control assignment
TRANSCRIPT
Control and Flight Dynamics – Autopilot Design
Aeronautical Engineering
Wednesday, 18 February 2015
Elliot Newman
@00320195
Word Count: 4370
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Contents Introduction ................................................................................................................................. 1
Objectives ................................................................................................................................ 1
Theory ..................................................................................................................................... 1
Longitudinal Dynamics ........................................................................................................... 1
Lateral Dynamics ................................................................................................................... 4
Lateral Modes........................................................................................................................... 5
Roll Damping......................................................................................................................... 5
Spiral Mode .......................................................................................................................... 6
Dutch Roll Mode ................................................................................................................... 6
Aircraft Actuator Influence..................................................................................................... 7
Autopilot Design ........................................................................................................................... 8
Longitudinal Systems ................................................................................................................ 8
Attitude Control .................................................................................................................... 8
Proportional Control .......................................................................................................... 8
Proportional-plus-Derivative Control .................................................................................. 9
Trial and Error Design Process .......................................................................................... 10
Altitude Hold Control........................................................................................................... 10
Lateral Systems....................................................................................................................... 11
Roll Control ......................................................................................................................... 11
Yaw Damper ....................................................................................................................... 13
Heading .............................................................................................................................. 15
Heading Hold Autopilot ....................................................................................................... 16
Results ....................................................................................................................................... 17
Longitudinal Results ................................................................................................................ 17
.Attitude Control Results...................................................................................................... 17
Trial and Error Process Results.......................................................................................... 18
Altitude Control Results ....................................................................................................... 19
Lateral Results ........................................................................................................................ 20
Roll Control Results ............................................................................................................. 20
Yaw Damper Results ............................................................................................................ 20
Heading Hold autopilot Results ............................................................................................ 21
Discussion & Conclusion.............................................................................................................. 23
Appendix.................................................................................................................................... 27
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Introduction
Flight control manifests itself in many forms, most notably systems dictating the orientation of the
aircraft. The pilot creates in input into the system, a change of heading or altitude, encompassed in
lateral and longitudinal dynamics, the system then reacts through a series of iterations to achieve
the desired conclusion. This process has to be refined in order to linearise a stable progression,
something that will be investigated thoroughly.
Objectives
The objectives initially are to focus on longitudinal control, an attitude and altitude hold autopilot,
analysing longitudinal dynamics to determine if the feedback behaviour is acceptable and iterate
accordingly. On the by mathematically modelling the system in Simulink, gain an appreciation for the
effects of important parameters to visually witness their effects.
Following the completion of this and by then applying the knowledge and skills developed, a heading
hold autopilot through investigating lateral dynamics, reefing the system to make it industry
applicable by controlling the maximum roll and yaw motions through damping.
Theory
The theory for these systems shall first be examined and appreciated, allowing for the residual errors
to be identified and factor these into our own designs at a later date.
Longitudinal Dynamics
The control systems of a modelled 747 are immensely complex and with this in mind, primarily we
shall focus on the short period approximation, generating a greater noesis and feel for the subject
moving forward:
Add that,
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Using the data acquired, the completed pole-zero map is illustrated in figure 1:
Figure 1: Pole zero map for Gθδe
As with designing a system or developing a solution to a problem, interpreting the needs in the
consumer is a paramount feature, understanding what behavioural patterns a pilot wants from the
aircraft. Unfortunately, there are legions of empirical data that indicate the pilots do not like operating
aircraft with the flying qualities generated by this combination of frequency and damping and
therefore signals the need to develop an acceptable relationship range for the natural f requency and
the damping ratio (figure 2):
Figure 2: ‘Thumb Print’ Criterion
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The ‘thumb print’ criterion, a concept developed during the 1950’s, signifies the acceptable region of
which the values can interact and is still applicable today. The graph illustrates that the primary
target value to aim for will be a natural frequency (ωn) of 3𝑟𝑎𝑑𝑠−1 and a damping ratio (ζ) of
approximately 0.6. As becomes apparent from comparing these values with those attained from the
formulas, the short period dynamics of the 747 are well outside the acceptable range, marked on
the graph, and therefore must be modified accordingly.
Figure 3 clearly illustrates the target pole locations on a pole-zero map when under these
approximated conditions:
Figure 3: Pole-zero map and target pole locations
Through investigating the graph, the shaded area represents the region of which the closed loop
dominant poles should be found and to accomplish this feat, we will require some feedback from
the system. There are two primary practices in achieving this which are:
Proportional Control.
Proportional-plus-Derivative Control.
Furthermore, we can employ the technique of plotting root-loci, which allow values of ωn and ζ to be
attained aiding the design process, then using Simulink to assess our designs.
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Lateral Dynamics
Obtaining the necessary values in this plane can be achieved using a procedure similar to that of the
longitudinal case, where we can develop the equations of motion, which include; dutch roll mode,
roll mode and spiral mode, using state space equations:
Extracting the numerical values for each of the modes can be achieved in Matlab, performing the
‘lat’ command on each of the matrices, e.g. Alat and Blat:
Alat =
-0.0558 0 -235.9000 9.8100
-0.0127 -0.4351 0.4143 0
0.0036 -0.0061 -0.1458 0
0 1.0000 0 0
Blat =
0 1.7188
-0.1433 0.1146
0.0038 -0.4859
0 0
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Encompassed in each of these state spaces equations are the behavioural patterns of all the lateral
modes with, for the Alat matrix; the top two rows being associated with the dutch roll mode, the
third line linked to the roll mode and the final row demonstrating the spiral mode. The Blat matrix
allows for an aero devices to be isolated being either; the ailerons for the first column or the rudder
by the second.
The true benefit of calculating these numerical values is to solve for the eigenvalues of matrix A,
relinquishing the modes of the system:
These are stable, although there is one very slow pole. The next step is to link these to each of the
three available modes:
Now, due to their increased complexity from longitudinal modes, before we can continue to the
design of each of the systems, we must first understand each of these modes and their effect on the
aircraft.
Lateral Modes
Roll Damping
This mode is heavily damped, increasing the ease of operation and the severity of the destabilising
effect of the roll. As the plane rolls, the wing going down has an increased α, the influence of wind is
effectively increased and this has an opposite affect for the neighbouring wing. This disparity
generates an imbalance in the incremental lift produced, more on the descending wing. This lift
differential creates a moment that tends to restore the equilibrium of the aircraft. After a
disturbance, the roll rate builds exponentially until the restoring moment balances this disturbance
and a stable roll is established, illustrated in figure 4:
Figure 4: Roll mode illustration.
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Spiral Mode
The spiral mode is the slowest of the lateral modes and is often unstable, from level flight consider a
small disturbance that creates a small roll angle φ > 0. This results in a small side slip 𝑣, as expected
the tail fin is now traveling through the air at an incidence angle β, creating extra tail lift, increasing
the yawing moment. The positive yawing moment further increases the side slip compounding the
situation which, if left unchecked, would cause the aircraft to commence a gradually diverging path
in roll, yaw and altitude, ‘spiralling towards the ground, visually demonstrated in figure 5:
Figure 5: Spiral mode destabilisation.
Dutch Roll Mode
The final lateral mode, the most complex, involves damped oscillation of yaw that couples into roll.
It manifests itself at a frequency close to the short period mode, although not as heavily damped
and therefore the fin has less effect than the horizontal tail plane. The term is coined from skating
circles, giving reference to the act of repeatedly skating from right to left on the outer edge of their
skates, imitating the aircrafts motion. Again we shall consider a disturbance from straight level flight,
where an oscillation in yaw ψ, with the fin providing the aerodynamic stiffness. The wings move back
and forth with respect to this yaw motion and the result is an oscillatory differential in lift/drag, as
depending on the direction and motion of the wing, the induced lift is either increased or reduced
accordingly. Adding to this motion is the roll, φ also oscillatory and is approximately lags the yaw by
90⁰ and therefore at all times the wing moving forwards during the cycle is the lowest. The final
result is an oscillating roll with side slip in the direction of the low wing. When witnessing the
phenomenon first hand it is clear to see the wing tip traces a figure of 8 in the sky during each time
period (figure 6):
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Figure 6: Dutch roll mode.
Aircraft Actuator Influence
As the B matrix implicitly explain the responses of the rudder and aileron inputs, their influence must
also be investigated. Due to the physical placement of the rudder, being quite high, it has a
significant influence on the aircrafts roll, whereas the ailerons affect the yaw by inducing drag
differentials. We shall view the impulse response of the two inputs:
Rudder input
Β shows a very lightly damped decay.
𝑝, 𝑟 clearly excited as well.
𝜑 oscillates around 2.5⁰.
Dutch-roll oscillations are clear.
Aileron input
Large impact on.
Causes large change to 𝜑.
Very small change to remaining variables.
Influence smaller than the rudder.
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Autopilot Design
The main objective of this assignment is to complete and test an autopilot system that can not only
control the longitudinal modes of attitude and altitude, but also the lateral effects of a heading hold
system, engaging roll and yaw limiters which shall be concluded in this section, beginning with the
longitudinal system.
Longitudinal Systems
Attitude Control
In this design the SPO transfer function model for a Boeing 747 is considered, an actuator for the
elevators is also added with a pole at −4. Using Simulink, the transient behaviour of the system
under proportional control shall be modelled.
Proportional Control
Figure 7: Simulink model for proportional controller
Using Matlab capabilities, the aim is to obtain a root-loci plot of this system, aiming to achieve a
point on the plot where the natural frequency is 3𝑟𝑎𝑑𝑠−1 and a damping ratio of 0.5 for this closed
loop system. Transferring the Simulink model (figure 7) into Matlab code for the command window
is demonstrated below:
n=4*[1.166 0.35]; a=[1 0]; b=[1 4]; c=[1 0.74 0.92]; d=conv(a,conv(b,c)); rlocus(n,d); axis([-6 2 -3 3]);
Constricting the axis to the desired region.
From inspecting the plot, the desired values could not be attained, alluding to the short comings of
the system, requiring a different approach. One approach to take is to replace the proportional
controller with a proportional-plus-derivative compensator.
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Proportional-plus-Derivative Control
Using this type of system will improve the performance of control, replacing the proportional control
with parameters k1 and k2. There are three different types of control feedback system which are:
Forwards loop (figure 8).
Feedback loop (figure 9).
Two loop control system (figure 10).
The proportional-plus-derivative compensator encompasses the two parameters k1 and k2, but there
are two design equations to solve (the angle and magnitude criteria), so any design is straight
forward. Each Simulink model is depicted below:
Figure 8: Forwards Loop.
Figure 9: Feedback Loop.
Figure 10: Two Loop system equivalent to the P-D above.
Determining the values for k1 and k2 becomes the next issue to address and the most practiced form
is trial and error, using converging iterations of the system to achieve the predetermined values of
ωn and ζ.
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Trial and Error Design Process
This method utilises the ease of conducting calculations using Matlab and Simulink, with the
preferred system to conduct the process being the forwards loop system (figure 8: 11).
By firstly, guessing a value for k2, the root-loci for this is then plotted and see if the roots pass
through the points:
𝑆 = −1.5 + 2.5981𝑖
𝑆 = −1.5 – 2.5981𝑖
The value of K2 selected was 0.975 and then the code used to obtain the final values for k1 and k2
was:
k2=0.975; x=[1 k2]; y=4*[1.66 0.35]; n=conv(x,y); a=[1 0]; b=[1 4]; c=[1 0.74 0.92]; d=conv(a,conv(b,c)); rlocus(n,d)
Using the resulting root-locus plot to determine these parameters.
Altitude Hold Control
The altitude control system is an extenuation of the attitude system, being fed into a new transfer
function to alleviate an altitude value, depicted by figure 11 below:
Figure 11: Transfer function.
By adding this into the attitude control system, coupled with a proportional controller on the
altitude loop, the system has been converted to the required altitude control system (figure 12):
Figure 12: Altitude control system.
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Using Simulink, obtaining a value for the parameter kh would complete the system.
Lateral Systems
The stability and modification of lateral dynamics can be controlled using a varie ty of different
feedback architectures, e.g. using integrators (figure 13):
Figure 13: Integral control.
Using the integrators, values of roll (𝑝) and yaw (𝑟) can be converted into roll rate (φ) and yaw rate
(ψ), looking for good sensor or actuator pairings to sustain suitable behaviour for the pilot. The block
Glat is comprised of a series of state space equations with the Matlab code depicted as appendix 1
(page 27).
Roll Control
When a desired bank angle is selected and input into the system, a roll controller is need to ensure
and maintain the accuracy at which the vehicle tracks request. In this situation, the ailerons are the
best actuator to use:
To obtain design value for KΦ and Kp, approximations of the roll mode must be made:
Which gives:
To fully encompass the design, add the aileron servo dynamics:
This generates a root loci plot that is typically demonstrated (figure 14:14) below:
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Figure 14: Typical root loci plot.
In the case of the system for the 747, it is design to achieve a damping ratio of 0.667 and a natural
frequency of 3𝑟𝑎𝑑𝑠−1 for the second order modes of the roll damper. The roll control system will
deploy a proportional controller as illustrated below (figure 15):
Figure 15: Proportional roll controller.
Then using root loci, the system shall be tested to see if the desired performance values can be
achieved, then using Simulink to simulate the system.
An improvement can be made on the controller by making it a proportional plus integral in the form
below (figure 16):
Figure 16: Proportional plus integral roll controller.
Again, root loci shall be used to determine values for k1 and k2, such that the system matches the
required performance characteristics.
Upon completion the final control system that will be implemented is below (figure 17:13):
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Figure 17: Completed roll controller.
Yaw Damper
In the case of a heading alteration, as the aircraft banks, the nose also ‘leans’ into the turn, known as
yaw. In order to avoid entering the pre-mentioned spiral mode (page 6), a yaw damper is need,
reducing the amount of natural yaw that can be induced in the aircraft. This can be subtly controlled
by altering the feedback in the control system:
Feedback only a high pass version of the 𝑟 signal.
High pass cuts of the low frequency content in the signal.
Steady state value of 𝑟 would not be fed back into the controller.
New yaw damper:
Figure 18: Frequency response of the washout filter.
This information then leads to the completion of the final yaw damper design (figure 19):
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Figure 19: Yaw damper control system.
The transfer function relating the rudder movement to the yaw rate is:
𝐺𝑙𝑎𝑡(𝑠)𝑑𝑟 𝑟⁄ =1.618(𝑠 + 0.6943)(𝑠2 − 0.2146𝑠 + 0.1678)
(𝑠 + 3.33)(𝑠 + 0.5613)(𝑠 + 0.007264)(𝑠2 + 0.06629𝑠 + 0.8978)
The washout filter has the following transfer function also:
𝐻𝜔(𝑠) = 𝑇𝑠
𝑇𝑠+1
Plotting a complex root loci of the system, finding kr and T such that the dominant complex roots
have a damping ratio of 0.83 and a natural frequency of 0.95𝑟𝑎𝑑𝑠−1 obtains a plot such as:
Figure 20: Complex root loci plot
With this aspect now completed the yaw damper can be added to the lateral control system and
allows the next feature to be designed, the heading autopilot.
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Heading
With the yaw damper completed and added to the system, controlling the heading (𝜓) is the next
characteristic to design. In order to achieve a heading change, first and foremost, the aircraft is going
to need to bank, resulting in a ‘coordinated turn’ with and angular rate �̇�.
The aircraft is banked to a predetermined angle 𝛷 so that the vector sum of 𝑚𝑔 and 𝑚𝑈0�̇� is along
the body of the 𝑧 − 𝑎𝑥𝑖𝑠. Summing up the body y-axis direction, this gives 𝑚𝑢0�̇�cos𝜙 = 𝑚𝑔 sin 𝜙
this will give an equation of:
tan 𝜙 =𝑈0 �̇�
𝑔
Since typically ϕ << 1, then:
𝜙 ≈𝑈0 �̇�
𝑔
Which gives the desired bank angle for a specified turn rate.
The issue with this is that 𝜓 tends to be a noisy signal to base the bank angle on, so a smoother
signal is generated through filtering it. By assuming that the desired heading is known 𝜓𝑑 and we
want 𝜓 to follow 𝜓𝑑 relatively slowly, then selecting the dynamics of 𝜏1 �̇� + 𝜓 = 𝜓𝑑 :
𝜓
𝜓𝑑=
1
𝜏1𝑠+1, with 𝜏1 = 15 − 20 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 depending on the situation.
A low pass filter that eliminates the higher frequency noise.
The filtered heading angle satisfies the equation:
�̇� =1
𝜏1(𝜓𝑑 − 𝜓)
Which can be used to create the desired bank angle for the aircraft:
𝜙𝑑 =𝑈0
𝑔�̇� =
𝑈0
𝜏1𝑔(𝜓𝑑 − 𝜓)
Now with all the individual aspects of the heading auto pilot designed and functional, the system can
be completed.
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Heading Hold Autopilot
By compiling the designed features into one, all-encompassing control system, the autopilot
controller is created (figure 21):
Figure 21: Heading hold autopilot system.
Before the system can be completed ready for operation, the final step is to analyse the effect of
closing the 𝜓 to 𝛷𝑑 loop. The parameter enclosed on the loop, 𝑈0/𝑇1 𝑔, has to be carefully selected
due to the sensitivity of the loop, too large and the system will go unstable. The value can be
determined using the conventional method of root loci, however the calculations can be quite
complicated. By assuming a value of 2, the performance can be investigated, before adding a roll
angle limiter, in the form of a saturation block, onto the path of 𝑈0/𝑇1𝑔 yielding a final design of
(figure 22):
Figure 22: Final heading hold autopilot design.
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Results
Determining certain characteristics, as well as conducting performance tests was a necessary
practice and the concluding data sets are depicted in this section.
Longitudinal Results
Attitude Control Results
Figure 23: Attitude control Simulink performance.
Figure 24: Attitude control root loci plot.
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Trial and Error Process Results
Figure 25: Trial and error root loci plot.
From the root loci plot, values for k1 and k2 were achieved at k1 = 2 and k2 = 0.45. Applying these
gains into the three forms of P-D control systems the transient responses can be evaluated in
Simulink:
Figure 26: Forwards loop transient response.
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Figure 27: Two loop transient response.
Altitude Control Results
Using the altitude gain kh as 2:
Figure 28: Altitude control Simulink performance.
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Lateral Results
Roll Control Results
Obtaining values of k1 = −20 and k2 = 1.5, the root loci plot is illustrated as (figure 29):
Figure 29: Roll control root loci plot.
Yaw Damper Results
Employing parameters of kr = 9.26 and T = 0.4:
Figure 30: Yaw damper root loci plot.
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Heading Hold autopilot Results
After meshing all the pretested systems, the final autopilot design was tested too yielding:
Figure 31: Heading hold root loci plot.
Figure 32: Heading orientation Simulink performance.
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Figure 33: Roll angle Simulink performance.
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Discussion & Conclusion
Overall, from the data set acquired, the responses from the autopilot systems was well within the
acceptable performance range, although rarely will a control system resonate with the required
performance parameters perfectly and therefore in the majority of cases ‘controllers’ are a
necessary fundamental. In this case, these characteristics could only be achieved with dampers and
limit controllers, hinting to their significance in system design.
The yaw damper helps in inhibit the maximum yaw angle experienced during flight, making
manoeuvre conditions more stable and easier to control for the pilot. Placing a ceiling on the
maximum yaw increases safety during certain flight conditions, such as heading alteration, where if
yaw and bank conditions increase above the maximum, the circumstances would be enough to
induce the spiral mode, which has clearly dangerous connotations. Another method of preventing
this phenomenon is constraining the maximum roll angle, controlled by the addition of a saturation
block. Maintaining this also enable a more pleasant flying experience for on board passengers, as the
‘banks’ experienced will have little effect on cabin conditions.
Pilots require flight control systems to react in a stable progressive manor after inputs, responding
with predictability before smoothly tracking the requested output. Clearly demonstrated with the
attained results is the response differences between the three varying P-D controllers; forwards
loop, feedback loop and two loop (figures 8, 9, 10: 9). Initially, the forwards loop rises at a steep
angle, reaches the required change and begins to oscillate below the required output before slowly
converging. This is in stark contrast to the most effective, and most complex of the three, the two
loop system. In this case there are no immediate changes initially, before the system gradually
converges on the required yield, providing the stable, controllable system the pilot’s desire.
Continuing with altitude response, the system is a special scenario, of which it is a positive feedback
system, which causes a change in the normal practices when conducting root loci plots. This occurs
when the flight dynamic systems have a non-minimum phase zeros and the system has to be
modelled as positive feedback. Certain fundamentals maintain; the number of branches, the
symmetry and the starting and ending points. The factors that charge are the fact that; on the real
axis, the root locus now exists to the left of an even number of poles or zeros and that the equations
to calculate the necessary criteria have subtle differences:
𝛷𝑙 = 360
𝑛−𝑚
𝜎 = ∑ 𝑝𝑖−∑ 𝑧𝑖
𝑛−𝑚
Also, calculating the angle criteria alters, with the equation becoming equal to 0 rather than the
conventional -180. The magnitude criteria remains the same easing calculations.
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Key to understanding the systems functionality is how it responds to changing parameters, affecting
the stability of the system. These tests will help a pilot determine the operational performance
boundaries of the aircraft and also help then utilise practical efficiencies during f light. Through
specific alterations, increasing, then decreasing gains an appreciation can be attained.
Beginning with gain increases for the value of Tau 1 located on the lowest feedback l oop of the
system (figure 22: 16), the Simulink plots revealed how the roll and the yaw responds to the
alteration within the system (Tau 1 = 8):
Figure 34: Roll gain change reaction.
Figure 35: Yaw gain change reaction.
From the results it is clear to see the unstable oscillating nature of the performance, each one begins
the manoeuvre in the conventional way until the incorrect gain value is fed back into the system and
begins to destabilise it. Continuing the analysis, it is clearly visible that the instability has a peak
amplitude, were the oscillations reach their maximum and continue for the time period. Visible in
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the periodic gain increases in the appendix (Appendices 2 and 3: 28,29), as the gain margin increases
the systems stability becomes more iritic, as when Tau 1 = 2, the system is barely affected, yet at 4,
clear unstable oscillations begin to occur as the system reverts to equilibrium after the manoeuvre.
Further investigation yields that the peak amplitudes demonstrated are intrinsically linked into the
magnitude of the gains, illustrated by contrasting the increasing gains graphs. Interestingly, as the
frequency of the yaw and roll are of a very similar time period, indicating the dutch role mode is
being induced, increasing in severity as the gain increases.
Now, investigating the effects of reducing the gain margin, which is a stark contrast to the effects of
the increase:
Figure 36: Roll gain change reaction.
Figure 37: Yaw gain change reaction.
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The most notable difference is there is no pronounced instability at all, the system appears reacts
perfectly, although, under closer inspection the subtle differences arise. The time period of the
exercise is far greater than normal, taking 10 times longer at 1600 seconds, this alleviates that, the
time period of any systems oscillation can be lowered by reducing the gain margin by the required
factor, a fact reinforced by appendix 4 (page 30), where Tau 1 is 0.5 and the total time take is twice
that of the design system. This factor also applies to the amplitude, increasing by the same
magnitude as its reduction. The only noticeable instability is at the peak of the roll, where the
amplitude is that at the peak, there is no stabilisation before beginning to return to level flight, the
change is quite sharp and could therefore destabilise the aircraft during flight.
Overall, the gain changes exercise alluded to the significance of a well-produced design to maximise
the efficiency of a system, alteration from this ‘sweet spot’ can cause either lethargic manoeuvre
response or a totally unstable, un-flyable aircraft. The fact that the performance changes alter either
side of the designed gain illustrates the success of the process undertaken and its industry
applications. Also, as when the gain is over the desired value instability occurs, it may be helpful to
introduce a safety margin into the design process to protect against this, by lowering the gain by 10-
20%, increasing the time period of modes and allowing the pilot valuable thinking time in the event
of any error.
The success of this project cannot be disputed as the results speak for themselves and mirror those
of predicted plots. Also, the synergy between value alterations, the fact that no anomalous results
are introduced during these changes, demonstrates their validity and reliability. To improve the
data, next time I would produce a consistent spread of gain alterations to enable a graphical
representation of the aforementioned trends witnessed.
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Appendix
Appendix 1 – Glat State Space Matlab Code
% B747 lateral dynamics %T= ???? Yv=-1.61e4;Yp=0;Yr=0; Lv=-3.062e5;Lp=-1.076e7;Lr=9.925e6; Nv=2.131e5;Np=-1.33e6;Nr=-8.934e6;
g=9.81;theta0=0;S=511;cbar=8.324;b=59.64; U0=235.9; m=2.83176e6/g;cbar=8.324;rho=0.3045; Iyy=.449e8;Ixx=.247e8;Izz=.673e8;Ixz=-.212e7;
Cyda=0;Cydr=.1146; Clda=-1.368e-2;Cldr=6.976e-3; Cnda=-1.973e-4;Cndr=-.1257;
QdS=1/2*rho*U0^2*S; Yda=QdS*Cyda;Ydr=QdS*Cydr;Lda=QdS*b*Clda;Ldr=QdS*b*Cldr; Nda=QdS*b*Cnda;Ndr=QdS*b*Cndr;
Ixxp=(Ixx*Izz-Ixz^2)/Izz; Izzp=(Ixx*Izz-Ixz^2)/Ixx; Ixzp=Ixz/(Ixx*Izz-Ixz^2);
Alat=[Yv/m Yp/m (Yr/m-U0) g*cos(theta0); (Lv/Ixxp + Ixzp*Nv) (Lp/Ixxp + Ixzp*Np) (Lr/Ixxp + Ixzp*Nr) 0; (Ixzp*Lv + Nv/Izzp) (Ixzp*Lp + Np/Izzp) (Ixzp*Lr + Nr/Izzp) 0; 0 1 tan(theta0) 0];
Blat=[1/m 0 0;0 1/Ixxp Ixzp;0 Ixzp 1/Izzp;0 0 0]*[Yda Ydr;Lda Ldr;Nda Ndr];
Clat= eye(4,4); D_lat =zeros(4,2); c=[0 0 1 0]
b=zeros(4,1) for n=1:1:4 b(n,1)=Blat(n,2) end d=0 sys1=ss(Alat,b,c,d) zpk(sys1) n1=[-T 0] d1=[T 1] sys2=tf(n1,d1)
sys3=sys1*sys2; n3=0.333 d3=[1 0.333]; sys4=tf(n3,d3) sys5=sys4*sys3 rlocus(sys5) axis([-2 1 -1.5 1.5])
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Appendix 2 – Tau 1 = 2
Figure 38: Roll gain change reaction.
Figure 39: Yaw gain change reaction.
P a g e | 29
Appendix 3 – Tau 1 = 4
Figure 40: Roll gain change reaction.
Figure 41: Yaw gain change reaction.
P a g e | 30
Appendix 4 – Tau 1 = 0.5
Figure 42: Roll gain change reaction.
Figure 43: Yaw gain change reaction.