Applications of control theory on operating and manipulating airplane

SUBJECT: 
APPLICATIONS OF CONTROL 
THEORY ON OPERATING AND 
MANIPULATING AIRPLANE 
MAKERS 
1.Nguyễn Phạm Hồng Phúc 41102617 DD11KSTD 
2.Thái Vương Khang 41101557 DD11KSVT 
3.Nguyễn Quốc Đăng 41100788 DD11KSVT 
4.Nguyễn Trọng Ngô Nhật Du 41100531 DD11KSVT 
Group 13 (TNDD-A) 
I. Airplane- How does it work? 
II. Airplane Pitch 
III. Demos by Matlab 
Primary Contents 
I. Airplane –How does it work? 
A.How can an airplane fly 
• Newton Law 
• Bernoulli Law 
• Thrust Forces 
• Turbojet 
• Turboprop 
• Angle of attack 
B.How to control airplane 
• Pitch, Yaw and Roll Plane 
• Pitch 
• Yaw 
• Roll 
• Control Surfaces 
 Newton Law 
A. How can an airplane fly? 
According to Newton law, Enable to fly : Lift ≥ Weight 
and Thrust ≥ Drag 
But, how do we get 
the lift forces? 
 Bernoulli Law 
Bernoulli law: 
1
2
𝜌𝑣2 + 𝜌𝑔𝑧 + 𝑝 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 
ρ: specific weight of the fluid 
𝑔: 𝑔𝑟𝑎𝑣𝑖𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛,
≈ 9.81 𝑘𝑔.𝑚/𝑠2 
 z: the height of the airplane 
 p: the static pressure 
A. How can an airplane fly? 
 Bernoulli Law 
The shape must be designed so that the surface 
above and below are different => v1 > v2 => P1 < P2. 
Besides, 𝐹𝑙𝑖𝑓𝑡 = 𝑃2𝑆2 − 𝑃1𝑆1 𝑎𝑛𝑑 𝑆1 ≈ 𝑆2 → 𝐹𝑙𝑖𝑓𝑡 > 0 
A. How can an airplane fly? 
 Bernoulli Law 
If we want 𝐹𝑙𝑖𝑓𝑡to be large enough to balance the gravity 
force, S and the operating velocity are usually large 
enough 
A. How can an airplane fly? 
 Thrust forces 
But how about 
thrust forces? 
Actually, we always 
face the drag forces 
when we move in 
fluid! 
Yes, We must create 
the thrust forces to 
move the plane 
horizontally. 
A. How can an airplane fly? 
 Turbojet 
Turbojet generates the thrust forces by pushing the air 
backward. It is efficient in generating speed more than 
sonic but not in saving power. 
A. How can an airplane fly? 
 Turbojet 
The rate of flow of fuel entering the engine is very 
small compared with the rate of flow of air. If the 
contribution of fuel to the nozzle gross thrust is 
ignored, the net thrust is: 
𝑭𝒕𝒉𝒓𝒖𝒔𝒕 = 𝒎 (𝑽𝒋 − 𝑽) 
With 
 𝑚 : the rate of flow of air through the engine; 𝑉𝑗: the 
speed of the jet (the exhaust plume); 
𝑉: is the true speed of the aircraft. 
A. How can an airplane fly? 
 TURBOPROP 
A turboprop engine is a type of turbine engine which 
drives an aircraft propeller using a reduction gear. It 
generates the thrust forces by rotating propeller. It is 
efficient in saving power but not in generating speed more 
than sonic. 
A. How can an airplane fly? 
In fluid dynamics, angle of attack (AOA, or (Greek letter alpha)) 
is the angle between a reference line on a body (often the chord 
line of an airfoil) and the vector representing the relative motion 
between the body and the fluid through which it is moving. 
𝛼 𝑤𝑖𝑡ℎ ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑐𝑎𝑛 ℎ𝑒𝑙𝑝 𝑢𝑠 𝑡𝑜 𝑐𝑜𝑛𝑡𝑟𝑜𝑙 𝑙𝑖𝑓𝑡 𝑓𝑜𝑟𝑐𝑒𝑠 
 Angle of attack 
A. How can an airplane fly? 
B. How to control airplane? 
 Pitch, yaw and roll 
plane 
Body Coordinates: 
The x-Roll-axis points through the nose of the 
craft. 
The y- Pitch -axis points toward right 
The z-Yaw axis points follow the right-handed 
rule. The origin is fixed at the center of mass. 
 Pitch 
 Pitch: nose up or down about an axis running from 
wing to wing. 
 Elevators (moving flaps on the horizontal tail) 
produce pitch. 
B. How to control airplane? 
 Yaw 
Yaw: nose left or right about an axis running up 
and down. 
A rudder on the vertical tail produces yaw. 
B. How to control airplane? 
 Roll 
Roll: rotation about an axis running from nose to 
tail. 
Ailerons (moving flaps on the wings) produce roll. 
B. How to control airplane? 
Control Surfaces 
Flight control surfaces of Boeing 727 
B. How to control airplane? 
Control Surfaces 
These rotations are produced by torques (or moments) 
about the principal axes. On an aircraft, these are 
produced by means of moving control surfaces, which 
vary the distribution of the net aerodynamic force about 
the vehicle's center of gravity 
B. How to control airplane? 
II. Airplane Pitch 
A.System Modeling 
• Physical setup and System Equations 
• Transfer function and state-space models 
• Design requirements 
• MATLAB representation 
B.PID Design 
Physical setup and 
system equations 
The basic coordinate axes and forces 
A. System Modeling 
𝛼 = 𝜇Ω𝜎 − 𝐶𝐿 + 𝐶𝐷 𝛼 + 
1
𝜇 − 𝐶𝐿
𝑞 − 𝐶𝑊 sin 𝛾 𝜃 + 𝐶𝐿 
𝑞 =
𝜇Ω
2𝑖𝑦𝑦
 𝐶𝑀 − 𝜂 𝐶𝐿 + 𝐶𝐷 𝛼 + 𝐶𝑀 + 𝜎𝐶𝑀 1 − 𝜇𝐶𝐿 𝑞
+ 𝜂𝐶𝑤 sin 𝜃 𝛿 
𝜃 = Ω𝑞 
the longitudinal equations of 
motion for the aircraft can be 
written as above 
For this system, the input will be 
the elevator deflection 
angle 𝜹 and the output will be 
the pitch angle 𝜽 of the aircraft. 
 Transfer function and 
state-space models 
These values are taken from the data from one 
of Boeing's commercial aircraft 
𝛼 = −0.313𝛼 + 56.7𝑞 + 0.232𝛿 
𝑞 = −0.0139𝛼 − 0.426𝑞 + 0.0203𝛿 
𝜃 = 56.7𝑞 
A. System Modeling 
𝑠𝐴 𝑠 = −0.313𝐴 𝑠 + 56.7𝑄 𝑠 + 0.232Δ 𝑠 
𝑠𝑄 𝑠 = −0.0139𝐴 𝑠 − 0.426𝑄 𝑠 + 0.0203Δ 𝑠 
𝑠Θ 𝑠 = 56.7𝑄(𝑠) 
𝑃 𝑠 =
Θ(𝑠)
Δ(𝑠)
=
1.151𝑠 + 0.1774
𝑠3 + 0.739𝑠2 + 0.921𝑠
The Laplace transform of the above equations are shown below 
After few steps of algebra, you should obtain the following transfer 
function. 
A. System Modeling 
 Transfer function and 
state-space models 
𝛼 
𝑞 
𝜃 
=
−0.313 56.7 0
−0.0139 −0.426 0
0 56.7 0
𝛼
𝑞
𝜃
+
0.232
0.0203
0
𝛿 
 𝑦 = 0 0 1
𝛼
𝑞
𝜃
State space Equations 
 Transfer function and 
state-space models 
A. System Modeling 
In summary, the design requirements are 
the following: 
Overshoot less than 10% 
Rise time less than 2 seconds 
Settling time less than 10 seconds 
Steady-state error less than 2% 
A. System Modeling 
Design Requirements 
Closed-loop diagram 
A. System Modeling 
MATLAB representation 
Oops! In fact, the steady-state 
error appears to be driven to 
zero and there is no overshoot in 
the response, though the rise-
time and settle-time 
requirements are not met 
Yes! We need a PID controller. Then we 
will come to PID Controller Design 
A. System Modeling 
B. PID Controller Design 
we will consider the following unity 
feedback system 
The output of a PID controller, equal to the 
control input to the plant, in the time-
domain is as follows: 
𝑢 𝑡 = 𝐾𝑃𝑒 𝑡 + 𝐾𝐼 𝑒 𝑡 𝑑𝑡 + 𝐾𝐷
𝑑𝑒(𝑡)
𝑑𝑡
B. PID Controller Design 
The effects of each of controller 
parameters, 𝐾𝑃, 𝐾𝐼, and 𝐾𝐷on a closed-loop 
system are summarized in the table below. 
CL 
RESPONSE 
RISE TIME OVERSHOOT 
SETTLING 
TIME 
S-S ERROR 
Kp Decrease Increase Small Change Decrease 
Ki Decrease Increase Increase Eliminate 
Kd 
Small 
Change 
Decrease Decrease No Change 
the table should only be used as a 
reference when you are determining the 
values for 𝐾𝑃, 𝐾𝐼, and 𝐾𝐷 
B. PID Controller Design 
After that, we use PID_Design function to 
determine 𝐾𝑃, 𝐾𝐼, and 𝐾𝐷.Then we have 
two controller as below: 
The PI controller helped reduce the steady 
error in the signal more quickly. But 
overshoot increases. 
0.56
( ) 1.00C s
s
 
 PI Controller 
 PID Controller 
 The generated PID controller is shown below. 
4.45
( ) 0.98 4.90sC s
s
  
 Overshoot = 5% < 10% 
 Rise time = 1.2 seconds < 2 seconds 
 Settling time = 5 seconds < 10 seconds 
 Steady-state error = 0% < 2% 
Therefore, this PID controller will provide the 
desired performance of the aircraft's pitch. 
B. PID Controller Design 
We use the PID controller, the optimal 
values of the gains KP, KI and KD differ 
per flight . The gains then depend on 
certain relevant parameters 
B. PID Controller Design 
III.DEMOS ON MATLAB 
Summary 
• Fundamentals of control systems are helpful to 
control flight ( landing, taking off, direction 
guidance as pitch, yaw and roll) 
• Using PID to help control efficiently and exactly 
in closed-loop system. 
• In addition, we can use many other control 
methods to control big system as airplane, 
factory: LQR, Fuzzy control 
Reference 
Assoc. Prof. Dr. Huynh Thai Hoang (2012) Fundamentals of Control 
Systems Lecture Series 
Website: 
tPitch§ion=SimulinkControl 
MATLAB (R2012a) Help 
Webpage: 
maticFlightControlFullVersion.pdf 
Some pictures for illustration from the Internet. 
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Goodbye 