PRÁCTICA 1 (2017)

Apunte Inglés
Universidad 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
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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. Pole-Zero 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 s-plane 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 s-plane obtained for the angle of attack is the next one. As it can be appreciated, the poles are on the left side s-plane 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. Pole-Zero 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 s-plane 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 s-plane 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 (fly-by-computer) controlling if any incidence happened in order to take control of it.
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