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Adaptive neuro-fuzzy inference system-based controllersfor smart material actuator modellingT L Grigorie and R M Botezâ
Ăcole de Technologie SupĂŠrieure, MontrĂŠal, 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 â a combination of back propagation and least-mean-square methods â to train thefuzzy inference systems. The controllersâ objective is to correlate each set of forces and electricalcurrents applied on the smart material actuator to the actuatorâs 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 SMAsâ experimen-tal testing and it takes advantage of the outstandingproperties of fuzzy logic, which allow the signalâsempirical 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 IFâTHEN form. Fuzzy logic is recommended forvery complex processes, when no simple mathemati-cal model exists, for highly non-linear processes andfor multi-dimensional systems.
âCorresponding author: Laboratory of Research in Active Controls,
Avionics and AeroServoElasticity LARCASE, Ăcole de Technologie
SupĂŠrieure, 1100, rue Notre-Dame Ouest, MontrĂŠal, QuĂŠbec 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 âfuzzy setsâ; the mappingprocess is called âfuzzificationâ. The control systemâsdecisions 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 âinferenceâ. 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 âdefuzzifica-tionâ. 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 systemsâ (FISs) training. To this end, a hybridmethod â a combination of back propagation andleast-mean-square (LMS) methods â 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 âŚC, for five load cases with forcesof 120, 140, 150, 180, and 190 N. The electrical cur-rents following the increasing-constantâdecreasing-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 SMAâs thermal inertia are used (the time valuesrequired for the SMA to recover its initial temperaturevalue (âź24 âŚC) 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 â 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 ofâIFâTHENâ logic rules is created; the IF part is calledthe âantecedentâ and the THEN part is called the âcon-sequentâ. 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 inputâoutput 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 âgenfis1â or âgenfis2â functions. The âgen-fis1â 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 âgenfis1â function uses generalizedBell-type membership functions for each input. Eachrule generated by the âgenfis1â function has one out-put membership function, which is, by default, of alinear type. It is also possible to create an FIS usingthe Matlab âgenfis2â 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 membershipfunctionsâ 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 modelsâ 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
Ďk =Mâ
j=1
exp(
â|uk â 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 Ďc1 its density measure. Next, thedensity measure for each data point uk is revised by
the formula
Ď â˛k = Ďk â Ďc1 exp
(â|uk â 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 inputâoutput 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 âgenfis1â function generates member-ship functions of the generalized Bell type, defined asfollows [2, 5]
Aiq(x) =
ââ1 +
âŁâŁâŁâŁâŁx â ciq
a
âŁâŁâŁâŁâŁ2b
ââ
â1
(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). Matlabâs âgenfis2â function generatesGaussian-type membership functions, defined withthe following expression [2, 5]
Aiq(x) = exp
âĄâŁâ0.5
(x â ci
q
Ď iq
)2â¤âŚ (4)
where ciq is the cluster centre and Ď i
q is the dispersionof the cluster.
The Sugeno fuzzy model was proposed by Takagi,Sugeno, and Kang to generate fuzzy rules from a giveninputâoutput 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
...
JAERO522 Proc. IMechE Vol. 223 Part G: J. Aerospace Engineering
658 T L Grigorie and R M Botez
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 =( âN
i=1 wi(x)yi)
( âNi=1 wi(x)
) (6)
where
wi(x) = Ai1(x1) Ă 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 Matlabâs âANFISâ 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 inputâoutput data and the FIS models generatedwith the âgenfis1â or âgenfis2â functions are considered.âANFISâ optimizes the membership functionsâ 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 âgenfis1â or âgenfis2â 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, Ď 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
Îľ = 12(ydes â y)2 (8)
where ydes is the desired output. The output of eachrule yi(x1, x2) is defined by
yi(t + 1) = yi(t) â kyâÎľ
âyi(9)
where ky is the step size.Starting from the Sugeno-systemâs output (equa-
tion (6)), modifying with equation (9) results in
âÎľ
âyi= âÎľ
âyâyâyi
(10)
with
âÎľ
ây= ydes â y,
âyâyi
= wi(x)âNi=1 wi(x)
(11)
Therefore, the output of each rule is obtained with theequation
yi(t + 1) = yi(t) â ky(ydes â y)wi(x)âN
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) â kaâÎľ
âaij
bij(t + 1) = bi
j(t) â kbâÎľ
âbij
cij (t + 1) = ci
j (t) â kcâÎľ
âcij
(13)
For a Gaussian-type membership function, the para-meters of the jth membership function of the ith fuzzyrule are calculated with
Ď ij (t + 1) = Ď i
j (t) â kĎ
âÎľ
âĎ ij
cij (t + 1) = ci
j (t) â kcâÎľ
âcij
(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 âkâ variablevalue. Depending on the value of âkâ, it can be decidedif a constant current value is part of an increasing vec-tor or part of a decreasing vector. The initial âkâ valueis equal to 1 when Controller 1 is used, and is equal to0 when Controllers 2, 3, or 4 are used.
Proc. IMechE Vol. 223 Part G: J. Aerospace Engineering JAERO522
Adaptive neuro-fuzzy inference system-based controllers 659
Fig. 3 The logical scheme for the four controllerâsintegration
4 THE INTEGRATED CONTROLLER DESIGN ANDEVALUATION
In the first phase, the âgenfis2â Matlab function [3] wasused to generate and train the FISs associated withthe four controllers in Fig. 3: âElongationFisâ (for the
current increase phase), âcElongationFisâ (for the con-stant phase of the current), âdElongationFisâ (for thedecrease phase of the current), and âd0ElongationFisâ(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 âANFISâ 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 âeval-fisâ 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(âdElongationFisâ) 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 âgenfis1â Matlabfunction [3] can be used to build and train the
Fig. 4 Training errors for the FISs generated and trained in the first phase
JAERO522 Proc. IMechE Vol. 223 Part G: J. Aerospace Engineering
660 T L Grigorie and R M Botez
Fig. 5 FISsâ evaluation as a function of the number of experimental data points in the first phase
remaining three FIS: âElongationFisâ, âcElongationFisâ,and âd0ElongationFisâ. 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 forâd0ElongationFisâ. 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 âElonga-tionFisâ, âcElongationFisâ, and âd0ElongationFisâ, 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 (âd0ElongationFisâ), 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 theâd0ElongationFisâ FIS. In this phase, two cases wereconsidered for the number of membership functions:case 1 â the mf numbers are 12 for the first input and 12for the second input; case 2 â 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 âANFISâ 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 âElongationFisâ, 0.265 95 per cent for âcElongation-Fisâ, 0.003 47 per cent for âdElongationFisâ, and 0.607 46per cent for âd0ElongationFisâ.
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 modelsâ 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
Proc. IMechE Vol. 223 Part G: J. Aerospace Engineering JAERO522
Adaptive neuro-fuzzy inference system-based controllers 661
Fig. 6 FISsâ 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
JAERO522 Proc. IMechE Vol. 223 Part G: J. Aerospace Engineering
662 T L Grigorie and R M Botez
Fig. 8 Training errors for the âd0ElongationFISâ generated and trained in the third phase
Fig. 9 FISâs 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
Proc. IMechE Vol. 223 Part G: J. Aerospace Engineering JAERO522
Adaptive neuro-fuzzy inference system-based controllers 663
can be achieved with the neuro-fuzzy methods onlywhen a higher quantity of experimental data is used.
To visualize the FISâs features, the Matlab âanfiseditâcommand [3] is used, followed by the FISâs importationon the interface level. The resulting surfaces for all four
final, trained FISs are presented in Fig. 11. The param-eters of the inputâs 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 inputâs 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 Ď/2 c Ď/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 â0.01 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 â â â 0.22 2 3.22 â â â 1.22 2 14.67 â â â â 6.78 2 164.72 5.4 2 64.90mf8 â â â 0.22 2 3.68 â â â 1.22 2 17.12 â â â â 6.78 2 178.29 5.4 2 75.72mf9 â â â 0.22 2 4.13 â â â 1.22 2 19.57 â â â â 6.78 2 191.86 5.4 2 86.53mf10 â â â 0.22 2 4.59 â â â 1.22 2 22.01 â â â â 6.78 2 205.42 5.4 2 97.35mf11 â â â 0.22 2 5.04 â â â 1.22 2 24.46 â â â â 6.78 2 218.99 5.4 2 108.17mf12 â â â 0.22 2 5.50 â â â 1.22 2 26.90 â â â â 6.78 2 232.56 5.4 2 118.99mf13 â â â â â â â â â â â â â â â â â â â 5.4 2 129.81mf14 â â â â â â â â â â â â â â â â â â â 5.4 2 140.62
JAERO522 Proc. IMechE Vol. 223 Part G: J. Aerospace Engineering
664 T L Grigorie and R M BotezTa
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0.5 âgenfis1â 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 âgenfis2â function, theparameters are one-half of the dispersion (Ď/2) andthe centre of the membership function (c). For our sys-tem, a set of 72 rules for âElongationFisâ and another72 for âcElongationFisâ, 6 rules for âdElongationFisâ and168 rules for âd0ElongationFisâ are generated.
Comparison of the FISsâ characteristics and mem-bership functionsâ 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 Ď parameters. According to the parameter valuesfrom Table 1, generating FISs with the âgenfis1â andâgenfis2â functions primarily results in the same valuesfor the a, b, and Ď/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 âgenfis1â function is used, or using thefuzzy subtractive clustering method if generating withthe âgenfis2â function.
For the âdElongationFisâ FIS (initially generated byusing the âgenfis2â function) the rules are of the type:if (in1 is in1cluster âkâ) and (in2 is in2cluster âkâ) then(out1 is out1cluster âkâ). For both of the inputs of thisFIS, six Gaussian-type mf were generated; within theset of rules they are noted by in âjâ cluster âkâ; j isthe input number (1/2), and k is the number of themembership function (1/6). The âdElongationFisâ 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 âgenfis1â function) the rules are of the type: if (in1is in1mf âkâ) and (in2 is in2mf âpâ) then (out1 is out1mfârâ). The number of output mf is k Ă p (r = 1/(k Ăp)) 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 âjâ mf ânâ; j is the input number (1/2) and nis the number of the membership functions. For âElon-gationFisâ and âcElongationFisâ, six membership func-tions for the first input (k = 6) and 12 membershipfunctions for the second input (p = 12 â r = 72) areproduced. The âd0ElongationFisâ results in 12 mem-bership functions for the first input (k = 12) and 14for the second input (p = 14 â r = 168). For exam-ple, the âElongationFisâ 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 (âElongationFisâ), Controller 2 (âcElongationFisâ),Controller 3 (âdElongationFisâ), and Controller 4
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Adaptive neuro-fuzzy inference system-based controllers 665
Fig. 12 Structure of the âdElongationFisâ FIS
Fig. 13 The structure of Controller 3
(âd0ElongationFisâ). 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 âGainâ block that gives their ratio is used; âTeâis the sample time in the experimental data and âT â isthe simulation sample time. In the schema, the con-stant âC â represents the maximum time considered forthe actuator to recover its initial temperature (âź24 âŚC)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 âElongationFisâ FIS
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666 T L Grigorie and R M Botez
Fig. 15 The integration model schema in Matlab/Simulink
Fig. 16 Elongations versus the number of experimental data points
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Adaptive neuro-fuzzy inference system-based controllers 667
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 actuatorsâ 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,
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668 T L Grigorie and R M Botez
four FISs were generated and trained to obtain fourneuro-fuzzy controllers: one controller for the currentincrease (âElongationFisâ), one for a constant cur-rent (âcElongationFisâ), one for the current decrease(âdElongationFisâ), and one controller for the nullcurrent, after its decrease (âd0ElongationFisâ). Theâgenfis1â and âgenfis2â 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 âElonga-tionFisâ, 0.265 95 per cent for âcElongationFisâ, 0.003 47per cent for âdElongationFisâ, and 0.607 46 per cent forâd0ElongationFisâ.
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 âgenfis1â, âgenfis2â, andâANFISâ functions already implemented in Matlab. Theuser need only assume the four FISâs training perfor-mances using the âanfiseditâ interface generated withMatlab.
Š 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 â 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 TakagiâSugeno fuzzy mod-elling algorithm by fuzzy c-regression models clustering
with a novel cluster validity criterion. IET Control TheoryAppl., 2007, 1(5), 1255â1265.
6 Mahfouf, M., Linkens, D. A., and Kandiah, S. FuzzyTakagiâSugeno 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 )
Îą angle of attack�δ actuator elongationďż˝t time variationÎľ cost functionĎ density measureĎ dispersionĎ i
q cluster dispersion
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