Assignmrnt of contorl system

8/7/2019 Assignmrnt of contorl system

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Hewlett-Packard

[Year]


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Adel Aljodaeme Page 2

CONTENTS

i Introduction...3

ii Description of the reconfigurable...................................................................... 3

iii Technical Specification .................................................................................5

1.0 Mathematical Modelling of the Pendulum system........6

2.0 Experimental Analysis...11

2.1determinations (Kr)12

2.3Determination (Kx).13

2.4 Determination of (Kp) .14

2.5 Determination of (Ka) .14

2.6Determination the Transfer Function, Gs(s) if the carriage..15

3.0Determination the Damping and Natural Frequency..16

3.1The Experimental Data..18

4.0 Test and System stability. 19

4.1Stability Test (upright)19

4.2 Stability Test - (Hanging Pendulum)22

5.0controllers Design..24

5.1System Stability Tests with (Operational Amplifier Controller).25

5.2Stability Test (upright)26

6.0Stability Test (when the pendulum is hanging with controller)29

7.0 Conclusion. 34

8.0References.. 35


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i) INTRODUCTION:

Remember when we were a child and we tried to keep a broomstick in balance when we hold

in our index finger or the palm of our hand? So what we did in that time. We had to constantly

adjust our hand as result to keep the object upright. Now an IVERTED PENDULUM does

basically the same idea in stated of it is limited for Pendulum to move only in one dimension,

while our hand could be move in any direction such as up, down, sideways, ect.

The Inverted Pendulum looks the same idea as the broomstick since both of them are an

inherently unstable system. Therefore nowadays the inverted pendulum offers typical example

for control system to prove a modern control theory.

Inverted Pendulum used for as a very good model in experiment as this very good example forthe attitude control of a space booster rocket and also automatic aircraft landing system ect .

The aim of this report is to balance a pole on mobile platform so that it can be move in only two

directions left or right.

ii) Description of the reconfigurable plant:

The reconfigurable that is experimented in this report is the Pendulum system. The figure (1)

below

Shows the pendulum system which is consists of two parts .the first part is the carriage module

and the second part is the control module .

Figure (1) the pendulum system


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There are two ways of using the Pendulum Control System , either as inverted pendulum or

overhead crane .In the first mode, control of unstable system ( the inverted pendulum) has to be

successful .Therefore to balance the Pendulum in the inverted position the rotate must be move

towards the falling pendulum . In the second mode, to keep the pendulum as crane we have to

turn the carriage upside down, so than the Pendulum will swings naturally into equilibrium

position with the centre of mass , in this case the pivot should now control as linear position

.Generally both experiment are foxing on how the feedback system function .


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iii) Technical Specification:

Table (2) shows Technical specification of the Pendulum from the Byronic Ltd .


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1.0 Mathematical Modelling of the Pendulum System

First we will start by determining the transfer function of the Pendulum, for each of the block in

the block diagram as shown below in figure (1.0)

Figure (1.0) block diagram

The second step is to design an analogue leadlag controller using two parallel RC plus. So now

the first thing is to model the dynamic equation for the system, than it can be transferred to the

S-Domain by using Laplace transforms.

Dynamic Equation:

The pendulum is defined in the diagram as above the figure (1.1).


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The carriage position (x) and (y) is the pendulum centre of gravity position as shown in the

figure above.

All the calculations below are base on the assumption so both the pendulum and the carriage

moves are in I ,j axis horizontally and vertically .

Now by the taking the moment from the above diagram and than equation them to zero, therefore

the dynamic equation for the pendulum can be in this formula.

The horizontal displacement of the pendulum can be in this form

Therefore (+sin) ( L) is the length of the pendulum which is the pivot centreline to the center of the mess

The vertical displacement for the pendulum is

(Lcos) j

Now the pendulum displacement can be represented as

=+sin+cos The total velocity which is the time dependent in different ion of displacement and

=v=

+

-

For the acceleration is the differentiation by using product of velocity

=

=a=

+

-

For the force of the pendulum can be calculated in the i, j axis. Moreover assuming T is

the tension in the pivot bar .

The horizontal force is the horizontal components of tension, T

sin

The vertical force of the pendulum is the horizontal components of tension ,T and free

body force on the pendulum due to gravity(g)

cosWhere (m)is the mass of the pendulum


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(f) is the total force on the pendulum , which is

=sincos By applying Newtons second law of motion, F=ma

sincos=m

By separating the I,j than we get two equations

sin = m

cos

=

Multiplying this equation bycos(1)sin = -m

Multiplying this equation by (2)- j -m j Substituting sincos in both equations (1,2) and removing the i, j components

m

sin

Simplifying further

m

sin


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sin=

But

+

sin=

Thus the above equation can be written as

+cossin=0 The above equation also can be written as a function of (f),and solving for the second

derivative of(t)

=

(3)

The non-linearity in f is sin and cos. By using Taylor to illustrated the equation Taylor series of sin

= + (a) + (a

+ (a)

Differentiations =sin, =cos, , For a=0 =0, =1, =0, Therefore:

Sin 0+1 + 0

-1

Sin- Taylor series expansion ofcos

=() + (a) + (a

+ (a)

Differentiations =cos , , =sinfor a=0 =1, =0,

, =0Hence:

cos 1+0 -1

+0

Sin1- sin


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Substituting in equation (3)

=

Talking Laplace transformer for the above equation

={()}20 (0)

By substituting the Laplace terms in (3)

2 ==-

[ - ] = -

= The transfer function for the pendulum, Gp(s), is therefore given by:

=


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2.0) Experimental Analysis:

In order to discover and solve any error for the Pendulum system, an experimental analysis has

been done on the pendulum rig as shown in the figure below (2.0)

Figure (2.0) Byronic Pendulum Rig and Control Unit

By doing several of experimental and measuring the conversion factors and built in gains, fine it

is possible to model the Pendulum system in conditions of the block diagram. As the figure

(2.1) shows the pendulum rig block diagram


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The transfer function Gs(s), for the pendulum carriage, was also carried out through the

experimental measurement. Furthermore the entire block diagram as shown above in figure (2.1)

has been determined to get the result for keeping the system in stable condition.

2.1 determinations (Kr):

The conversion factor of the output voltage is (Kr) and it gets the signal form the set point slider

which was built in board of the Control Unit System Diagram. The function of the slider

controller is to control position of the pendulum carriage. (Kr)is very important in order to

control the system because as soon as the carriage move will cause lateral forces to be exerted

on the pendulum .

For finding (Kr) value which is the output of the slider, first the control unit connected to the

Oscilloscope (Picoscope) after that we can see the output signal from the slider is shown on the

computer screen. the function of the computer is to get accurate voltage reading for using to

calculate the average voltage of (Kr).

By moving the slider through the rang of scalar positions, than we manage to get values of the

voltage as shown below in the table (2.2)

Table (2.2) Experimental values of Kr

Slider position

(V-in )

Voltage Out

(V -out )

Kr

(V-out/V-in)

-10 -9.98 0.998

-8 -9.285 1.160625

-6 -7.232 1.205333

-4 -4.781 1.19525

-2 -2.484 1.242

0 -0.481

2 1.575 0.7875

4 3.872 0.968

6 6.2495 1.041583

8 8.71 1.08875

10 9.922 0.9922

Average (Kr)= 1.067924


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The above table can be illustrated that a value of Kr in (0) position was not mentioned as this

when its dividing by zero than producing infinity value, as this happen it will distorting the

result .

2.3 Determination (Kx):

(Kx)The conversion factor between the movement which is on the set point of the slider and the

physical movement of the pendulum carriage along the track. This value was important as this

provided us the block diagram to understand the hysteresis of the carriage movement.

The target of this stage was to get the value of (Kx) first by connecting the set point of the slider

to the carriage and than measure the carriage movement each time when the slider move left or

right. In the beginning we set the slider in the zero position in the mean time we put a mark on

the carriage track the next step was to move the slide each time along the scale rang in the same

time the carriage position was measured the length along the carriage track as the table

(2.3).below described.

Table (2.3) Experimental values of Kx

Slider position

(V)

Carriage

Movement

(Inches )

Kx

(Inches/V)

-10 -7 0.7

-8 -6.92 0.865

-6 -6.4 1.066667

-4 -4.32 1.08

-2 -2.36 1.118

0 0

2 1.8 0.9

4 4.2 1.05

6 6.6 1.1

8 7.8 0.975

10 8 0.8

Average (Kx)= 0.971667


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2.4 Determination of (Kp):

(kp)is used for as conversion factor between the output voltage which is produced by different

angles of the pendulum rod movement .

The aim of calculating (kp) was to fine the relationship between the angular position of the

pendulum and the output voltage, each time the voltage was measured by changing the angular

of the pendulum. In addition that every measurement was taken from the centre of the pivot

point of the pendulum rod to make sure the angular was correctly.

All the angles reading were measured in degrees and form the all reading we took the average.

Note that the readings of the angular were converted to radians to keep all the calculations in the

same units. Table (2.4) below shows the date reading of angular.

Pendulum

Angle

(Degrees )

Pendulum

Angle

(Radians)

Output

voltage

(V)

Kp

(V/Radian)

-25 -0.43633 -1.555 3.56379749

-20 -0.34907 -1.126 3.22575239

-10 -0.174553 -0.5064 2.90145827

0 0 0.8012

10 0.174533 1.392 4.5905378520 0.349066 1.615 3.98778625

25 0.436332 0.9 3.70130736

Average (Kp)= 3.66177

Table (2.4) Experimental Values of (Kp)

2.5 Determination of (Ka):

We used (ka) as the constant value to calculate the position of the centre of gravity for the

system. (Ka) was required to determine the carriage movement in order to keep the pendulum

upright.


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The location of the (Ka) was in the feedback loop of the block diagram this constant was

calculated by using this formula as shown.

Vy=Vx+KaV

By using the above formula we managed to calculate (Ka) and by formative the scale of (Vy)

than had the same voltage output when the carriage is moved on the other hand the centre of the

gravity fixed in the same position.

Vx= 0

Ka= =

, =2.469

To get the (Vy) voltage by contacting with Picoscopeto fatal (L) on the pendulum control, the

value of (Ka) was 2.469.

2.6 Determination the Transfer Function, Gs(s) if the carriage:

In order to find the Gs(s) the Picoscope was used for to produce a square wave output therefore

the Picoscope was connected to the pendulum carriage .the rang of the wave was set from 1HZ

up to 1.4 HZ hence that happened the carriage movement was (+/-2 Inches ).

The velocity feedback and servo motor gain were adjusted to eliminate from the overshoot than

we got damping ratio (Zeta, ) of 0.7.

When the critically damped response was solved, the picoscope was switched to sinusoidal

wave.

Furthermore, the Picoscope overlaying was also used to measure the output of the carriage .Now

by doing this experimental we got the natural frequency and it was (4.9HZ) or (30.788 radians/s).

We assumed that the motor servo and the carriage transfer function are the second order system

and the value of the natural frequency and damping were the input of the original formula for a

second order transfer function as below.

Gs(s) = =


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3.0 Determination the Damping and Natural Frequency of the pendulum

Gp(s):

In order to run the Pendulum correctly in the block diagram , therefore the transfer function

which is determined previously needed to be improved by including a term to account of the

pendulum damping .this step was carried out by turning the pendulum upside down and

connected to the Pioscope in order to terminal K. This process created a plot which was showing

the angular position of the pendulum , as it was free swing .The frequency of the pendulum and

the damping could be determined by measuring the amplitude of the peaks , which is the time

between peaks and number of cycle in sample part .The table(3.0)below display all the values .

Table (3.0) Pendulum Response Data

The damping natural frequency (D), and the damping ratio (,) can be calculated by using

the following formulae.

=1n

property Value

X1 459mV

X2 414.7mV

X3 360.9mV

X4 316.6mV

t1 1.022seconds

t2 0.97seconds

n 3cycle


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,=

=

=0.0197

= =

= 6.343 Radians/s

From the above equations the damped natural frequency was calculated , due to natural thick

damping (I,e Air resistance , ect ) since this damping ratio is very small therefore can be

assumed that equal to the natural frequency .

The equation below can calculate the frequency to acceleration due to the gravity and the length

of the pendulum rod.

The transfer function Gp(s) became:

GP(s) =

=

=

For modelling the pendulum when it upside down (hanging position) it necessary to change

sign to positive on the numerator, because of the effects of the gravity on the system.


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3.1 The Experimental Data:

The table (3.1) below shows all the values which are obtained through the experimental

methods, and also including all measured frequency, and damping constant.

Table (3.1) all the values of Experimental data

Gs(s) =

Gs(s) =

property Value Units

Kr 1.067924 None

Kx 0.971667 Inches /V

Kp 3.66177 V/ Radian

Ka 2.469 None

30.788 Rads/s 0.7 None 6.344 Rads/s 0.0197 None

L 10 Inches

g 386.22 Inches/s

X1 459 mV

X2 414.7 mV

X3 360.9 mV

X4 316.6 mV

t1 1.011 Second

t2 0.97 Secondn 3 None(cycle)


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4.0 Test and System stability:

To test the system stability and also find the range of (kc) value witch keep the system stable in

both cases the upright and hanging positions. Therefore, a Simulink model has been created as

the original block diagram.

The model as shown blow in figure ( 4.0) has been modified by adding some features , such as

there was no Gc(s) block comparing with the original one , also the step input has been

modified pt the model to simulate the effect of the pendulum .

Saturation block has been added to the carriage feed forward loop to measure the limited

movement range of the carriage (the drive belt length).

Figure 4.0 the System Model ( Simulink)

4.1 Stability Test (upright):

In order to test the pendulum stability first we have to check the root locks, which was built in

the Simulink model by using the compensator design tools. This option allow us to change the

value of (Kc) in order to get stability in the system. The figure(4.1) below shows the root locus

plot for the pendulum when it is in upright position without controller.

From the experiment the range of the (Kc) values which keep the stable condition was between

(0.00339 to 4.15).


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The response of the (Kc) in (4.15) has got a large settling time , in 5000 seconds and this is not

good result because for practical purpose . On the other side the (Kc) value (0.00339) in

approximately was only 30 second which is looks better than the pervious on, however it is still

high.

Figure 4.1 root locus plot in the Upright Position


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Figure 4.2Step Response for Kc Value of ( 0.00339 )

Figure 4.3Step Response for Kc Value of 4.15


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4.2 Stability Test - (Hanging Pendulum):

Now in this stage we used the same method as the previous one to test the stability by using the

Simulink model of the pendulum but in hanging potion . Hence the sign was changed of the

numerator on the pendulum of the transfer function to the positive.

In the figure blow (4.4) shows that the range of the root locus plot can be changed to get the

value of the (Kc) which is between (0.00339 to 1.28 x 1017

.), and this result was expected since

the pendulum is on hanging position because the effects of the gravity causing the mass of the

pendulum to stay in the middle position naturally.

We notified that when the pendulum in hanging position the step response show that the gain is

increased the settling time in the same time of the system was dramatically decreased to 0.2

second and the maximum amplitude for both side (Kc) is only 0.1.

Figure(4.4)Root Locus for the Pendulum (Hanging Position)


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Figure(4.5 )Step Response ( Kc) Value (0.00339)

Figure(4.6)Step Response( Kc) Value (1.28 x 1017)


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5.0 controllers Design:

In this stage the aim of the design controller is to improve the time response of the system in the

form of an operation al amplifier. The amplifier was used a set of patroller resistor and capacitor

circuits as in figure (5.0) below.

Figure (5.0) Inverted Pendulum Compensator

The transfer function of the circuit

Gc(s) = -

Both of the resistance and capacitance were readied in order to get the value, than we could

complete the transfer of function

R1=R2=100 K

C1= = 1.047 F

47K 47x 103

pF 0.047 F

103K 10x103

0.01 F

0.33F 0.33F 0.33 F

0.33F 0.33F 0.33 F

0.33F 0.33F 0.33 F

0.33F 0.33F 0.33 F


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C2=1 F

0.1 F =1.1F

By using these values in transfer of function Gc(s):

Gc(s) = -

=

5.1 system Stability Tests with (operational Amplifier controller

In the system stability was used the same block diagram for testing and once again created the

root locus plot for both situation (upright hanging ). The system with transfer function as

shown in figure (5.1) below having a positive numerator, in the mean time the derivation

indicates should be minus so that means the pendulum control unit inversed the gain of the

controller. That is why in order to Simulink the correct controller ,the block diagram must have

this inversion by getting a positive numerator .

Figure(5.1)Block diagram


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5.2 Stability Test (upright):

In the upright position it can be seen in the figure (6.1)below that the values of root locus plot

for (Kc)which keep the system in stable condition were between ( 0.000232 to 0.164) and

(79.9 to 4.9 x 107

).

Figure5.2Root Locus Plot for Upright Position


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Figure 5.3Step Response Plot for (Kc) value of 4.9 x 107

Figure(5.4)Step Response Plot for Kc Value 79.9


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Figure(5.5)Step Response for Kc Value( 0.000232)


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Figure(5.6)Step Response for Kc Value (0.164)

6.0 Stability Test ( when the pendulum is hanging with controller

In this level the pendulum was in hanging position with block diagram. as before the root locus

plot was used with the compensator design tool . below as shown in the figure (6.7) how the

system is stable with possibility of the range of (Kc) ob the plot .During the test the range of the

value of (Kc) was between (8.33 x 10-5

to 2.19 x 106)

.

Significantly, the hanging position really stable and also faster response time even when we

used the value of (Kc)in the middle of the range as shown below in the figures .the reason of

that te dominant close loop poles were near to the zero point on the real axis .Note that when we

removed the poles further a way it became unstable again .


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Figure(6.0)Root Locus Plot for Hanging Pendulum with Controller


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Figure (6.1)Step Response Plot for Kc Value( 8.33 x 10-5)

Figure(6.2)Step Response Plot for Kc Value (2.19 x 10-6)


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From the above experimental that by using mid range values of (Kc) a greatly improved the

response. As shown below in the figure (6.10) that settling time has been reduced to 15 second

only ,in the same time the amplitude was maintaining of (1). In figure (6.11) was reduced the

settling time while the amplitude decreased to approximately (0.75).generally speaking that we

got a good improvement for the response since we achieved a faster response and less dramatic

response.

Figure(6.3)Step Response for Hanging Pendulum with Controller for Kc value of 0.162


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Figure (6.4)Step Response Plot for Kc Value (0.572)


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7.0 Conclusion:

It can be concluded that the aim of this modelling. Intensification and simulation of the

inverted pendulum since this experimental provided a big chance to design controller for

the system which has a good dynamic behaviour .The experience of working on the

inverted pendulum is great since it is an ideal exercise to appear talent as Control

Engineering.

In fact the practical work helps us to understand and develop an insight into the design of

the control system.

The inverted pendulum is highly unstable system, therefore our requirement was to

control the system in both ways(hanging and upright position) with and without any

compensator .in this experiment we have got an idea how the power of MATLAB and

Simulik is very important since all designing would not have been possible without

these kind of tools.

The planning for the inverted pendulum was like that , First of all we had to make the

mathematical modelling and then apply the Taylor series than finally we managed to getthe equation , the second step was to design a compensator in order to used in the

Simulike model which is keep the system in stable condition . Thirdly the controller has

been design and tune every block , from the root locus we tried to find the stable values

of gains witch keep the system in stable .from experimental we found that some values of

give us a high settling time and also overshooting . Generally speaking the inverted

pendulum is unstable system but we tried to make stable and finally we managed to do

that.


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8.0) References

1. Golten JW, Verwer AA, "Control System Design and Simulation", McGraw

Hill, 19

2. Charles W. Anderson, Learning to Control an Inverted Pendulum Using

Networks

3. Victor Williams, Kiyotoshi Matsuoka, Learning to Balance the Inverted

Pendulum using Neural Network, Kitakushu Japan

4. Dr. Rashid_ Ali , control system Lecture Note Herefordshire university5. http://www.engin.umich.edu/group/ctm/examples/examples.html

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Assignmrnt of contorl system