1. Introduction The ball and beam system is one of the most enduringly popular and important laboratory models for teaching control system engineering. It is widely used because it is simple to understand as a system and yet the control techniques studied cover many important lessons and modern design methods. It has a very “important property” – it is open loop unstable .

2. What it is A steel ball rolling on the top of a long beam. The beam is mounted on the output shaft of an

electric motor and so the beam can be tilted about its centre axis by applying an electrical control signal to the motor amplifier.

The position of the ball on the beam can be measured using a special light sensor.

PCMATLAB 7.5

GUI

SignalProcessing

Unit

DCMotor

Ball and BeamSystem

IncrementalEncoder

CCD Camera

Servo Amplifier

MF 624 Card

Analog Output

Encoder Input

DigitalI/O

MotorControlSignal

EncoderSignal

BallPosition

EncoderSignal

PC ACTUATOR SYSTEM

VideoOutput

m Ib

u(t)

m, Iw y l

l

lw

x

K

r

α

b

mg

ψ

m: Mass of the ball u(t) : Force of the drive mechanics

g : Gravity l : Radius of force application

r : Roll radius of the ball Lw : Radius of beam

Ib :Inertia moment of the ball x: Ball coordinates with respect to the beam

Iw Inertia moment of the beam y: Ball coordinates with respect to the beam

M : Mass of the beam ψ : Angle of the ball to the beam

b: Friction coefficient of the drive mechanics a : Angle of the beam to the horizontal

K : Stiffness of the drive mechanics

2 22

1( ) ( ) sin( )b

b

Im x mr I a mxa mg a

r r

2 2 2 2 1( ) (2 ) cos( ) ( ) cosb w bmx I I a mxx bl a Kl a mr I x mgx a u t l a

r

t t t

t t t

x = f x , u ,

y = g x ,u ,

1

2

3

1

x x Position of the ball

x x Velocity of the ball

x a Angle of the beam

x a Angular velocity of the beam

10 0

, ,

0 0 0 ,

with

and x and u any 0 0 0 0x , u x , u

0

x = Ax + bu,

y = Cx

δf δfA = b =

δx δu

x

10 0

, ,

0 0 0 ,

with

and x and u any 0 0 0 0x , u x , u

0

x = Ax + bu,

y = Cx

δf δfA = b =

δx δu

x

1

2

3

1

x x Position of the ball

x x Velocity of the ball

x a Angle of the beam

x a Angular velocity of the beam