Design of Digital Pole placement Controller for a Ball & Beam system

GRD Journals | Global Research and Development Journal for Engineering | International Conference on Innovations in Engineering and Technology (ICIET) - 2016 | July 2016

e-ISSN: 2455-5703

Design of Digital Pole Placement Controller for a Ball & Beam System 1S.

Nagammai 2S. Latha 1 Department of Electronics and Instrumentation Engineering 2Department of Electrical and Electronics Engineering 1 K.L.N. College of Engineering, Pottapalayam, Sivagangai 630612, India 2Thiagarajar College of Engineering, Madurai 625015, India Abstract The Ball and Beam system is a nonlinear unstable system due to this property it is used as a bench mark problem for many researchers. The proposed work focuses on the design of a digital pole placement controller for a Ball and Beam system (BBS) connected with servo motor which is an open loop unstable system due to the presence of multiple poles at the origin. A one degree of freedom (DOF) and 2-DOF digital pole placement controller is designed using the discrete transfer function for the specified constraints namely peak overshoot and rise time. The digital controller designed is capable of providing system stability and also provides good set point tracking and disturbance rejection. The simulation result shows the feasibility of using the proposed controller for control of an unstable Ball and Beam system.

Keyword- BBS, 1 DOF Controller, 2 DOF Controllers, Pole Placement __________________________________________________________________________________________________

I. INTRODUCTION The BBS mimics the real control problems namely stabilization of an airplane during landing, rocket and aircraft vertical takeoff. The BBS has two degrees of freedom namely rolling of the ball back and forth on the beam, and the other is beam rotation. It has nonlinear dynamics and under-actuated characteristics. The nonlinear property becomes insignificant when the beam deflects a small angle from the horizontal position which leads the mathematical modelling simpler. It is an ideal example for examining the performance of classical, modern and advanced control theory. Many researchers have done work on this laboratory test bench. Hauser et al proposed feedback linearization controller [1]. Pang et al designed a Linear Quadratic Regulator (LQR) for stabilization [2]. H.Verrelst et al used Neural Network for stabilization of BBS [3],S.K Oh et al proposed Fuzzy based cascade control [4] and Chang et al designed a tracking control strategy using a fuzzy sliding-mode controllers [5]. Further all the intelligent controllers are knowledge based and the stability of the system is not guaranteed. In this proposed work controller is designed for positioning and stabilization of the ball along the beam by manipulating the voltage applied to the servomotor. Thus the system under consideration is a single input single output system (SISO).

The paper is organized as follows. Section II gives details about the mathematical modelling of BBS and servomotor. In Section III the concepts of 2 DOF pole placement controllers is presented. In Section IV the design of pole placement controller for a desired specification is discussed. The simulated results are given in section V. Finally conclusion is given in section VI.

II. MATHEMATICAL MODELLING A. Modelling of Ball and Beam System

Fig. 1: Schematic of ball and beam system

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Design of Digital Pole Placement Controller for a Ball & Beam System (GRDJE / CONFERENCE / ICIET - 2016 / 078)

The Ball and Beam system (BBS) is driven by the servomotor and the schematic is shown in Fig. 1. The beam consists of a steel rod in parallel with a nickel-chromium wire-wound resistor forming a track upon which a metal ball slides. One end of the beams is connected to the servomotor through a lever arm and gear train while the other end is fixed.

The two forces influencing the motion of the ball are Ftx Force due to translational motion

Frx

Force due to ball rotation The force due to translational motion is

Ftx  m

d2x  mx dt 2

The force due to rotational motion is T J d b J d (vb / R) J Frx  r    2x R R dt R dt R J 

2 mR 2 5

Where the moment of inertia of the Ball is, Substituting for the moment of inertia of the ball, we get

Frx 

2 mx 5

Applying the Newton’s second law for forces along the inclination, we have

Frx  Ftx  mg sin 



2 mx  mx  mg sin  5

or

x

5 g sin  7

For small angle, sin    , and (1) becomes x

5 g 7

(1)

(2)

The arc distance traversed by the gear at radius r is equated to the arc distance traversed by the beam at radius L , hence r  L or

r L 5 gr x   K 7L

 

X (s) K   (s) S 2

Where  is Beam angle  is Angle of gear m is Mass of the Ball R is Radius of the Ball

(3)

B. Modelling of Servomotor The armature voltage equation is, Va  t   La

dia  t  dt

 Ra ia  t   K m m

The torque balance equation is, K g K m ia  t   J L  BL ia  t  

J L  BL K g Km

Ignoring armature inductance the armature voltage equation becomes,

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Design of Digital Pole Placement Controller for a Ball & Beam System (GRDJE / CONFERENCE / ICIET - 2016 / 078)

 J   BL  Va  t   Ra  L   K m Kg  K g K m  The transfer function of the servo motor including inertia and friction of load shaft is  Km K g

 (s) Va ( s)



Ra J eq  2  Beq  K m2 K g2  S   J eq  Ra J eq  

   S    

(4)

Or equivalently,

 (s) Va ( s )



am S ( S  bm )

(5)

Where

am 

 Km K g Ra J eq

,

bm 

Beq J eq



 K m2 K g2 Ra J eq Parameter Beam length (L) in meter Lever arm offset (r) in meter Acceleration due to Gravity ( g ) in m/S2

Value 0.1675 0.0254 9.84

Armature resistance ( Ra ) in Ohms

2.6

Motor voltage constant ( K m )

0.00767

gear ratio ( K g )

70

Efficiency (  ) Equivalent moment of inertia ( J eq )Kg

m2

Equivalent viscous friction  B  in Nm/rad/sec eq

0.8 0.0023 4×10-3

Table 1: System parameters

The transfer function of the BBS connected with servo motor is, Kam X (s) 73  3  3 Va ( s ) S ( S  bm ) S ( S  40) Substituting the system parameters tabulated in Table I, the overall transfer function of the system is given as below: X (s) 73 (6)  3 Va ( s ) S ( S  40) The transfer function indicates that the open loop system is unstable due to the presence of multiple poles at origin.

III. CONCEPTS OF 2 DOF POLE PLACEMENT CONTROLLER The pole placement controller is considered as the basis of all controllers and the idea of designing such a controller is to assign closed loop poles at the desired locations.

Consider a process or a plant output in discrete form as given below, Y (z

With system function of the form,

1

)  H (z

1

)U ( z

1

)

1 B( z ) H ( z 1 )  z k A( z 1 )

(7.a)

(7.b)

It is desired to design a controller such that the plant output y tracks the set point ysp in the following manner:

Y (z

1

)z

k Br 1 Ysp ( z ) cl

(7.c)

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Design of Digital Pole Placement Controller for a Ball & Beam System (GRDJE / CONFERENCE / ICIET - 2016 / 078)

 

 cl

Where is the characteristic polynomial obtained based on desired closed loop specification. And trajectory tracking is achieved.

Referring to Fig.2, R( z 1 )U ( z 1 )   T ( z 1 )Ysp ( z 1 )  S ( z 1 )Y ( z 1 ) (7.d)

cl Br

so that

S

The two DOF controller shown in Fig.2 has two parts namely, a feedback component R that helps in ensuring internal stability T Y and a feed forward component R that helps in tracking set point sp .

Fig. 2: Schematic of two degrees of freedom pole placement controller 1

1 1 Where R ( z ) , S ( z ) and T ( z ) are polynomials in

z 1 and which are to be determined.

S The one DOF controller has only the component R in the forward path as shown in Fig.3.

Fig. 3: Schematic of one degree of freedom pole placement controller

Further simplifying equation (2.a),(2.b) and (2.d), we get

Y   z k

TB Ysp AR  z  k BS

(7.e)

For set point tracking equation (2.e) is equated to equation 2.c) hence we arrive at, B BT  r AR  z  k BS  cl

(7.f) In general degree of Br < deg of B, so that the desired closed loop transfer function is of order lower than given by BT. This is achieved by cancelling common terms between numerator and denominator. In view of this, A and B is factored as good and bad terms as,

A  Ag Ab

and

B  Bg Bb

If we let, S  Ag S1 T  Ag T1 R  Bg R1 Now, equation (7.f) becomes,

Bg Bb Ag T1 Ag Ab Bg R1  z  k Bg Bb Ag S1



Br

cl

(8) Cancelling the good common terms and on equating the numerator and denominator parts in the equation (8) we get, Br  BbT1 (9) Ab R1  z  k Bb S1  cl

(10)

Equation (10) is solved to get the polynomials R1 and S1 .

Let the error signal e(n) be represented as,

e(n)  r n cos n

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Design of Digital Pole Placement Controller for a Ball & Beam System (GRDJE / CONFERENCE / ICIET - 2016 / 078)

And from the desired specifications on the transient response the allowable range for ‘r’ and ‘ω’ is determined. 



2N



 and for a small undershoot in e(n), r   where

For a small fall time in error signal e(n) the angular frequency specified overshoot.

Y ( z) The transfer function relating the set point sp and the actual output Y ( z ) is given by,

TY ( z ) 

1  r cos     r 2  r cos   z 1 1  2 z 1r cos   r 2 z 2





is the

 ( z 1 ) cl ( z 1 )

Hence the desired transfer function is,

TY ( z 1 ) 

Y ( z)  ( z 1 )  z 1 Ysp ( z ) cl ( z 1 )

(11)

IV. DESIGN OF DIGITAL POLE PLACEMENT CONTROLLERS In order to track the desired trajectory by the ball the following specifications are considered: Overshoot  = 5%, Settling time (ts) ≤ 0.5 secs, Sampling time = 0.01 secs.

1) Step 1: The discrete version of the continuous time transfer function given in equation (6) is obtained as H(z) .

H(z)  z 1

 2.82x10

8

 2.9x107 z 1  2.7x107 z 2  2.21x10 8 z 3 

1  3.69z 1  5.01z 2  3.01z 3  0.67z 4

 zk

B(z) A(z)

(12)

Now, A and B is factored into good and bad parts as given: A  Ag Ab Ag  1  0.982 z 1  1  0.69 z 1  Ab  1  z 1  1  1.01z 1  B  Bg Bb Bg  2.83  10 8 1  10.25 z 1  9.5 z 2  0.8 z 3 

Bb  1 2) Step 2: From the desired specifications on the transient response the allowable range for ‘r’ and ‘ω’ is determined.

The rise time is chosen as at least one third of the settling time i.e, 0.15 sec N ≤ rise time / Ts =15 

0.1047

Overshoot constraint: r     0.05   0.905 3) Step 3: The desired characteristic polynomial is determined as:

cl ( z 1 )  1  1.8z 1  0.819z 2 A R z 4) Step 4: The Aryabhatta’s identity as given below is used to determine the terms S1 and R1. b 1 1  z 1  1  1.01z 1  R1  z 1 S1  1  1.8 z 1  0.819 z 2

k

Bb S1  cl

5) Step 5: The polynomials for controller are determined as,

S(z)= Ag ( z )S1 = 0.137  0.344 z 1  0.02875 z 2  0.0794 z 3



R(z) = B g ( z ) R1 =107 0.283+ 0.591z 1  0.33 z 2  0.026z 3



T(z) = Ag ( z)T1 = 0.1  0.15z 1  0.056 z 2

 T(z) = 0.01  0.0147 z 1  0.0055 z 2

V. SIMULATION RESULTS The performance of the proposed control algorithms are analyzed by simulating the 1 DOF and the 2 DOF pole placement controller structure shown in Fig.2 and Fig.3 using MATLAB. The discrete plant transfer function is obtained using a zero order

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Design of Digital Pole Placement Controller for a Ball & Beam System (GRDJE / CONFERENCE / ICIET - 2016 / 078)

hold circuit. The feedback controller term

S  z

R z

and the feed forward filter term

T  z Rz

are evaluated for simulation. The open

loop response of the BBS connected with servo motor is shown in Fig.4, which indicates that the steady displacement is not achieved in finite time and the system dynamics is nonlinear. The performance of 1 DOF and 2 DOF pole placement controllers is shown in Fig.5 for step change in voltage variation. It indicates that the 1 DOF controller exhibits excess overshoot compared to 2 DOF controllers. In order to show the disturbance rejection capability a disturbance input is given at 1 & 4 secs respectively as indicated in Fig.6. The variation of armature voltage is shown in Fig.7 which indicates that the excitation voltage is damped oscillatory until the ball tracks the desired trajectory. This is because the lever arm rotates in a bidirectional way within 1800 angle variation. The stabilization response of the ball and beam system from its initial position of 2 Cm is shown in Fig.8.

Fig. 4: Open loop step response of BBS connected with servo motor

Fig. 5: Trajectory tracking response to set point change in voltage variation

Fig. 6: Regulatory response to disturbance input

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Design of Digital Pole Placement Controller for a Ball & Beam System (GRDJE / CONFERENCE / ICIET - 2016 / 078)

Fig. 7: Variation of manipulated variable (armature voltage)

Fig. 8: Stabilization response of BBS connected with servo motor

VI. CONCLUSIONS The 1 DOF and 2 DOF pole placement controller algorithms are demonstrated for a modelled BBS connected with servo motor.

The time response shows that the settling time, overshoot and rise time constraints are almost met for the 2 DOF controller compared to the 1 DOF controller. The performance summary given in Table II indicates that, the 2 DOF pole placement controller offers less ISE and IAE value. Further 1 DOF pole placement controller exhibits more overshoot compared to 2 DOF pole placement controllers and which is an undesirable phenomenon.

Parameter 1 DOF PID Controller 2 DOF PID Controller Peak time tp 0.15 0.24 Rise time tr 0.04 0.1 Settling time ts 0.51 0.4 % overshoot 27.8 7.5 ISE 95.67 95.21 IAE 95.60 95.10 Table 2: Comparison of Time domain Specifications and Performance Indices

REFERENCES [1] J. Hauser, S. Sastry, and P. Kokotovic, “Nonlinear control via approximate input-output linearization the ball and beam example,” Institute of Electrical and Electronics Engineers, Transactions on Automatic Control, vol. 37, no. 3, pp. 392–398, 1992.

[2] Z.H Pang, G. Zheng and C.X. Luo, “Augmented State Estimation and LQR Control for a Ball and Beam System”, Proc. of the 6th IEEE Conference on Industrial Electronics and Applications., pp.1328-1332, June 21-23, 2011.

[3] H. Verrelst, K. Van Acker, J. Suykens, B. Motmans, B. De Moor and J. Vandewalle, “Neural Control Theory: Case Study for a Ball and Beam System”, Proc. of the European Control Conference Brussels, Belgium, July –4, 1997.

[4] S.K Oh, H.J. Jang and W. Pedrycz, “The Design of a Fuzzy Cascade Controller for Ball and Beam System: A Study in Optimization with the Use of Parallel Genetic Algorithms”, Engineering Applications of Artificial Intelligence, Vol. 22, pp.

261–271, 2009.

[5] Y.H. Chang, C.W. Chang, C.W. Tao, H.W. Lin and J.S. Taur, “Fuzzy Sliding-mode Control for Ball and Beam System with Fuzzy ant Colony Optimization”, Expert Systems with Applications, Vol. 39, No. 3, pp. 3624-3633, 2012.

[6] S. Nagammai, S.V. Prasanna Kumar, S. Latha, “Design of Digital Controller for an Unstable CSTR Process” Journal of Electronic Design Technology, Volume 5, Issue 3, pp.235-240, 2014.

[7] “Ball and Beam Experiment and Solution,” Quanser Consulting, 1991

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