf3

Where f3(t): ret): 8(t): g: m: M: L:

Angle of Beam Position of Ball Servo gear angle Acceleration due to gravity Ball Mass Beam Mass Beam Length

The parameters of servo motor are;

Rm: 1m: Cm: Cg: d: II: Cb:

Resistance of armature of the motor Motor's Moment of Inertia Motor's Torque Constant Gear Ratio Radius of lever arm attached to servo motor Beam's Moment of Inertia Constant of Back EMF

(1)

(2)

C}, C2, C3 and C4 are constants for the system and their relations are given below:

C = RmxfmXL + I 1 CmXCbxd 1 L (CmCb Rmfm) Cz = d Rm

+ Cb + CmCg Cm C3 = 1 +R m

C4 = 7/5 Vinet) = Motor inputvoltage u(t) = C3Vin(t): Control Input to the Ball and Beam System

A. Equilibrium point of the system

System's equilibrium points can be described using equations (1) and (2) by putting rand /3 equal to zero. As, in equilibrium ball is stationary and beam angular velocity becomes equal to zero. So, randpbecomes zero. Hence;

(mgr+ �Mg) cosf3eq =ueq 9 sinf3eq = 0

(3)

(4)

Practically beam angle is limited to- n/2 < f3 < n/2. So, we can say that beam angle at equilibrium, f3eq is zero. Hence,

1 LM r: =-U --eq mg eq Zm (5)

So, equilibrium point must satisfy f3eq=O and req=rd, where rd is our desired ball position.

B. Open Loop Analysis

From (1) and (2) we can make a simple state space mode by assuming states of the system as;

(6)

(7)

(8)

(9)

(10)

y represents desired output. So, system model can be represented as; x = {(x)

Linearizing the system around equilibrium and calculating the eigen values, it is found that one root is lying on the right half plane. So, the open loop system is unstable as expected.

III. DESIGN OF STATIC SLIDING MODE CONTROLLER

For static SMC, let the regulation error be;

(11)

Where rd is constant and beam angle error is;

ep = Xz -f3e = X3 (12) Whilef3eq is beam angle at equilibrium which is zero. Choosing the sliding surface as,

Differentiating Sl with respect to time, we have:

Substituting X4 and xz from (7) and (8) we have,


Substituting the following control law,

u = (2mXIXZ + CZ)X4 + (mgxl + � Mg) cos X3 +

( X1X� -gSinX3) Z - blX4 + b3xz + bz C4 (mXl + Cl) + -(mxf + cl)(Klsl + Kzsign(sl)) (16)

Implies,

This ensures that S151 is less than zero which guarantees switching so that finite time reachability phenomenon for sliding mode may occur from any starting condition SI (0), provided that the constants KI & K2 are chosen strictly positive. b j, b2 and b3 are sliding surface parameters and can be found using Routh-Hurwitz criterion for stability, which gives the condition bl > 0, bz < 0, b3 < 0 andb3 > blbz for convergence, which is verified by the simulations.

Although, the designed control law effectively stabilizes the system but has a serious issue with the control signal as shown in Figure 5 which may damage the actuator. Therefore, it is not practically applicable. It raises the need of an improvement.

IV. DEVELOPMENT OF DYNAMIC SLIDING MODE

CONTROLLER

Dynamic sliding mode control (DSMC) technique is a welldeveloped for smoothening the chattering caused by SSMC. Infect, SSMC reduces the system order when system reaches sliding phase but DSMC does not reduces the system order on the sliding surface. It uses the full order system on its sliding surface. The detailed designed process for DSMC is discussed below. Defining the error sliding surface as,

Now, differentiating S2 with respect to time, we have:

Where .. _ f Zmx,X2 + 1 . x4 - - 2 U 1 U (mxi+Cl) (mxi+Cl)

Where, f =

zmx1x22

(h) - 1 Z (mxi +Cl) (mxi +Cl)

h = (2mXIXZ + CZ)X4 + (mgxl + � Mg) cos X3 z=<p + �-p

(20)

(21)

(22)

(23)

(24)

(25)

(26)

We have, 5z = -K3SZ - K4sign(sz) (31)

This guarantees that Sz5z < O. So, trajectories related to it, converges to zero in finite time from any given initial condition Sz (0), if K3 and K4 are strictly positive. Parameters ai, a], a3 and a4 are chosen using Routh-Hurwitz criterion for stability; it is found that, choosing al > 0, az > 0,

a3 < 0, a4 < 0, alaZ + J!...a3 > 0, afa4 > alaZa3 + J!...a� C4 C4 guarantees convergence of the system.

V. RESULTS

The controllers designed in Section III and IV is simulated using MATLAB & Simulink. Parameters of the system are taken from the Quanser Ball on Beam apparatus [12]. Following Table 1 shows the used parameters values for simulation purpose.

T bl 1 B II d B a e a an m .q L M Rm 1m

Cm

Cq d 11 Cb

S t P earn lYS em arame ers O.064K.q 9.81m/sz

0.43m 0.15K.q 90.

-4 Nm 7.35 X 10 d/� z

Nm 0.0075-:::1 75

0.03m O.OOlKg mZ

V

ra S

0.5625 ----;ij; ra S

A. Simulation results for static SMC

The controller parameters are chosen such that roots of the polynomial on the sliding surface are -2 ±j2 and-l0 with the Routh-Hurwitz criterion conditions described previously in Section II. So, bl = 14, bz = -6.85, b3 = 11.42, Kl = 30,Kz = 80

Figure 2 shows the simulation results of ball position (XI) and its velocity (X2) with respect to time using static SMC with the matched disturbance having unit amplitude and unit frequency. It is clear that desired output ball position is


Cristy Reyes

Resaltado

being achieved within five seconds and system is tracking well while successfully rejecting the disturbances.

E Q)

0.4

0.2

g 0 .l!l tJ) o

-02

5

� 1\

i\ /

;

]0 15 20 Time(sec)

25

Figure 2: ball position (x,) vs. time (I) with Static SMC

B. Simulation results for the dynamic SMC

-··xl -x2 �

30

The controller parameters are taken such that the polynomial P2 roots are [-1 ± j, -30, -40] with the conditions described by Routh-Hurwitz criterion as mentioned in Section IV.

So,a! = 72, az = 1342, a3 = -362.5, a4 = -342.5, K3 = 80, K4 = 180. Following Figure 3 represents

dynamic SMC simulation results of ball position (Xl) and its velocity (X2) with the unit amplitude and frequency matched disturbance. As shown from the Figure 3 that system is robust for the matched disturbances under the designed control law and ball position xlis tracking the desired reference rd.

...... 5 '"

O.4,----,----r----,----.----.---,------" --'xl -x2

0.2 .----�--l - rd

� 0 iii (5 - O . 2 P'--"-:=-----J.·/

-0.40 5 10 15 20 25 30 Time(sec)

Figure 3: Ball position (x,) vs. time (I) with Dynamic SMC

C Comparison of Static SMC & Dynamic SMC

Above simulation shows that both the controller work well but the main advantage and superiority of dynamic SMC is its greatly reduced chattering. Following Figure 4 shows the controller efforts of both schemes which clearly indicates that static SMC (SSMC) scheme controller effor ujis having lots of chattering as compared to dynamic SMC (DSMC) scheme controller effoer U2. Also, it can be seen that control signal is ocillating and it is only due to matched disturbance oscillations. To counter the sinousisal disturbance, controller is responding accordingly.Figure 5 is just a zoom version of Figure 4to elaborate the chattering more clearly.

t::

1 :� .................... III-�� !fl-IO �

J:! -20 -0 § -30

u

-40

-500 2 4 6 8 10

t:: !fl � ... � g c:: -I 8

Time(sec) Figure 4: Controller Efforts of Static SMC & Dynamic SMC

9.5 9.6 ! ,

9.7 9.8 Time(sec)

9.9

Figure 5: Zoom Version of Both Controller Efforts

D. Comparison with Practical PD controller [12]

10

The designed controllers (SSMC & DSMC) were compared with practical PD controller [12] which was developed using simplified system dynamics. Practical PD controller designed by [12] was put under matched disturbances of having unit magnitude and frequency as used for earlier controllers. Simulated result for the output is shown in Figure 6 .

I r---�--�--�--�--�--�

...... � � � :;; 0 5 " "/-\" :' \ 03 . /\ : \

,-E___ : \ II '\. ,: \ - .f \. , . .: • '"

........... '·'1 .•. "....

---"--�---l i 0, ! \ .. ':.! \/ -0 .. _ .... .. ,\ .f \ , \ \" .• 1 " \,-,: 5

,.,' �-.. 10 15 20

Time(seconds) 25

Figure 6: Ball Position with Practical PD controller

30

As, clearly depicted from the Figure 6, that the designed controller is unable to cater the disturbances but the SSMC and DSMC counters it well. This is a big drawback of PD controller which raised the need of an improvement.

VI. CONCLUSION

In this paper, SSMC and DSMC techniques are discussed for the controller development of the ball and beam system and compared with practical PD controller under matched disturbances. From the simulation it can be concluded that practical PD controller is unable to reject the disturbances while SSMC and DSMS effectively counter them.

Proceedings of 2012 9th International Bhurban Conference on Applied Sciences & Technology (IBCAST) 35 Islamabad. Pakistan, 9th - 12th January, 2012

Moreover, static SMC leaves the system with lots of unacceptable chattering. These unavoidable chattering can damage the actuator eventually. Although SSMC stabilizes the system well but its chattering phenomenon makes it impractical for implementation. The solution to this problem was provided by dynamic sliding SMC which removes chattering effectively while rejecting the disturbances.

VII. REFERENCES

[1] Ball and Beam Experiment and Solution, Quanser Consulting. 1991.

[2] S. Boyd, L. Ghaoui, E. Feron, and V. Balakrishna, " Linear Matrix Inequalities in Systems and Control Theory. Philadelphia, PA: SIAM, 1994.

[3] T. Hu, Z. Lin, W. Jiang, and P. E. Allaire, "Constrained control design for magnetic bearing systems," ASME Journal of Dynamic Systems, Measurement, and Control, vol. 127, pp. 601--{)16,2005.)

[4] T. Hu and Z. Lin, "On enlarging the basin of attraction for linear systems under saturated linear feedback," Systems and Control Letters, vol. 40, pp. 59-69, 2000.

[5] F. Bianchini, "Set invariance in control - a survey," Automatica, vol. 35, pp. 1747-1767, 1999.

[6] A. D. Mahindrakar and V. Sankaranarayanan, " Stateconstrained stabilization of beam-balance systems," International Journal of Robust and Nonlinear Control, vol. 18, pp. 333-350, February 2008.

[7] Hauser, J. Sastry, S., Kokotovic, P.,"Nonlinear Control via Approximate input -output linearization: the Ball and beam example," IEEE Trans/Autom. Control. Pentice 398(1992)

[8] Naif B. Almutairi Mohamed Zribi,"On the sliding mode control of a Ball on a Beam system,"Nonlinear Dyn (2010) 59: 222-239

[9] J. Yang, Zhong-hua Wang ," Adaptive Robust Dissipative Design on Stability Control for Ball and Beam System," IEEE 2nd International Conference on Computer and Automation Engineering (ICCAE), 2010, Singapore , pp. 68-71 (2006)

[10] M.F. Rahmat, H. Wahid, N. Abdul Wahab, "Application of Intelligent Control in a ball and beam control system," International Journal on smart sensing and intelligent systems vol. 3, no. I, march 2010

[II] Yuhong Jiang C. Mc Corkell and R. B. Zmood, "Application of Neural Networks for Real Time Control of a Ball-Beam System", Proceedings of. IEEE International Conference on Neural Networks, 1995)

[12] Quanser. SRV02 User Manual.

Proceedings of 2012 9th International Bhurban Conference on Applied Sciences & Technology (IBCAST) Islamabad, Pakistan, 9th - 12th January, 2012

36

Documents

Static & Dynamic Sliding Mode Control of Ball and Beam System