PRÁCTICA 1 (2017)
Apunte InglésUniversidad  Universidad Politécnica de Cataluña (UPC) 
Grado  Ingeniería de Aeronavegación  3º curso 
Asignatura  Control y Guiaje 
Año del apunte  2017 
Páginas  10 
Fecha de subida  25/06/2017 
Descargas  2 
Subido por  areig 
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MATLAB TUTORIAL #1:
Study of transfer functions using
Matlab
CONTROL AND GUIDANCE
Alba Martín, Anna Reig  6GX32  UPC
INDEX
1
BUSINESS JET
The Business Jet flies in cruise at 40000ft, with a 400kt velocity (U0). Its transfer functions
for the velocity and the angle of attack are:
𝑢 (𝑠) −379𝑠 2 + 271888𝑠 + 24033
=
𝛿𝑒 (𝑠)
∇
3
(
)
∝ 𝑠
−42𝑠 − 11939𝑠 2 − 89𝑠 − 79
=
𝛿𝑒 (𝑠)
∇
With ∇ = 676𝑠 4 + 1359𝑠 3 + 5540𝑠 2 + 57𝑠 + 46
1. Poles of the aircraft
To know the stability of a system, we have to focus our attention on its poles. From the
theory, it is known that when the poles of a system have negative real part, the system is
dynamically stable. Otherwise, the system is unstable. However, if the system only has
imaginary poles, its behaviour corresponds to the one of an oscillator. As it can be seen,
the zeros of a system don’t influence its stability.
As the denominator is the same for the velocity and for the angle of attack, the poles are
going to be equals. Using Matlab, these are the values found:
Poles: P1: 0.0041 + 0.0912i system dynamically stable
P2: 0.0041  0.0912i system dynamically stable
P3: 1.0010 + 2.6774i system dynamically stable
P4: 1.0010  2.6774i system dynamically stable
This leads to say that the system characterizing the velocity as well as the one for the
angle of attack are dynamically stable.
2. Modes, time constant τ and damping factor ζ
As before, the time constant τ and damping factor ζ only depend on the denominator so
their values are going to be the same for the velocity and for the angle of attack and thus
the derivate modes from these ones.

Time constant
241.57s (from pole 1 and 2) and 0.9990s (from pole 3 and 4)
Damping factors
0.0454 (from pole 1 and 2) and 0.3502 (from pole 3 and 4)
Modes
From the pole 1 and 2, the obtained time constant (241.57s) and the damping
factor (0.0454) lead to have a system with a high frequency and a low damping
factor for a really short time. This means it is difficult to know its existence and
that the damping happens in a fast way without effort from the point of view of
the pilot. So, the system is in Short Periode mode.
2
On the contrary, for the poles 3 and 4 it is obtained a low frequency time constant
(0.9990s) and a high damping factor (0.3502) which mean having a Pughoid mode.
As the sinusoid tends to decrease but does not mitigate, due to the high value of
the time constant and the damping factor, the airplane tends to a sinusoidal flight.
Consequently, the Phugoid mode is the one that remains and dominates.
3. PoleZero Map

Velocity
Zeros: 717.4709 and 0.0884
Poles: 0.0041 + 0.0912i
0.0041  0.0912i
1.0010 + 2.6774i
1.0010  2.6774i
The splane obtained for the velocity is the following one. As it can be appreciated,
the poles are on the left side s plane which means having a stable system.
Figure 1. Poles and zeros for velocity function

Angle of attack
Zeros: 284.2545
0.0037 + 0.0813i
0.0037  0.0813i
Poles: 0.0041 + 0.0912i
0.0041  0.0912i
1.0010 + 2.6774i
1.0010  2.6774i
3
The splane obtained for the angle of attack is the next one. As it can be
appreciated, the poles are on the left side splane which means having a stable
system.
Figure 2. Poles and zeros for the angle of attack
Consequently, these two systems describing the aircraft behaviour allow us to say that
as both are stable systems, the aircraft will be stable as well.
4. Displacement of 2º of the elevator, the aircraft response
In order to particularly analyse the stability of the aircraft, the system is studied for a
displacement of 2º of the elevator. It is needed to differentiate the response of the velocity
and the angle of attack as shown below.
Figure 3. Business Jet responses for a displacement of a 2º of the elevator
4
For both, the velocity and the angle of attack the system response is underdamped,
this is beneficial for a business jet because it is more comfortable for the passengers
as initial oscillation vanishes with time. However, its manoeuvrability is reduced which
incapacitates the aircraft to abruptly change its heading for instance or to translate a
quick thought (of a pilot) in an action on the plane. Finally, looking at both plots, the
phugoid mode already mentioned, can be appreciated as a sinusoid that mitigates
slowly with time. Nevertheless, the short period is that short that it is impossible to
see it as it disappears almost instantaneously.
5. Final values of angle of attack and total velocity
In order to know the final values on steady state for both variables the system will be
analysed by two different methods.

Transfer function evaluation on s=0
This first method is based on a theoretical equation studied in Control and Guidance
classes. The method consists on evaluating the transfer function on its limit at s=0,
what would correspond to steady state.
𝑆𝑦𝑠𝑡𝑒𝑚(𝑡 = ∞) = lim 𝑠 ∗ 𝑆(𝑠)
𝑠→0
For the case of velocity it is possible to obtain a final value of 18.24m/s.
𝑉 (𝑠 = 0) = 𝑠 ∗
0,0349066
−379𝑠 2 + 271888𝑠 + 24033
∗
𝑠
676𝑠 4 + 1359𝑠 3 + 5540𝑠 2 + 57𝑠 + 46
𝑉 (𝑡 = ∞) = 18.24
𝑚
+ 400
𝑠
For the case of angle of attack we obtain a final value of −3.43°.
𝛼 (𝑠 = 0) = 𝑠 ∗
0,0349066
−42𝑠 3 − 11939𝑠 2 − 89𝑠 − 79
∗
𝑠
676𝑠 4 + 1359𝑠 3 + 5540𝑠 2 + 57𝑠 + 46
𝛼 (𝑡 = ∞) = −0.0599 𝑟𝑎𝑑 → −3.43°
To know the final values of both, it is needed to multiply the transfer function per the
step 𝛿𝑒 (𝑠) that is 2º, which corresponds to 0.0349066 rad and substituting s=0.

Time constant * 5
In this case, the objective is to evaluate the final variation of both functions when time
is five times the time constant. With that we achieve a steady state that does not tend
to infinite but the function is already stable as to consider a final value.
This method has been applied with Matlab in the same code where the poles, zeros,etc
have been computed. Doing that we obtain the following values:
𝑉(5 ∗ 𝑡𝑖𝑚𝑒𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 ) = 18.3366
𝑚
,
𝑠
𝛼 (5 ∗ 𝑡𝑖𝑚𝑒𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 ) = −3.4266°
As it can be seen, by means of two different methods the same final values have been
obtained for the velocity and angle of attack, which validates our results as correct
5
JET FIGHTER
The Jet Fighter flies in cruise at 45000ft, with a 516kt velocity (U0). Its transfer functions
for the velocity and the angle of attack are:
𝑢(𝑠) −247𝑠 3 + 51𝑠 2 − 218196𝑠 − 68073
=
𝛿𝑒 (𝑠)
∇
∝ (𝑠) −35𝑠 3 + 5786𝑠 2 + 11.5𝑠 + 22.5
=
𝛿𝑒 (𝑠)
∇
With ∇ = 871𝑠 4 + 608𝑠 3 − 9065𝑠 2 − 43𝑠 − 43
1. Poles of the aircraft
As the denominator is the same for the velocity and for the angle of attack, the poles are
going to be equals. Using Matlab, these are the values found:
Poles: P1: 0.0025 + 0.0688i system dynamically stable
P2: 0.0025  0.0688i system dynamically stable
P3: 2.8994 + 0.0000i system dynamically unstable
P4: 3.5924 + 0.0000i system dynamically stable
This leads to say that the system characterizing the velocity as well as the one for the
angle of attack are dynamically unstable due to the positive real part of the third pole, that
as explained before develops on an instability.
2. Modes, time constant τ and damping factor ζ
As before, the time constant τ and damping factor ζ only depend on the denominator so
their values are going to be the same for the velocity and for the angle of attack and thus
the derivate modes from these ones.

Time constants
395.6389s and 0.2784s
Damping factors
0.0367 (from pole 1 and 2) and 1 (from pole 4)
Modes
For the first two poles we obtain a dynamically stable system within a low frequency
time constant (395.6389s) and a Phugoid mode for the damping factor (0.0367).
The third one the system is dynamically unstable as the real part of the pole is on
the RSP (Right Side Plane). Finally, the fourth develops in a dynamically stable
system with a high frequency time constant (0.2784s) and a short period mode for
the damping factor (1).
Even though most of the poles are stable, just one unstable does the system
develop unstable, that is why it is very important to maintain negative real parts if
a totally stable aircraft is desired.
6
3. PoleZero Map

Velocity
Zeros: 0.3119
0.2592 + 29.7234i
0.2592 + 29.7234i
Poles: 0.0025 + 0.0688i
0.0025  0.0688i
2.8994 + 0.0000i
3.5924 + 0.0000i
The splane obtained for the velocity is the following:
Figure 4Poles and zeros for velocity function

Angle of attack
Zeros: 165.3163
0.0010 + 0.0624i
0.0010 – 0.0624i
Poles: 0.0025 + 0.0688i
0.0025  0.0688i
2.8994 + 0.0000i
3.5924 + 0.0000i
7
The splane obtained for the angle of attack is the following:
Figure 5 Poles and zeros for the angle of attack
4. Displacement of 2º of the elevator, the aircraft response
In order to particularly analyse the stability of the aircraft, the system is studied for a
displacement of 2º of the elevator. It is needed to differentiate the response of the velocity
and the angle of attack as shown below.
Figure 6 Step response for a Fighter Jet
If we analyse the stability of a Jet Fighter it is easy to see that the system develops
unstable under the excitation, this can be beneficious because it increments the
manoeuvrability of the aircraft. This means that a little movement on surface controls
has a big impact on aircraft movement on all dimensions.
8
The instability of a fighter jet can be recovered thanks to a computer installed on the
aircraft that helps to manage control surface as well as to be able to control the aircraft in
case the pilot loses the control.
CONCLUSIONS
The main objective of this study was to determine the stability and other important
parameters in two completely different aircrafts. By means of the Matlab software and
using some simple functions, it has been possible to determine the poles and zeros, time
constants and damping factors of the transfer functions for velocity and angle of attack of
both aircraft.
Once we had all that information and the plots shown all over this report we have analysed
the different modes each aircraft could have and finally determined some conclusions.
For Business Jet, it has been demonstrated by exposing the system to an excitation that
the aircraft is dynamically stable. This is an advantage for this type of vehicles because
they are mostly designated to transport people all over the world so they do not want to
experiment any type of perturbation or instability.
About the Fighter Jet we could conclude that it is a dynamically unstable aircraft and this
is the desired. Fighters are mostly designed for wars or acrobatics, where it is very
important to have total control of the surfaces and also a quick response. This is the main
reason why this type or aircraft are already designed to be unstable, even though there is
always a computer (flybycomputer) controlling if any incidence happened in order to take
control of it.
9
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