Control of Quadcopter Roll

ASSIGNMENT
Module Code
Module Name
Course
Department

ESD 527
Embedded Control System
M.Sc in Real Time Embedded Systems
Computer Engineering

Name of the Student

Bhargav Rajivbhai Shah

Reg. No

CHB0911001

Batch

Full-Time 2011.

Module Leader

Viswanath K Reddy

M.S.Ramaiah School of Advanced Studies
Postgraduate Engineering and Management Programmes(PEMP)

POSTGRADUATE ENGINEERING AND MANAGEMENT PROGRAMME – (PEMP)

MSRSAS - Postgraduate Engineering and Management Programme - PEMP

#470-P Peenya Industrial Area, 4th Phase, Peenya, Bengaluru-560 058
Tel; 080 4906 5555, website: www.msrsas.org

i

Declaration Sheet
Student Name

Bhargav Rajivbhai Shah

Reg. No

CHB0911001

Course

Real Time Embedded Systems

Batch

Full-Time2011

Module Code

ESD 527

Module Title
Module Date

Embeddede Control System
to

Module Leader

Viswanath K Reddy

Batch Full-Time2011

5.11.2011

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Declaration
The assignment submitted herewith is a result of my own investigations and that I have conformed to the
guidelines against plagiarism as laid out in the PEMP Student Handbook. All sections of the text and
results, which have been obtained from other sources, are fully referenced. I understand that cheating and
plagiarism constitute a breach of University regulations and will be dealt with accordingly.

Signature of the student

Date

Submission date stamp
(by ARO)

Signature of the Module Leader and date

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ii


Abstract
____________________________________________________________________________
The history of micro air vehicles really began with the development of model airplanes
in the 19th century and the development of radio controlled model airplanes in the 20th century.
The development of miniature radio receivers and control components in the 1990s also had a
large impact on the ability to design a very small flying vehicle. Once the aerodynamics and
control of small aircraft models with a mass less that 100 grams were better understood, the
micro-air-vehicle was born. As the first part of this assignment a brief essay prepared on
Construction, working principal of single rotor and quad rotor MICAVs. Comparison of the
single rotor and quad rotor MICAV, application and roadmap is discussed. Based on this work,
MICAV is suggested for low altitude costal area surveillance.
There are three possible motions any aircraft can take in space named as roll, yaw and
pitch .Mathematical model is derived in for roll control systems part b. Step response of the
mathematical model is created and analyzed. To get desire time response form that roll control
system PID controller is designed. Open loop step response of PID and close loop response of
whole mechanism with PID is plotted.
To Digitalize designed PID ,z-transform of

PID controller is derived with three

different methods in part c. Apart from that , open loop step response of PID is plotted,
analyzed and compared with the step response of analog controller which is plotted in part b. C
program for the PID controller is created and tested with different test cases. By the end of the
section direct form1 and direct form 2 realization of PID is done.

iii

Contents
____________________________________________________________________________

Contents
Declaration Sheet ......................................................................................................................... ii
Abstract ....................................................................................................................................... iii
Contents ........................................................................................................................................iv
List of Figures ..............................................................................................................................v
List of Tables ...............................................................................................................................vi
List of Symbols .......................................................................................................................... vii
PART-A ........................................................................................................................................8
CHAPTER 1 Quad rotor v/s Single rotor .................................................................................8
1.1 Introduction.....................................................................................................................8
1.2 Construction and working principle of single and Quad rotor MICAVs .......................8
1.3 Comparison of the single rotor and quad rotor MICAVs ...............................................9
1.4 Applications ..................................................................................................................10
1.5 Roadmap .......................................................................................................................10
1.6 Recommendation of a suitable MICAV for coastal surveillance .................................10
PART-B ......................................................................................................................................11
CHAPTER 2 Modeling and Simulation of Quad rotor .........................................................11
2.1
Introduction ..................................................................................................................11
2.1 literature survey on the relative speed of the rotor for correcting the roll....................11
2.2 FBD for roll of the quad rotor.......................................................................................12
2.4 Mathematical model .....................................................................................................13
2.3 Assumptions of motor ...................................................................................................15
2.4 Transfer function based assumed parameter .................................................................15
2.5 Mathematical model simulation and analysis ...............................................................16
2.6 Design and implementation of PID controller ..............................................................17
2.7 Testing of PID controller ...............................................................................................19
2.8 Conclusion .....................................................................................................................20
PART-C ......................................................................................................................................21
CHAPTER 2 Digitalization of PID ..........................................................................................21
3.1 Introduction ...................................................................................................................21
3.2 Digitalizing of controller ...............................................................................................21
3.3 Modeling of digital controller .......................................................................................22
3.4 Comparison of analog and digital controller responses ................................................23
3.5 Implementation of digital controller in c .......................................................................24
3.6 Test cases .......................................................................................................................24
3.7 Responses of two implemented controller structures ....................................................26
3.8 Conclusion .....................................................................................................................27
CHAPTER 4 ..............................................................................................................................28
4.1 Learning outcome ..............................................................................................................28
REFRENCE ...............................................................................................................................29
BIBILOGRAPGY .....................................................................................................................30
Appendix ....................................................................................................................................31

iv


List of Figures
____________________________________________________________________________
Figure 1. 1 Straight Rotations & Tilted Rotations ......................................................8
Figure 1. 2 Rotor Structure of Quad rotor[6] ..............................................................9
Figure 2. 1 Construction of quad rotor......................................................................11
Figure 2. 2 Phenomena of Roll motion .....................................................................12
Figure 2. 3 Block diagram of the rite rotor with brushless motor[3] ........................13
Figure 2. 4 Step response of roll control system .......................................................17
Figure 2. 5 Close loop response of roll control system with Kp...............................18
Figure 2. 6 Close loop response roll control system with Kp and Ki ......................18
Figure 2. 7 Close loop response roll control system with Kp,Ki and Kd .................19
Figure 2. 8 Testing of PID ........................................................................................20
Figure 3. 1 Step response of controller .....................................................................23
Figure 3. 2 Responses of analog and digital controller.............................................23
Figure 3. 3 Response Ramp exited digital PID ........................................................25
Figure 3. 4 Response of Impulse exited digital PID .................................................25
Figure 3. 5 Configuration of filter parameter in fdatool ...........................................26
Figure 3. 6 Step exited direct form 1 ........................................................................26
Figure 3. 7 Step exited direct form 2 ........................................................................27

v


List of Tables
____________________________________________________________________________
Table 2. 1 Parameters of motor ...................................................................................................15
Table 2. 2 Relation between roll angel and input voltage ...........................................................16

vi


List of Symbols
____________________________________________________________________________

Symbol

Description

MAV

Micro Air Vehicle

VTOL

Vertical take-off and landing

MEMs

Micro-electromechanical Systems

FDB

Free Body Diagram

RPM

Revolutions per Minute

BLDC

Brush Less Direct Current

ZOH

Zero Order Hold

PID

Praposnal, integrator and differentiator
.

vii

PART-A
CHAPTER 1 Quad rotor v/s Single rotor
v

1.1 Introduction
MAVs are defined as small flying systems which are designed for performing useful
operations. The concept to design micro air vehicle came in lime light over the past few year, with
the purpose of carrying law altitude surveillance. To fulfill the utility of law altitude aerial
surveillance, the aircrafts that are small and slow as the law of aerodynamics is essential. This kind
of small vehicle is known as the micro air vehicles. As per the pattern of the landing and takeoff
there are two arrangement of the rotor is possible. The first one is the tilt rotor or single rotor in
the
which the takeoff and landing is possible with the horizontal axis [4]. Tilt-rotor aircraft feature the
ng
rotor
ability to hover like a helicopter, enabling a vehicle to loiter directly over a target and to fly at high
speeds. The anther one is a quad rotor, in which the four rotors are used to control the spee and
speed
altitude of the vehicle. This type of MA is capable to land or take off along with vertical axis.
MAVs
r
VTOL MAVs are also gaining popularity, mainly because of their ability to quietly linger in one
s
popularity,
spot for an extended period of time [
[4].

1.2 Construction and working principle of single and Quad rotor MICAVs
Single rotor: A single rotor MAV is the type of the micro version of the helicopter. Which fly
MAV
with thrust that is produced by the rotation of the only one main rotor which is mounted above the
body. The lifting force is produced by the main rotor As they spin in the air and produced the lift.
duced
rotor.
Each blade of rotor produces an equal share of the lifting force. The weight of a helicopter is
equal
divided evenly between the rotor blades on the main rotor system. The straight rotations of the main
rotation
rotor as shown in figure 1.1 are responsible for the producing the vertical thrust. By tilting the
direction of the main rotor as shown in the figure 1.1 produce the horizontal thrust.

Figure 1. 1 Straight Rotations & Tilted Rotations

8


With effect of only single rotor, if the rotor is rotating in the clock wise direction so the whole body
moves in the anti clock wise direction as per the Newton's third law of motion states. To avoid this
unnecessary rotation contradictory torque to main rotor is required. Usually a small rotor is used on
the tail to produce a contradictory thrust that prevents the body from the rotation. Sometimes whole
body can rotate by the thrust produced you the tail rotor.
Quad rotor:
The quad-rotor MAV consists of a rigid cross frame equipped with four rotors placed in the
four corners of a planar square, those ones placed oppositely rotate in the same direction, while the
perpendicular ones rotate reversely. A revolution or a torque control can be realized from group of a
motor and the rotor mounted on it to ensure simple and efficient actuation, which can practically be
realized within a local control loop. The attribute control in the quad rotor is carried out by
changing the speed of the four motors.

Figure 1. 2 Rotor Structure of Quad rotor[6]
The arrangement of the motors is shown in the figure 1.1. Pitch and roll angles are controlled using
moments generated by differential thrust between rotors on opposite sides of the vehicle, and the
yaw angle is controlled using the difference in reaction torques between the pitch and roll rotor
pairs.

1.3 Comparison of the single rotor and quad rotor MICAVs
•

A phenomena of the tilting of the rotor while running at high speed is mechanically
complicated as compared changing the speed of the rotors in the quad rotor

•

In single rotor one main rotor and one tail rotor rotate so the vibrations produced on the
body is less as compared to four rotors in quad rotor design

•

Four rotors are used in the quad rotor so the cost and weight is more as compared to the
single rotor

•

In the quad rotor each individual rotor has s small diameter as compared to the single rotor,
which is having less kinetic energy during the flight. Which reduce damage if rotor hits
anything .

9


1.4 Applications
1) A law cost spying

2) Search for survivors[4]

3) Urban traffic management[4]

4) Ares surveillance[4]

5) Pipeline inspection
6) Used by the military for recognizance and search and rescue operations[5]

1.5 Roadmap
In 1907, the Breguet Brothers constructed the first quad-rotor named Gyroplane. The flight was a
good work to show the principle of a quad rotor. In 1922, Georges de Bothezat built a quad rotor
with a rotor located at each end of a truss structure of intersecting beams, placed in the shape of a
cross. Experimental aircrafts X-19 and Bell X-22A are also designed as quad rotor aircrafts. In time
due to the tremendous improvements in manufacturing techniques and innovations in metallurgical
material knowledge more precise and smaller sensors can now be produced. The Microelectromechanical Systems (MEMs) technology now allows the production of machine components
such as gears with sizes in 10ି଺ meter range. Using this MEMs technology very small
accelerometers, gyros and magnetometers are also produced, which caused the production of
smaller strap down inertial navigation systems. As a result of this improvement in technology very
small quad rotors are developed around the world.

1.6 Recommendation of a suitable MICAV for coastal surveillance
The quad rotor MICAVs has very stable performance in the windy condition due to the four
rotors. Driving flexibility is more in quad rotor .Moreover the due to four rotor structure of the quad
rotor it can stay at one place for long time as compare to the single rotor ,which can provide
accurate surveillance of costal area. So, quad rotor type MICAVs are recommended for costal
surveillance.

1.7 Conclusion
On the basis of above I can conclude that single rotor MAV has a less power consumption due to
the single rotating wing but for good stability in the windy condition and ability to hover for a long
time on same pace quad rotor is the suitable.

10


PART-B
CHAPTER 2 Modeling and Simulation of Quad rotor
________________________________________________________________________________

2.1 Introduction
In this part of assignment the literature survey on the relative speed of the rotor and FBD is
created. Based on the FDB mathematical model is created. Mathematical model is simulated using
the Matlab and analyzed the Quad rotor model for stability.

2.1 literature survey on the relative speed of the rotor for correcting the roll
A Quad rotor is an aerial vehicle that generates lift with four rotors. The MAV is controlled
by varying the rpm and not by using any mechanical actuators like in a helicopter. This makes it
particularly suitable for MAVs. The MAV requires active control of six degrees of freedom to fly. The
layout of a quad rotor is shown in the figure.

Figure 2. 1 Construction of quad rotor
The Quad Rotor layout is shown Figure 2.1. There are two arms, each having motors at its ends.
The motors 1 and 3, which are mounted on the same arm, rotate in the clockwise direction while the
motors 2 and 4, mounted on the second arm, rotate in the anti-clockwise arrangement. Both motors
at opposite ends of the same arm should rotate in same direction to prevent torque imbalance during
linear flight.
Altitude Motion
The throttle movement is provided by increasing (or decreasing) the speed of all the rotors by the
same amount. It provides a vertical force to the quad rotor.

11


Roll motion
The roll movement is provided by increasing (or decreasing) the left rotor speed and at the same
time decreasing (or increasing) the right rotor speed. It leads to a torque with respect to the central
axis as shown in Fig 2.2 which makes the quad rotor roll. The overall vertical thrust is the same as
in hovering.

Figure 2. 2 Phenomena of Roll motion
Pitch Motion
The pitch movement is provided by increasing (or decreasing) the front rotor speed and at the same
time decreasing (or increasing) the back rotor speed. It leads to a torque with respect to the central
axis.
Yaw Motion
The yaw movement is provided by increasing (or decreasing) the front-rear rotor speed and at the
same time decreasing (or increasing) the left-right couple. It leads to a torque which makes the quad
rotor turn in horizon level.

2.2

FBD for roll of the quad rotor
A FBD is a pictorial representation of the force acting on the body of interest. It included all

the forces acting on a body.FBD consists of sketch of body &arrows which shoes the force applied
on it. For drawing FBD assume one condition in which at t=0 speed of a right and left rotor is 50
rpm and aircraft is in hovering condition. At this condition Roll angle with respect to the ground
axis is zero. In stable state at t=1 aircraft got some problem and it tilted to right side and to roll
angle with respect to the ground axis is changed to 30 degree. At this movement still both rotors
rotating at the same speed. Sometimes this condition is hazardous for aircraft. To overcome this tilt
and make roll angle to zero there must be sudden increase in upward torque from right rotor. For

12


increasing the produced torque by the right rotor speed of the rotor should be increase. So, the
needed increment in speed is proposal to the roll angle. At t=2 speed of the right rotor is increased
by the 20 rpm. Sudden change in rotor speed produced more lift at right side and roll angle will
recover .Once roll angle reached to the 0 degree or near to that speed of the right motor should be
decreased again. So, by increasing and decreasing the speed of one rotor, roll angle can be
controlled. Figure 2.3 shows the free body diagram of the brush less D.C motor. At the electrical
side of a free body, diagram stator is represented as three phase why connection and input voltage
to motor is given to by why connection network.

Figure 2. 3 Block diagram of the rite rotor with brushless motor[3]
At other side torque which is produced by the motor is represented by the upwards arrow. Load is
the wing of the rotor .which has a angular displacement and friction.

2.4 Mathematical model
Mathematical model of the brushless DC motor is almost same as brushed DC motor.
Only by considering one condition of which phase is affecting the output of BLDC. The phase only
affected the resistive and the inductive load of the BLDC[3].
The transfer function for DC motor is:
G(s) =

ఠ೘
௏ೞ

= ௦మ ௃௅ା௦௄

௄೟

೑ ௅ା௦ோ௃ା௄೑ ோା௄೐ ௄೟

Where,
ܶ௘ =an electrical tourqe
‫=ܥ‬a friction constent
J=rotor innertia
߱௠ =the angular velocity
ܶ௟ =the suspended mechanical load

13


Where the electrical tourqe and the back emf is written as
e=݇௘ ߱௠ and ܶ௘ =݇௧ ߱௠
where ݇௘ is back emf constant and ݇௧ ݅‫ݐ݊ܽݐݏ݊݋ܿ ݁ݍݎݑ݋ݐ ܽ ݏ‬
G(s) =

ఠ೘
௏ೞ

=

௄೟

௦మ ௃௅ା൫௄೑ ௅ାோ௃൯௦ା௄೑ ோା௄೐ ௄೟

Considering the following assumption.
1. Due to brushless in the nature friction constent is small for the brushless motor, ‫ܭ‬௙ tends to
zero.
2. Rj>>‫ܭ‬௙ L
3. ݇௘ ݇௘ >>‫ܭ‬௙ ܴ
So, the final transfer function is written is
G(s) =

ఠ೘
௏ೞ

௄

೟
= ௦మ ௃௅ାା௦ோ௃ାା௄

೐ ௄೟

By multiplying the top and bottom equation by:
ܴ
1
∗
‫ܭ‬௘ ‫ܭ‬௧ ܴ
Finally above equation became:
G(s) =

ఠ೘
௏ೞ

=

భ
಼೐
ೃ಻
ಽ
ೃ಻
௦మ
∗ ା௦
ାଵ
಼೐ ಼೟ ೃ ಼೐ ಼೟

From the above equation mechanical time constant and electrical time constant is
߬௠ =

ܴ‫ܬ‬
‫ܭ‬௘ ‫ܭ‬௧

߬௘ =

‫ܮ‬
ܴ

So above equation became
G(s) =

ఠ೘
௏ೞ

= ௦మ ∗ఛ

భ
಼೐

೘ ∗ఛ೐ ା௦∗ఛ೘ ାଵ

Construction of the D.C brushless motor will affect only mechanical and the electrical co offecient
of the D.C motor transfer function
So mechanical and the electrical time constent for D.C brushless motor is
߬௠ = ∑

ܴ‫ܬ‬
‫ܴ∑ܬ‬
=
‫ܭ‬௘ ‫ܭ‬௧ ‫ܭ‬௘ ‫ܭ‬௧

߬௘ = ∑

‫ܮ‬
‫ܮ‬
=
ܴ ∑ܴ

Due to symmetrical and three phase arrangement

14


߬௠ =
߬௘ =

(3ܴ)‫ܬ‬
‫ܭ‬௘ ‫ܭ‬௧
‫ܮ‬
3∗ܴ

Therefore the equation for the brushless motor is obtain by
G(s) =

2.3

ఠ೘
௏ೞ

= ௦మ ∗ఛ

భ
಼೐

೘ ∗ఛ೐ ା௦∗ఛ೘ ାଵ

Assumptions of motor

Assumptions are taken from the Data sheet of the EC45 flat 45mm brushless motor.
Table 2. 1 Parameters of motor
Motor Data

Unit

Value

1

Nominal voltage

V

12.0

2

No load speed

rpm

2500

3

Nominal speed

rpm

1200

4

Nominal torque

mNm

30

5

Terminal resistant phase to phase

Ω

4.5

6

Terminal impedance phase to phase

H

3

7

Torque constant

mNm/A 2

8

Mechanical time constant

ms

2.25

9

Rotor inertia

gܿ݉ଶ

5

2.4 Transfer function based assumed parameter
By substituting the values to the general transfer function for the BLDC motor. The electrical
time constant

߬௘ =

‫ܮ‬
3∗ܴ

Where,
R=Rɸ=4.5Ω
‫ܬ‬௥௢௧௢௥ =5 gܿ݉ଶ =5*10ିଷK݃݉ଶ
߬௠ =2.25 ms
L=3H
߬௘ =

3
3 ∗ 4.5

15


߬௘ = 0.2222
For mechanical time constant
߬௠ =
‫ܭ‬௘ =

(ଷோ)௃
௄೐ ௄೟

=2.25

(3ܴ)‫ି01 ∗ 5 ∗ 5.4 ∗ 3 ܬ‬ଷ
=
= 15
߬௠ ‫ܭ‬௧
2.25 ∗ 2

There for final transfer function is
‫= )ݏ(ܩ‬

0.0666
0.5‫ ݏ‬ଶ + 2.25ܵ + 1

Above transfer function is for the stable condition of the MICAV. For unstable condition if the roll
angle of the rotor with respect to ground axis is 30 degree at right hand side, so speed of the right
rotor should be increased by some factor of roll angle. The speed of motor is depend upon the input
voltage .Assumed relation between the roll angle, rpm to overcome roll angle and voltage to
achieve rpm is given by table 2.2.In the case of 30 degree roll angle with respect to right rotor,
speed of the right motor should be increased by the 30 rpm to overcome roll angle. To achieve this
rpm input voltage should increased by 3.
Table 2. 2 Relation between roll angel and input voltage
Roll angle

Voltage to achieve rpm

10

10

1

20

20

2

30

30

3

40

40

4

50

50

5

60

2.5

Rom to overcome roll angle

60

6

Mathematical model simulation and analysis
Simulation of the open-loop transfer function and step response of right side motor is show

by figure 2.4.From the plot , if 1V is applied to the system the motor can only achieve 0.7 rad/sec. It
also takes 8.9 second to reach its final speed (steady state). Amount of overshoot in step response is
0%.Required overshoot should less than 10%, steady state time is 2 sec and peak overshoot is less
the 10 %.PID controller is required to regulate the input of system for which it responds under
require parameter. In this condition of the motor time taken by to MICAV to recover the roll angle
is
ଷ଴∗଼.ଽ ௥௣௠∗௦௘௖

T=

଺.଻௥௣௠

= 39 ‫ܿ݁ݏ‬

16


1 rad/sec =9.6 rpm, so speed achieve by motor is by applying 1 V is 0.7 ras/sec=6.7 rpm[1].

2.6 Design and implementation of PID controller
PID controller is added to the roll control system to get the desired output response. First,
only the proportional control (Kp) in the controller is considered. The closed-loop transfer function
of the roll control system with a proportional control is obtained, as.

Figure 2. 4 Step response of roll control system
‫= )ݏ(ܩ‬

0.5‫ ݏ‬ଶ

0.0666 ∗ ‫ܭ‬௣
+ 2.25‫ܭ + 1( + ݏ‬௣ )

Initially the proportional controller is set at 150.Figure 2.5 shows the response of the system
using only proportional controller at kp=145.Graph shows that peak overshoot is at 0% and
settling time is also increased by the around 16 second.
To achieve desire performance of the roll control system integral controller is being
designed with the proportional controller. The response of close loop transfer function of the roll
control system is shown in the figure2.6. In the plot, Overshoot is 1% that is under required range.
Settling time is approximately 14 seconds that is doesn’t match specifications. Match system
performance with the requirement PID controller is designed.

17


Figure 2. 5 Close loop response of roll control system with Kp

Figure 2. 6 Close loop response roll control system with Kp and Ki
Proportional control (Kp), the integral control (Ki) and the differential control in the controller are
considered in the roll control system. The closed-loop transfer function of the roll control system
with the PID controller is obtained, as
‫ܭ‬ௗ ܵ ଶ + ‫ܭ‬௣ ‫ܭ + 6660.0( + ݏ‬௜ )
‫= )ݏ(ܩ)ݏ(ܪ‬
(‫ܭ‬ௗ + 0.5)ܵ ଶ + (2.25+‫ܭ‬௣ )ܵ + 1 + ‫ܭ‬௜
18


0.4541ܵ ଶ + 9.729‫214.3 + ݏ‬
‫= )ݏ(ܩ)ݏ(ܪ‬
0.5ܵ ଷ + 2.704ܵ ଶ + 10.73ܵଵ + 3.412
Figure 2.7 shows the response of the roll control system using PID controller with values of
kp=145.9316,Ki=51.1785 and Kd= 6.8114. Graph shows that at this values of the PID controller
peak overshoot is deceased to the 8.8% And settling time is also decreased by the around 2 second.
By comparing the step response open loop transfer function without PID with step response of close
loop transfer function with PID controller, 1V is directly given to that right motor without PID
controller so motor it will rotates at the constant speed of 0.7rad/sec. To achieve this speed motor
will take 8.9 sec . At this same input condition with PID controller same motor will rotate at 1
rad/sec. To achieve this speed same motor will take 2 sec. After PID time taken to recover 30
degree roll angle is

૜૙∗૛ ࢘࢖࢓∗࢙ࢋࢉ

T=

ૢ.૞࢘࢖࢓

= ૟. ૜ ࢙ࢋࢉ.

1 rad/sec =9.6 rpm

Figure 2. 7 Close loop response roll control system with Kp,Ki and Kd

2.7 Testing of PID controller
Testing of PID control is being done by providing the step input to the controller. A figure 2.8 show
the output of the output of the analog PID controller .it can observed from figure 2.8 which has
setting time 11.8 seconds .
Note: The PID transfer function is not proper. Matlab will not allow this. By switching the
numerator and denominator the step command can be fooled into giving the right answer.

19


Figure 2. 8 Testing of PID

2.8 Conclusion
On the basis of above analysis of responses I can conclude that, by using the PID controller time
taken to the recover roll angle can be optimized. By combining of the two unstable system one
stable system can be possible.

20


PART-C
CHAPTER 2 Digitalization of PID
________________________________________________________________________________

3.1 Introduction
In this part of assignment, digitalization of the analog PID controller is done using three
methods is done and matlab/simmulink model is created. Response of the analog and digital
controller is compared. Direct form 1 and direct form 2 is created for digital controller and c
program is written.

3.2 Digitalizing of controller
1. Zero Order Hold (ZOH) method:
It is assumed that the system has a zero order hold element on the input side of the system.
This is the case when the physical system is controlled by a computer via a DA converter
(digital to analog). Zero order hold means that the physical input signal to the system is held
fixed between the discrete points of time[2].
PID controller transfer function
௄

H(s)=‫ܭ‬௣ + ௌ೔+‫ܭ‬ௗ ܵ
௄೏ ௌ మ ା௄೛ ௌା௄೔

H(s)=

ௌ

Bys subtitling values of Kp,Kd and Ki
଺.଼ଵଵସௌ మ ାଵସହ.ଽଷଵ଺ௌାହଵ.ଵ଻଼ହ

H(s)=
G(z)=[
G(z)=[

ௌ

ଵି௘ ೄ೅

G(z)=[
G(z)=[

ௌ

ଵି௘ ೄ೅

ଵ

][
][

ଵି௘ ೄ೅
ଵ

ଵି௭ షభ
ଵ


]

௦మ

଺.଼ଵଵସ

][

ଵ

଺.଼ଵଵସ

][

]

ௌ

ଵ

+

+

ଵସହ.ଽଷଵ
ௌ

ଵସହ.ଽଷଵ௭
(௭ିଵ)భ

+

ହଵ.ଵ଻଼ହ

+

ହଵ.ଵ଻଼ହ்௭

ௌమ

]

(௭ିଵ)మ

]

At T=1
G(z)=[

G(z)=[

ଵି௭ షభ
ଵ

଺.଼ଵଵସ

][

ଵ

+

ଵସହ.ଽଷଵ௭
(௭ିଵ)భ

+

ହଵ.ଵ଻଼ହ௭
(௭ିଵ)మ

]

௭ିଵ ଺.଼ଵଵସ൫௭ మ ିଶ௭ାଵ൯ାଵସହ.ଽଷଵ൫௭ మ ି௭൯ାହଵ.ଵ଻଼ହ௭
௭

][

(௭ିଵ)మ

]

21

ଵହଶ.଻ସଶସ൫௭ మ ൯ିଵ଴଼.ଷ଻ହଷ(௭)ା଺.଼ଵଵସ

௒(௭)

G(z)= ௑(௡) = [

௭ మ ି௭

]

By multiplying denominator and numerator with 1/‫ ݖ‬ଶ ,
(z)=

௒(௡)

௑(௡)

ଵହଶ.଻ସଶସାଵ଴଼.ଷ଻ହଷ൫௭ షభ ൯ା଺.଼ଵଵସ௭ షమ

=[

ଵି௭ షభ

]

By cross multiplying,
Y(n)-Y(n-1)= 152.7424ܺ(݊) + 108.3753ܺ(݊ − 1) + 6.8114ܺ(݊ − 2)
Y(n)=152.7424ܺ(݊) + 108.3753ܺ(݊ − 1) + 6.8114ܺ(݊ − 2) + ܻ(݊ − 1)
2

Bilinear transformation:
This method is based on an integral approximation where the integral is interpreted as the
area between the integrand and the time axis, and this area is approximated with trapezoids.
This is the Tustin’s method .but with a modification so that the frequency response of the
original continuous-time system and the resulting discrete-time system has exactly the same
frequency response at one or more specified frequencies [2].

H(s)=

ௌ

ଶ ௓ିଵ

S=்(௓ାଵ),Sampling time T=1;
ଶ ௓ିଵ

H(்(௓ାଵ)) =

ସ∗଺.଼ଵଵସቀ

ೋషభ మ
ೋషభ
ቁ ାଵସହ.ଽଷଵ଺∗ଶ(
)ହଵ.ଵ଻଼ହ
ೋశభ
ೋశభ
ೋషభ
ଶ(
)
ೋశభ

௒(௭) ଷ଻଴.ଶହ଻ଷ௭ మ ାହ଻.଼ହହ଼௭ିଶଵଷ.ସ଴ଽଵ

=

ଶ௭ మ ିଵ

௑(௭)

3.

Backward differentiation method

H(s)=

ௌ

௭ିଵ

S=

்

Sampling time T=1;
H(

௓ିଵ
ଵ

)=
H(

଺.଼ଵଵସ((௭)మ ିଶ௭ାଵ) ାଵସହ.ଽଷଵ଺∗ଶ(௭ିଵ)ାହଵ.ଵ଻଼ହ
௭ିଵ

௓ିଵ
ଵ

)=

଺.଼ଵଵସ((௭)మ ାଵଷଶ.ଷ଴ଶ଺௭ି଼ଵ.ଵଶସଵ
௭ିଵ

3.3 Modeling of digital controller
Zoh method is chosen for matlab modeling of the PID controller. By step excitation of PID
system alone the response of the filter is shown by figure 3.1. Graph shows the open loop response
of the PID controller. Due to open loop in nature amount of overshoot is infinity. Settling time of
the response is 12 seconds.

22


Figure 3. 1 Step response of controller

3.4 Comparison of analog and digital controller responses
Figure 3.2 show the comparison of the step response of the analog and digital controller which is
developed in the previous section.

Figure 3. 2 Responses of analog and digital controller
From figure, settling time of analog controller is 11.2 second with final value of 0.3 but in the case
of the digital controller it is being increased by the 12 second with the final value of 0.3. Settling

23


time of the analog controller is 11.2 second, for digital controller settling time is increased by
0.8.Rise time of the analog controller 0.000541 second but for digital controller rise time is 0
second. Peak amplitude of the analog controller is 0.0065 with occurs at the time of 0.197 seconds
but for digital system peak amplitude is 0.0049 second which occurs at 1 second.

3.5

Implementation of digital controller in c

As per z-transform equation derived by the zoh method c algorithm has been created. filter equation
is Y(n)=152.7424ܺ(݊) + 108.3753ܺ(݊ − 1) + 6.8114ܺ(݊ − 2) + ܻ(݊ − 1) .
****************************************************************************
#include<stdio.h>
Void main(void)
{
int input[]={1,1,1,1,1,1};
int output[];

// input sample string
// filtered output sample string

int i;
for(i=0;i<=size of(input);i++)
{
If(i>=2)
{
output[i]=152.7424*input[i]+108.3753*input[i-1]+6.8114 input[i-2]+output[i-1];
Elseif(i=1)
{
Output[i]=152.7424*input[i]+108.3753*input[i-1]+output[i-1];
Elseif(i=0)
{
Output[i]=152.7424*input[i]
}}}}
At the end output is ={152.7424,413.8601,681.7892,949.7183,1217.6474,1485.5765}

3.6

Test cases

There are three input condition is possible for filter .Based on the input condition there are three
test cases are designed.
1. Input of step signal
By providing the step input to the filter, the input string will be input={0 ,1 ,1, 1, 1, 1}.

24


Figure 3.1 shows the response of the filter by inserting step input.
2. Input of ramp signal
By providing the ramp input to the filter, the input string will be input={01,2,3,4,5,6}.
Figure 3.3 shows the open-loop response of the PID exited with ramp signal.

Figure 3. 3 Response Ramp exited digital PID
3. Input of impulse signal
By providing the step input to the filter, the input string will be input=1.
Figure 3.4 shows the open-loop response of the PID exited with impulse signal.

Figure 3. 4 Response of Impulse exited digital PID

25


3.7 Responses of two implemented controller structures
Direct form 1 and the direct form2 is a different pictorial representation of the filter equation.
Digital PID structure is implemented on simmulink to develop direct form 1 and direct form
2.Fdatool is used to get direct form 1 and direct form2.figure 3.3 shows the configuration of the
filter parameter in fdatool. The output of the fdatool is simmulink subsystem with direct form 1 and
direct form 2.

Figure 3. 5 Configuration of filter parameter in fdatool

Figure 3. 6 Step exited direct form 1

26


Figure 3. 7 Step exited direct form 2
Figure 3.3 and 3.4 shows the direct form realization of digital PID which is generated by fdatool.
By exiting this filter realization with unit step, output of the system is shown by graph in figures.
Left graph shows input of the digital filter as the right once represents the output response of the
filter.

3.8 Conclusion
By doing above work i can conclude that, by converting the analog PID into the digital the overall
response of the system will almost be same. Accuracy of the digital resource is gained by
decreasing the sampling time.

27


CHAPTER 4
Module learning outcomes
______________________________________________________________________________

4.1 Learning outcome
The module was well taught by module leader. This module helped me to understand the
following things.

•

To identify free body diagram of system on the basis of FBD derivation of mathematical
model of system

•

Analyzing the time and frequency response of the system

•

To design of PID controller to obtain desire response of system

•

To convert the analog PID in to digital by z-transform and analyze the response of the same

•

To make efficient use of matlab/simmulink for control theory

28


REFRENCE
________________________________________________________________________________
[1] http://www.convertunits.com/from/radian/second/to/RPM
[2] http://techteach.no/publications/articles/discretization/discretization.pdf
[3]http://amrita.academia.edu/AdityaSreekumar/Papers/590760/Design_and_Implementation_of_th
e_Closed_Loop_Control_of_a_Quad_Rotor_UAV_for_Stability
[4] http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA521374
[5] http://illumin.usc.edu/162/the-quadrotors-coming-of-age/
[6] www.ijcaonline.org/volume11/number10/pxc3872181.pdf
[7] http://www.winlab.rutgers.edu/~samar/public/mimo_sensing_aces05.pdf

29


BIBILOGRAPGY
________________________________________________________________________________
1. Viswanath K Reddy, Embedded Control System, Course Notes – M.S.Ramaiah School of
Advanced Studies, Bangalore, Feburary, 2011
2. www.mathworks.com
3. control system engineering by Norman Nise

30


Appendix
________________________________________________________________________________
%% for step respounce of open loop system
close all
clear all
J=5;
p=3;
Kt=2;
R=4.5;
L=3;
tm=2.25;
te=L/(p*R);
Ke=(3*R*J)/(tm*Kt)
num=1/Ke
den=[tm*te tm 1]
a=tf(num,den)
//openloop TF of motor
figure(1)
step(num,den)
// Step respounce of motor
title('open loop transfar function')
grid on
%%% for PID controller
close all
clear all
J=5;
p=3;
Kt=2;
R=4.5;
L=3;
tm=2.25;
te=L/(p*R);
Ke=(3*R*J)/(tm*Kt)
num=1/Ke
den=[tm*te tm 1]
a=tf(num,den)
Kp=145.9316;
Ki=51.1785;
Kd=6.8114;
figure(2)
po=tf([Kd, Kp, Ki],[1 0]);
// TF of PID
s=feedback(a*po,1)
step(s)
// step resource of PID
title('PID Control with small Ki, Kd and Kp')
%%% plotting of analog and digital controller respounce
Kp=145.9316;
Ki=51.1785;
Kd=6.8114;
figure(1)
step([1 0],[Kd Kp Ki])
title(' respounce of Analog controller')
po=tf([Kd, Kp, Ki],[1 0]);
[denasz,numasz]=c2dm([1 0],[Kd, Kp, Ki],1,'zoh')
// z-transform of PID
figure(2)
dstep(denasz,numasz,101)
[x1]=dstep(denasz,numasz,101)
figure(3)
t=0:.12:12
stairs(t,x1)
// steap ounce of digital PID
title('respounce of Digital controller')
%%%%%

31

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