655 Adaptiveneuro-fuzzyinferencesystem-basedcontrollers ...

655

Adaptive neuro-fuzzy inference system-based controllersfor smart material actuator modellingT L Grigorie and R M Botez∗

École de Technologie Supérieure, Montréal, Quebec, Canada

The manuscript was received on 6 February 2009 and was accepted after revision for publication on 15 May 2009.

DOI: 10.1243/09544100JAERO522

Abstract: An intelligent approach for smart material actuator modelling of the actuation linesin a morphing wing system is presented, based on adaptive neuro-fuzzy inference systems. Fourindependent neuro-fuzzy controllers are created from the experimental data using a hybridmethod – a combination of back propagation and least-mean-square methods – to train thefuzzy inference systems. The controllers’ objective is to correlate each set of forces and electricalcurrents applied on the smart material actuator to the actuator’s elongation. The actuator experi-mental testing is performed for five force cases, using a variable electrical current. An integratedcontroller is created from four neuro-fuzzy controllers, developed with Matlab/Simulink softwarefor electrical current increases, constant electrical current, electrical current decreases, and fornull electrical current in the cooling phase of the actuator, and is then validated by comparisonwith the experimentally obtained data.

Keywords: smart material actuator, neuro-fuzzy controller, simulation, modelling, testing

1 INTRODUCTION

The aim of this article is to obtain a reliable, easy-to-implement model for smart material actuators(SMAs), with direct applications in the morphing wingproject. Based on adaptive neuro-fuzzy inference sys-tems, an integrated controller is built to model theSMAs used in the actuation lines of a wing. This modeluses the numerical values from the SMAs’ experimen-tal testing and it takes advantage of the outstandingproperties of fuzzy logic, which allow the signal’sempirical processing without the use of mathemati-cal analytical models. Fuzzy logic systems can emu-late human decision-making more closely than manyother classifiers through the processing of expert sys-tem knowledge, formulated linguistically in fuzzy rulesin an IF–THEN form. Fuzzy logic is recommended forvery complex processes, when no simple mathemati-cal model exists, for highly non-linear processes andfor multi-dimensional systems.

∗Corresponding author: Laboratory of Research in Active Controls,

Avionics and AeroServoElasticity LARCASE, École de Technologie

Supérieure, 1100, rue Notre-Dame Ouest, Montréal, Québec H3C

1K3, Canada.

email: [email protected]

The input variables in a fuzzy control system areusually mapped into place by sets of membershipfunctions (mf) known as ‘fuzzy sets’; the mappingprocess is called ‘fuzzification’. The control system’sdecisions are made on the basis of a fuzzy rules set,and are invoked using the membership functions andthe truth values obtained from the inputs; a processcalled ‘inference’. These decisions are mapped into amembership function and truth value that controlsthe output variable. The results are combined to givea specific answer in a procedure called ‘defuzzifica-tion’. Elaboration of the model thus requires a fuzzyrules set and the mf associated with each of theinputs [1, 2].

The ability and the experience of a designer inevaluating the rules and the membership functionsof all of the inputs are decisive in obtaining a goodfuzzy model. However, a relatively new design methodallows a competitive model to be built using a combi-nation of fuzzy logic and neural-network techniques.Moreover, this method allows the possibility to gener-ate and optimize the fuzzy rules set and the parametersof the membership functions by means of fuzzy infer-ence systems’ (FISs) training. To this end, a hybridmethod – a combination of back propagation andleast-mean-square (LMS) methods – is used, in whichexperimentally obtained data are considered. Already

JAERO522 Proc. IMechE Vol. 223 Part G: J. Aerospace Engineering

656 T L Grigorie and R M Botez

implemented in Matlab [1, 3], the method is easy touse, and gives excellent results in a very short time.

2 ACTUATOR EXPERIMENTAL TESTING

The SMA testing was performed using the bench testin Fig. 1 at Tamb = 24 ◦C, for five load cases with forcesof 120, 140, 150, 180, and 190 N. The electrical cur-rents following the increasing-constant–decreasing-zero values evolution were applied on the SMA in eachof the five cases considered for load forces. In eachof the cases to be analysed, the following parame-ters were recorded: time, the electrical current appliedto the SMA, the load force, the material temperature,and the actuator elongation (measured using a linearvariable differential transformer (LVDT)).

To model the SMA, the present authors built an inte-grated controller based on adaptive neuro-FISs. Theexperimental elongation-current curves obtained inthe five load cases are shown in Fig. 2. One can observethat all five of the obtained curves have four distinctzones: electrical current increase, constant electricalcurrent, electrical current decrease, and null electrical

Fig. 1 The SMA bench test

Fig. 2 Elongation versus the current values for differentforce values for four cases

current in the cooling phase of the SMA. Four FISs areused to obtain four neuro-fuzzy controllers: one forthe current increase, one for the constant current, onefor the current decrease, and one controller for thenull current (after its decrease). For the first and thethird controllers, inputs such as the force and the cur-rent are used, whereas for the second and the fourth,inputs such as the force and the time values reflect-ing the SMA’s thermal inertia are used (the time valuesrequired for the SMA to recover its initial temperaturevalue (∼24 ◦C) are used for the four controller). Finally,the four obtained controllers must be integrated intoa single controller.

The reasoning behind the design of the first and thethird controllers is that, from the available experimen-tal data, two elongations for the same values of forcesand currents are used (see Fig. 2). Due to the experi-mental data values, these data cannot be representedas algebraic functions; therefore, it is impossible touse the same FIS representation. Matlab producesan interpolation between the two elongation valuesobtained for the same values of forces and currents,which cannot be valid for our application.

The constant values, namely the null values of thecurrent before and after the current decrease phaseshould not be considered as inputs in the second andfourth controllers because they are not suggestive forthe characterization of the SMA elongation. The val-ues of the actuator temperatures may appear to bevery suggestive in these phases, but the temperaturemust be a model output. For these phases the time val-ues are very suggestive, as they represent a measure ofthe actuator thermal inertia. Time is the second inputof the third controller, and so time is also the secondinput of the second and the fourth controllers – sinceforce was considered as the first input (the time val-ues must be considered when the current becomesconstant or null).

3 THE PROPOSED METHOD

Fuzzy controllers are very simple conceptually and arebased on FISs. Three steps are considered in an FISdesign: an input, the processing, and then an out-put step. In the input step, the controller inputs aremapped into the appropriate mf. Next, a collection of‘IF–THEN’ logic rules is created; the IF part is calledthe ‘antecedent’ and the THEN part is called the ‘con-sequent’. In this step, each appropriate rule is invokedand a result is generated. The results of all of the rulesare then combined. In the last step, the combinedresult is converted into a specific control output value.

Considering the numerical values resulting from theSMA experimental testing, an empirical model can bedeveloped, which is based on a neuro-fuzzy network.The model can learn the process behaviour based on

Proc. IMechE Vol. 223 Part G: J. Aerospace Engineering JAERO522

Adaptive neuro-fuzzy inference system-based controllers 657

the input–output process data by using an FIS, whichshould model the experimental data.

Using methods already implemented in commer-cial software, an FIS can be generated simply withthe Matlab ‘genfis1’ or ‘genfis2’ functions. The ‘gen-fis1’ function generates a single-output Sugeno-typeFIS using a grid partition on the data (no cluster-ing). This FIS is used to provide initial conditions forANFIS training. The ‘genfis1’ function uses generalizedBell-type membership functions for each input. Eachrule generated by the ‘genfis1’ function has one out-put membership function, which is, by default, of alinear type. It is also possible to create an FIS usingthe Matlab ‘genfis2’ function. This function generatesan initial Sugeno-type FIS by decomposing the oper-ation domain into different regions using the fuzzysubtractive clustering method. For each region, a low-order linear model can describe the local processparameters. Thus, the non-linear process is locally lin-earized around a functioning point by using the LSM.The obtained model is then considered valid in theentire region around this point. The limitation of theoperating regions implies the existence of overlappingamong these different regions; their definition is givenin a fuzzy manner. Thus, for each model input, sev-eral fuzzy sets are associated with their membershipfunctions’ corresponding definitions. By combiningthese fuzzy inputs, the input space is divided intofuzzy regions. A local linear model is used for each ofthese regions, whereas the global model is obtainedby defuzzification with the gravity centre method(Sugeno), which performs the interpolation of thelocal models’ outputs [1, 3].

Based on the goal of finding regions with a highdensity of data points in the featured space, the sub-tractive clustering method is used to divide the spaceinto a number of clusters. All of the points with thehighest number of neighbours are selected as centresof clusters. The clusters are identified one by one, asthe data points within a pre-specified fuzzy radius areremoved (subtracted) for each cluster. Following theidentification of each cluster, the algorithm locatesa new cluster until all of the data points have beenchecked. If a collection of M data points, specified byl-dimensional vectors uk , k = 1, 2, . . . , M , is consid-ered, a density measure at data point uk can be definedas follows

ρk =M∑

j=1

exp(

−|uk − uj|(rm/2)2

)(1)

where rm is a positive constant that defines the radiuswithin the fuzzy neighbourhood and contributes tothe density measure. The point with the highest den-sity is selected as the first cluster centre. Let uc1 be theselected point and ρc1 its density measure. Next, thedensity measure for each data point uk is revised by

the formula

ρ ′k = ρk − ρc1 exp

(−|uk − uc1|

(rn/2)2

)(2)

where rn is a positive constant, greater than rm, thatdefines a neighbourhood where density measures willbe reduced in order to prevent closely spaced clustercentres. In this way, the data points near the first clus-ter centre uc1 will have significantly reduced densitymeasures, and therefore cannot be selected as subse-quent cluster centres. After the density measures foreach point have been revised, then the next clustercentre uc2 is selected and all the density measures areagain revised. The process is repeated until all the datapoints have been checked and a sufficient number ofcluster centres generated. When the subtractive clus-tering method is applied to an input–output data set,each of the cluster centres are used as the centres forthe premise sets in a singleton type of rule base [4].

The Matlab ‘genfis1’ function generates member-ship functions of the generalized Bell type, defined asfollows [2, 5]

Aiq(x) =

⎛⎝1 +

∣∣∣∣∣x − ciq

a

∣∣∣∣∣2b

⎞⎠

−1

(3)

where ciq is the cluster centre defining the posi-

tion of the membership function, a and b are twoparameters that define the membership functionshape, and Ai

q (i = 1, N ) are the associated individ-ual antecedent fuzzy sets of each input variable (N =number of rules). Matlab’s ‘genfis2’ function generatesGaussian-type membership functions, defined withthe following expression [2, 5]

Aiq(x) = exp

⎡⎣−0.5

(x − ci

q

σ iq

)2⎤⎦ (4)

where ciq is the cluster centre and σ i

q is the dispersionof the cluster.

The Sugeno fuzzy model was proposed by Takagi,Sugeno, and Kang to generate fuzzy rules from a giveninput–output data set [6]. In our system, for each ofthe four FISs (two inputs and one output), a first-ordermodel is considered, which for N rules is given by [5, 6]

Rule 1 : If x1 is A11 and x2 is A1

2, then y1(x1, x2)

= b10 + a1

1x1 + a12x2

...

Rule i : If x1 is Ai1 and x2 is Ai

2, then yi(x1, x2)

= bi0 + ai

1x1 + ai2x2

...



Rule N : If x1 is AN1 and x2 is AN

2 , then yN(x1, x2)

= bN0 + aN

1 x1 + aN2 x2

(5)

where xq(q = 1, 2) are the individual input variablesand yi(i = 1, N ) is the first-order polynomial functionin the consequent. ai

k(k = 1, 2, i = 1, N ) are parame-ters of the linear function and bi

0(i = 1, N ) denotes ascalar offset. The parameters ai

k , bi0(k = 1, 2, i = 1, N )

are optimized by the LSM.For any input vector, x = [x1, x2]T, if the singleton

fuzzifier, the product fuzzy inference, and the centreaverage defuzzifier are applied, then the output of thefuzzy model y is inferred as follows (weighted average)

y =( ∑N

i=1 wi(x)yi)

( ∑Ni=1 wi(x)

) (6)

where

wi(x) = Ai1(x1) × Ai

2(x2) (7)

wi(x) represents the degree of fulfilment of theantecedent, i.e. the level of firing of the ith rule.

The adaptive neuro-FIS calculates the Sugeno-typeFIS parameters using neural networks. A very simpleway to train these FISs is to use Matlab’s ‘ANFIS’ func-tion, which uses a learning algorithm to identify themembership function parameters of a Sugeno-typeFIS with two outputs and one input. As a starting point,the input–output data and the FIS models generatedwith the ‘genfis1’ or ‘genfis2’ functions are considered.‘ANFIS’ optimizes the membership functions’ param-eters for a number of training epochs, determinedby the user. With this optimization, the neuro-fuzzymodel can produce a better process approximationby means of a quality parameter in the trainingalgorithm [3]. After this training, the models may beused to generate the elongation values correspondingto the input parameters.

To train the fuzzy systems, ANFIS employs a back-propagation algorithm for the parameters associatedwith the input membership functions, and LMS esti-mations for the parameters associated with the outputmembership functions. For the FISs generated usingthe ‘genfis1’ or ‘genfis2’ functions, the membershipfunctions are generalized Bell type or Gaussian type,respectively. According to equations (3) and (4), inthese types of membership functions, a, b, and c,respectively, σ and c, are considered variables andmust be adjusted. The back-propagation algorithmmay, therefore, be used to train these parameters. Thegoal is to minimize a cost function of the followingform

ε = 12(ydes − y)2 (8)

where ydes is the desired output. The output of eachrule yi(x1, x2) is defined by

yi(t + 1) = yi(t) − ky∂ε

∂yi(9)

where ky is the step size.Starting from the Sugeno-system’s output (equa-

tion (6)), modifying with equation (9) results in

∂ε

∂yi= ∂ε

∂y∂y∂yi

(10)

with

∂ε

∂y= ydes − y,

∂y∂yi

= wi(x)∑Ni=1 wi(x)

(11)

Therefore, the output of each rule is obtained with theequation

yi(t + 1) = yi(t) − ky(ydes − y)wi(x)∑N

i=1 wi(x)(12)

If a generalized Bell-type membership function isused, the parameters for the jth membership functionof the ith fuzzy rule are determined with the followingequations

aij (t + 1) = ai

j (t) − ka∂ε

∂aij

bij(t + 1) = bi

j(t) − kb∂ε

∂bij

cij (t + 1) = ci

j (t) − kc∂ε

∂cij

(13)

For a Gaussian-type membership function, the para-meters of the jth membership function of the ith fuzzyrule are calculated with

σ ij (t + 1) = σ i

j (t) − kσ

∂ε

∂σ ij

cij (t + 1) = ci

j (t) − kc∂ε

∂cij

(14)

After the four controllers (Controller 1 for increasingcurrent, Controller 2 for constant current, Controller 3for decreasing current, and Controller 4 for null cur-rent) have been obtained, they must be integrated,resulting in the logical scheme in Fig. 3.

The decision to use one of the four controllersdepends on the current vector types (increasing,decreasing, constant, or zero) and on the ‘k’ variablevalue. Depending on the value of ‘k’, it can be decidedif a constant current value is part of an increasing vec-tor or part of a decreasing vector. The initial ‘k’ valueis equal to 1 when Controller 1 is used, and is equal to0 when Controllers 2, 3, or 4 are used.



Fig. 3 The logical scheme for the four controller’sintegration

4 THE INTEGRATED CONTROLLER DESIGN ANDEVALUATION

In the first phase, the ‘genfis2’ Matlab function [3] wasused to generate and train the FISs associated withthe four controllers in Fig. 3: ‘ElongationFis’ (for the

current increase phase), ‘cElongationFis’ (for the con-stant phase of the current), ‘dElongationFis’ (for thedecrease phase of the current), and ‘d0ElongationFis’(for the null values of the current obtained after thedecrease phase). The FISs are trained for differentepochs (100 000 for the first FIS, 200 000 epochs forthe second and the last FISs, and 10 000 epochs forthe third) using the ‘ANFIS’ Matlab function. Figure 4displays the deviation between the neuro-fuzzy mod-els and the experimentally obtained data for differenttraining epochs, defining the quality parameter fromthe training algorithm. A rapid decrease in the devi-ation between the experimental data and the neuro-fuzzy model is apparent for all four FISs in terms ofthe quality parameter within the training algorithmover the first 103 training epochs. Evaluating each ofthe four FISs for the experimental data using the ‘eval-fis’ command, the characteristics shown in Fig. 5 wereobtained. The means of the relative absolute valuesof the errors for all four FISs are 0.410 85, 5.597 08,0.003 47, and 3.523 28 per cent for ElongationFis,cElongationFis, dElongationFis, and d0ElongationFis,respectively. The error obtained for the third FIS(‘dElongationFis’) is very good, and so this FIS will beconsidered for implementation in the Simulink inte-grated controller. The first, second, and fourth FISshave large error values and so the generating methodmust be changed.

During the second phase, the ‘genfis1’ Matlabfunction [3] can be used to build and train the

Fig. 4 Training errors for the FISs generated and trained in the first phase



Fig. 5 FISs’ evaluation as a function of the number of experimental data points in the first phase

remaining three FIS: ‘ElongationFis’, ‘cElongationFis’,and ‘d0ElongationFis’. The number of membershipfunctions considered for each of these is 6 for thefirst input and 12 for the second input. The num-ber of training epochs considered for the three FISsare 10 000 for the first and second FISs, and 1000 for‘d0ElongationFis’. Following the evaluation of thesethree trained FISs for experimental data, the charac-teristics depicted in Fig. 6 were obtained. The evo-lution of the training errors is represented in Fig. 7.Evaluation of these three FISs gives the followingvalues of the mean of the relative absolute errors:0.269 09, 0.265 95, and 1.662 42 per cent for ‘Elonga-tionFis’, ‘cElongationFis’, and ‘d0ElongationFis’, res-pectively.

The errors obtained in the second phase for the firstand the second FISs are very good, and so these FISscan be implemented in the Simulink-integrated con-troller. For the last FIS (‘d0ElongationFis’), the errorvalues are still too large, and so the number of mem-bership functions used to generate it must be adjusted.Therefore, a third phase of FISs building and train-ing is reserved to obtain a better solution for the‘d0ElongationFis’ FIS. In this phase, two cases wereconsidered for the number of membership functions:case 1 – the mf numbers are 12 for the first input and 12for the second input; case 2 – the mf number is 12 forthe first input, and 14 for the second. A number of 4000training epochs were considered in the first case and1000 in the second. The training errors for both cases,

after training with the ‘ANFIS’ function, are presentedin Fig. 8, and the evaluation as a function of the num-ber of experimental data points is shown in Fig. 9. Themeans of the relative absolute error values for the twocases are 0.748 44 and 0.607 46 per cent, respectively.Since the errors in the second case are lower, that isthe configuration that was chosen to be implementedin a Simulink-integrated controller.

The final values of the relative absolute errors for thefour generated and trained FISs are 0.269 09 per centfor ‘ElongationFis’, 0.265 95 per cent for ‘cElongation-Fis’, 0.003 47 per cent for ‘dElongationFis’, and 0.607 46per cent for ‘d0ElongationFis’.

Representing the elongations (those obtainedexperimentally and by using the four FIS models) asfunctions of electrical current for the first and thirdFISs, and as a function of time for the other two FISs,produces the graphics in Fig. 10. The curves are rep-resented for all five cases of the SMA load. One caneasily observe that, through training, the FISs modelthe experimental data very well, and the SMA hasdifferent thermal constants, depending on the forcevalue.

A good overlapping of the FIS models’ elongationswith the elongation experimental data is clearly visi-ble in Fig. 10. This superposition is dependent on thenumber of training epochs, and improves as the num-ber of training epochs is higher. Because the trainingerrors of all of the trained FISs ultimately take constantvalues, an improved approximation of the real model



Fig. 6 FISs’ evaluation as a function of the number of experimental data points in the secondphase

Fig. 7 Training errors for the three FISs generated and trained in the second phase



Fig. 8 Training errors for the ‘d0ElongationFIS’ generated and trained in the third phase

Fig. 9 FIS’s evaluation as a function of the number of experimental data points for the third phase

Fig. 10 FIS evaluations as functions of current or time



can be achieved with the neuro-fuzzy methods onlywhen a higher quantity of experimental data is used.

To visualize the FIS’s features, the Matlab ‘anfisedit’command [3] is used, followed by the FIS’s importationon the interface level. The resulting surfaces for all four

final, trained FISs are presented in Fig. 11. The param-eters of the input’s membership functions for eachof the four FISs before and after training are shownin Tables 1 and 2, respectively. For the generalizedBell-type membership functions, produced with the

Fig. 11 The surfaces produced for all four of the final trained FISs

Table 1 Parameters of the FIS input’s membership functions before training

ElongationFis cElongationFis dElongationFis d0ElongationFis

Force (N ) Current (A) Force (N) Time (s) Force (N) Current (A) Force (N) Time (s)

a b c a b c a b c a b c σ/2 c σ/2 c a b c a b c

mf1 7.72 2 120.19 0.22 2 0.50 9.11 2 119.39 1.22 2 0 16.4 142.2 0.97 5 6.78 2 83.31 5.4 2 0mf2 7.72 2 135.65 0.22 2 0.95 9.11 2 137.61 1.22 2 2.44 16.4 141.9 0.97 0 6.78 2 96.88 5.4 2 10.81mf3 7.72 2 151.11 0.22 2 1.40 9.11 2 155.83 1.22 2 4.89 16.4 203.8 0.97 5.5 6.78 2 110.45 5.4 2 21.63mf4 7.72 2 166.56 0.22 2 1.86 9.11 2 174.05 1.22 2 7.33 16.4 214.4 0.97 0 6.78 2 124.02 5.4 2 32.45mf5 7.72 2 182.02 0.22 2 2.31 9.11 2 192.27 1.22 2 9.78 16.4 177.9 0.97 −0.01 6.78 2 137.58 5.4 2 43.26mf6 7.72 2 197.48 0.22 2 2.77 9.11 2 210.49 1.22 2 12.23 16.4 186.6 0.97 5 6.78 2 151.15 5.4 2 54.08mf7 – – – 0.22 2 3.22 – – – 1.22 2 14.67 – – – – 6.78 2 164.72 5.4 2 64.90mf8 – – – 0.22 2 3.68 – – – 1.22 2 17.12 – – – – 6.78 2 178.29 5.4 2 75.72mf9 – – – 0.22 2 4.13 – – – 1.22 2 19.57 – – – – 6.78 2 191.86 5.4 2 86.53mf10 – – – 0.22 2 4.59 – – – 1.22 2 22.01 – – – – 6.78 2 205.42 5.4 2 97.35mf11 – – – 0.22 2 5.04 – – – 1.22 2 24.46 – – – – 6.78 2 218.99 5.4 2 108.17mf12 – – – 0.22 2 5.50 – – – 1.22 2 26.90 – – – – 6.78 2 232.56 5.4 2 118.99mf13 – – – – – – – – – – – – – – – – – – – 5.4 2 129.81mf14 – – – – – – – – – – – – – – – – – – – 5.4 2 140.62


664 T L Grigorie and R M BotezTa

ble

2Pa

ram

eter

so

fth

eF

ISin

pu

t’sm

emb

ersh

ipfu

nct

ion

afte

rtr

ain

ing

Elo

nga

tio

nFi

scE

lon

gati

on

Fis

dE

lon

gati

on

Fis

d0E

lon

gati

on

Fis

Forc

e(N

)C

urr

ent(

A)

Forc

e(N

)T

ime

(s)

Forc

e(N

)C

urr

ent(

A)

Forc

e(N

)T

ime

(s)

ab

ca

bc

ab

ca

bc

σ/2

cσ

/2c

ab

ca

bc

mf1

9.27

2.1

112.

46.

513.

088

0.47

9.31

2.04

119.

50.

92.

0−0

.216

.39

142.

20.

784.

756.

80.

883

.35.

72.

50.

5m

f29.

292.

412

9.7

5.85

93.

851

12.5

8.92

1.14

137.

41.

32.

32.

413

.25

139.

20.

431.

036.

90.

596

.86.

02.

111

.2m

f39.

452.

814

8.7

7.03

61.

078

25.2

9.79

2.76

155.

01.

21.

74.

516

.38

203.

81.

025.

357.

12.

011

0.7

3.9

3.1

19.9

mf4

9.46

1.0

166.

66.

391

2.00

538

.39.

574.

3217

6.4

1.8

1.9

8.0

16.7

221

4.1

0.97

1.3e

-77.

61.

512

3.7

5.4

4.8

31.5

mf5

6.62

2.9

182

6.39

2.00

6951

.17.

901.

6419

3.4

1.4

3.3

9.9

12.0

517

7.9

0.97

1.2e

-66.

13.

913

7.8

5.8

3.6

43.6

mf6

9.51

4.2

200

6.39

142.

0036

63.9

102.

4020

9.5

1.1

3.2

12.6

16.4

818

6.6

1.04

5.22

6.3

1.3

151.

44.

8−0

.354

.0m

f7–

––

6.39

172.

002

76.7

––

–0.

61.

415

.1–

––

–7.

52.

816

3.5

4.5

4.1

64.6

mf8

––

–6.

3918

2.00

1289

.4–

––

1.0

2.6

16.6

––

––

3.8

4.2

177.

66.

71.

775

.0m

f9–

––

6.39

182.

0007

102.

2–

––

1.6

2.5

19.2

––

––

5.2

8.4

191.

55.

91.

786

.2m

f10

––

–6.

3919

2.00

0511

5.0

––

–1.

32.

022

.0–

––

–7.

22.

620

4.8

5.6

1.9

97.1

mf1

1–

––

6.39

192.

0003

127.

8–

––

1.5

1.8

24.3

––

––

7.9

1.7

219.

05.

61.

610

8.2

mf1

2–

––

6.39

22.

002

140.

6–

––

1.4

0.6

27.0

––

––

6.8

0.7

232.

75.

50.

711

9.0

mf1

3–

––

––

––

––

––

––

––

––

––

5.7

1.1

129.

7m

f14

––

––

––

––

––

––

––

––

––

–5.

51.

514

0.5 ‘genfis1’ function, the parameters are the membership

function centre (c) defining their position, and a, b thatdefine their shape. For the Gaussian-type membershipfunctions, generated with the ‘genfis2’ function, theparameters are one-half of the dispersion (σ/2) andthe centre of the membership function (c). For our sys-tem, a set of 72 rules for ‘ElongationFis’ and another72 for ‘cElongationFis’, 6 rules for ‘dElongationFis’ and168 rules for ‘d0ElongationFis’ are generated.

Comparison of the FISs’ characteristics and mem-bership functions’ parameters before and after train-ing, from Tables 1 and 2, indicates a redistribution ofthe membership functions in the working domain anda change in their shapes, by modification of the a, b,and σ parameters. According to the parameter valuesfrom Table 1, generating FISs with the ‘genfis1’ and‘genfis2’ functions primarily results in the same valuesfor the a, b, and σ/2 parameters for all of the member-ship functions that characterize an input. A secondaryresult is the separation of the working space for therespective input using a grid partition on the data (noclustering) if the ‘genfis1’ function is used, or using thefuzzy subtractive clustering method if generating withthe ‘genfis2’ function.

For the ‘dElongationFis’ FIS (initially generated byusing the ‘genfis2’ function) the rules are of the type:if (in1 is in1cluster ‘k’) and (in2 is in2cluster ‘k’) then(out1 is out1cluster ‘k’). For both of the inputs of thisFIS, six Gaussian-type mf were generated; within theset of rules they are noted by in ‘j’ cluster ‘k’; j isthe input number (1/2), and k is the number of themembership function (1/6). The ‘dElongationFis’ FIShas the structure shown in Fig. 12, whereas the cor-responding controller (Controller 3) has the structurepresented in Fig. 13.

For the other three FISs (initially generated by usingthe ‘genfis1’ function) the rules are of the type: if (in1is in1mf ‘k’) and (in2 is in2mf ‘p’) then (out1 is out1mf‘r’). The number of output mf is k × p (r = 1/(k ×p)) and is equal to the number of rules. For thesethree FISs, generalized Bell-type membership func-tions were generated; within the sets of rules they arenoted by in ‘j’ mf ‘n’; j is the input number (1/2) and nis the number of the membership functions. For ‘Elon-gationFis’ and ‘cElongationFis’, six membership func-tions for the first input (k = 6) and 12 membershipfunctions for the second input (p = 12 → r = 72) areproduced. The ‘d0ElongationFis’ results in 12 mem-bership functions for the first input (k = 12) and 14for the second input (p = 14 → r = 168). For exam-ple, the ‘ElongationFis’ FIS has the structure shown inFig. 14, whereas the corresponding controller (Con-troller 1) has the same structure as Controller 3 (seeFig. 13).

Each of the four FISs is imported at the fuzzy con-troller level, resulting in four controllers: Control-ler 1 (‘ElongationFis’), Controller 2 (‘cElongationFis’),Controller 3 (‘dElongationFis’), and Controller 4



Fig. 12 Structure of the ‘dElongationFis’ FIS

Fig. 13 The structure of Controller 3

(‘d0ElongationFis’). These four controllers are inte-grated using the logical scheme given in Fig. 3; theMatlab/Simulink model in Fig. 15 is the result.

In the Matlab/Simulink model shown in Fig. 15, thesecond input of Controller 2 and that of Controller 4(time) are generated by using integrators, starting fromthe moment that these inputs are used in Controller2 or Controller 4 (the input of the Gain block is 0 ifthe schema decides not to work with one of the Con-trollers 2 or 4). It is possible that the simulation sample

time may be different from the sample time used inthe experimental data acquisition process, and there-fore the ‘Gain’ block that gives their ratio is used; ‘Te’is the sample time in the experimental data and ‘T ’ isthe simulation sample time. In the schema, the con-stant ‘C ’ represents the maximum time considered forthe actuator to recover its initial temperature (∼24 ◦C)when the current becomes 0 A.

Evaluating the integrated controller model (seeFig. 15) for all five experimental data cases producesthe results shown in Figs 16 and 17. These graphicsshow the elongations versus the number of experi-mental data points and versus the applied electricalcurrent, respectively, using the experimental data andthe integrated neuro-fuzzy controller model for theSMA. A good overlapping of the outputs of the inte-grated neuro-fuzzy controller with the experimentaldata can be easily observed.

Fig. 14 Structure of the ‘ElongationFis’ FIS



Fig. 15 The integration model schema in Matlab/Simulink

Fig. 16 Elongations versus the number of experimental data points



Fig. 17 Elongations versus the applied electrical current

Fig. 18 Three-dimensional evaluation of the integrated neuro-fuzzy controller

The same observation can be made from the three-dimensional characteristics of the experimental dataand the neuro-fuzzy modelled data in terms of temper-ature, elongation, and force, depicted in Fig. 18(a), andin terms of current, elongation, and force, depicted inFig. 18(b).

The mean values of the relative absolute errors of theintegrated controller for the five load cases of the SMA,based on adaptive neuro-FISs, are 0.459 97 per cent for120 N, 0.502 95 per cent for 140 N, 0.513 19 per cent for150 N, 0.716 09 per cent for 180 N, and 0.507 75 per centfor 190 N. The mean value of the relative absolute errorbetween the experimental data and the outputs of theintegrated controller is 0.54 per cent.

5 CONCLUSIONS

In this article, an integrated controller based on adap-tive neuro-FISs for modelling smart material actuatorswas obtained. The direct application of this con-troller is in a morphing wing system. The general aimof the smart material actuators’ desired model is tocalculate the elongation of the actuator under theapplication of a thermo-electro-mechanical load fora certain time. Therefore, the smart material actua-tors were experimentally tested in conditions close tothose in which they will be used. Testing was per-formed for five load cases, with forces of 120, 140,150, 180, and 190 N. Using the experimental data,



four FISs were generated and trained to obtain fourneuro-fuzzy controllers: one controller for the currentincrease (‘ElongationFis’), one for a constant cur-rent (‘cElongationFis’), one for the current decrease(‘dElongationFis’), and one controller for the nullcurrent, after its decrease (‘d0ElongationFis’). The‘genfis1’ and ‘genfis2’ Matlab functions were used togenerate the initial FISs, and the adaptive neuro-IFStechnique was then used to train them.The final valuesof the relative absolute errors for the four generatedand trained FISs were 0.269 09 per cent for ‘Elonga-tionFis’, 0.265 95 per cent for ‘cElongationFis’, 0.003 47per cent for ‘dElongationFis’, and 0.607 46 per cent for‘d0ElongationFis’.

Each of the four obtained and trained FISs wereimported at the fuzzy controller level, resulting in fourcontrollers. Finally, these four controllers were inte-grated by using the logical scheme given in Fig. 3;resulting in the Matlab/Simulink model for the inte-grated controller shown in Fig. 15. The integratedcontroller performances were evaluated for all fiveload cases; the values obtained for the mean rela-tive absolute errors were 0.459 97 per cent for 120 N,0.502 95 per cent for 140 N, 0.513 19 per cent for150 N, 0.716 09 per cent for 180 N, and 0.507 75 percent for 190 N. Thus, the mean value of the rela-tive absolute error between the experimental dataand the outputs of the integrated controller was 0.54per cent.

A particular advantage of this new model is itsrapid generation, thanks to the ‘genfis1’, ‘genfis2’, and‘ANFIS’ functions already implemented in Matlab. Theuser need only assume the four FIS’s training perfor-mances using the ‘anfisedit’ interface generated withMatlab.

© Authors 2009

REFERENCES

1 Sivanandam, S. N., Sumathi, S., and Deepa, S. N. Intro-duction to fuzzy logic using MATLAB, 2007 (Springer,Berlin, Heidelberg).

2 Kosko, B. Neural networks and fuzzy systems – a dynam-ical systems approach to machine intelligence, 1992(Prentice Hall, New Jersey, NJ).

3 Matlab fuzzy logic and neural network toolboxes, avail-able from http://www.mathtools.net/MATLAB/Books/Neural_Network_and_Fuzzy_Logic/.

4 Khezri, M. and Jahed, M. Real-time intelligent patternrecognition algorithm for surface EMG signals. BioMed.Eng. OnLine, 2007, 6, 45. DOI: 10.1186/1475-925X-6-45.

5 Kung, C. C. and Su, J. Y. Affine Takagi–Sugeno fuzzy mod-elling algorithm by fuzzy c-regression models clustering

with a novel cluster validity criterion. IET Control TheoryAppl., 2007, 1(5), 1255–1265.

6 Mahfouf, M., Linkens, D. A., and Kandiah, S. FuzzyTakagi–Sugeno Kang model predictive control for processengineering, 1999, p. 4 (IEE, Savoy place, London WCPROBL, UK).

APPENDIX

Notation

a, b parameters of the generalized bellmembership function

aik parameters of the linear function

(k = 1, 2, i = 1, N )

Aiq associated individual antecedent fuzzy sets

of each input variable (i = 1, N )

bi0 scalar offset (i = 1, N )

c cluster centreci

q cluster centre (q = 1, 2)

Cp pressure coefficientF forcei electrical currentk variable for neuro-fuzzy controller selectionky step sizel dimension of the data vectorsM number of data pointsN number of rulesrm radius within the fuzzy neighbourhood,

contributes to the density measureRe Reynolds numbert timeT temperature of the smart material actuatorTamb ambient temperatureucj centre of the jth clusteruk data vectorsV speedwi degree of fulfilment of the antecedent, i.e.

the level of firing of the ith rulex input vectorxq individual input variables (q = 1, 2)

y output of the fuzzy modelyi first-order polynomial function in the

consequent (i = 1, N )

α angle of attack�δ actuator elongation�t time variationε cost functionρ density measureσ dispersionσ i

q cluster dispersion


655 Adaptiveneuro-fuzzyinferencesystem-basedcontrollers ...

Documents

Transcript of 655 Adaptiveneuro-fuzzyinferencesystem-basedcontrollers ...