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.