This document describes the design of a pitch controller for a Boeing aircraft. It provides the mathematical model and transfer function for the aircraft's pitch dynamics. Five different controller designs are analyzed: 1) Digitized PID, 2) Direct method using closed-form equations, 3) Direct method using Diophantine equations, 4) Pole placement using Ackerman's formula, and 5) Optimal control. For each design, the response is simulated and analyzed against requirements for overshoot, rise time, settling time, and steady-state error.

1. AIRCRAFT PITCH
EECE 682
Computer Control Of Dynamic System
Project Report
Boeing Aircraft- Pitch Controller
Example: Dynamics, Modeling, Simulation, Analysis
Instructor:
Dr. Adel Ghandakly
Dept. Electrical and Computer Engineering
California State University, Chico
Submitted By:
Nasser Al Ahbabi
AIRCRAFT PITCH

2. BOEING AIRCRAFT- PITCH CONTROLLER
Example: Dynamics, Modeling, Simulation, Analysis
by
Nasser Al Ahbabi
California State University, Chico.
NOVEMBER 2014
Abstract
Though airplane has a number of important factors, its stability
and control is a key design parameter that must be met. In an
airplane
the stability is defined in three angles i.e. pitch, yaw, and roll.
In this paper I have focused on the pitch. The system transfer
was

3. AIRCRAFT PITCH
obtained through analyzing the various parameter involved in
the pitch control. In all the designs, I considered the design
parameter
requirements i.e. the percentage overshoot, steady state error,
settling time, and rise time of Boeing aircraft. The designs of
pitch
controller using various techniques have been implemented on
the system transfer function. I have provided an extension of to
these
techniques by using MATLAB/Simulink models that plays an
important role in monitoring the results of designed controllers.
In
addition, I have also provided a descriptive analysis of the
system response to the designed controllers and their
conclusions.
Keywords: Aircraft, Pitch, Ackerman, Digitized PID,
Diophantine, Optimal Control, Controller , Simulink and
MATLAB design.
CONTENTS:

4. INTRODUCTION
� INTRODUCTION
MATHEMATICAL MODEL
� BOEING AIRCRAFT: PHYSICAL SETUP AND SYSTEM
EQUATIONS
� TRANSFER FUNCTION AND STATE-SPACE MODEL
� DESIGN REQUIREMENTS:
CONTROLLER DESIGN
AIRCRAFT PITCH
� DESIGN 1 : DIGITIZED PID
� DESIGN 2 : DIRECT METHOD ( CLOSED FORM)
� DESIGN 3 : DIRECT METHOD ( DIOPHANTINE)
� DESIGN 4 : POLE PLACEMENT (ACKERMAN’S
FORMULA)
� DESIGN 5 : OPTIMAL CONTROL
CONCLUSION
REFERENCES

5. 1. INTRODUCTION
Aircrafts are perfect and good examples of a Controller system.
They possess unique characteristics that make their controller
design a
more challenging problem. On linearization of the model we can
attain results with simplified controller designs.
Major parameter in the design of aircrafts entails the horizontal
speed, pitch control and the throttle. The throttle controls the
main
motor revolutions per minute; the pitch controls the magnitude
of the motor thrust. There are two inputs that are independent;
the
longitudinal input and the lateral cyclic input. These controls
An aircraft in flight is free to rotate in three dimensions: pitch,
nose up or down about an axis running from wing to wing, yaw,
nose
left or right about an axis running up and down; and roll,
rotation about an axis running from nose to tail. In this project,
only one
dimension is considered. These two inputs control the angles of
roll and pitch.
In this project, I have used MATLAB based approach for
simulation and design by applying four various controller
design techniques;
Digitized PID, Direct Methods (Closed form and with
Diophantine) and pole placement (Ackerman's Formula) and
Optimal Control. I
AIRCRAFT PITCH

6. used Simulink in modeling these controller designs on the
system transfer function for Direct method (Diophantine) and
Optimal
Control. I analyzed their response as per the design
requirements.
2. MATHEMATICAL MODEL
2.1 Boeing Aircraft: Physical setup and system equations
The equations governing the motion of an aircraft are a very
complicated set of six nonlinear coupled differential equations.
However,
under certain assumptions, they can be decoupled and linearized
into longitudinal and lateral equations. Aircraft pitch is
governed by
the longitudinal dynamics. In this example we will design an
autopilot that controls the pitch of an aircraft.
The basic coordinate axes and forces acting on an aircraft are
shown in the figure given below.
AIRCRAFT PITCH
Assuming that the aircraft is in steady-cruise at constant
altitude and velocity; thus, the thrust, drag, weight and lift

7. forces balance each
other in the x- and y-directions. We will also assume that a
change in pitch angle will not change the speed of the aircraft
under any
circumstance (unrealistic but simplifies the problem a bit).
Under these assumptions, the longitudinal equations of motion
for the
aircraft can are:
ά=μΩσ [-(CL +CD) α + 1 q – (CW sinϒ) θ+CL]
(μ-CL)
q·=μΩ [[CM-η(CL +CD)] α
+[CM(1-μCL)]q+ (ηCW sinϒ)δ )
2iyy
θ·=Ωq
Where;
α = Angle of attack.
q=Pitch rate.
θ=Pitch angle.
δ=Elevator deflection angle.
μ=ρSḉ
4m
ρ= Density of air.
s= Platform area of the wing.
ḉ=Average chord length.
m=Mass of the aircraft.
Ω=2U
ḉ
U=Equilibrium flight speed.
CT = Coefficient of thrust.
CD=Coefficient of drag.
CL= Coefficient of lift.
CW= Coefficient of weight.
CM= Coefficient of pitch moment.
ϒ= Flight path angle.

8. σ=Constant.
iyy = Normalized moment of inertia.
η=μσCM=Constant.
In this system, the input will be the elevator deflection angle δ
and the output will be the pitch angle θ of the aircraft.
2.2 Transfer function and state-space model
The linearized equations governing the motion of a Boeing's
commercial aircraft are given by;
dα(t) =ά=-0.313α(t) +56.7q(t) +0.232δ(t)
AIRCRAFT PITCH
dt
dq(t) = q·=-0.0139α(t) - 0.426q(t) +0.0203δ(t)
dt
dθ(t) = θ·=56.7q(t)
dt
Transfer Function
The transfer function of the above system is obtained by taking
the Laplace transform of the above modeling equations
assumimg zero
initial conditions. This gives;
sA(s)=-0.313A(s) +56.7Q(s) +0.232∆(s)
sQ(s)=-0.0139A(s) -0.426Q(s) +0.0203∆(s)
sθ(s)=56.7Q(s)
The open-loop transfer function obtained by carrying about few
steps of algebra on the above equations giving;

9. P(s)=Θ(s) = 1.151s + 0.1774
∆(s) s3 + 0.739s2 + 0.921s
State Space
The above equation can be written as matrices as;
AIRCRAFT PITCH
2.3 Design requirements:
In this project, I chose some design criteria in which I designed
a feedback controller in response to a step command of pitch
angle.
The actual pitch angle overshoots less than 10%, has a rise time
of less than 2 seconds, a settling time of less than 10 seconds,
and a
steady-state error of less than 2%. For example, if the reference
is 0.2 radians (11 degrees), then the pitch angle will not exceed
approximately 0.22 rad, will rise from 0.02 rad to 0.18 rad
within 2 seconds, will settle to within 2% of its steady-state
value within 10
seconds, and will settle between 0.196 and 0.204 radians in
steady-state.
In summary, the design requirements are the following.
x Overshoot less than 10%
x Rise time less than 2 seconds
x Settling time less than 10 seconds
x Steady-state error less than 2%

10. AIRCRAFT PITCH
3. THE CONTROLLER DESIGN
The figure above shows a simple pitch controller design block
model for the Boeing Aircraft. Taking the system transfer
function
derived above I will be designing the controller through the
following various techniques;
� Digitized PID
� Direct Methods (Closed Form and with Diophantine)
� Pole Placement (Ackerman’s Formula)
� Optimal Control
AIRCRAFT PITCH
3.1) AIRCRAFT PITCH: THE DIGITIZED PID
CONTROLLER:

11. A proportional-integral-derivative controller (PID controller) is
a control loop feedback mechanism (controller) widely used in
controlling systems in industries. A PID controller calculates an
error value as the difference between a measured process
variable and
a desired set point. The controller attempts to minimize the
error by adjusting the process through use of a manipulated
variable. A
block diagram of a PID controller in a feedback loop is as
shown below
AIRCRAFT PITCH
The closed-loop transfer function of the system with a PID
controller is:
X(s) = Kds2 + Kps + Ki .
F(s) s3 + (10 + Kd)s2 + (20 +Kp)s + Ki
The effects of each of controller parameters Kp, Kd and Ki, on
a closed-loop system are summarized in the table below.
CL RESPONSE RISE TIME OVERSHOOT SETTLING TIME S-
S ERROR
AIRCRAFT PITCH

12. After several trial and error runs, the gains Kp =350, Ki=300,
and Kd= 50 provided the desired response giving the following
step
response;
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Step Response
Time (seconds)
Am
pl
itu
de
Kp Decrease Increase Small Change Decrease
Ki Decrease Increase Increase Eliminate
Kd Small Change Decrease Decrease No Change

13. AIRCRAFT PITCH
Increasing the derivative gain Kd in a PID controller often helps
reduce overshoot. Adding a derivative control reduces the
oscillation
in the response a sufficient amount that the other gains can be
increased to reduce the settling time.
C(s)= 4.4545 (1 + 0.22s + 1.1s2) ≈ 4.45 + 0.98 + 4.90s
s s
This transfer function is a PID compensator with Ki = 4.45, Kp
= 0.98, and Kd = 4.90. The resulting closed-loop step response
is
shown below.
AIRCRAFT PITCH
This response meets all of the requirements except for the settle
time which at 12.6 seconds is a little larger than the given
requirement
of 10 seconds. The proportional gain was increased in order to
reduce the system's settle time. Increasing Kp means that I may
no
longer achieve the minimum possible performance metric,
however, i did that in order to decrease the resulting settle time.
Specifically, l changed Kp so that it equals 2. The resulting PID
controller is shown below.

14. AIRCRAFT PITCH
The PID controller response shown below meets all of the given
requirements as summarized below.
� 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.
AIRCRAFT PITCH
PID controller Response
AIRCRAFT PITCH
3.2) AIRCRAFT PITCH: DIRECT METHOD - CLOSED
FORM:
This method is a direct analytical technique for design. In order
to stabilize a control system and eventually meet the given
design
requirements, a feedback controller is added and hence we have
both the forward and feedback path controllers. The figure
below
illustrates the control architecture we will employ.

15. P(s)=Θ(s) = 1.151s + 0.1774
∆(s) s3 + 0.739s2 + 0.921s
The closed-loop transfer function for the above with the
controller C(s) simply set equal to one can be generated using
the MATLAB
command feedback.
I scaled the response to model the fact that the pitch angle
reference is a 0.2 radian (11 degree) step. Running my m-file at
the
command line produced the plot shown below in I added the
annotations for the rise time, settling time and final value.
AIRCRAFT PITCH
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.
I used the MATLAB commands pole and zero to reveal the
poles and zeros of the closed-loop transfer function (shown on
closedform.m file)
Assuming the closed-loop transfer function has the form Y(s) /
R(s), the output Y(s) in the Laplace domain is calculated as
follows
where R(s) is a step of magnitude 0.2.
Y(s) = 1.151s + 0.1774

16. R(s) s3 + 0.739s2 + 0.921s
AIRCRAFT PITCH
Y(s) = 1.151s + 0.1774 R(s)
s3 + 0.739s2 + 0.921s
Y(s) = 0.2 (1.151s + 0.1774 )
s3 + 0.739s2 + 0.921s
The above expression can be expressed as partial fraction as,
Y(s) = 0.2 _ 0.08 81 . _ 0.1121s + 0.08071
s s + 0.08805 s2 + 0.6509s + 2.015
The inverse Laplace transform of the above expression is taken
to generate the corresponding time domain expression shown
below.
y(t)=0.2 – 0.0881e-0.08805t – e-0.3255t(0.1121cos(1.3816t) +
0.0320 sin(1.3816t))
Using the above equation, I generated the graph below in matlab
AIRCRAFT PITCH
0 10 20 30 40 50 60 70
-0.05
0
0.05

17. 0.1
0.15
0.2
time (sec)
pi
tc
h
an
gl
e
(ra
d)
Closed-loop Step Response
AIRCRAFT PITCH
3.3) AIRCRAFT PITCH: DIRECT METHOD ( DIOPHANTINE)
Controller design by
Diophantine method is based on the Diophantine Equation.
When given the paramount and key polynomials D(z) and G(z)
each
having an order n and a H(z) polynomial of the order (2n-1) the
Diophantine equation is given by;

18. d(z)D(z)+g(z)G(z)=H(z)
From the mathematical modeling, the transfer function is given
as;
P(s)=Θ(s) = 1.151s + 0.1774
∆(s) s3 + 0.739s2 + 0.921s
AIRCRAFT PITCH
AIRCRAFT PITCH
3.4) AIRCRAFT PITCH: POLE-PLACEMENT ACKERMAN
The continuous-time state-space model of the aircraft pitch
dynamics was derived as;
Where the input is elevator deflection angle δ and the output is
the aircraft pitch angle θ. For a step reference of 0.2 radians,
the
following are the design criteria;
� Overshoot less than 10%

19. � Rise time less than 2 seconds
� Settling time less than 10 seconds
� Steady-state error less than 2%
Discrete state-space
In designing a control system a sampled-data model of the plant
was generated. I used MATLAB to generate this model from a
continuous-time model using the c2d command. The c2d
command requires three arguments: a system model, the
sampling time (Ts)
and the type of hold circuit. In this design I assumed a zero-
order hold (zoh) circuit
In choosing a sample time, it is desired that the sampling
frequency be fast compared to the dynamics of the system in
order that the
sampled output of the system captures the system's full
behavior, that is, so that significant inter-sample behavior isn't
missed. One
measure of a system's "speed" is its closed-loop bandwidth. A
good rule of thumb is that the sampling time be smaller than
1/30th of
the closed-loop bandwidth frequency. Thus, to be sure of a
small enough sampling time, I used a sampling time of 1/100
sec/sample.
The after running commands (shown in m-file) in MATLAB, I
obtained four matrices representing the sampled-data state-
space
model. Hence, the discrete-time state-space model is;
AIRCRAFT PITCH

20. Controllability: Before designing the controller I had to verify
the controllability of the system. For the system to be
completely state
controllable, the controllability matrix must have rank n where
the rank of a matrix is the number of independent rows (or
columns).
The controllability matrix of a discrete-time system has the
same form as a continuous-time system.
C=[|A|AB|A2B|. . .|An-1B|]
Since the controllability matrix is 3x3, its rank must be 3. The
MATLAB command rank was used to calculate the rank of a
matrix.
Adding the following additional commands in an m-file and
running in the MATLAB command window will produce the
following
output.
co = ctrb(sys_d);
Controllability = rank(co)
Controllability =
3
Therefore, the system is completely state controllable since the
controllability matrix has rank 3
The schematic of a discrete full-state feedback control system is
shown below, where q-1 is the delay operator (not the aircraft's
pitch
rate q). NB: I assumed that D = 0(Hence not shown).

21. AIRCRAFT PITCH
where
K = control gain matrix
x = [ alpha, q, theta ]' = state vector
theta_des = reference (r)
delta = theta_des - K x = control input (u)
theta = output (y)
Substituting the state-feedback law δ(k)=θdes(k) – K x(k) for
δ(k) to the state-space equations leads to the following
assuming that all
of the state variables are measured;
x(k+1)=(A-BK)x(k)+Bθdes (k)
θ(k)=Cx(k)
Linear Quadratic Regulator (LQR) method is used to find the
control matrix (K). The discrete version of the same LQR
method is
used. This type of control technique optimally balances the
system error and the control effort based on a cost specified that
defines
the relative importance of minimizing errors and minimizing
control effort. In the case of the regulator problem, it is
assumed that the
reference is zero. To use this LQR method, two parameters are
defined: the state-cost weighted matrix (Q) and the control
weighted
matrix (R). For simplicity, I chose the control weighted matrix
equal to 1 (R=1), and the state-cost matrix (Q) equal to pC'C.
By
employing the vector C from the output equation means I only
considered those states in the output in defining the cost. In this
case, θ
is the only state variable in the output. The weighting factor (p)

22. was chosen by trial and error in order to modify the step
response to
achieve the given requirements. Since we have a single input
system, R is a scalar.
AIRCRAFT PITCH
Control matrix (K) is found by employing the MATLAB
command dlqr which is the discrete-time version of the lqr
command. I
chose a weighting factor p =50. From matlab, the values of Q
and K are found as;
Q =
0 0 0
0 0 0
0 0 50
K =
-0.6436 168.3611 6.9555
The stair-step response is generated as shown below ;
AIRCRAFT PITCH
Examination of the above demonstrates that the rise time,
overshoot, and settling time are satisfactory. However, there is
a large
steady-state error but this can be corrected by introducing a
precompensator to scale the overall output.
Adding Precompensator

23. Unlike other design methods, the full-state feedback system
does not compare the output to the reference; instead, it
compares all
states multiplied by the control matrix (K x) to the reference as
shown below.
In obtaining the desired output, the reference input is scaled so
that the output does equal the reference in steady state by
introducing a
precompensator scaling factor, Nbar. The basic schematic of the
state-feedback system with scaling factor (Nbar) is shown
below.
0 1 2 3 4 5 6 7 8 9 10
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
time (sec)
pi
tc
h
an

24. gl
e
(ra
d)
Closed-Loop Step Response: DLQR
AIRCRAFT PITCH
The generated the stair-step response is;
AIRCRAFT PITCH
From the plot above, the Nbar factor eliminates the steady-state
error and all design requirements are satisfied.
0 1 2 3 4 5 6 7 8 9 10
0
0.05

25. 0.1
0.15
0.2
0.25
time (sec)
pi
tc
h
an
gl
e
(ra
d)
Closed-Loop Step Response: DLQR with Precompensation
AIRCRAFT PITCH
3.5) AIRCRAFT PITCH: OPTIMAL CONTROL
In optimal control, the given state space model is converted to a
discrete state space model. I did this using the function c2dm
that
requires 6 parameters ( A,B, C, D, the sampling time Ts and the
state space matrix). I implemented this in Matlab using the

26. function
'dlqr'.
Optimal control
AIRCRAFT PITCH
AIRCRAFT PITCH
4) CONCLUSION
Among various numbers of important factors, an airplane’s
stability and control is major design parameter that must be met.
In an
airplane the stability is defined in three angles i.e. pitch, yaw,
and roll. This paper only focused to control the pitch.
Overshoot, rise
time, steady-state error and settling time of a pitch are
minimized and regulated during the designing of the controllers
using different
methods and techniques. MATLAB played an important role in
analyzing and comparing the results of the designed controller.
I
successfully designed, tested, implemented and analyzed the
various controller design techniques on the Pitch Control of the
Boeing
Aircraft.

27. AIRCRAFT PITCH
5) REFERENCES
[1]
http://wikis.controltheorypro.com/index.php?title=Aircraft_pitc
h_Example
[2] S. Kamalasadan and A. A. Ghandakly, “Multiple fuzzy
reference model adaptive controller design for pitch-rate
tracking,” IEEE
Transactions on Instrumentation and Measurement, vol. 56, no.
5, pp. 1797–1808, 2007. View at Publisher · View at Google
Scholar ·
View at Scopus

28. [3] S. Kamalasadan and A. A. Ghandakly, “A neural network
parallel adaptive controller for fighter aircraft pitch-rate
tracking,” IEEE
Transactions on Instrumentation and Measurement, vol. 60, no.
1, pp. 258–267, 2011. View at Publisher · View at Google
Scholar ·
View at Scopus
[4] S. E. Talole, A. Ghosh, and S. B. Phadke, “Proportional
navigation guidance using predictive and time delay control,”
Control
Engineering Practice, vol. 14, no. 12, pp. 1445–1453, 2006.
View at Publisher · View at Google Scholar · View at Scopus
[5] M. Pachter, P. R. Chandler, and L. Smith, “Maneuvering
flight control,” Journal of Guidance, Control, and Dynamics,
vol. 21, no.
3, pp. 368–374, 1998. View at Scopus