Adaptive Neuro-fuzzy Controllers for an Open-loop Morphing Wing System

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Adaptive neuro-fuzzy controllers for anopen-loop morphing wing systemT L Grigorie,R M Botez, andAV Popov

cole de Technologie Suprieure, Laboratory of Research in Active Controls, Avionics and AeroServoElasticity LARCASE,Montral, Quebec, Canada

The manuscript was received on 7 December 2008 and was accepted after revision for publication on 10 June 2009.

DOI: 10.1243/09544100JAERO487

Abstract: A new method for the realization of two neuro-fuzzy controllers for a morphing wingdesign application is presented here. The controllers main function is to correlate each set ofpressure differences, calculated between the optimized and the reference airfoil, with each of theairfoil deformations produced by the actuators system. The pressures are calculated at differentchord positions and will also be measured during wind tunnel tests. During a first identificationphase, the two fuzzy inference systems (FISs) from the controllers structure are generated for 16flightconditionscharacterizedbyMachnumbersandanglesofattack.Next,theFISareoptimizedwiththeMatlabfunctionadaptiveneuro-fuzzyinferencesystem(ANFIS)bytrainingoverdifferentepochs. Finally, the controllers are validated for the other 33 flight conditions of the open-loopmorphingwingsystem.Thisisthefirsttimethatsuchamethodofrelatingthepressuredifferencesto airfoil displacements has been conceived and used in an open-loop morphing wing controllersystem.

Keywords: morphing, neuro-fuzzy controller, simulation, modelling, testing

1 INTRODUCTION

Recently, morphing wing system studies have bran-ched outinto newresearch directions. Extremely com-plex and catalogued as inter- and multidisciplinarystudies, morphing wing studies continue to push thescience, up to the extreme boundaries of mathematicsand physics. These multidisciplinary studies thereforerequireknowledgeinthefollowingdisciplines:aerody-

namics and computational fluid dynamics, aeroelas-ticity, automatic control, intelligent materials, signaldetection using the latest miniaturized sensors, highcomputer-time calculations, wind tunnel and flighttesting, instruments, and signal acquisition thesesignals have such speed that they are raising seri-ous problems for the existing calculus technology [1].Consequently, real-time system functioning is con-ditioned (in addition to other factors) by obtaining

Corresponding author: Dpartement de Gnie de la Production,Quebec University, 1100, rue Notre-Dame Ouest, Montral, Quebec

H3C 1K3, Canada.

email: [email protected]

the best data processing algorithms, easy to imple-ment software within the command and control unit.Fuzzy logic theories, which offer remarkable facilities,may therefore be used in these algortihms. They facil-itate signal processing by allowing empirical modelsto be designed based on experimental data; thus, thecomplex mathematical calculus currently in use canbe avoided. In addition, fuzzy logic can be used tomodel highly non-linear, multidimensional systems,

including those with parameter variations, or wherethe sensors signals are not accurate enough for othermodels [2, 3].

In order to conceive such a model, a fuzzy setmust be designed, which may be given by the orig-inal fuzzy logic theory conceived by Lotfi A. Zadeh.The most serious problem arises from the determina-tion of a complete set of rules and the membershipfunctions corresponding to each input. The manyattempts to reduce errors and to optimize the modelare time-consuming and, very often, the results arefar from what was expected. A modern design methodallows fuzzy model design to be completed in a rel-atively short time interval. The adaptive neuro-fuzzyinference system (ANFIS) design technique allows thegeneration and optimization of the set of rules and the

JAERO487 Proc. IMechE Vol. 223 Part G: J. Aerospace Engineering


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966 T L Grigorie, R M Botez, and A V Popov

membership functions parameters by use of neuralnetworks [4]. Moreover, the ANFIS design techniquealready implemented in Matlabs neuro-fuzzysoftwaretools should be relatively easy to use.

In the following paragraphs are mentioned someof the most important applications of neural networkand fuzzy logic theories in the aerospace field.

Neural network methods have been used in theaeronautical industry for the following applications:the detection and identification of structural dam-age [5], fault diagnostics (detection and isolation)leading to compensationof control surfacefailures [6],the modelling of aerodynamic characteristics fromflight data [7, 8], generalized reference models for sixdegrees of freedom motion simulation using globalaerodynamic models, including unsteady aerodynam-ics and dynamic stall [911], the detection of unan-

ticipated effects such as icing [12], and autopilotcontrollers and advanced control laws for applicationssuch as carefree manoeuvring [13, 14]. Neural networkmethods have also been used for model identificationpurposes, based on flight flutter tests. A good exam-ple is the method proposed in reference [15] for theprediction of damping ratios during flight flutter testsusing a neural network trained on model data. Thatmethod was compared to a simple statistical extrap-olation approach and was focussed on studying noiseeffects.

To date, fuzzy logic theories have been used less

extensively than neuralnetwork theories forthe designof active control systems and for the identificationand validation of an aircraft model. A new methodfor the conversion of aerodynamic forces from thefrequency to the Laplace domain was validated onan F/A-18 aircraft for aeroservoelasticity studies [16].The F/A-18 aircraft non-linear model was identifiedfrom flight flutter tests by use of fuzzy logic [17]and a combination of fuzzy logic and neural networkmethods [18].

A neuro-fuzzy controller design, simulation, andvalidation for a morphing wing application, using

Fuzzy Logic Toolbox and Matlab/Simulink, is pres-ented in this study. A hybrid learning algorithm isused to tune the fuzzy inference systems (FISs) fromthe controllers structure; the least-squares and the

back propagation gradient descent methods are com-bined for the FIS membership functions (mf) param-eters training to emulate a given training dataset.Another dataset is used to validate and test the robust-ness of the neuro-fuzzy controllers.

2 OPEN-LOOP ARCHITECTURE OF THEMORPHING WING SYSTEM

The functional diagram of the morphing wing systemusing an implemented controller is presented in Fig. 1.The reference airfoil is a modified WTEA type. Thetwo shape memory alloy (SMA) actuators that deflect

the airfoils (from the reference to the optimized) areshown in the lower left side of Fig. 1, and the neuro-fuzzy controllers algorithm conceived here are shownin the upper right corner of Fig. 1. It is importantto mention that the present study is entirely numer-ical and is done with the aim to integrate it into thereal-time controller, to be used and then validated ina wind tunnel. Thirty pressure sensors are installedon the wing at different chord positions, as shown inFigs 1 and 2. These sensors are used to detect the air-foil transition points position from their pressuredata[1]. Their installation is shown non-dimensionally on

the wing airfoil between its leading and trailing edgesin Fig. 2.The equipment presented in Figs 1 and 2 will be

installed soon in a wind tunnel; experimental datawill be compared to numerical results obtained withthe present authors new algorithm for 49 flight casesdependent on seven attack angles varying between 1and 2, and on seven Mach numbers between 0.2 and0.35. An optimized airfoil was determined for each ofthe 49 flight cases.

The airfoil deformation is realized by using two SMAactuators installed at 25.3 and 47.6 per cent from its

Fig. 1 Open-loop architecture of the morphing wing system with the neuro-fuzzy controllers

Proc. IMechE Vol. 223 Part G: J. Aerospace Engineering JAERO487


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Adaptive n euro-fuzzy c ontrollers for a n o pen-loop m orphing wing system

Fig. 2 Pressure sensors arrangement on the morphingairfoil

leadingedge.Foreachflightcase,theactuationsystemwith twoSMA actuatorsmust ensurethetwo referenceairfoil vertical displacements dY1 and dY2 until theoptimized airfoil is obtained.

For each flight case, the pressures detected by the 30sensors on the reference airfoil and on the optimizedairfoil are determined, and then one set of 30 pressuredifferences between the optimized and the referenceairfoil (dp1 dp30) is calculated. In addition, for eachflight case, the two vertical displacements of the refer-ence airfoil with respect to the optimized airfoil (dY1and dY2) correspond to each set of 30 pressure differ-ences (dp1 dp30), and are generated by the actuationsystem. The role of the neuro-fuzzy controllers is tofind a correspondence between each set of pressure

differences detected by the 30 sensors and each of thetwo displacements given by the two SMA actuators.Therefore, two controllers are required: dY1 = f1(dp1,dp2, . . . , dp30) anddY2 = f2(dp1, dp2, . . . , dp30).Thetwocontrollers identification is performed for 16 flightcases, while their validation is performed for 33 flightcases. Thus, neuro-fuzzy controllers are built for all 49flight cases.

3 FUZZY LOGIC CONTROLLERS DESIGN

For all 49 flight cases, the pressure differences(dp1 dp30) between the optimized and the referenceairfoils are traced for each sensor, as shown in Figs 3to 9, for seven incidence angles: = 1, 0.5, 0,0.5, 1, 1.5, and 2 and for seven Mach numbers:

M = 0.2 (Fig. 3), M = 0.225 (Fig. 4), M = 0.25 (Fig. 5),M = 0.275 (Fig. 6),M = 0.3 (Fig. 7),M = 0.325 (Fig. 8),and M = 0.35 (Fig. 9). The vertical deflection differ-ences (dY1 anddY2) are depicted in Fig. 10 as functionsof Mach number for each of the seven incidenceangles.

By analyzing the pressures differences (dp1 dp30)and vertical position differences (dY1 and dY2) for all

49 flight cases, 16 identification cases (I1 I16) and33 validation cases (V1 V33) are chosen for the con-trollers identification and validation; these cases areshown in Table 1.

0 5 10 15 20 25 30-600

-400

-200

0

200

400

600

800

1000alpha1

alpha2

alpha3

alpha4

alpha5

alpha6

alpha7

Sensor number

dp[N/m2]

Fig. 3 dp1 dp30 for M = 0.2 and various

0 5 10 15 20 25 30-1000

-500

0

500

1000

1500alpha1

alpha2

alpha3

alpha4

alpha5

alpha6

alpha7

Sensor number

dp[N/m2]


0 5 10 15 20 25 30-1000

-500

0

500

1000

1500alpha1

alpha2

alpha3

alpha4

alpha5

alpha6

alpha7

Sensor number

dp[N/m

2]


The 16 identification cases are selected in orderto obtain the most robust controllers, which wouldcontain the generated FISs covering the maximumdomain for the 30 pressure differences (dp1 dp30)



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0 5 10 15 20 25 30-1500

-1000

-500

0

500

1000

1500

2000alpha1

alpha2

alpha3

alpha4

alpha5

alpha6

alpha7

Sensor number

dp[N/m2]


0 5 10 15 20 25 30-1500

-1000

-500

0

500

1000

1500

2000

2500alpha1

alpha2

alpha3

alpha4

alpha5

alpha6

alpha7

Sensor number

dp[N/m2]


and for the vertical displacements between airfoils(dY1 anddY2). As shown in Table 2, each identificationcase is chosen because it maximizes or minimizesthe values of dp1 dp30, dY1 and dY2. The minimumand maximum values of the indexed parameter aredenoted with indices of min and max, respectively.

0 5 10 15 20 25 30-1500

-1000

-500

0

500

1000

1500

2000

2500

3000alpha1

alpha2

alpha3

alpha4

alpha5

alpha6

alpha7

Sensor number

dp[N/m2]


0 5 10 15 20 25 30-1500

-1000

-500

0

500

1000

1500

2000

2500

3000

3500alpha1

alpha2

alpha3

alpha4

alpha5

alpha6

alpha7

Sensor number

dp[N/m2]


The maximum and minimum values of dp1 dp30,dY1 and dY2 are covered by 15 identification cases(with no case I9), which can be observed in Table 2.From Table 1, it is obvious that these 15 cases cover allof the values of Mach number and incidence anglesexcept for M = 0.225 and = 1. For this reason,

alpha1

alpha2

alpha3

alpha4

alpha5

alpha6

alpha7

0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.362

3

4

5

6

7

8

Mach number

dY1

[mm]

Mach number

dY2

[mm]

2

3

4

5

6

7

8

0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36

Fig. 10 dY1 and dY2 as functions ofM for various



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Table 1 Identification (I1 I16) and validation(V1 V33) flight cases

Mach 1 2 3 4 5 6 7

() 0.2 0.225 0.25 0.275 0.3 0.325 0.35

1 1 I1 V1 V2 I2 I3 I4 V32 0.5 V4 V5 V6 V7 I5 V8 V93 0 V10 V11 V12 V13 V14 V15 I64 0.5 I7 V16 V17 V18 V19 I8 V205 1 V21 I9 V22 V23 V24 V25 V266 1.5 V27 V28 I10 I11 V29 V30 I127 2 V31 V32 V33 I13 I14 I15 I16

Table 2 Correlation of the identification cases with theextreme values of dp1 dp30, dY1, and dY2

Maximized or minimized parametersIdentification cases (dp1 dp30, dY1, and dY2)

I1 dp2min dp5min, dp11max dp20maxI2 dp1minI3 dp30min, dp8max dp10maxI4 dp29minI5 dY1min, dY2minI6 dp28minI7 dp6minI8 dp7maxI10 dp22max, dp23maxI11 dp21maxI12 dp9min, dp12min, dp24max dp28maxI13 dY1max, dp7min, dp8min, dp10minI14 dY2maxI15 dp21min, dp23min dp26min, dp29max,

dp30maxI16 dp11min, dp13min dp20min, dp22min,

dp27min, dp1max dp6max

the I9 (M = 0.225 and = 1) identification case isconsidered.

For the16 identification cases,two Sugeno-typeFISsare generated using the Matlab genfis2 function [4].Therefore, the working field is separated into regionswhere, for each region, the local parameters to be pro-

cessed are represented by a low-order linear model.The non-linear process is locally linearized around afunctioning point by using the least-squares method,and the obtained model is considered valid in theentire region around this point.

The limitation of the operating regions impliesthe existence of overlapping among these differentregions so their definition is given in a fuzzy man-ner. Thus, for each model input, several fuzzy setsare associated with their corresponding definitions oftheir mf. By combining these fuzzy inputs, the inputspace is divided into fuzzy regions. For each of these

regions, a local linear model is used, while the globalmodel is obtained by defuzzification with the gravitycentre method (Sugeno), which is used to interpolatethe local models outputs [2, 4].

The Sugeno fuzzy model was proposed by Takagi,Sugeno, and Kang to generate the fuzzy rules from agiven inputoutput dataset [19]. For the system pre-sented in this paper, a first-order model is considered

for both FISs, and for N rules is given by [19, 20]Rule 1: Ifx1 is A11 and x2 is A

12 and . . . and x30 is A

130,

then y1(x1, x2, . . . ,x30) = b10 + a

11x1 + a

12x2 +

+ a130x30,

...

Rule i: Ifx1 is Ai1 and x2 is Ai2 and . . . and x30 is A

i30,

then yi(x1, x2, . . . ,x30) = bi0 + ai1x1 + a

i2x2 +

+ ai30x30,

...

Rule N: Ifx1 is AN1 and x2 is AN2 and . . . and x30 is A

N30,

then yN(x1, x2, . . . ,x30) = bN0 + aN1 x1 + a

N2 x2 +

+ aN30x30

(1)

where xq(q = 1,30) are individual input variables,Aiq(i = 1,N) are the associated individual antecedentfuzzy sets of each input variable, and yi(i = 1,N) isthe first-order polynomial function in the consequent.

aik(k = 1,30, i = 1,N) are parameters of the linearfunction and bi0(i = 1,N) denotes a scalar offset. Theparameters aik, b

i0(k = 1,30, i = 1,N) areoptimizedby

the least-squares method.For any input vector x = [x1,x2, . . . ,x30]T, if the sin-

gleton fuzzifier, the product fuzzy inference, and thecentre average defuzzifier are applied, the output ofthe fuzzy model y is inferred as follows (weightedaverage)

y =

Ni=1 w

i(x)yi

Ni=1 w

i

(x

) (2)

where wi(x) is the degree of fulfillment of theantecedent; that is, the level of firing of the ith rule

wi(x) = Ai1(x1) Ai2(x2) A

i30(x30) (3)

The Matlab genfis2 function generates Gaussian-typemembership functions, defined as [3, 18]

Aiq(x) = exp

0.5

x ciq

iq 2 (4)

where ciq is the cluster center and i

q is the dispersionof the cluster.



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in1

in1cluster1

in1clusterk

in1cluster30

.

.

.

.

.

.

Inputs Input mf Rules Output mf

.

.

.

.

.

.

1 ou t1c lus ter1

inj

in1cluster1

in1clusterk

in1cluster30

.

.

.

.

.

.

.

.

.

.

.

.

dp1

dpj

in30

in1cluster1

in1clusterk

in1cluster30

.

.

.

.

.

.

.

.

.

.

.

.

dp30

...

.

.

.

k

30

...

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

out1clusterk

out1cluster30

Agregated

Output1out1

Output

Fig. 11 Structure of the dY1Fis and the dY2Fis

in1

inj

dp1

dpj

in30 dp30

.

.

.

.

.

.

out1

dY1

or dY2

Sugeno

FIS

Fig. 12 Structure of the controllers

Thus, for the system presented here, a set of 16rules is obtained, which are of the following type:if (in1 is in1clusterk) and (in2 is in2clusterk) and. . . and (inj is inj cluster k) and . . . and (in30 is

in30clusterk) then (out1 is out1clusterk). For eachinput, out of the 30 inputs, 16 Gaussian-type mf aregenerated; within the set of rules, they are denoted byinj cluster k; where j is the input number (1 30),and k is the number of the membership function(1 16).

The two FISs generated for dY1 and dY2 displace-ments, denoted by dY1Fis and dY2Fis, have thestructure shown in Fig. 11, while the correspondingcontrollers have the structure presented in Fig. 12.

4 DESIGN OF ADAPTIVE FUZZY LOGIC

CONTROLLERS

Adaptive fuzzy logic controllers can be obtained byoptimizing the previously generated FISs (dY1Fis anddY2Fis). The neural networks are used to optimizethe FIS mf parameters. Therefore, two neuro-fuzzymodels (trained FISs) are obtained to control the dY1and dY2 deflections based on the dp1 dp30 pressuredifferences, by use of Matlabs ANFIS function [4].

Matlabs ANFIS function uses an adaptive learn-ing algorithm to identify and modify the membershipfunction parameters for the two Sugeno-type FISs

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Training epochs number

ANFISerror

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11


ANFISerror

0.045

0.05

0.055

0.06

0.065

0.07

0.075

0.08

0.085

0.09

0.095


ANFISerror

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14


ANFISerror

0 1000 2000 3000 4000 5000 6000 7000 8000 900010000

0 10 20 30 40 50 60 70 80 90 1001 2 3 4 5 6 7 8 9 10

0 100 200 300 400 500 600 700 800 900 1000

Fig. 13 The ANFIS training errors for dY1Fis over 10, 102, 103, and 104 training epochs



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0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1


ANFISerro

r

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14


ANFISerror

0.06


ANFISerror

0

0.02

0.04

0.08

0.1

0.12

0.14


ANFISerror

1 2 3 4 5 6 7 8 9 10 0 10 20 30 40 50 60 70 80 90 100

0 100 200 300 400 500 600 700 800 900 1000 0 1000 2000 3000 4000 5000 6000 7000 8000 9000100000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Fig. 14 The ANFIS training errors for dY2Fis over 10, 102, 103, and 104 training epochs

2

3

4

5

6

7

8

All 16 airfoil identification cases for 100 training epochs

Data

FIS model

2

3

4

5

6

7

8

dY1

[mm]


2

3

4

5

6

7

8

Data

FIS model

2

3

4

5

6

7

8


Data

FIS model

2

3

4

5

6

7

8

0 2 4 6 8 10 12 14 160 2 4 6 8 10 12 14 16

0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16


Data

FIS model

dY1

[mm]

dY1

[mm]

dY1

[mm]

Fig. 15 The inputoutput data (dY1) and the outputs of trained dY1Fis over different trainingepochs



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with 30 inputs and one output [4]. This is a hybridalgorithm that uses a combination of the gradientmethod along with the least-squares method. Aninputoutput dataset and the FIS models generated

with genfis2 are considered as a starting point; then,the membership functions parameters are optimizedwith the ANFIS algorithm for a number of trainingepochs (set by the user). This optimization is real-ized by means of a quality parameter in the trainingalgorithm [4], which is there in order to improve theprocess approximation performed by the neuro-fuzzymodel. The data from the 16 identification cases areused as the inputoutput dataset.

Thetrainingerrors,definedbythequalityparameterin the training algorithm, are shown in Figs 13 and 14for the dY1Fis and the dY2Fis, respectively. Thesetwo FISs are trained over 10, 102, 103, and 104 training

epochs. The post-training FIS evaluations are shownin Figs 15 and 16; the star shapes indicate the deflec-tions dY1 and dY2 from the inputoutput data, whilethe dY1 and dY2 deflections obtained from the fuzzymodels (the outputs of dY1Fis and dY2Fis trainedFIS, respectively) are shown as circles.

Figures 13 and 14 show decreasing oscillations inthe training errors over the first 3200 training epochsfor dY1Fis, with similar decreases over the first 2700training epochsfor dY2Fis. After these intervals, oscil-

lations with approximately constant amplitudes forbothFISscanbeobserved.Therefore,atrainingofover104 epochs gives the best results.

Figures 15 and 16 reiterate the same observationsas those obtained for Figs 13 and 14. There is an over-lapping of the neuro-fuzzy models outputs and thedY1 and dY2 values from the inputoutput data. Thissuperposition is dependent on the number of train-ing epochs, and improves as the number of epochsincreases.

Table3presentsthenumericalvaluesoftheabsolutemaximum relative errors between the inputoutputdata and their neuro-fuzzy models, for displacements

dY1 and dY2. The errors are evaluated over 10, 102, 103,and104 trainingepochs.Theseerrorsdecreasewiththeincrease in thetrainingepochs number;the minimumvalues of these errors are obtained for 104 trainingepochs: 0.5305 percent fordY1Fis and0.6753percentfor dY2Fis.

2

3

4

5

6

7

8

dY2

[mm]


Data

FIS model

2

3

4

5

6

7

8


Data

FIS model

2

3

4

5

6

7

8


Data

FIS model

2

3

4

5

6

7

8

9

0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16

0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16


Data

FIS model

dY2

[mm]

dY2

[mm]

dY2

[mm]

Fig. 16 The inputoutput data (dY2) and the outputs of trained dY2Fis over different trainingepochs



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Table 3 The absolute maximum relative errors betweenthe inputoutput data and their neuro-fuzzymodels

Training epochs 10 100 1000 10 000

dY1Fis 2.2842% 1.1196% 0.8657% 0.5305%dY2Fis 2.1716% 1.1872% 0.8165% 0.6753%

The variations of the relative errors between theinputoutput data and their neuro-fuzzy models forall 16 identification cases are shown in Figs 17 and 18 the different training epochs 10, 102, 103, and 104 areconsidered for the FIS training. Both diagrams showthat the training of the two FISs diminished the mostrelative error for case 6 (I6M = 0.35, = 0) and theleast for case 12 (I12M = 0.35, = 1.5). The maxi-

mum values of the absolute relative error were foundwith case 4 (I4M = 0.325, = 1) for dY1 and forcase 11 (I11M = 0.275, = 1.5) for dY2.

Twofuzzylogiccontrollersareconceived;theirmod-els give relative errors smaller than 0.7 per cent forall 16 identification flight cases (see the black curves

0 2 4 6 8 10 12 14 16-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

10 training epochs

100 training epochs

1000 training epochs


All airfoil 16 identification cases

Relativeerrors

Fig. 17 Relative errors for dY1Fis

0 2 4 6 8 10 12 14 16-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

10 training epochs

100 training epochs



All airfoil 16 identification cases

Relativeerrors

Fig. 18 Relative errors for dY2Fis

with six-pointed star symbols in Figs 17 and 18). Theserelative errors are obtained by training the originalFISs, generated from the identification data, over 104

training epochs.

5 VALIDATION OF ADAPTIVE FUZZY LOGICCONTROLLERS

The validation and testing of the two controllersrobustness is performed by using the Simulink dia-gram shown in Fig. 19. This diagram also shows thecontrollers implementation. The data to be validatedare those corresponding to the 33 validation flightcases presented in Table 1 (V1 V33). Figures 20and 21show the controllers evaluation for these data, whichconsists of the validation data and the outputs of the

two neuro-fuzzy models (the trained FISs) for all 33validation flight cases. There is very good overlappingofthesetwodatacategoriesforbothcontrollers,whichis clearly visible in Figs 20 and 21.

The values of the absolute relative errors are shownin Fig. 22, where they are at their maximum at 2.5158per cent for controller 1 and at 1.0887 per cent for con-troller 2. The maximal values are obtained for the 20thvalidation case, V20, for controller 1 and for the 23rdvalidation case , V23, for controller 2.

dY2

To Workspace1

dY1

To Workspace

dp

Signal FromWorkspace

Fuzzy LogicController 2

Fuzzy LogicController 1

Fig. 19 Simulink implementation of the two controllers

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

dY1

[mm]

All 33 airfoil validation cases

Data

FIS model

0 5 10 15 20 25 30 35

Fig. 20 The validation of controller 1



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dY2

[mm]

All 33 airfoil validation cases

Data

FIS model

0 5 10 15 20 25 30 352.5

3

3.5

4

4.5

5

5.5

6

6.5

Fig. 21 The validation of controller 2

0 5 10 15 20 25 30 35-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03controller1

controller2

All airfoil 33 validation cases

Relativeerrors

Fig. 22 The relative errors obtained when validating thecontrollers for the 33 validation flight cases

From Fig. 22, it can be seen that the absolute valuesof the relative errors exceed 1 per cent in three valida-tion cases for controller 1 and in two validation casesforcontroller2,whilemostoftheabsolutevaluesoftherelative errors remain smaller than 0.5 per cent; there-

fore, it can be said that the two controllers are robustand accurate.

6 CONCLUSIONS

Two robust neuro-fuzzy controllers were obtained fora morphing wing application, which relates the airfoildisplacements caused by the actuator system to thepressure differences detected by the sensors systemon the reference airfoil and on the optimized airfoils.

The two controllers were trained for 16 identifi-cation flight cases and validated for 33 other flight

cases. The final FISs training was performed on104 training epochs. The maximum absolute valuesof the relative errors among the identification dataand those obtained for the controllers outputs were

0.5305per cent for the first controller (correspondingto the displacement dY1) for the fourth identificationcase, I4 (M = 0.325, = 1),and0.6753percentforthe second controller (corresponding to the displace-

ment dY2) for the 11th identification case, I11 (M=

0.275, = 1.5) (see the black curves with six-pointedstar symbols in Figs 17 and 18, respectively).

The validation of the two controllers led to max-imum absolute values of the relative errors that areless than 2.6 per cent, more precisely, 2.5158 per centfor controller 1 and 1.0887per cent for controller 2.The maximum values were obtained for the 20th val-idation case, V20, (M = 0.35, = 0.5) for controller1 (black curve with circle symbols in Fig. 22) and forthe 23rd validation case, V23, (M = 0.275, = 1 forcontroller 2 (grey curve with square symbols in Fig.22). Finally, it can be concluded that the results pre-

sented here are extremely useful for morphing wingapplications.

ACKNOWLEDGEMENTS

The authors would like to thank Dr Mahmoud Mamouand Dr Mahmood Khalid from the NSERC (NationalSciences and Engineering Research Council) for mod-elling the WTEA-TE1 airfoils, and Dr Octavian Trifufor the 49 optimized airfoils shapes. The authors alsowant to thank Mr George-Henri Simon from Thales

Avionics for initiating this project, and Mr PhilippeMolaret from Thales Avionics and Mr Eric Laurendeaufrom Bombardier Aeronautics for their collaborationon this article. The authors want to acknowledge thefinancial support of Thales Avionics and BombardierAerospace, as well as that of CRIAQ (Consortium ofResearch in the Aerospace Industry in Quebec), andNSERC (National Sciences and Engineering ResearchCouncil of Canada).

Authors 2009

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APPENDIX

Notation

aik parameters of the linear function(k = 1,30, i = 1,N)

Aiq associated individual antecedent fuzzy setsof each input variable (i = 1,N)

bi

0 scalar offset (i = 1,N)ciq cluster centre (q = 1,30)dpj jth sensor pressure differences between

the optimized and the reference airfoil(j = 1,30)

dY1 vertical displacement between optimizedand reference airfoil produced by the firstSMA

dY2 vertical displacement between optimizedand reference airfoil produced by the secondSMA

FIS fuzzy inference systemIl lth identification case (l = 1,16)

M Mach numberN number of rulesVm mth validation case (m = 1,33)wi degree of fulfillment of the antecedent,

i.e. the level of firing of the ith rulex input vectorxq individual input variables (q = 1,30)y output of the fuzzy modelyi first-order polynomial function in the

consequent (i = 1,N)

airfoil incidence anglei

q

dispersion of the cluster



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Documents

Adaptive Neuro-fuzzy Controllers for an Open-loop Morphing Wing System