Ils Report v5

8/3/2019 Ils Report v5

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Numerical Methods in Aerospace Engineering 3

Instrument Landing System Lateral BeamGuidance System

Jekaterina Sazonova

1004263


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Propeller Lab Report Page 1

1 Variable Notation

Summary


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2 Abstract

Summary

3 IntroductionThe Instrument Landing guidance System is designed to help aircraft pilots align their planes

with the center of a landing line during final approach. The ground equipment of the ILS

consists of two directional transmitters that send out radio beams from either side of the

runway's centerline. The radio pulses are picked up by instruments on the plane and then

processed and converted into precise directional and altitude information. The ILS is tied into

an automatic pilot, whereby localiser and glideslope transmitters guide the plane into position

while those on the aircraft control airspeed by means of an automatic control.

Figure 1: ILS Diagram

This is the technical report that outlines the development of the state space model of the ILS

and analyses the simulation of the system for the later beam guidance. Using MATLAB and

Simulink, the code and the block diagram were generated for the simulation. The aircraft

dynamics is then considered and analysed to represent the change in the range of the aircraft.

The ILS can be used for automated landing. In order to achieve this direction, descent andspeed have to be controlled through automated control systems. The general principle of

completely automated landing systems is to feed information from the localiser and

glideslope beams into the autopilot of the aircraft.

From the initial conditions, using the appropriate mathematical equations, the state space

model should be developed and then described through the state variables. The Lateral BeamGuidance System takes a value for the reference angular error between the aircraft and the

centre line and compares it with the actual angular error. The comparison of these values is

performed by the coupler.The resulting heading angle is then fed into the autopilot togenerate an appropriate heading for the aircraft.


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4 Equation Based Simulation of ILS Model

The most important part of the lateral beam guidance system is the lateral autopilot and itsinteraction with the lateral dynamics of the aircraft. The Lateral Beam Guidance System takes

a value for the reference angular errorPref and compares it with the actual errorP. The

resulting commanded heading angle ]c is then fed into the lateral autopilot:

Where (m) is the lateral displacement and R is the range of the aircraft. and are

parameter values found in the Table 1. The Lateral Autopilot has numerous components that

provide the actuator commands to the ailerons. When heading angle ]c goes into the autopilot

it passes through directional gyro and gains a parameter Kd. Then, it is a representation of the

roll angle , which passes through vertical gyro and gains Kv. The output is roll rate p(rad/s):

The roll rate passes through roll gyro and gains Kr. These generate the signal for the aileron

deflection (rad), for the banking turn:

Where is the time variable and is the comparison error:

Where is parameter value. The dynamics of the aircraft are formed through two

relationsh

ips: th

e conversion of aileron deflection to roll rate p and th

e conversionbetween the roll angle and the heading angle . These form the following twoequations:

Where g is the acceleration due to gravity, VT is the velocity of the aircraft and us theparameter value.

From the equations listed above, the state space model of the system can be developed.

First, the state variables should be defined:


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The derivatives:Substitute into the state expression:

In order to calculate the states at each time step we need to integrate the system state

derivatives. The algorithms of integration will depend on the step size of the simulation, as

will their accuracy. Developing the MATLAB code, which is found in the Appendix, the

Euler method is used to integrate the state space model. Euler is a first order numerical

integration algorithm which approximates the integral as the first two terms of the Taylors

Series.

This is the DERIVATIVE SECTION of the simulation used for the ILSassignment.%% The current time, state and input values are passed to thefunction as arguments% and the function returns the state derivative.function xdot = system_model(x,psi);

global kd kv kr tau Ka Ta g Vt % global parameters transferred frommain program

xdot(1,1) = 1/tau*((psi-x(3))*kd*kv-x(4)*kv-x(2)*kr-x(1)); %StateSpace equation for x(1)

xdot(2,1) = 1/Ta*(Ka*x(1)-x(2));xdot(3,1) = (g/Vt)*x(4); %g is the value of the gravityxdot(4,1) = x(2);xdot(5,1) = x(3)*Vt;

To analyse the system, the variables can be plotted versus time, Figure 2:


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Figure 2: Euler Integrated Model Simulation.

From the Figure 2 it can be observed that system follows sinusoidal pattern and oscillation

amplitude tends to zero or steady state. Amplitude is defined from the initial conditions. The

rate or how fast the system tends to steady state is defined by variable Gc and aircraft flight

speed .For comparison, this model can be integrated by other method like Runge-Kutta 4. Fourth

Order Runge-Kutta approximates the first 5 terms of the Taylors Series. It does this by

making 4 predictions of the state derivatives or slopes of the time interval. These take the

form of an evaluation of the slope at the start, 2 predictions of the slope in the middle and

then a final prediction of the slope at the end. Because the system does not go to the higher

order, the difference in simulation is not noticeable. The Runge-Kutta is more precise method

and stores more information, because the number of predictions is higher than in Euler

method, therefore it takes longer time to simulate the system. The graphs almost match each

other, Figure 3:


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Figure 3: The Comparison of Runge-Kutta 4 and Euler Integration Methods.

5 Simulink Block Diagram

Simulink is an environment for multidomain simulation and Model-Based Design for

dynamic and embedded systems. It provides an interactive graphical environment and a

customizable set of block libraries that let you design, simulate, implement, and test a variety

of time-varying systems.

In this section, the state space model was simulated in Simulink, Figure 4:

Figure 4: Simulink System Diagram.


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The Lateral Beam Guidance System initially sends the reference Pref =0 to the system. To

generate the actual value of P]c gains Gc value in a coupler and then feds it in to the

autopilot, Figure 5:

Figure 5: Autopilot.

The Autopilot forms 3 loops around the aircraft roll and heading dynamics. The commanded ]c

gains the value of and and then passes through the Aileron servos, Figure 6:

Figure 6: Aileron Servo.

The comparison error is then divided by time constant and being integrated to getthe value for the aileron deflection , which is then passed to the Roll and Heading Servos:

Figure 7: Roll and Heading Servos.


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Where aileron deflection is converted to roll rate p, roll angle and heading angle .Once the heading has been changed so that aircraft approaches the centre line of the runway,

it directly influences the lateral speed. This is used to calculate the lateral displacement YR,

Figure 4. As YR is divided by range value R, the estimation of the angular error between the

aircraft and the centre line can be found.

The Simulink should produce same output as in MATLAB and. Therefore, validate the

model. The output graphs should match:

Figure 8: Simulink Graphs of the Model Simulation.


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6 The Performance of the ILS System

The performance for the ILS Guidance System depends on many factors. However, the

important thing to consider is the actual aircraftheading speed VT and the localiser transmitter

parameters that affects the state variables of the autopilot.

This is the section of the report that investigates the system performance based on the aircraftheading speed VT and compensator parameter value GC , which directly proportional to the

command heading of the aircraft.

If GC value is increased by approximately 25%, from 48 to 60, it can be observed that the system

starts to be unstable. The amplitudes of the oscillations increase, therefore, steady state is

impossible to achieve, Figure 9:

Figure 9: The Simulation of the Model, Gc value of 60.


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Theoretically, the increase of the aircraft speed VT should have the same result on the system, as

there is an increase in momentum. If the value of the Gc is kept as the initial value of 48 and VT

is increased from 60 (m/s) to 75 (m/s), from Figure 10, the continuous sinusoidal graphs with

the constant amplitude values can be observed within the 100 s time domain. Therefore, system

does not reach steady state:

Figure 10: The Simulation of the Model, VT value of 75 (m/s).

To make the system more stable, these parameters should be decreased. More rapid changes in

a system are observed with the VT, as it directly affects the aircraftheading. Decreasing Gc by

half and leaving VT with the nominal value, the results are in Figure 11:


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Figure 11: The Simulation of the Model, Gc value of 24 and VT value of 60 (m/s).

The obvious change can be noticed in the simulation. It approaches zero value much faster,

comparing to the initial results. The system reaches the steady state at the time of

approximately of 30 s of the simulation.


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Figure 12: The Simulation of the Model, Gc value of 12 and VT value of 45 (m/s).

As the values of Gc and VT continue to decrease, the simulation become more stable and reaches

steady state much faster. From the Figure 12, with Gc value of 12 and VT value of 45 (m/s), the

steady state is reached at approximately 30 s, the number of oscillation decreased till 2 and does

not go negative, therefore, vector direction does not change for YR.


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As the integral value appears in the model, the simulation becomes more unstable. hasan inverse effect on the model and it becomes harder to reach the steady state.

Figure 14: The Simulation of the Model, Gc value of 48 and value of value of0.1.

From the previous section, the optimal value of Gc of 6 was recognized. The smaller is ,the better is the performance of the system. should tend to zero for the optimalperformance of the system.


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8 Interpolation

So far the longitudinal dynamics of the aircraft have not been considered. To incorporate

these dynamics, the range of the aircraftR can be varied. Having the information about the

range R change with time and using the appropriate interpolation method, the residuals canbe determined.

Interpolation is estimation of a value between two known data points. From there, the

difference between the numerical value of one particular observation and the theoretical

result can be found.

The basic idea of interpolation is that there is npairs of points and we want to find a function

P(x) or polynomialthat for eachxi is equal toyi. The degree of the polynomial depends on

the number of points that needs to be interpolated. There are different ways of the

interpolation: Newton Interpolation and Divided difference Form, Lagrange Interpolation,

Chebyshev-Gauss interpolation and others.

Time(s) 0 10 20 30 40 50 60 70 80 90 100

Range(m) 6000 4673 3639 2834 2207 1719 1339 1043 812 632 4932

Figure 15: Range Data.

Given a set ofn+1 data points (xi,yi), i = 0, 1, 2, ., n where thexi are alldifferent and theyi are sampled from an underlying functiony =f(x), the nth order

interpolating polynomial can be expressed as:

The general approach to interpolate data with Newton Divided Difference Method is based

on the evaluation of finite divided differences. Given a set of n+1 data points [xi,f(xi)], i = 0,

1, 2, ., n the finite divided differences of various orders are defined in Equations below:


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0 th order f[xi] = f(xi)

1 st order f[xi, xj,] =

,

2 nd order f[xi, xj, xk] =

,

n th order f[xn, xn-1, ,x1, x0] =

Note how the divided differences are obtained recursively from two divided differences of

order one less. When there are 3 data points (n = 2), the divided differencesf[x0],f[x1,x0],

and f[x2,x1,x0] are identical to the coefficients b0, b1, and b2 and bn.

Figure 16: Table of Finite Divided Differences for a Functionf(x).

In general, with n + 1 data points there are n +1 zero order divided differences, n first order

divided differences, n-1 second order divided differences, etc. up to one nth order divided

difference. The coefficients bi, i = 0, 1, 2,, n of the interpolating polynomial are the first

finite divided-difference entries in their respective columns.

To calculate the divided differences in MATLAB, two nested loops and general formula is

required to create a interpolation function:

function [coef] = newtoninterp(x,y)% Plot the Newton interpolation polynomial P(x) that satisfies P(xi) = yi% coef is the first divided difference for each levelx=0:10:100;

y=[6000 4673 3639 2834 2207 1719 1339 1043 812 632 493];nx = length(x);ny = length(y);if nx~=ny, disp('x and y have different length'); return; end

% calculate the divided differencesdd = zeros(nx,nx);dd(:,1) = y;

for i=2:nx % i is the level (the columns)for j=1:nx-i+1 % the divided differences in level i


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% Newton Divided Difference general formula:dd(j,i) = ( dd(j,i-1) - dd(j+1,i-1) ) / ( x(j) - x(j+i-1) );

endend

coef = dd(1,:);

end

Which give us the values

for the residuals:

Columns 1 through 3

6.000000000000000e+003 -1.327000000000000e+002 1.464999999999999e+000

Columns 4 through 6

-1.066666666666663e-002 5.416666666666551e-005 -8.333333333329854e-008

Columns 7 through 9

-4.166666666667676e-009 1.190476190476453e-010 -2.480158730159310e-012

Columns 10 through 11

4.684744268078674e-014 -8.267195767197460e-016

Figure 17: Interpolated Polynomial.


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9 Conclusion

In conclusion, propellers withhigh pitchwill be more efficient at a lower RPM than a similar

propeller with lower pitch. In this experiment the pitch 18 propeller performs the best at lower

frequencies (60 to 80Hz), the pitch 15 propeller performs better at the higher frequencies (80 to

100Hz) and it is expected that the pitch 10 propeller would perform the best at even higher

frequencies. It was also observed that each propeller performed the best with power

consumption in the range of 20 to 24W.

10 Reference List


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Appendix


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Reference List:

Thomson, G. P., 1987.Applied Aerodynamics. Hodder and Stroughton: London.

Scanlan, Robert H., 1951. Introduction to the study of aircraft vibration and

flutter. New York : Macmillan.

Propeller Laboratory Handout. 2010. The University of Glasgow.

References for the Figures:

Figure 1: Propeller Laboratory Handout. 2010. The University of Glasgow.

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