Enhancements of an auto- thrust function using fuzzy logic

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Enhancements of an

auto-thrust function using fuzzy logic

G U S T A V Ö M A N L U N D I N

Master of Science Thesis

Stockholm, Sweden

2014

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Enhancements of an auto-thrust

function using fuzzy logic

G U S T A V Ö M A N L U N D I N

Master’s Thesis in Systems Engineering (30 ECTS credits) Master Programme in Aerospace Engineering (120 credits) Royal Institute of Technology year 2014 Supervisor at Airbus was Matthieu Barba

Supervisor at KTH was Per Engvist Examiner was Per Engvist

TRITA-MAT-E 2014:59 ISRN-KTH/MAT/E--14/59--SE

Royal Institute of Technology

School of Engineering Sciences KTH SCI

SE-100 44 Stockholm, Sweden URL: www.kth.se/sci

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Acknowledgements

First of all I would like to thank Matthieu Barba for giving me the opportunity to work with this project and for patiently guiding me through the jungle that is aircraft control, in addition to being a good supplier of ideas and a sounding board for my own. For their shared thoughts and fruitful discussions regarding the A/THR and manual piloting, I thank engineers Jean Muller, Fabien Guignard, and Lilian Ronceray, as well as test pilots Jacques Rosay and Martin Scheuermann.

Furthermore, I thank service manager Emmanuel Cortet together with the rest of EYCDR for welcoming me into the service and inspiring me in my day-to-day work during my six months at Airbus Operations S.A.S.

I would also like to thank Per Enqvist and the School of Engineering Sciences (SCI) at the Royal Institute of Technology (KTH) for helping me through the administrative circus and for giving me the opportunity to study abroad and to write my master's thesis in France.

Last but not least, my ever helpful and cheerful colleagues with whom I shared the oce during my time at Airbus; namely Victor Gibert, Josep Boada-Bauxell, Alexandre Desbiez, Vincent Bassien-Capsa, and Marcos Medrano.

Toulouse, October 5, 2014 Gustav Lundin

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Abstract

This master's thesis aims to investigate how fuzzy logic can be used to adapt the tuning of a speed control law during certain conditions such as turbulence. The objective is to lower the speed overshoot caused by the auto-thrust function as well as the general engine agitation. The main modications studied are direct lowering of the closed loop gains, hybridisation and ltering of the longitudinal acceleration estimation. Finally, saturations or limits on the control signal as well as on the coordination with the longitudinal control law are studied in order to cope with the possible consequences of a softer control law.

To detect the turbulence, an already existing turbulence detector is used. In addition, a wind gradient detector is designed in order to increase the gain during such wind conditions to counter ramp errors.

It is found that a general lowering of the closed loop gain in combination with a slow hybridisa-tion, all proportional to the detected turbulence level, together with a limitation of the coordination gives a satisfactory result. In scenarios including severe turbulence and wind gradients, the forced limits are shown to be indispensable.

A conclusion is drawn that the fuzzy tuning is better adapted to turbulent conditions but that the wind gradient detection and the forced limits must be studied further. It is also concluded that the coupling between the closed loop gain and the acceleration hybridisation can be interesting to investigate. Moreover, additional realistic scenarios should be simulated in order to further validate the design.

For future studies on the subject; it is recommended that the controller tuning is validated with the help of expert knowledge. Alternatively, the tuning could be handled by an ANFIS (Adaptive Neuro Fuzzy Inference System). Finally the tuning of the controller should be validated for a wider range of ight points, most importantly the forced limits since the engine response varies a lot between dierent points in the ight envelope.

Keywords: Fuzzy logic, aircraft speed control, turbulence

Note: Some tuning values, gures, and graph scales have been removed in this version due to condentiality reasons.

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List of Figures

1 Upper: Typical severe turbulence representation (pilot agreed); Lower: Up-chirp

oscil-lating wind (0.009→0.06Hz). . . 9

2 Comparison of wind- and A/THR-caused speed deviations during turbulence and oscil-lating wind. . . 10

3 Engine agitation during turbulence and oscillating wind. . . 11

4 Fuzzied input membership functions for input 'sky'. . . 12

5 Fuzzied output membership functions for output 'weather'. . . 13

6 FIS output function. . . 13

7 Example of an (optimistic) FIS (rule 1 weighted by 0.2). . . 14

8 FIS output function (rule 1 weighted by 0.2). . . 14

9 Implementation of the FIS in the (very) simplied aircraft model. . . 15

10 Simplied model of longitudinal aircraft dynamics. . . 16

11 Singular values for the sensitivity function SGSC, in approach. . . 18

12 Comparison of dierent τ in medium turbulence. . . 21

13 Comparison of dierent τ in severe turbulence. . . 22

14 Bode diagram comparing complementary high pass lters with dierent τ. . . 23

15 Nx,est hybridisation coecients as a function of turbulence (TURBLVLDET). . . 24

16 Implemented band-stop and band-pass lter. . . 25

17 Comparison of rst and second order complementary washout lter. . . 26

18 Membership functions for ω as a function of turbulence (TURBLVLDET). . . 27

19 Comparison of dierent gains in medium turbulence. . . 28

20 Comparison of dierent gains in severe turbulence. . . 28

21 Singular values corresponding to the gain at dierent turbulence levels in approach. . 29

22 Fuzzy gain as a function of turbulence. . . 30

23 Bode diagram of the adaptive band stop-/pass lter for dierent ω. . . 31

24 Wind gradient detection overview. . . 32

25 Evaluating simulation of the wind gradient detector. . . 33

26 Limits of ˙N1,c during speed deviations in approach. . . 34

27 Limits of ˙N1,c near the edges of the ight envelope. . . 35

28 Consistent speed deviation conrmation. . . 36

29 Limits due to consistent speed deviations, triggering and implied ˙N1,c-limits. . . 36

30 Glide slope deviation scenario. The dashed line shows the target glide slope. . . 37

31 Limit of ˙N1,c when in underspeed and over the glide slope. . . 38

32 Limit of ˙N1,c when in overspeed and under the glide slope. . . 38

33 Classical A/THR limits on Nz,c. . . 39

34 Fuzzy A/THR limits on Nz,c for light,medium, andsevereturbulence. . . 40

35 Implemented limit on |Nz,c| as a function of turbulence (TURBLVLDET) for dierent speed deviations. . . 40

36 'Pilot agreed turbulence' and detected turbulence level (TURBLVLDET). . . 42

37 Turbulence classication in approach for fuzzy controller input. . . 43

38 Closed loop simulator SIMBOX, simulink based controller. . . 43

39 Closed loop simulator SIMPA, compiled C-based controller. . . 44

40 ATHR-caused speed deviation as a function of the turbulence level. . . 45

41 Typcal time series (m = 330t) for VCAS during medium (top) and severe (bottom) turbulence. . . 46

42 Fan speed standard deviation as a function of the turbulence level. . . 46

43 Typical time series (m = 330t) for N1during medium (top) and severe (bottom) turbulence. 47 44 Mean closed loop pulsation. . . 47

45 Approach Perpignan: WX,0, TURBLVLDET, and KWINGRADDET. . . 48

46 Approach Perpignan: VCAS and N1. . . 49

47 Recorded severe turbulence: WX,0, TURBLVLDET, and KWINGRADDET. . . 50

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49 Oscillating wind: WX,0 and TURBLVLDET. . . 51 50 Oscillating wind: VCAS, ∆VCAS,A/T HR, and N1. . . 52

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List of Tables

1 Implemented Nx,est hybridisation time constants. . . 22

2 Baseline ω as a function of ight phase. . . 27

3 Gain as a function of turbulence level. . . 29

4 Wind gradient detector, turbulence acceleration threshold. . . 32

5 Wind gradient detector, conversion from wind acceleration to severity factor. . . 32

6 Examples of the ratio σ(Nz,est) σ(Nz,c) for dierent turbulence severities. . . 40

7 Wind speed standard deviation as a function of the turbulence level. . . 41

8 Summary of fuzzy A/THR modications. . . 57

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1 Symbols and acronyms

Acronym Full form

A/THR Auto-thrust

GSC Generic Speed Controller

FL Fuzzy Logic

FLC Fuzzy Logic Controller

FIS Fuzzy Inference System

MF Membership Function

FDBK Feedback

DSPDCASDMD Speed deviation in A/THR (VT GT-VCAS)

DCASTGT Speed deviation (VCAS-VT GT)

GSDEV Glide Slope Deviation

Symbol Description Unit

N1 Engine fan speed (percentage of max) [%]

m Aircraft mass [kg]

g Gravitational acceleration [m/s2_]

VT AS Aircraft true airspeed [m/s]

VCAS Aircraft calibrated airspeed [m/s]

VT GT A/THR reference speed [m/s]

VGN D Aircraft ground speed [m/s]

VLS Lowest selectable speed [m/s]

VM O Maximum operating speed [m/s]

Wx,0 Longitudinal wind speed (rel. to the aircraft) [m/s]

S Aircraft reference area (wing surface area) [m2]

ρ Air density [kg/m3_]

ρ0 Air density at Standard Sea Level conditions [kg/m3]

Cx Total drag coecient [-]

Cz Lift coecient [-]

F, c Engine thrust, commanded [N]

f Lift-to-drag ratio [-]

γ Flight path angle (rel. to the ground) [rad]

θ Total aircraft pitch angle [rad]

Nz,c Vertical load factor, feed forward [-]

Nz,F DBK Vertical load factor, feedback [-]

Nx,est Longitudinal load factor, estimated [-]

ω Closed loop pulsation [rad/s]

ζ Closed loop damping coecient [-]

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2 Introduction

The speed is one of the most critical parameters to control when ying an aeroplane. Naturally, the speed monitoring therefore requires a lot of attention from the pilot. Airbus has developed a generic speed controller (GSC) with two objectives, to attain a certain speed and to maintain it. There are two ways to control the speed of the aeroplane, either by the engines using the throttles or by the elevators using the stick. The GSC is basically a linear PID-controller together with a feed forward term to coordinate the thrust control with the longitudinal control law, i.e. the elevators.

For a linear controller it is well known that tracking and disturbance must be weighted against each other due to the relation between the sensitivity function and the complementary sensitivity function. The speed control law in use today is heavily tuned towards reference following. It is known that such a control law can have an unsatisfactory behaviour regarding disturbance attenuation. Although the current tuning gives a robust behaviour for speed acquisition and maintenance it comes at the cost of large control signals in some situations.

This master's thesis aims to study whether the tuning of a speed control law during turbulence can be improved by the use of fuzzy logic. Possible gains of a control law based on fuzzy logic could be to get an adaptive control whose parameters are optimised for the current circumstances. Presumably it will be possible to reduce the control signals under certain conditions while maintaining the robustness of the current tuning.

Initially, the reader will be introduced to the equations governing the speed dynamics and the structure of the generic speed controller as well as the concept of fuzzy logic. Thereafter, the control strategy during approach and cruise and some proposed solutions will be discussed. Finally the results will be presented, along with a discussion regarding the advantages, disadvantages, and possible future development.

3 Problem

3.1 Problem description

When ying through turbulent air, where the wind speed varies in an oscillatory manner as depicted in Figure 1 for the case of pure turbulence and an analytic oscillating wind, there are two main issues that aect the speed control.

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Figure 1: Upper: Typical severe turbulence representation (pilot agreed); Lower: Up-chirp oscillating wind (0.009→0.06Hz).

The rst one is that the A/THR creates deviations in the speed during oscillating winds in addition to the deviations caused by the wind itself. These oscillations are found notably around 0.03Hz for the current A/THR, as can be concluded from the lower plot in Figure 2. Note that in the gure below, WX,0 > 0 is dened as a headwind in the aircraft frame of reference to and will therefore lead to an increase of the airspeed.

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Figure 2: Comparison of wind- and A/THR-caused speed deviations during turbulence and oscillating wind.

The second problem is that the A/THR sometimes over-agitates the engines during turbulence, which can be seen in the upper plot in Figure 3. This kind of behaviour can wear out the engines faster than if they would remain close to their stationary speed for the current ight condition. Since turbulence phenomena are usually near zero-mean (or small in comparison to the amplitude), it might be interesting to dampen the A/THR response when ying in such wind conditions. Furthermore, traces of Auto-thrust Induced Oscillations, AIO, can be seen in the lower gure of Figure 3 as an increase of the engine speed amplitude.

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Figure 3: Engine agitation during turbulence and oscillating wind.

In the A/THR in use today, the issue of AIO is handled by a feedback consisting of a series of binary activation logics and non-linear ltering, called simply ANTIAIO. Although robust in performance, it has been proven rather complicated to tune. Therefore, this study also aims to see if it is feasible to replace the ANTIAIO with a controller based on fuzzy logic.

The main issue when dampening the A/THR is that it must still be able to compensate quickly when the aircraft is struck by a non-null average wind (e.g. a wind gradient or gust). Hence these winds must be detected and distinguished from the turbulence in order to adapt the A/THR response accordingly.

3.2 Objective

The objective of the adaptive A/THR can be summarised as follows: 1. Requirements: While tracking the reference airspeed closely:

(a) Limit speed deviations caused by the A/THR. (b) Limit engine agitation.

2. Prerequisites: To achieve the above, it is imperative to: (a) Detect turbulence and wind gradients.

(b) Adapt the control law accordingly.

4 Method

This section presents a short introduction to fuzzy logic and some of its applications along with equations governing the GSC. This is followed by a reasoning concerning the control strategy and nally a presentation of the actual uses of fuzzy logic in the A/THR.

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4.1 Fuzzy logic - What? How? Why? Where?

The central control concept in this master's thesis is fuzzy logic (FL), implemented in fuzzy logic controllers (FLC), through the use of fuzzy inference systems (FIS). The FIS engine used is the Fuzzy Logic Toolbox in MATLAB provided by The MathWorksr.[2]

4.1.1 What is fuzzy logic (FL)?

Fuzzy logic is essentially human reasoning in mathematical form. Imagine looking out the window at the sky, trying to judge the weather outside is suitable for an outdoors activity (e.g. ying, to use an unrelated example). If the sky is clear blue and the sun is shining, it is most likely 'good'; if it is cloudy, it might be 'acceptable'; nally, if it is raining, it is likely 'bad'. However, we all know that there are certain degrees of cloudy and rainy. If this would be fuzzied on a scale from 0 to 10, where 0 is rainy and 10 is clear, it might look like the curves in Figure 4.

Figure 4: Fuzzied input membership functions for input 'sky'.

This function is called the antecedent and can be seen as the possibility of a certain input variable ('sky') belonging to a certain fuzzy set made up by {rainy, cloudy, clear}, called membership functions. Moreover, a simple model of the human reasoning could be dened by the following rules:

1. If 'the sky is rainy', then 'the weather is bad'.(weight = 1)

2. If 'the sky is cloudy', then 'the weather is acceptable'. (weight = 1) 3. If 'the sky is clear', then 'the weather is good'. (weight = 1)

The set {bad, acceptable, good}, called the consequent, is fuzzied in the same manner as the antecedent. They might be represented by the membership functions in Figure 5. Moreover, the weight added to each rule represents how prioritised or important a certain rule is. For example, an optimist might put a lower weight on rule number 1, while a pessimist would put less weight on rule number 3.

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Figure 5: Fuzzied output membership functions for output 'weather'.

These three components yield the FIS, depicted as the output function curve below in Figure 6.

Figure 6: FIS output function.

An interpretation of the above graph could be that when starting to rain heavily or it gets very clear, the weather is most possibly bad or good respectively. On the other hand , a smaller change in the cloudiness of an already cloudy sky will not change the perception of the weather as much. 4.1.2 How does it work?

As mentioned in the previous section, fuzzy logic is an intuitive form of logic. This means that every statement is always true to a certain degree (which can of course be zero). A membership function as depicted in Figure 4 will take a truth value between 0 and 1 depending on how much the input is a member of that set. The rules are then resolved to relate the antecedents to the consequents. The consequent MF of a rule will be true to the same degree as its corresponding MF for the antecedent.

By introducing some refreshing mathematical notations; a MF A of an input x ∈ U is denoted µA(x). In the example above, this gives the set of MFs {µrainy(sky), µcloudy(sky), µclear(sky)}, and sky ∈ [0 10].

For example, if the input 'sky' is judged as 2.5, the fuzzication gives µrainy(2.5) = 0.5, µcloudy(2.5) = 0.5, and µclear(2.5) = 0. The rules above then give the consequent as

• The weather is bad to a degree of 0.5. • The weather is acceptable to a degree of 0.5.

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• The weather is good to a degree of 0.

Each truth value will then be multiplied by its corresponding rule weight. Thereafter, the consequent of each rule will be represented by the surface covered by the corresponding MF integrated along the vertical axis from 0 to the weighted truth value. Since multiple rules aect the same output, the rules must be aggregated to get a weighted output that represents the fuzzy inference. Normally, this is done by calculating the position of the mean center of gravity of the total surface (centroid method); however any other method can be used that better suits the controller design (such as bisector or middle-of-maximum). An illustration of this simple example is seen in Figure 7 below.

Figure 7: Example of an (optimistic) FIS (rule 1 weighted by 0.2). This gives in turn the FIS output function seen in Figure 8.

Figure 8: FIS output function (rule 1 weighted by 0.2).

In case a rule includes logical operators (AND, OR, NOT) relating the MFs µA(x), µB(x), the total truth value of its antecedent is calculated by the following relations [1]:

µA∪B(x) = max[µA(x), µB(x)] µA∩B(x) = min[µA(x), µB(x)]

µ6A(x) = 1 − µA(x)

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4.1.3 Why use a fuzzy logic controller (FLC)?

Depending on the nature and complexity of the controlled system or the stated performance require-ments, it might be more or less suitable to use a FLC. For a simple SISO system or nearly decoupled linear MIMO systems, it is likely that a simpler controller can be easily tuned to give a satisfactory behaviour. However when the system complexity increases or the underlying dynamics become less clear, it is possible that a controller based on for example FL is a better choice. As an example, any system which is manually controlled by a human with no deeper understanding of the system dynam-ics, can be suitable for fuzzy control. In fact, one of the main applications of FLCs is where a human behaviour and reasoning would normally give a satisfactory result, e.g. robots, automatic steering, etc. In this master's thesis, a typical example is studied. It is well known that a pilot is more than capable of controlling the speed of an aircraft without knowing neither the exact dynamics of the engines, nor the value of every single parameter that aects the speed. Furthermore, the pilot can for example have knowledge and experience of how the winds are up ahead and so has the ability to anticipate the control in a way a normal linear controller cannot.

As described in section 2, the A/THR reacts somewhat opposite to a human pilot during turbulence by trying to compensate for the oscillating airspeed rather than not reacting and instead watching the airspeed so that it stays within acceptable limits. The problem is of course that the pilot's workload is very high during some ight phases, which is one reason why the A/THR is used. Since there is a quite signicant discrepancy between the pilot's and the A/THR's behaviour, it might be interesting do make the A/THR behave more "human" under certain circumstances.

4.1.4 Where to use fuzzy logic?

In the eld of aircraft speed control, a fuzzy logic controller can either be implemented as a complete speed controller or as an adaptive part of a more classical controller. This master's thesis discusses the latter example, i.e. fuzzy logic is used to augment a linear controller. However there exists research on the use of a FLC to control the glide slope and vertical speed of an aircraft during the descent and landing.[3]

The adaptation of the controller parameters can be done either by feedback or feed forward. In the case of this study, the general approach is feed forward, although the speed deviation is used as an input. The implementation is illustrated by the simplied model in Figure 9.

Figure 9: Implementation of the FIS in the (very) simplied aircraft model. 4.2 Generalised speed control (A/THR)

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Figure 10: Simplied model of longitudinal aircraft dynamics. Recalling that the dynamic pressure q = 1

2ρV 2

T AS and starting from the drag equation, assuming a small ight path angle, gives the equation

m ˙VT AS = −1 2ρSV

2

T ASCx+ F − mgγ. (1)

Together with the assumption of stationary ight (i.e. no acceleration in Z) mg = L = 1 2ρSV 2 T ASCz⇒ 1 2ρSV 2 T AS = mg Cz (2) Equation 1 becomes m ˙VT AS = −mg Cx Cz |{z} 1/f +F − mgγ = −mg f + F − mgγ. (3)

The command to the engine is given on the form ˙ Fc= k1∆V | {z } Direct + k2V˙T AS | {z } Damping + mg 2 VT AS Nz,c | {z } F eed f orward (4) where the three parts represent a proportional compensator (∆V = VT GT − VCAS), a desired damping and a compensation for the vertical load factor commanded by the longitudinal law. Furthermore, the speed control works under the following assumptions:

• Constant aircraft mass and lift-to-drag ratio. • Small FPA (sin γ ≈ γ).

• Ideal engine response ( ˙F = ˙Fc).

• Perfect longitudinal control (Nz = Nz,c, normally _NN_z,cz = _{1+τ p}1 but τ is negligible in comparison to the engine time constant).

Time derivation of Equation 3 gives

m ¨VT AS = − ˙ mg f | {z } =0 + ˙F − mg ˙γ. (5)

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Which, together with the expression for the vertical load factor Nz= VT AS˙γ

g ⇒ ˙γ =

gNz

VT AS (6)

yields the expression

m ¨VT AS = ˙F − mg 2 VT AS

Nz. (7)

Under the ideal engine assumption above, Equation 4 inserted in Equation 7 gives

m ¨VT AS = k1∆V + k2VT AS˙ . (8)

By taking the Laplace transform of this equation, its frequency domain representation becomes

mp2VT AS = k1∆V + k2pVT AS. (9)

Hence the speed transfer function can be dened as G := VCAS

Vtgt =

k1

mp2_{− k}_{2p + k1}. (10)

remembering that VCAS =q_ρρ

0VT AS. For an harmonic damped oscillator, the transfer function is

H = ω

2

p2_{+ 2ζωp + ω}2. The coecients k1 and k2 can then easily be identied as

k1 = mω2

k2 = −2mζω.

Together with Equation 4, this yields the nal expression for the (open loop) command as ˙ Fc mg = 1 VT AS ω2 g VT AS∆V − 2ωζ g VT ASVT AS˙ + gNz,c . (11)

The current GSC uses an ω of X in approach and a ζ of X, this gives the closed loop transfer function GGSC and the related sensitivity function SGSC = 1 − GGSC below.

(Removed for condentiality reasons) (12)

Since disturbance attenuation is one of the main topics studied in this project, it can be interesting to look at the singular values of the sensitivity function. These can be seen in Figure 11 for the frequency range 0.1→1Hz.

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Figure 11: Singular values for the sensitivity function SGSC, in approach.

From the singular values, it is clear that the overshoot during the oscillating wind shown in section 3.1, Figure 2, is caused by bad disturbance attenuation due to the tuning as described in section 2. 4.3 Control strategy

The main advantage of using a FLC instead of a classic controller is that it can easily adapt to dierent conditions without the need of several separate control laws. As mentioned above, the control parameters can easily adapt according to the current state; however the question is when and how to change the parameters. One large disadvantage on the contrary is that such a controller needs to be tuned with the help of expert knowledge (e.g. pilots and control law experts in this case) to be optimally tuned, it is not always an easy task to tune it with respect to classical control parameters. Furthermore, a FLC does not give any guarantee of stability or robustness for the closed loop system; however it is not said that these criterion cannot be achieved with an adequate tuning.

The main topics studied is the case of a turbulent approach with or without wind gradients. Since the current A/THR is well tuned for reference tracking, the fuzzy logic controller will not alter the control law during speed changes or in calm air when a larger control signal ( ˙N1) is more preferable. This can be formulated as the following logical statement:

• If ' no speed change is being eectuated' and 'no wind gradient present', then 'fuzzy A/THR tuning is used', else 'classic tuning is used'.

This essentially means that as soon as the aircraft switches from speed tracking to speed hold, the A/THR switches to fuzzy tuning.

Since the aircraft speed response is non-linear and non-symmetric, it could be interesting to include some parameters in the fuzzy controller that are not used in the classic A/THR. The parameters which might be interesting will depend on the current ight phase (i.e. cruise, approach).

4.3.1 Approach specic strategy

During the approach phase, the aircraft is descending at a certain glide slope. If the current ight path angle would be too steep, the pilot must perform a nose up action, and vice versa if the slope would

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be too at in order to get back to the reference glide slope. This allows the pilot to also control the speed in some cases without using the throttle.

Nonetheless it is imperative to ensure a minimum level of performance when the aircraft is near the edges of its ight envelope. In approach the aircraft is most likely to get close to or below VLS. This is handled by the 'guaranteeing principle' for the thrust derivative described in 4.4.3.

4.3.2 Cruise specic strategy

During cruise at altitude and speed hold it is not possible to control the speed with the ight path angle as it would lead to the a conict between the two laws. However, according to pilot feedback (see Appendix A) it is less important to have a very responsive A/THR during cruise since the speed is relatively far from both VLS and VM O. Therefore larger speed variations, up to a certain level, can be allowed in order to reduce engine agitation. Conclusively, this means that more authority is given to pacifying the A/THR (by modifying Nx,est and reducing ω) than ensuring that the speed deviation is kept small.

4.4 Application of fuzzy logic in the A/THR

The idea of using fuzzy logic in the speed control is to tune the response of the A/THR such that its behaviour is more intuitive and it corresponds better to that of a human pilot. There are two main applications where fuzzy logic can be used to modify the control law, one in the derivative term of Equation 11 and one directly in the closed loop transfer function in Equation 10. In the derivative term it is in the estimation of the longitudinal load factor Nx. Obviously the frequency content of the estimated acceleration will aect the frequency content of the control signal. For example, a very turbulent Nx estimation will lead to a heavily oscillating thrust. From Equation 11, the feedback part can be isolated as the two rst terms on the right hand side, i.e.

˙ Fc mg ! F DBK = ( Nx = ˙ VT AS g ) = ω 2 g ∆V − 2ωζNx. (13)

It is clear that reducing ω will directly reduce the A/THR agitation since the control signal is pro-portionally reduced. Furthermore, it is interesting to imitate the pilot's behaviour in situations where either the engines are not used to maintain the speed or where the pilot would have commanded more or less thrust than the A/THR in order to avoid excessive under- or overspeeds. This is done by under certain conditions implying certain limits directly on the thrust command derivative, given as ˙N1,c. 4.4.1 Nx estimation hybridisation

The main idea concerning of the Nx estimation is to combine air- and ground accelerations by using a complimentary lter, in order to get an estimation of the real acceleration.

General principle

The relation between the airspeed and the ground speed can be written as

VT AS = VGN D+ Wmean+ v, (14)

where the wind is represented by an average wind, Wmean, plus a white noise, v, representing the turbulence. Derivation with respect to time yields

˙

VT AS = ˙VGN D+ ˙Wmean | {z }

=0

+w. (15)

To get a state space representation, the state is chosen as x = ˙VT AS, the input as u = ¨VGN D, and the output as y = ˙VT AS+ w. Under the assumption that the aeroplane acceleration only depends on the ground acceleration, the system becomes

˙

x = V¨T AS = ¨VGN D = Ax + Bu

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where A = 0, B = 1, and C = 1. Since the output disturbance w is considered to be Gaussian white noise, the state is optimally estimated with a Kalman lter on the form

˙ˆx = Aˆx |{z}

=0

+Bu + Ka(y − C ˆx) . (17)

With Equation 16 inserted, this becomes

p ˙Vest = p ˙VGN D+ Ka ˙VT AS− ˙Vest

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Rewritten on transfer function form, this gives ˙ Vest = p p + Ka ˙ VGN D+ Kap p + KaVT AS = p p + Ka ˙VGN D+ KaVT AS . (19)

The lter gain Ka in is inversely proportional to the time constant τ as τ = 1/Ka. In addition, the horizontal load factor is Nx= ˙V /g. Equation 19 can therefore be written as:

Nx,est= p p +1_τ ˙ VGN D+1 τVT AS 1 g (20) Nx,est fuzzication

For the load factor estimation above, there are at two ways to use a fuzzy approach. First of all, the washout time constant can be modied to alter the ratio of air/ground frequency content in the signal. This could be useful in turbulence where the high frequency content of the air acceleration signal increases but that of the ground acceleration remains relatively slow varying.

In addition to the fuzzy time constant, a band stop/pass-lter can be implemented at the input of VGN D and VT AS in order to attenuate the auto-throttle induced oscillation described in section 3. Finally, it is possible to replace the rst order washout lter in Equation 20 with one of second order in order to get a steeper frequency cut-o.

Washout time constant hybridisation Given the high pass lter

H(p) = p

p +1_τ (21)

there are two ways to modify the "eective" time constant. One is to directly change the time constant via a command of the form "If 'turbulence is XX' then 'τ is YY'". This can be easily implemented in a fuzzy controller and gives a simple control architecture. However, this approach implies that the lter must be reinitialised every time the time constant changes, which can be often by the nature of the fuzzy controller. Thus it might prove hard to implement this approach in practice. Another way is to run the signal through several washout lters in parallel and then use the fuzzy controller to determine how much "weight" should be given to each washed signal. I.e if the signals {s1, . . . , sN|si ∈ R} are a discrete set of washed signals and the coecients {c1, . . . , cN | 0 ≤ ci ≤ 1,P_kck = 1}, then the resulting signal will be

sres= N X

i=1

cisi. (22)

For example, the time constants τ = {5, 10 15} are used, giving the Nx estimations: Nx,5s, Nx,10s, and Nx,15s. The total estimation becomes

Nx,est = c5sNx,5s+ c10sNx,10s+ c15sNx,15s

The coecients c5s, c10s, and c15s can in turn be related to a specic turbulence severity, e.g. light turbulence gives {1, 0, 0}, medium gives {0, 1, 0}, and severe gives {0, 0, 1}. Any intermediary level would give a combination of the three, respecting that the sum must be 1.

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The advantage of this method is that it is much easier to implement and is more robust due to the use of constant time constants in parallel. It is however slightly more computational heavy as it will calculate all washed signals regardless of whether they are in use or not for the moment. By dening the set of dierent hybridisations as {medium, medium − slow, slow, very slow}, the fuzzy rules governing the hybridisation can be expressed as follows.

• If 'no turbulence' then 'the hybridisation is medium'.

• If 'the turbulence is light' then 'the hybridisation is medium-slow'. • If 'the turbulence is medium' then 'the hybridisation is slow'. • If 'the turbulence is severe' then 'the hybridisation is very slow'.

In order to determine which τ is appropriate to use for each turbulence level, a series of simulation with a xed τ were performed and evaluated for speed maintenance and engine agitation. For medium turbulence, the curves in Figure 12 were obtained.

Figure 12: Comparison of dierent τ in medium turbulence. Furthermore, the curves obtained during severe turbulence are shown in Figure 13.

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Figure 13: Comparison of dierent τ in severe turbulence.

A τ of 100 seconds is clearly too large, since it corresponds to a cuto frequency of 0.01Hz, the signal will be dominated by the ground acceleration, i.e. the inertial acceleration of the aeroplane. Since this acceleration is loosely coupled with the air acceleration, it risks giving a very sluggish response in the airspeed, which is seen in the gure above during severe turbulence.

For the other τ above, the curves are almost identical, which basically means that an arbitrary τ in the span of those values can be used. However, since the engine agitation is a secondary objective, as described in section 3, a smaller τ is preferable. The τ chosen for the hybridisation are seen below in Table 1.

Medium Medium-slow Slow Very slow

τ [s] τ0 2τ0 4τ0 10τ0

Table 1: Implemented Nx,est hybridisation time constants. These dierent τ correspond to the Bode diagrams in Figure 14.

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Figure 14: Bode diagram comparing complementary high pass lters with dierent τ.

It is reasonable to assume that a lower frequency limit for wind classed as turbulence is around 0.03Hz (referring to Figure 1). With this in mind, it is quite clear that a time constant of 50s gives a signal which is heavily dominated by ground content. Although since the disturbance attenuation is relatively large at low frequencies, it is possible that such a high time constant is unnecessary. Conclusively, Figure 15 shows the membership functions of the respective hybridisation coecients as a function of the detected turbulence level.

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Figure 15: Nx,est hybridisation coecients as a function of turbulence (TURBLVLDET). Band-stop/pass lter

The idea of using a band-stop/pass lter is to attenuate the 0.03 Hz oscillation which causes airspeed overshoots in the classic A/THR and compensate the lost frequency content with the ground speed. By starting with the transfer function ( ˙VGN D, ˙VT AS) → ˙Vest from Equation 20 and redening ˙VT AS := GBSV˙T AS+ GBPV˙GN D, the transfer becomes

˙ Vest = p p +1_τ ˙ VGN D+ 1 τ p +1_τ GBSV˙T AS+ GBPV˙GN D = p p +1_τ ˙ VGN D+ GBS VT AS τ + GBP VGN D τ . (23)

A bode diagram of the band-stop (GBS) and band-pass (GBP) lters of Equation 23 are shown in Figure 16 below.

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Figure 16: Implemented band-stop and band-pass lter. The respective transfer functions are

GBS = (Removed for condentiality reasons) (24)

GBP = (Removed for condentiality reasons) (25)

(26) Just as for the nominal case, the complementary lter could optionally be replaced by a higher order one.

Higher order washout lter

As mentioned above, it might be interesting to use a higher order washout lter for the complementary lter. This is possible due to the fact that the high pass lter

H2(p) = p

2

p2_{+ 2ζωp + ω}2. (27)

has the relative degree zero, i.e. the output must be derived zero times in order to see the input signal.[4] When using a higher order washout, the frequency cut-o will be much more distinct and thus the damping will be greater at the lower frequencies, see Figure 17 below. However, this might not be favourable during for example a wind gradient as it is desirable that the aeroplane reacts quickly to such phenomena. Note that the cut-o frequency is the same in both cases. However, it is important to notice that the cut-o frequency in Equation 21 is dened as ωc = 1/τ and in Equation 27 as ωc= 2π/τ. This means that the 'eective' τ will not be the same in the two cases.

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Figure 17: Comparison of rst and second order complementary washout lter.

Clearly the problem with the substitution of a rst order high pass lter for a second order one is that more phase lead is introduced. This could potentially be compensated for with a phase lagging all-pass lter. Although this approach might work in practice, a more proper solution would be to derive a second order Kalman estimation. Such a solution would also require information about the derivative of the disturbance (i.e. wind acceleration). This means that the assumption of a mean wind Wmean is no longer valid and that the wind must be added as a state equation in Equation 16.

The implementation of a higher order washout focuses on when to switch between the rst and second order hybridisation. This requires that both turbulence and wind gradients are detected with precision since a second order lter will radically dampen the estimated acceleration signal at lower frequencies (where wind gradients usually reside).

4.4.2 Closed loop gain modication

Since the regulation of the Nx estimator is not applied directly on the closed loop transfer function, it will not aect the overall performance of the control law as much as modifying ω but rather help to cancel out any anomalies in the control signal. Therefore it is interesting to modify the closed loop dynamics directly in order to handle the overall response of the control law. The two parameters to alter as described above are the pulsation and the damping. For the aeroplane to have a suciently fast response to speed changes, the gain must be high, although in turbulence the gain should be reduced to avoid engine agitation. Furthermore in wind gradients where the speed relative to the air changes slower than during turbulence, the gain must also be high in order to maintain the commanded speed. This boils down to a few simple rules regarding the general adaptation of the gains:

• If 'no turbulence' then 'the gain is high'. • If 'wind gradient' then 'the gain is high'. • If 'turbulence' then 'the gain is lowered'.

These rules can be directly translated into fuzzy rules, which is very handy for the implementation. In addition, the damping will be left rather unchanged (0.7 normally, a little higher during approach or in high lift conguration), thus the gain will be proportional to the pulsation ω. An obvious problem with this control is the scenario of a wind gradient in turbulence. The turbulence can quite easily be

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detected since it has high frequency content whereas the wind gradient does not and might not be detected in the case of this unfortunate combination.

For the case of turbulence, it is important that it is detected and classied accordingly. The turbulence level is basically the normalised dierence between the largest and smallest measured wind acceleration during a short period of time. The signal is then ltered to get a good estimation of the turbulence severity.[5] However, since the signal is ltered, the estimation has a convergence time (depending on the lter) before it delivers a just estimation. For a rst order lter at τ seconds, this time is roughly 3τ seconds.

In general there are three levels of turbulence: light, medium, severe. The turbulence level depends on the standard deviation of the assumed Gaussian wind speed. In addition to these three, extreme turbulence is essentially the case of severe turbulence at high altitude. To counter these levels of turbulence, it can be suitable to associate each level with a corresponding gain, i.e. in order not to paralyse the control as soon as there is a little turbulence. A simple and intuitive way to do this is according to the following rules:

• If turbulence is light, then the gain is medium. • If turbulence is medium, then the gain is low. • If turbulence is severe, then the gain is very low These rules correspond to the membership in Figure 18.

Figure 18: Membership functions for ω as a function of turbulence (TURBLVLDET). The baseline gains chosen for approach, high-lift, and clean are shown in

Approach (APP) High lift (HYP) Cruise (CRZ)

ω0 [rad/s] 0.16 0.14 0.1

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For the denition of the fuzzy set {high, medium, low, very low}, simulations with a xed ω were performed, as seen in Figure 19 and Figure 20. For medium turbulence, the following curves were acquired:

Figure 19: Comparison of dierent gains in medium turbulence. In addition, the curves obtained during severe turbulence were the following:

Figure 20: Comparison of dierent gains in severe turbulence.

It is clear that the both the speed deviation and the engine agitation are improved when the gain is lowered to a certain limit. From the above curves, the following gains can be derived as appropriate for the dierent levels of turbulence.

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Turbulence None Light Medium Severe

Gain High Medium Low Very low

ω [rad/s] ω0 0.75ω0 0.5ω0 0.375ω0

Table 3: Gain as a function of turbulence level.

As an illustration, singular values of the sensitivity functions to Equation 10, i.e. |S| = |1 − G|[4], for the above ω are shown in Figure 21.

Figure 21: Singular values corresponding to the gain at dierent turbulence levels in approach. The peak frequency of the sensitivity function is directly proportional to ω. As described in the problem section, the problem frequency for the classic A/THR is around 0.03Hz, which also is the frequency in the Nx estimation around which the band stop/pass lter is centred. This means that it could also be interesting to adjust the centre frequency as a function of the current ω, indirectly as a function of turbulence. Not doing so means that the double hybridisation would only be optimal for the nominal gain case during approach and its impact would be degraded as ω changes due to turbulence. This further means that the engine agitation would not be lowered as much during severe turbulence as during lower levels of turbulence. Another factor which is aected by the frequency shift of the singular values is the Nx estimation. Lower frequency content is amplied more at higher levels of turbulence and is also included to a greater extent due to the increase of the complementary high pass lter time constant. This means that the total "amplication" at low frequencies risk being worsened by these two factors, and thus further motivates additional ltering in order to reduce the eect on the airspeed.

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Figure 22: Fuzzy gain as a function of turbulence.

Since the fuzzy controller will be deactivated as soon as a wind gradient is detected, the gain will automatically be high. It might however be interesting to increase the gain in situations of excessive speed deviations, since lowering the engine agitation is a secondary objective.

Adaptation of Nx,est hybridisation due to gain lowering

As mentioned above, a possible issue when lowering the closed loop gain is that the hybridisation using band stop/pass lters, meant to dampen the resonance peak in the sensitivity function, becomes less well adapted. Since the resonance frequency is proportional to the closed loop pulsation, a formulae to alter the centre frequency can be written as

ωadapt= ωresω

ω0 (28)

where ωresis the resonance frequency to the sensitivity function in Equation 12 (i.e. the nominal centre frequency which varies with the ight phase: cruise, high lift, approach), ω is the current gain, and ω0 is the nominal gain as per Table 2. Together with the general transfer functions for band-stop/pass, this gives HBP = 2kpζωadaptp p2_{+ 2ζω} adapt+ ω2adapt , HBS= ks(p 2_{+ ω}2 adapt) p2_{+ 2ζω} adapt+ ωadapt2 (29) where ks and kp are arbitrary constants related to the maximum/minimum lter gain. It is clear the the band-stop lter has its zeros on the imaginary axis, therefore it can be interesting to add a rst order term as in Equation 26 in order to get LHP zeros and to "soften" the frequency response. Adding the term 0.03ω

ω0 to the nominator in HBS (the term ω/ω0 maintains the relation between the real and

imaginary part of the zeros for all ω) and choosing k = 1 yields the Bode diagram below for dierent modied ω.

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Figure 23: Bode diagram of the adaptive band stop-/pass lter for dierent ω.

Theoretically this should give a better adapted hybridisation, especially for very low ω. As an example, in severe turbulence (ω = 0.06) the static hybridisation will take 0.537 parts of the air acceleration around the resonance peak (which has moved to 0.0088Hz) while the adaptive hybridisation will use 0.108 parts. It can however be argued whether the low frequencies are already dominated by the ground accelerations and thus the adaptation will have very little impact on the closed loop. Wind gradient detection

Both the Nx estimation and the closed loop gain modication require that superposed winds during turbulence are detected in order for the speed control law to remain well adapted to the current conditions. It is therefore interesting to investigate by which parameters such a wind is distinguished and how to estimate its acceleration. As mentioned above, a wind gradient during turbulence can be dicult to detect and anticipate. However, with the help of the turbulence detector, it should be possible to get a rough estimation of the current wind condition with the following procedure:

Dene the wind speed as

Vw,est = VGN D− VT AS

To get a distinction between the low and high frequency contents of the wind, it is ltered in parallel at 1 and 10 seconds. Since the interesting content is mostly at low frequencies, the ltered wind is pseudo-derived using a high pass lter at 1 second.

The amplitude of the 10 seconds ltered signal is then compared to the presumed turbulence acceleration amplitude given the estimated turbulence level. If the ltered signal amplitude is larger than the turbulence induced one, it is assumed that a wind gradient is detected. The wind gradient acceleration is then taken as the 1 second ltered wind acceleration. Below in Figure 24 is a schematic overview of the wind gradient detector.

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Figure 24: Wind gradient detection overview.

The lookup table TURBLVLDET TO DVWPTEST for the wind acceleration threshold uses the values in Table 4.

TURBLVLDET 0 0.2 0.5 1 1.5 2

DVWPTEST 0 0.3 0.5 0.6 0.7 0.8

Table 4: Wind gradient detector, turbulence acceleration threshold.

The lookup table WINGRADACC TO KWINGRAD for converting the wind gradient acceleration into a corresponding severity factor uses the values in Table 5

WINGRADACC 0 0.3 3

KWINGRADDET 0 0 1

Table 5: Wind gradient detector, conversion from wind acceleration to severity factor.

As a concluding example, Figure 25 shows a simulation that illustrates the function of the wind gradient detector.

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Figure 25: Evaluating simulation of the wind gradient detector.

There are two major drawbacks with this method. Firstly, it's error is heavily dependent on the turbulence estimation. This can be problematic during the transient phases where turbulence is present but has not yet been detected due to the "settling time" of the turbulence detector. Since this means that the turbulence threshold will be close to zero in the wind gradient detector, any turbulence will be detected as a wind gradient. This further implies that the closed loop gain will be kept on a higher level than the turbulence detector would normally propose. On the bright side, the A/THR response will not be worse than that of the classical since the gain will never be higher than the baseline.

Secondly, the estimated wind gradient acceleration is strongly inuenced by the turbulence acceler-ation. A consequence is that an increasing wind amplitude might not be detected until it has surpassed the amplitude of the turbulence. This makes the detection rather slow in severe turbulence, as shown by Figure 25 where the detection is lagging behind by around 5 seconds.

4.4.3 Limiting the command: ˙N1,c limits

With the above approach of pacifying the A/THR, there is a risk that the command becomes too slow in the case of heavy turbulence and wind gradients and hence does not react fast enough when the aircraft approaches VLS or VM O or when it gets too far from its target. It is therefore imperative to introduce boundary conditions for the command as a function of the speed in comparison to the three aforementioned speeds VLS, VM O, and VT GT. In a turbulent approach, which is the case primarily studied in this master's thesis, it is more important to privilege overspeeds than underspeeds since the speed is close to VLS, according to pilot input (see Appendix A). During cruise, larger speed deviations can normally be accepted, however close to VM O the overspeed limit should be tighter instead.

The control signal limitations should be activated at the same premises as the fuzzy A/THR tuning, with the exception that the control signal should not be limited in calm air. Formulated as a logical rule this can be written as:

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A problem that can arise when applying forced limits is that it can easily create an over- or undershoot of the target speed. This is due to the linear control law (Equation 11) not being capable of compen-sating the thrust deviation (compared to the linear response) caused by the forced limits. The forced limits must therefore be applied tight enough on VCAS to avoid too large deviations, yet loose enough to avoid shooting past VT GT. One solution is to have a tighter limit on over-/underspeed where the A/THR is not allowed to command an acceleration/deceleration and a looser limit which forces the engines to decelerate/accelerate. Formulated as a fuzzy rule, this so called "guaranteeing principle" can be written as:

• If 'the underspeed is small' then 'the thrust derivative is at least slightly negative'. • If 'the underspeed is large' then 'the thrust derivative is greater than zero'.

• If 'the overspeed is small' then 'the thrust derivative is at max slightly positive'. • If 'the overspeed is large' then 'the thrust derivative is less than zero'.

The values of the forced thrust derivative for large speed deviations should correspond to the value which the classical A/THR would have commanded. Using adequate values, the guaranteeing principle is illustrated in Figure 26 below in the case of an approach.

Figure 26: Limits of ˙N1,c during speed deviations in approach.

As can be seen in the above gure, the lower limit is closer to zero than the upper limit. This is due to the fact that the speed during approach equals VLS + 5kt. Moreover, a rule of thumb is to allow 1/3 of the speed deviation in underspeed and 2/3 in overspeed when ying close to VLS, which is reected by the above limits. In the normal case where the aircraft is not ying near VLS, e.g. during cruise, the two facing curves above would be symmetrical.

In addition to the guaranteeing principle for speed deviations, the limits when surpassing VLS or VM O are formulated as:

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• If 'the speed is conrmed (2 seconds) over VM O', then the thrust derivative is less than zero'. The above rules are illustrated in Figure 27 below.

Figure 27: Limits of ˙N1,c near the edges of the ight envelope.

In case the linear law does not manage to compensate the steady state error during turbulence, i.e. the speed variations are not centred around zero, an additional condition is needed in order to avoid ending up in a general overspeed or underspeed. This condition reects the pilot's reaction to slightly adjust the throttle when noticing a stationary speed deviation and was described during the pilot discussions (see Appendix B). In fuzzy terms, this can be expressed as

• If 'the speed deviation is consistently greater than zero', then 'the thrust derivative is at the maximum slightly positive'.

• If 'the speed deviation is consistently less than zero', then 'the thrust derivative is at the minimum slightly negative'.

This is implemented by integrating the speed deviation and activating the limits when the integration reaches a certain threshold (ex. 2kt overspeed during 10 seconds). The integration is reset to zero if the speed deviation or the speed derivative changes sign, or if they have the same sign. Once the threshold has been surpassed, the limits are deactivated only when the speed deviation and the speed derivative get the same sign again. An illustration of these limits are shown below in Figure 28. Note that DSPDCASDMD is positive in underspeeds and negative in overspeeds.

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Figure 28: Consistent speed deviation conrmation.

The eect of these limits are seen in Figure 29 for an example of severe turbulence.

Figure 29: Limits due to consistent speed deviations, triggering and implied ˙N1,c-limits. As can be seen in the gure above, the main function of of these limits is to compensate for the sluggishness of the modied linear law in the event of a consistent speed deviation. The tendency of the linear law in this case is likely due to the speed deviation term in Equation 11 is linear in ω2_{. This} means that in severe turbulence, the weight of ∆V is heavily reduced leading to a slower attainment of the reference speed.

Conclusively, the objective of the forced limits is to assure a certain increase or decrease of the thrust when the speed strays from the target. In addition, the application of these limits is softened by the use of several levels of implication (e.g. 'don't accelerate' to 'decelerate'). If the conditions

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described above are not met, the limits are set to their default values.

4.4.4 Limiting the command: Adaptation to glide slope deviations

According to test pilots (see Appendix B), the A/THR risks compensating too much when the longitu-dinal control is tracking a given glide slope (GS) during a turbulent approach. The most obvious cases are the combinations of underspeed and over glide or over speed and under glide where it often suces to do a nose down or up action respectively to regain the target speed and the reference glide slope. Since this is the natural reaction of the longitudinal control, the A/THR risks overcompensating the speed, causing an over- or undershoot and working against the pilot (manual or automatic).

The hypothesis in this case is that the longitudinal control (pitch up/down) will naturally compen-sate the speed under certain conditions. In order to nd in which conditions the longitudinal control will suce, start with the expression of the total energy given a certain ight point (VCAS, VT GT, h, m)

Etot = 1 2mV

2_{+ mgh.} ₍₃₀₎

Dene two points A and B, as in Figure 30 below, where A is the deviated point and B is at the target speed and glide slope.

Figure 30: Glide slope deviation scenario. The dashed line shows the target glide slope. Given that the glide slope deviation is small (ρA ≈ ρB), the energy dierence between the two points can be written as

∆E = 1 2m V

2

CAS− VT GT2 + mg∆h. (31)

For the longitudinal law alone to be able to attain and retain the target speed and glide slope, this dierence must obviously be zero. However, since the zero condition is not likely to be met in practice, it is more realistic to check if the energy dierence is greater than zero, i.e. whether it is possible to reach the target values (not necessarily in steady state).

The limits would logically be applied to ˙N1,C as for the guaranteeing principle. It is easy to write a fuzzy statement based on the above reasoning:

• If 'in underspeed' and 'over the glide slope' and 'the total energy dierence is equal to or greater than zero', then ˙N1,C is upper limited.

• If 'in overspeed' and 'under the glide slope' and 'the total energy dierence is equal to or greater than zero', then ˙N1,C is lower limited.

Since it has not been investigated how much the longitudinal law can compensate speed deviations, a small margin is left around zero. Figure 31 and Figure 32 illustrate the implemented limits. The curves change very little with variations in the glide slope and speed variations, which is why the gures below only show the curves for a single value of DCASTGT and GSDEV respectively.

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Figure 31: Limit of ˙N1,c when in underspeed and over the glide slope. Additionally, for the case of overspeed and under the glide slope, the following applies:

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4.4.5 Limiting the command: Feed forward Nz,c limits

A problem that can arise when softening the A/THR response according to the above techniques (lowering of ω and modication of Nx,est) is that the feed forward term (gNz,c) in Equation 11 becomes too inuential. Since this term is used to compensate for any longitudinal command (e.g. pitch up or down) in order to maintain the speed, it will naturally be proportional to the glide slope deviation in approach. For simplicity, the ratio between the commanded vertical load factor Nz,AT and the feed forward one, Nz,c, can be called the relative authority.

In calm air or in very light turbulence this is not an issue due to the size of the feedback. However in heavier turbulence it is clear that the engine agitations will be caused primarily by the feed forward commanded by the longitudinal law since Nz,AT will be small due to ω being lowered, hence it can be interesting to limit the feed forward under certain conditions.

The current limits on Nz,c are dened according to Figure 33 below

Figure 33: Classical A/THR limits on Nz,c.

According to the above reasoning, the feed forward command should be limited when the relative authority becomes too small, that is when the feedback becomes small or the feed forward becomes large. Since the amplitude of the feedback term in the fuzzy controller will increase with the turbulence severity but will decrease due to ω being lowered, it can be reasonable to assume that it will follow a similar curve as the classic A/THR. Additionally, the feed forward term will increase monotonically with the turbulence level. Therefore, the feed forward should be limited tighter at low turbulence levels and should gradually be loosened with the turbulence in order to maintain roughly the same relative authority for all levels of turbulence. This can be formulated with the following fuzzy rules:

• If 'the turbulence is light' then 'the Nz,c limit is tight'. • If 'the turbulence is medium' then 'the Nz,c limit is loose'. • If 'the turbulence is severe' then 'the Nz,c limit is very loose'. • If 'the speed deviation is excessive' then 'the Nz,c limit is nominal'.

These rules are illustrated in gure Figure 34 below, using adequate values (not the actual tuned values) for the limits.

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Figure 34: Fuzzy A/THR limits on Nz,c for light,medium, and severeturbulence.

A possible issue when limiting the feed forward can be that the glide slope deviation becomes too large during heavy turbulence. Therefore the feed forward needs a certain authority to compensate the longitudinal law commanding Nz which give a non-negligible impact on the speed.

Taking the above observations into account, the limit as seen in Figure 35 on the absolute value of Nz,c was implemented.

Figure 35: Implemented limit on |Nz,c|as a function of turbulence (TURBLVLDET) for dierent speed deviations.

In the above FIS, the speed deviation is used directly as an input, although it can be discussed whether it would be better to use the glide slope deviation instead. For this application the speed deviation is used since it makes it easier to tune the FIS over a larger ight envelope. As a concluding example, Table 6 shows the relative authority expressed as the ratio between the standard deviations of Nz,AT and Nz,c, for three dierent levels of turbulence. The fuzzy controller includes the modication for ω and Nx,est as described in sections 4.4.2 and 4.4.1.

Turbulence Light Medium Severe

Classic 2.16 2.33 1.86

Fuzzy 1.37 0.99 0.85

Fuzzy (Nz,c limited) 2.25 3.09 2.37

Table 6: Examples of the ratio σ(Nz,est)

σ(Nz,c) for dierent turbulence severities.

With the implemented limits, the ratio between the feedback and the feed forward are slightly increased which means that the modications of Nx,est and ω will maintain their impact on the engine

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agitation during turbulence as desired.

4.5 Simulation,validation, and evaluation 4.5.1 Variables for evaluation

As mentioned above, the objective of the fuzzy logic based A/THR is to reduce the speed overshoot caused by the engines, as well as the engine agitation. The A/THR-caused speed deviation is given by the absolute value of the calibrated airspeed minus the target speed and the projected wind in the aeroplane frame of reference, i.e

∆VCAS,A/T HR= |∆Vtotal− Vwind| = |VCAS − VT GT − VGN D+ VT AS|. (32) Intuitively, it is the mean value of this variable that is interesting to look at during turbulence.

For the engine agitation it is mainly the variation of the engine fan speed that is important, i.e. the standard deviation of N1. However, in a scenario which is not purely turbulent (e.g. when a wind gradient strikes or a speed change is commanded), neither the mean value of N1, nor of ∆VCAS,A/T HR, will be centred around a certain steady state value. This means that such scenarios must be evaluated qualitatively using entire time series.

4.5.2 Scenarios for validation and evaluation

The main interest is to improve the A/THR response during turbulence. Hence dierent levels of turbulence will make up the base in all simulations. The turbulence input is modelled as a zero-mean Gaussian distribution and tuned in accordance with pilot experience to give a realistic aircraft response. At low altitude, the standard deviations corresponding to the dierent levels are as follows.

Turbulence (TURBLVL) σx [kt] σy [kt] σz [kt] σp [deg/s]

None (0) 0 0 0 0

Light (1) 1 1.5 1.5 0.6

Medium (2) 2.5 3 3 1

Severe (3) 3 4.5 4.5 2

Table 7: Wind speed standard deviation as a function of the turbulence level.

This yields the following wind speed (in the aircraft frame of reference) and the corresponding turbulence severity detection when simulating on SIMPA.

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Figure 36: 'Pilot agreed turbulence' and detected turbulence level (TURBLVLDET).

Furthermore, the following fuzzication was used to classify the turbulence in approach for all fuzzy controllers using turbulence as an input.

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Figure 37: Turbulence classication in approach for fuzzy controller input.

In addition to pure turbulence, it is also interesting to simulate more realistic scenarios, in particular cases in which a wind gradient is hidden in turbulence.

4.5.3 Simulation environments

Two dierent simulation environments were used during the course of the project. For the preliminary results and testing of dierent techniques and parameters, an environment called SIMBOX, consisting of a simulink model of the primary ight computer connected to a ight dynamics simulator was used, see Figure 38.

Figure 38: Closed loop simulator SIMBOX, simulink based controller.

For the nalising of the tuning and simulation of the recorded winds, a closed simulator called SIMPA (Simulation Pilote Automatique) was used, see Figure 39. This required the simulink models to be converted into C-code and compiled before they could be used for simulation.

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Figure 39: Closed loop simulator SIMPA, compiled C-based controller.

An issue when switching from one simulator to the other is that SIMPA does not use the same precision as SIMBOX regarding for example the comparison of two 'identical' values (e.g. when checking speed changes). SIMBOX uses the standard smallest number handled by MATLAB, = 2.2204∗10−16 while SIMPA works with tolerances of ≈ 10−5_.

5 Results

5.1 Non-retained solutions

The modications below were not implemented in the nal controller for various reasons. 5.1.1 Higher order Nx estimation

Due to problems with modelling the the second order Kalman lter and the increased phase lead, the second order Nx estimation has not been implemented.

5.1.2 Variable stop/pass frequency in the Nx,est hybridisation

Due to problems tuning the lters properly and uncertain eects on the closed loop system, the proposition of using a varying centre frequency for the band stop/pass lters has not been implemented. 5.1.3 Glide slope deviation adaptation

Due to diculties when regulating the implied limits, the adaptation to glide slope deviations has not been implemented in the nal controller design.

5.2 Simulation results

The below simulations were conducted on the platform SIMPA. In all simulations of the "Fuzzy A/THR", the following modications were in use:

Modication In use Var. in SIMPA

Nx,est hybridisation BADISATHRFUZZY

- Nx,est fuzzication Yes

- Band-stop/pass lter Yes

- Higher order washout lter No

Closed loop gain modication BADISATHRFUZZYGAIN

- ω lowering in turbulence Yes

- Adaptation of Nx,est hyb. due to gain lowering No ˙

N1,c and Nz,c limits BADISATHRFUZZYLIM

- ˙N1,c limits Yes

- Nz,c limits in turbulence Yes

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Additionally, the aircraft model used was an A380-800 with Rolls Royce engines. 5.2.1 Generic turbulent approach

Flight point: z0=4000ft, CONF=3, CG=36.5%, γglide = −3◦. Simulation sweep: m=uniform(260t,400t), TURBLVL={0; 1; 2; 3} The injected wind is depicted in Figure 36 in 4.5.2.

ATHR-caused speed deviation

The average speed deviations caused by the ATHR, as per Equation 32, are seen below in Figure 40 as a function of the turbulence level.

Figure 40: ATHR-caused speed deviation as a function of the turbulence level.

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Figure 41: Typcal time series (m = 330t) for VCAS during medium (top) and severe (bottom) turbu-lence.

Engine agitation

The standard deviation of N1 for dierent levels of turbulence is shown in Figure 42.

Figure 42: Fan speed standard deviation as a function of the turbulence level.

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Figure 43: Typical time series (m = 330t) for N1 during medium (top) and severe (bottom) turbulence. The mean ω in these simulations can be seen in Figure 44.

Figure 44: Mean closed loop pulsation. 5.2.2 Approach: Perpignan, recorded wind 2010-01

Flight point: z0=4000ft, CONF=3, CG=36.5%, m=330t γglide= −3◦. The wind used in the scenario is depicted below in Figure 45.

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Figure 45: Approach Perpignan: WX,0, TURBLVLDET, and KWINGRADDET. This wind yields the results seen below in Figure 46.

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Figure 46: Approach Perpignan: VCAS and N1. 5.2.3 Approach: Recorded severe turbulence

Flight point: z0=4000ft, CONF=3, CG=36.5%, m=330t γglide= −3◦. The wind used in the scenario is depicted below in Figure 47.

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Figure 47: Recorded severe turbulence: WX,0, TURBLVLDET, and KWINGRADDET. This wind yields the results seen below in Figure 48.

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5.2.4 Oscillating wind - No ANTIAIO

Flight point: z0=4000ft, CONF=3, m=330t, CG=36.5%, level ight.

The wind used in this scenario is shown in Figure 49. The frequency sweep is 0.009Hz - 0.06Hz.

Figure 49: Oscillating wind: WX,0 and TURBLVLDET. This wind yields the results seen below in Figure 50.

Enhancements of an auto- thrust function using fuzzy logic

Enhancements of an

auto-thrust function using fuzzy logic

G U S T A V Ö M A N L U N D I N

Master of Science Thesis

Stockholm, Sweden

Enhancements of an auto-thrust

function using fuzzy logic

G U S T A V Ö M A N L U N D I N

Acknowledgements

Contents

List of Figures

List of Tables

1 Symbols and acronyms

2 Introduction

3 Problem

4 Method

5 Results