Position Control of PP ball

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Position Control of a
Ping-Pong Ball
Position Control of a Ping-Pong Ball Within a Glass
Tube Using a P.I.D. Control System
James Goddings St. No.3131147
11/22/2015
Project Supervisor: Shyamal Mondal
Dynamics and System Modelling (ENG_6_451)

Table of Contents
Aim........................................................................................................................................................ 1
Apparatus ............................................................................................................................................. 1
Hardware .......................................................................................................................................... 1
Software ........................................................................................................................................... 4
PID Controller Theory ........................................................................................................................... 4
Proportional Controller..................................................................................................................... 4
Differential Controller....................................................................................................................... 5
Integral Controller............................................................................................................................. 6
Experimental Procedure ....................................................................................................................... 7
Open Loop System ............................................................................................................................ 7
Open Loop Results ........................................................................................................................ 7
Closed Loop System .......................................................................................................................... 7
Proportional Control......................................................................................................................... 8
Proportional Gain Results ........................................................................................................... 10
Integral Control............................................................................................................................... 11
Integral Gain Results ................................................................................................................... 13
Differential Control ......................................................................................................................... 14
Integral Gain Results ................................................................................................................... 15
Changing Demand Position............................................................................................................. 15
Changing Demand Position Results............................................................................................. 17
Calibration of Ultrasonic Range Finder Scale .................................................................................. 18
Conclusion .......................................................................................................................................... 20
References ............................................................................................................................................ 1

Position Control of a Ping-Pong Ball James Goddings 3131147
1
Aim
The aim of this investigation was to achieve stable position control of a Ping-Pong ball using
A PID controlled DC fan adjusting airflow through a glass tube.
This investigation will achieve the objective of familiarising the operator with the following:
 PID control systems
 Matlab Simulink Models and Real Time Windows Target
 Use of Feedback and Feed-Forward Control
 Use of the Sensoray 626 Interface Card
 Use of the FR04 Ultrasonic Range Finder
Apparatus
Hardware
Figure 1.1 Ping-Pong ball in a glass tube apparatus

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Figure 1.2 Close-up of base of glass tube showing fan impeller assembly, vent and motor
controller.
The hardware consists of:
A Ping-Pong ball constrained vertically within a close fitting glass tube, with a fan at the base
and a position sensor at the top; as shown in Figure 1.1.
The fan is a Mini 9V DC model with an impeller mounted on a common shaft above it,
drawing air in a side vent and propelling it upward through the glass tube.
The position sensor is an FR04 ultrasonic range finder that consists of 2 elements, a
transmitter and a receiver. Activated by a 50 ms period pulse from one of the counters on a
Sensoray 626 interface card, the transmitter emits a 20μs ultrasonic pulse which travels at
the speed of sound down the tube, rebounds off the ball, and returns at the speed of sound
to the receiver.
Simultaneous to the ultrasonic pulse, an electrical signal in the form of a 40 kHz series of
pulses is output back to the Sensoray 626 interface card. When the ultrasonic echo is
received by the receiver the electrical output pulses are stopped, therefore the time it takes
for the ultrasonic pulse to return (time of flight) can be measured by counting the number
of electrical pulses received by the Sensoray 626 interface card. (See Figure 1.3)

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Figure 1.3 Graphical representation of the ultrasonic range finder signal input and output
traces.
“The Sensoray model 626 is a Multifunction Input/Output Interface card.
Some of the features include:
• 48 digital I/O channels.
• 20 of the digital I/O channels have edge detection and interrupt
capability.
• 7 of the digital Outputs can be used as counter overflow outputs.
• Digital I/O connectors are industry standard pinouts.
• Watchdog timer with several selectable reset periods that can reset the
PCI bus.
• Six 24 bit up/down counters arranged in 3 pairs with:
• Inputs that can be driven in various modes (1x, 2x, 4x) from
incremental encoders inputs, the digital inputs, the paired counter’s
overflow, the system clock or software driven.
• Can generate an interrupt on counter overflow or encoder/digital input
index.
• Can be preloaded/cleared on an overflow.
• Output of second counter can be captured on the overflow of the first.
• Can be used as a programmable periodic interrupt generator.
• Counter circuitry can be battery backed up to prevent count loss during
a power failure.
• Charge control of the backup Ni-Cad battery.

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• 16 differential analog inputs (14 bit resolution).
• 4 analog outputs (13 bit resolution) with remote sense inputs to
compensate for any external output resistance.”(Sensoray, 2004)
The Sensoray 626 interface card acts as a conduit between the hardware system and the
Matlab Simulink software, facilitating the control of the hardware system through models
created in Simulink by the operator. A single digital input on the interface card is used to
count the 40 kHz electrical pulse output from the ultrasonic sensor, this is converted to a
voltage output by one of the interface card’s digital to analogue converters and relayed to
the Simulink model as a voltage.
The Simulink model interprets the voltage from the interface card as a position using
predetermined range information and returns an output voltage to one of the interface
cards analogue inputs, this is digitised through an analogue to digital converter and
converted back through a digital to analogue converter to a 0-10V DC output to the motor.
Software
Matlab is a long running mathematical software package originally designed with the
intention of facilitating the processing of matrix calculations. Over its 30year history it has
been developed into a very powerful software package capable performing complex
mathematical calculations and of modelling and simulating complex systems. Simulink is a
powerful plug-in tool developed as part of Matlab to enable graphical modelling of
electronic hardware and process controllers within the Matlab environment, with the
capability of simulating these models and components and outputting signals to an interface
in real time using the Real Time Windows Target.
PID Controller Theory
The PID controller is a tuning controller that is very useful as it does not rely on the operator
having prior knowledge of the underlying mathematical model of the system being
controlled. In effect it combines 3 separate controllers that each implements different
mathematical operators on the system in the form of variable gains.
Proportional Controller
The proportional controller’s purpose is to increase the speed of a system’s response and
reduce the steady state error of the system response. The output of the system is measured
and fed back to the input of the system where the error between the output and the
demanded input value is multiplied by a gain proportional to the error.

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Figure 1.4 Block diagram of a proportional controller applied to a plant system. (Satar, 2013)
𝐸𝑞𝑛. 1.1 𝑓( 𝑡) = 𝐾𝑝(𝑥 𝑑( 𝑡) − 𝑥( 𝑡)) = 𝐾𝑝 𝑒(𝑡)
There are downsides to proportional control; it reduces the damping’s effect of the overall
system, resulting in the system becoming less stable. At a point called the critical gain, the
proportional gain overcomes the damping of the system, moving the system poles to the
right hand side of the imaginary axis of the s-plane, causing the system to become unstable.
Proportional gain can also cause overshoot of the demand input, as a rule of thumb, an
overshoot of up to 25% is tolerated in return for the increased system response rate.
In the case of this investigation the proportional gain will act as a proportional multiplier to
the error in the Ping-Pong ball’s displacement from the demanded position.
Differential Controller
The differential controller’s purpose is to increase system response stability; this is achieved
by adding another feedback loop affecting the derivative (rate of change) of the primary
control variable. This increases damping, moving the system poles away to the left from the
imaginary axis of the s-plane, thereby increasing stability, and is often used in conjunction
with proportional control to alleviate the instability introduced by proportional gain.
Figure 1.5 Block diagram of a differential controller in conjunction with a
proportional controller applied to a plant system. (Satar, 2013)
𝐸𝑞𝑛. 1.2 𝑓( 𝑡) = 𝐾𝑝(𝑥 𝑑( 𝑡) − 𝑥( 𝑡)) − 𝐾𝑑 𝑥̇(𝑡)
Kp
Plant
System
+
-
χd(t) χ(t)
Kp
Plant
System-
χd(t)
χ(t)
+
-
+
dχ(t)
dt
Kd
e(t) f(t)
e(t) f(t)

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In the case of this investigation the differential gain will act as a multiplier to the error in the
Ping-Pong ball’s velocity from the demanded velocity (zero for a fixed position), again too
much gain will overwhelm the system’s response to the demand input becoming the
overriding influence, causing instability.
Integral Controller
The integral controller’s purpose is to decrease the steady state error of the system
response to zero. This is achieved by integrating the error in the primary control variable
between the output and demand and feeding this forward as a ramp up of the control
signal. This is often used in conjunction with proportional control and differential control to
reduce any steady state error to zero. Integral control increases the type number of the
closed loop system by one, making the model more complex.
Figure 1.6 Block diagram of an integral controller in conjunction with a
differential controller and a proportional controller applied to a plant system.
(Satar, 2013)
𝐸𝑞𝑛. 1.3 𝑓( 𝑡) = 𝐾𝑝(𝑥 𝑑( 𝑡) − 𝑥( 𝑡)) + 𝐾𝑖 ∫(𝑥 𝑑( 𝑡) − 𝑥( 𝑡)) 𝑑𝑡 − 𝐾𝑑 𝑥̇(𝑡)
In the case of this investigation the integral gain will act as a multiplier to the error in the
Ping-Pong ball’s displacement from the demanded position. Integral gain is very effective at
reducing steady state error when a system has a stable or slow changing system response,
however with a dynamic rapidly changing system response the integrator can “wind up”
causing instability.
Kp
Plant
System-
χd(t)
χ(t)
+
-
+
dχ(t)
dt
Kd
+
e(t) f(t)
Ki ∫dt

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Experimental Procedure
Open Loop System
The first part of the investigation is a simple test to familiarise the operator with the
procedures of building and operating Matlab Simulink models and compiling them for
export to the Real Time Windows Target environment for use with the Sensoray 626
Interface Card, the FR04 Ultrasonic Range Finder and the Ping-Pong ball in a glass tube/fan
arrangement.
An existing open loop Simulink model is opened in Matlab which enables a voltage gain
value to be applied to the fan controller and the resulting Ping-Pong ball position to be
displayed in the form of a value with an arbitrary scale. This value is an output from the
ultrasonic range finder that had been linearised by the Sensoray 626 interface card to have
a range between 0-10V to correspond with the 0-10V scale of the output voltage to the fan.
(The Ping-Pong balls resting position at the bottom of the tube corresponds to a value of
1.3V)
Open Loop Results
This resulted in Table 1.1 providing knowledge of voltages required to initiate the Ping-Pong
ball climbing the glass tube, hovering stationary and falling in a controlled manner.
Fan Voltage [V] P-P Ball Observation
6.1 Started to vibrate
6.85 Started to spin
7.57 Started to lift off
7.65 Upward movement
7.5
Controlled downward
movement
7.55 Hovers stationary
Table 1.1 Observed Voltages required for initiating movement of the Ping-Pong ball
Closed Loop System
The second part of the investigation involves designing and implementing a Matlab Simulink
PID control model to control the position of the Ping-Pong ball in the glass tube. This is done
in a methodical manner introducing proportional control, then integral control, then
differential control to establish and optimise control of the ball’s position.
When sufficient control is established, the arbitrary scale resulting from the output signal
from the ultrasonic range finder is calibrated against the measuring tape to the right of the
glass tube.
Figure 1.7 Shows the Simulink model used to apply PID control to the Ping-Pong ball in a
glass tube system through the interface card.

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Figure 1.7 Closed Loop Matlab Simulink model with PID block.
Proportional Control
For the PID investigation, an initial demand position of 8 on the arbitrary range finder scale
is selected. The proportional controller gain is then increased in steps to determine its effect
on the control of the Ping-Pong balls position and the ability of the control system to track
the demand position.
For all of the charts following:
Yellow Line = Demand Position
Cyan Line = Ping –Pong Ball’s Position
Magenta Line = Fan Controller Voltage

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Chart 1.1 Demand Position = 8, Proportional Gain = 1.0, Steady State Position = 1.5, Steady
State Output Voltage = 6.7V
Chart 1.2 Demand Position = 8, Proportional Gain = 2.0, Steady State Position =
4.1(Oscillating 1.2), Steady State Output Voltage = 7.7V (Oscillating 2.2V)
Position/OutputVoltage
Time (s)
Time (s)

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Chart 1.3 Demand Position = 8, Proportional Gain = 4.0, Steady State Position =
5.8(Oscillating 2.4), Steady State Output Voltage = 9.8V (Oscillating 6V)
Proportional Gain Results
Charts 1.1-1.3 show that increasing the proportional gain increased the speed of the system
response and decreased the error in the steady state response, however it also decreased
stability of the system and introduced oscillations at values above 1.5. A proportional gain of
1.2 was chosen to continue the investigation as it maintained stability whilst offering an
increase in the speed of the system response and reduction the error of the steady state
response.
Time (s)

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Integral Control
Chart 2.1 Demand Position = 8, Proportional Gain = 1.2 Integral Gain = 0, Steady State
Position = 1.9, Steady State Output Voltage = 7.7V
Chart 2.2 Demand Position = 8, Proportional Gain = 1.2, Integral Gain -= 0.1, Steady State
Position = 7.7, Steady State Output Voltage = 7.7V, Rise Time = 27s
Time (s)
Time (s)

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Time (s)
Time (s)

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Position = 8.0(Oscillating 0.6), Steady State Output Voltage = 7.8V, Rise Time = 2.5s,
Overshoot =20%
Integral Gain Results
It is evident from Charts 2.1-2.5 that increasing the integral gain reduces the lead-lag of the
system, thereby increasing the speed of the system response and decreasing the error in the
steady state response by ramping up the gain in the system more the longer an error
between the demand and system response existed; however it also decreased stability of
the system and introduced overshoot and oscillations at values above 0.8. An integral gain
of 0.8 in conjunction with a proportional gain of 1.2 was chosen to continue the
investigation as these values offered a rapid system response and reduction in the steady
state error. The integral gain did introduce a 20% overshoot, however this is commonly
accepted prior to introducing differential control, as differential control can reduce or
eliminate overshoot.
Time (s)

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Differential Control
Chart 3.1 Demand Position = 8, Proportional Gain = 1.2, Integral Gain -= 0.8, Differential
Gain =0.02, Steady State Position = 8.0 (Oscillating 1.2), Steady State Output Voltage = 7.6V,
Rise Time = 2.5s, Overshoot =15%
Rise Time = 3s, Overshoot =12%
Time (s)
Time (s)

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Rise Time = 4s, Overshoot =2%
Integral Gain Results
Charts 3.1-3.2 show that increasing the differential gain was reducing the overshoot of the
system response, however instability and oscillations were being introduced, increasing the
error in the steady state response. Upon consultation with the laboratory supervisor, Mehdi
Zahir he suggested a review of the integral control procedure and the integral gain final
value. This review resulted in an optimised system shown by Chart 3.3, where the integral
gain had been reduced considerably, and the differential gain increased, producing a system
response that was fast, stable and with minimal steady state error.
Changing Demand Position
Having arrived at the PID system in Chart 3.3 it was realised that this system is optimised for
a demand position of 8.0, a large difference from the rest position of 1.3, but what if the
demand position was changed?
A further investigation is embarked upon to ascertain if the systems response would be
suitable for all demand positions.
Time (s)

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Gain =0.5, Steady State Position = 6.0, Steady State Output Voltage = 7.8V, Rise Time = 4s,
Overshoot =2%
Gain =0.5, Steady State Position = 4.0, Steady State Output Voltage = 7.7V, Rise Time = 7s
Time (s)
Time (s)

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Changing Demand Position Results
It is evident from Charts 4.1-4.3 that decreasing the demand position, and thereby the
difference from the rest position of 1.3 to the demand position, resulted in decreasing
Time (s)
Time (s)

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response times for the PID system implemented. Adjusting the differential gain in Chart 4.4
improved this slightly, however further investigation would have been required to obtain
proportional, integral and differential gains more suitable for a variety of demand positions,
and varying demand positions, such as a time dependant cyclic demand instead of a step
demand. Unfortunately due to the limited time of the lab session this further work lay
outside of the scope of the investigation.
Calibration of Ultrasonic Range Finder Scale
When sufficient control was established, the arbitrary scale resulting from the output signal
from the ultrasonic range finder was calibrated against the measuring tape to the right of
the glass tube. This involved using the Simulink model in figure 1.7 with the PID values
shown in Chart 4.3 and setting a range of range finder demand values, then reading off the
corresponding steady state position of the Ping-Pong ball on the measuring tape to the right
of the glass tube. This resulted in Table 1.2
Position Demand
Range Finder
Output
Measured
Position
[V] [V] [mm]
1.4 1.4 1310
1.5 1.5 1300
3 3 1110
4.5 4.5 890
6 6 680
7.5 7.5 470
9 9 260
10 10 110
Table 1.2 System response to demand values, and the corresponding steady state position of
the Ping-Pong ball on the measuring tape to the right of the glass tube.
This table was charted, Chart 5.1, and the mathematical relationship evaluated relating the
arbitrary scale resulting from the output signal from the ultrasonic range finder to the
measuring tape to the right of the glass tube. A new Simulink model with a block facilitating
the input of a demand value for the position of the Ping-Pong ball in mm was developed,
Figure 1.8, and tested. This allowed for a demand to be tracked effectively by the Ping-Pong
ball upon a user input related to the measuring tape.

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Chart 5.1 Relationship between the arbitrary scale resulting from the output signal from the
ultrasonic range finder and the corresponding steady state position of the Ping-Pong ball on
the measuring tape to the right of the glass tube.
Figure 1.8 Simulink model facilitating the input of a demand value for the position of the
Ping-Pong ball in mm corresponding to a position on the measuring tape of the glass tube.
y = -0.0072x + 10.8465
R² = 0.9997
0
2
4
6
8
10
12
0 200 400 600 800 1000 1200 1400
RangeFinderOutputPosition[V]
Position on Measuring Tape [mm]

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Conclusion
The aim of this investigation was to achieve stable position control of a Ping-Pong ball using
A PID controlled DC fan adjusting airflow through a glass tube. This was achieved and the
PID system was tuned to control the system adequately to allow sufficient control of the
Ping-Pong ball’s position within the tube to accurately calibrate it to a physical scale.
The optimum PID control algorithm derived from his investigation was most dependent on
proportional and derivative gains, and was most effective at tracking larger changes in
positional demand to the system.
There is still room for improvement of the PID system developed in responding to small
demand changes; further testing would consist of investigating this and the systems
response to other demand types, i.e. cyclic demands.
The system was very dynamic with little damping (underdamped) due to the main dynamic
force acting on the system in the form of turbulent airflow being variable and transient in its
nature. This results in the system having some poles close to the imaginary axis of the s-
plane, which is evident from the system becoming unstable with the addition of larger
proportional and integral gains. The other dynamic force, gravity, remains constant and has
a predictable effect which enables the system to be controlled as it results in the system
having some poles close to, or situated on, the real axis.
The investigation was successful in achieving the objectives of familiarising the operator
with Matlab Simulink Models and Real Time Windows Target, Use of Feedback and Feed-
Forward Control, Use of the Sensoray 626 Interface Card and Use of the FR04 Ultrasonic
Range Finder. It was especially informative in the use and tuning of PID controllers on
dynamic systems, which the author feels he has a much better understanding of as a result
of carrying out this work.

A-1
References
Ashwini Miryala, Kyle Scarlett, Zachary Zell, Brandon Kountz, (2006), The University of
Michigan Chemical Engineering Process Dynamics and Controls Open Textbook - PID
Downslides & Solutions, available at
https://controls.engin.umich.edu/wiki/index.php/PIDDownsides accessed on 14/11/2015
LSBU (2015), London South Bank University School of Engineering –Control Systems Lab
Room 405 – PID Position of a Ping Pong Ball in a glass tube.
Satar Dr. T. (2013), Principles of Control Lecture Notes - PID Control
Sensoray (2004), Sensoray Model 626 PCI Multifunction I/O Board Revision F available at
http://www.sensoray.com/downloads/man_626_1.0.5.pdf accessed on 14/11/2015

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