Adaptive neuro-fuzzy inferene system-based ontrollers for smart material atuator modelling T L Grigorie and R M Botez Éole de Tehnologie Supérieure, Montréal, Quebe, Canada The manusript was reeived on 6 February 009 and was aepted after revision for publiation on 15 May 009. DOI: 10.143/09544100JAERO5 655 Abstrat: An intelligent approah for smart material atuator modelling of the atuation lines in a morphing wing system is presented, based on adaptive neuro-fuzzy inferene systems. Four independent neuro-fuzzy ontrollers are reated from the experimental data using a hybrid method a ombination of bak propagation and least-mean-square methods to train the fuzzy inferene systems. The ontrollers objetive is to orrelate eah set of fores and eletrial urrents applied on the smart material atuator to the atuator s elongation. The atuator experimental testing is performed for five fore ases, using a variable eletrial urrent. An integrated ontroller is reated from four neuro-fuzzy ontrollers, developed with Matlab/Simulink software for eletrial urrent inreases, onstant eletrial urrent, eletrial urrent dereases, and for null eletrial urrent in the ooling phase of the atuator, and is then validated by omparison with the experimentally obtained data. Keywords: smart material atuator, neuro-fuzzy ontroller, simulation, modelling, testing 1 INTRODUCTION The aim of this artile is to obtain a reliable, easyto-implement model for smart material atuators (SMAs), with diret appliations in the morphing wing projet. Based on adaptive neuro-fuzzy inferene systems, an integrated ontroller is built to model the SMAs used in the atuation lines of a wing. This model uses the numerial values from the SMAs experimental testing and it takes advantage of the outstanding properties of fuzzy logi, whih allow the signal s empirial proessing without the use of mathematial analytial models. Fuzzy logi systems an emulate human deision-making more losely than many other lassifiers through the proessing of expert system knowledge, formulated linguistially in fuzzy rules in an IFTHEN form. Fuzzy logi is reommended for very omplex proesses, when no simple mathematial model exists, for highly non-linear proesses and for multi-dimensional systems. Corresponding author: Laboratory of Researh in Ative Controls, Avionis and AeroServoElastiity LARCASE, Éole de Tehnologie Supérieure, 1100, rue Notre-Dame Ouest, Montréal, Québe H3C 1K3, Canada. email: ruxandra.botez@etsmtl.a JAERO5 The input variables in a fuzzy ontrol system are usually mapped into plae by sets of membership funtions (mf) known as fuzzy sets ; the mapping proess is alled fuzzifiation. The ontrol system s deisions are made on the basis of a fuzzy rules set, and are invoked using the membership funtions and the truth values obtained from the inputs; a proess alled inferene. These deisions are mapped into a membership funtion and truth value that ontrols the output variable. The results are ombined to give a speifi answer in a proedure alled defuzzifiation. Elaboration of the model thus requires a fuzzy rules set and the mf assoiated with eah of the inputs [1, ]. The ability and the experiene of a designer in evaluating the rules and the membership funtions of all of the inputs are deisive in obtaining a good fuzzy model. However, a relatively new design method allows a ompetitive model to be built using a ombination of fuzzy logi and neural-network tehniques. Moreover, this method allows the possibility to generate and optimize the fuzzy rules set and the parameters of the membership funtions by means of fuzzy inferene systems (FISs) training. To this end, a hybrid method a ombination of bak propagation and least-mean-square (LMS) methods is used, in whih experimentally obtained data are onsidered. Already Pro. IMehE Vol. 3 Part G: J. Aerospae Engineering
656 T L Grigorie andrmbotez implemented in Matlab [1, 3], the method is easy to use, and gives exellent results in a very short time. ACTUATOR EXPERIMENTAL TESTING The SMA testing was performed using the benh test in Fig. 1 at T amb = 4 C, for five load ases with fores of 10, 140, 150, 180, and 190 N. The eletrial urrents following the inreasing-onstantdereasingzero values evolution were applied on the SMA in eah of the five ases onsidered for load fores. In eah of the ases to be analysed, the following parameters were reorded: time, the eletrial urrent applied to the SMA, the load fore, the material temperature, and the atuator elongation (measured using a linear variable differential transformer (LVDT)). To model the SMA, the present authors built an integrated ontroller based on adaptive neuro-fiss. The experimental elongation-urrent urves obtained in the five load ases are shown in Fig.. One an observe that all five of the obtained urves have four distint zones: eletrial urrent inrease, onstant eletrial urrent, eletrial urrent derease, and null eletrial urrent in the ooling phase of the SMA. Four FISs are used to obtain four neuro-fuzzy ontrollers: one for the urrent inrease, one for the onstant urrent, one for the urrent derease, and one ontroller for the null urrent (after its derease). For the first and the third ontrollers, inputs suh as the fore and the urrent are used, whereas for the seond and the fourth, inputs suh as the fore and the time values refleting the SMA s thermal inertia are used (the time values required for the SMA to reover its initial temperature value ( 4 C) are used for the four ontroller). Finally, the four obtained ontrollers must be integrated into a single ontroller. The reasoning behind the design of the first and the third ontrollers is that, from the available experimental data, two elongations for the same values of fores and urrents are used (see Fig. ). Due to the experimental data values, these data annot be represented as algebrai funtions; therefore, it is impossible to use the same FIS representation. Matlab produes an interpolation between the two elongation values obtained for the same values of fores and urrents, whih annot be valid for our appliation. The onstant values, namely the null values of the urrent before and after the urrent derease phase should not be onsidered as inputs in the seond and fourth ontrollers beause they are not suggestive for the haraterization of the SMA elongation. The values of the atuator temperatures may appear to be very suggestive in these phases, but the temperature must be a model output. For these phases the time values are very suggestive, as they represent a measure of the atuator thermal inertia. Time is the seond input of the third ontroller, and so time is also the seond input of the seond and the fourth ontrollers sine fore was onsidered as the first input (the time values must be onsidered when the urrent beomes onstant or null). Fig. 1 The SMA benh test 3 THE PROPOSED METHOD Fig. Elongation versus the urrent values for different fore values for four ases Fuzzy ontrollers are very simple oneptually and are based on FISs. Three steps are onsidered in an FIS design: an input, the proessing, and then an output step. In the input step, the ontroller inputs are mapped into the appropriate mf. Next, a olletion of IFTHEN logi rules is reated; the IF part is alled the anteedent and the THEN part is alled the onsequent. In this step, eah appropriate rule is invoked and a result is generated. The results of all of the rules are then ombined. In the last step, the ombined result is onverted into a speifi ontrol output value. Considering the numerial values resulting from the SMA experimental testing, an empirial model an be developed, whih is based on a neuro-fuzzy network. The model an learn the proess behaviour based on Pro. IMehE Vol. 3 Part G: J. Aerospae Engineering JAERO5
Adaptive neuro-fuzzy inferene system-based ontrollers 657 the inputoutput proess data by using an FIS, whih should model the experimental data. Using methods already implemented in ommerial software, an FIS an be generated simply with the Matlab genfis1 or genfis funtions. The genfis1 funtion generates a single-output Sugeno-type FIS using a grid partition on the data (no lustering). This FIS is used to provide initial onditions for ANFIS training. The genfis1 funtion uses generalized Bell-type membership funtions for eah input. Eah rule generated by the genfis1 funtion has one output membership funtion, whih is, by default, of a linear type. It is also possible to reate an FIS using the Matlab genfis funtion. This funtion generates an initial Sugeno-type FIS by deomposing the operation domain into different regions using the fuzzy subtrative lustering method. For eah region, a loworder linear model an desribe the loal proess parameters. Thus, the non-linear proess is loally linearized around a funtioning point by using the LSM. The obtained model is then onsidered valid in the entire region around this point. The limitation of the operating regions implies the existene of overlapping among these different regions; their definition is given in a fuzzy manner. Thus, for eah model input, several fuzzy sets are assoiated with their membership funtions orresponding definitions. By ombining these fuzzy inputs, the input spae is divided into fuzzy regions. A loal linear model is used for eah of these regions, whereas the global model is obtained by defuzzifiation with the gravity entre method (Sugeno), whih performs the interpolation of the loal models outputs [1, 3]. Based on the goal of finding regions with a high density of data points in the featured spae, the subtrative lustering method is used to divide the spae into a number of lusters. All of the points with the highest number of neighbours are seleted as entres of lusters. The lusters are identified one by one, as the data points within a pre-speified fuzzy radius are removed (subtrated) for eah luster. Following the identifiation of eah luster, the algorithm loates a new luster until all of the data points have been heked. If a olletion of M data points, speified by l-dimensional vetors u k, k = 1,,..., M, is onsidered, a density measure at data point u k an be defined as follows ρ k = M j=1 ( exp u ) k u j (r m /) (1) where r m is a positive onstant that defines the radius within the fuzzy neighbourhood and ontributes to the density measure. The point with the highest density is seleted as the first luster entre. Let u 1 be the seleted point and ρ 1 its density measure. Next, the density measure for eah data point u k is revised by JAERO5 the formula ρ k = ρ k ρ 1 exp ( u ) k u 1 (r n /) () where r n is a positive onstant, greater than r m, that defines a neighbourhood where density measures will be redued in order to prevent losely spaed luster entres. In this way, the data points near the first luster entre u 1 will have signifiantly redued density measures, and therefore annot be seleted as subsequent luster entres. After the density measures for eah point have been revised, then the next luster entre u is seleted and all the density measures are again revised. The proess is repeated until all the data points have been heked and a suffiient number of luster entres generated. When the subtrative lustering method is applied to an inputoutput data set, eah of the luster entres are used as the entres for the premise sets in a singleton type of rule base [4]. The Matlab genfis1 funtion generates membership funtions of the generalized Bell type, defined as follows [, 5] A i q (x) = x 1 i q + a b 1 where q i is the luster entre defining the position of the membership funtion, a and b are two parameters that define the membership funtion shape, and Aq i (i = 1, N) are the assoiated individual anteedent fuzzy sets of eah input variable (N = number of rules). Matlab s genfis funtion generates Gaussian-type membership funtions, defined with the following expression [, 5] A i q (3) ( ) x i q (x) = exp 0.5 (4) σq i where q i is the luster entre and σ q i is the dispersion of the luster. The Sugeno fuzzy model was proposed by Takagi, Sugeno, and Kang to generate fuzzy rules from a given inputoutput data set [6]. In our system, for eah of the four FISs (two inputs and one output), a first-order model is onsidered, whih for N rules is given by [5, 6] Rule 1 : If x 1 is A 1 1 and x is A 1, then y1 (x 1, x ) = b 1 0 + a1 1 x 1 + a 1 x. Rule i :Ifx 1 is A i 1 and x is A i, then yi (x 1, x ) = b i 0 + ai 1 x 1 + a i x. Pro. IMehE Vol. 3 Part G: J. Aerospae Engineering
658 T L Grigorie andrmbotez Rule N :Ifx 1 is A N 1 and x is A N, then yn (x 1, x ) = b N 0 + an 1 x 1 + a N x (5) where x q (q = 1, ) are the individual input variables and y i (i = 1, N) is the first-order polynomial funtion in the onsequent. a i k (k = 1,, i = 1, N) are parameters of the linear funtion and b0 i (i = 1, N) denotes a salar offset. The parameters a i, k bi 0 (k = 1,, i = 1, N) are optimized by the LSM. For any input vetor, x =[x 1, x ] T, if the singleton fuzzifier, the produt fuzzy inferene, and the entre average defuzzifier are applied, then the output of the fuzzy model y is inferred as follows (weighted average) ( N ) i=1 wi (x)y i y = ( N ) (6) i=1 wi (x) where w i (x) = A i 1 (x 1) A i (x ) (7) w i (x) represents the degree of fulfilment of the anteedent, i.e. the level of firing of the ith rule. The adaptive neuro-fis alulates the Sugeno-type FIS parameters using neural networks. A very simple way to train these FISs is to use Matlab s ANFIS funtion, whih uses a learning algorithm to identify the membership funtion parameters of a Sugeno-type FIS with two outputs and one input. As a starting point, the inputoutput data and the FIS models generated with the genfis1 or genfis funtions are onsidered. ANFIS optimizes the membership funtions parameters for a number of training epohs, determined by the user. With this optimization, the neuro-fuzzy model an produe a better proess approximation by means of a quality parameter in the training algorithm [3]. After this training, the models may be used to generate the elongation values orresponding to the input parameters. To train the fuzzy systems, ANFIS employs a bakpropagation algorithm for the parameters assoiated with the input membership funtions, and LMS estimations for the parameters assoiated with the output membership funtions. For the FISs generated using the genfis1 or genfis funtions, the membership funtions are generalized Bell type or Gaussian type, respetively. Aording to equations (3) and (4), in these types of membership funtions, a, b, and, respetively, σ and, are onsidered variables and must be adjusted. The bak-propagation algorithm may, therefore, be used to train these parameters. The goal is to minimize a ost funtion of the following form ε = 1 (y des y) (8) Pro. IMehE Vol. 3 Part G: J. Aerospae Engineering where y des is the desired output. The output of eah rule y i (x 1, x ) is defined by y i (t + 1) = y i ε (t) k y (9) y i where k y is the step size. Starting from the Sugeno-system s output (equation (6)), modifying with equation (9) results in ε = ε y (10) y i y y i with ε y = y des y, y w i (x) = y i N i=1 wi (x) (11) Therefore, the output of eah rule is obtained with the equation y i (t + 1) = y i (y des y)w i (x) (t) k y N (1) i=1 wi (x) If a generalized Bell-type membership funtion is used, the parameters for the jth membership funtion of the ith fuzzy rule are determined with the following equations a i j (t + 1) = ai j (t) k ε a a i j b i j (t + 1) = bi j (t) k ε b b i j i j (t + 1) = i j (t) k ε i j (13) For a Gaussian-type membership funtion, the parameters of the jth membership funtion of the ith fuzzy rule are alulated with σ i i j (t + 1) = σj (t) k ε σ σj i i j (t + 1) = i j (t) k ε i j (14) After the four ontrollers (Controller 1 for inreasing urrent, Controller for onstant urrent, Controller 3 for dereasing urrent, and Controller 4 for null urrent) have been obtained, they must be integrated, resulting in the logial sheme in Fig. 3. The deision to use one of the four ontrollers depends on the urrent vetor types (inreasing, dereasing, onstant, or zero) and on the k variable value. Depending on the value of k, it an be deided if a onstant urrent value is part of an inreasing vetor or part of a dereasing vetor. The initial k value is equal to 1 when Controller 1 is used, and is equal to 0 when Controllers, 3, or 4 are used. JAERO5
Adaptive neuro-fuzzy inferene system-based ontrollers 659 Fig. 3 The logial sheme for the four ontroller s integration 4 THE INTEGRATED CONTROLLER DESIGN AND EVALUATION In the first phase, the genfis Matlab funtion [3] was used to generate and train the FISs assoiated with the four ontrollers in Fig. 3: ElongationFis (for the urrent inrease phase), ElongationFis (for the onstant phase of the urrent), delongationfis (for the derease phase of the urrent), and d0elongationfis (for the null values of the urrent obtained after the derease phase). The FISs are trained for different epohs (100 000 for the first FIS, 00 000 epohs for the seond and the last FISs, and 10 000 epohs for the third) using the ANFIS Matlab funtion. Figure 4 displays the deviation between the neuro-fuzzy models and the experimentally obtained data for different training epohs, defining the quality parameter from the training algorithm. A rapid derease in the deviation between the experimental data and the neurofuzzy model is apparent for all four FISs in terms of the quality parameter within the training algorithm over the first 10 3 training epohs. Evaluating eah of the four FISs for the experimental data using the evalfis ommand, the harateristis shown in Fig. 5 were obtained. The means of the relative absolute values of the errors for all four FISs are 0.410 85, 5.597 08, 0.003 47, and 3.53 8 per ent for ElongationFis, ElongationFis, delongationfis, and d0elongationfis, respetively. The error obtained for the third FIS ( delongationfis ) is very good, and so this FIS will be onsidered for implementation in the Simulink integrated ontroller. The first, seond, and fourth FISs have large error values and so the generating method must be hanged. During the seond phase, the genfis1 Matlab funtion [3] an be used to build and train the Fig. 4 Training errors for the FISs generated and trained in the first phase JAERO5 Pro. IMehE Vol. 3 Part G: J. Aerospae Engineering
660 T L Grigorie andrmbotez Fig. 5 FISs evaluation as a funtion of the number of experimental data points in the first phase remaining three FIS: ElongationFis, ElongationFis, and d0elongationfis. The number of membership funtions onsidered for eah of these is 6 for the first input and 1 for the seond input. The number of training epohs onsidered for the three FISs are 10 000 for the first and seond FISs, and 1000 for d0elongationfis. Following the evaluation of these three trained FISs for experimental data, the harateristis depited in Fig. 6 were obtained. The evolution of the training errors is represented in Fig. 7. Evaluation of these three FISs gives the following values of the mean of the relative absolute errors: 0.69 09, 0.65 95, and 1.66 4 per ent for ElongationFis, ElongationFis, and d0elongationfis, respetively. The errors obtained in the seond phase for the first and the seond FISs are very good, and so these FISs an be implemented in the Simulink-integrated ontroller. For the last FIS ( d0elongationfis ), the error values are still too large, and so the number of membership funtions used to generate it must be adjusted. Therefore, a third phase of FISs building and training is reserved to obtain a better solution for the d0elongationfis FIS. In this phase, two ases were onsidered for the number of membership funtions: ase 1 the mf numbers are 1 for the first input and 1 for the seond input; ase the mf number is 1 for the first input, and 14 for the seond. A number of 4000 training epohs were onsidered in the first ase and 1000 in the seond. The training errors for both ases, after training with the ANFIS funtion, are presented in Fig. 8, and the evaluation as a funtion of the number of experimental data points is shown in Fig. 9. The means of the relative absolute error values for the two ases are 0.748 44 and 0.607 46 per ent, respetively. Sine the errors in the seond ase are lower, that is the onfiguration that was hosen to be implemented in a Simulink-integrated ontroller. The final values of the relative absolute errors for the four generated and trained FISs are 0.69 09 per ent for ElongationFis, 0.65 95 per ent for Elongation- Fis, 0.003 47 per ent for delongationfis, and 0.607 46 per ent for d0elongationfis. Representing the elongations (those obtained experimentally and by using the four FIS models) as funtions of eletrial urrent for the first and third FISs, and as a funtion of time for the other two FISs, produes the graphis in Fig. 10. The urves are represented for all five ases of the SMA load. One an easily observe that, through training, the FISs model the experimental data very well, and the SMA has different thermal onstants, depending on the fore value. A good overlapping of the FIS models elongations with the elongation experimental data is learly visible in Fig. 10. This superposition is dependent on the number of training epohs, and improves as the number of training epohs is higher. Beause the training errors of all of the trained FISs ultimately take onstant values, an improved approximation of the real model Pro. IMehE Vol. 3 Part G: J. Aerospae Engineering JAERO5
Adaptive neuro-fuzzy inferene system-based ontrollers 661 Fig. 6 FISs evaluation as a funtion of the number of experimental data points in the seond phase Fig. 7 Training errors for the three FISs generated and trained in the seond phase JAERO5 Pro. IMehE Vol. 3 Part G: J. Aerospae Engineering
66 T L Grigorie andrmbotez Fig. 8 Training errors for the d0elongationfis generated and trained in the third phase Fig. 9 FIS s evaluation as a funtion of the number of experimental data points for the third phase Fig. 10 FIS evaluations as funtions of urrent or time Pro. IMehE Vol. 3 Part G: J. Aerospae Engineering JAERO5
Adaptive neuro-fuzzy inferene system-based ontrollers an be ahieved with the neuro-fuzzy methods only when a higher quantity of experimental data is used. To visualize the FIS s features, the Matlab anfisedit ommand [3] is used, followed by the FIS s importation on the interfae level. The resulting surfaes for all four 663 final, trained FISs are presented in Fig. 11. The parameters of the input s membership funtions for eah of the four FISs before and after training are shown in Tables 1 and, respetively. For the generalized Bell-type membership funtions, produed with the Fig. 11 The surfaes produed for all four of the final trained FISs Table 1 Parameters of the FIS input s membership funtions before training ElongationFis Fore (N ) mf1 mf mf3 mf4 mf5 mf6 mf7 mf8 mf9 mf10 mf11 mf1 mf13 mf14 ElongationFis Current (A) Fore (N) delongationfis Time (s) Fore (N) Current (A) a b a b a b a b σ / σ / 7.7 7.7 7.7 7.7 7.7 7.7 10.19 135.65 151.11 166.56 18.0 197.48 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.50 0.95 1.40 1.86.31.77 3. 3.68 4.13 4.59 5.04 5.50 9.11 9.11 9.11 9.11 9.11 9.11 119.39 137.61 155.83 174.05 19.7 10.49 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 0.44 4.89 7.33 9.78 1.3 14.67 17.1 19.57.01 4.46 6.90 16.4 16.4 16.4 16.4 16.4 16.4 14. 141.9 03.8 14.4 177.9 186.6 0.97 5 0.97 0 0.97 5.5 0.97 0 0.97 0.01 0.97 5 JAERO5 d0elongationfis Fore (N) Time (s) a b a b 83.31 96.88 110.45 14.0 137.58 151.15 164.7 178.9 191.86 0 18.99 3.56 0 10.81 1.63 3.45 43.6 54.08 64.90 75.7 86.53 97.35 108.17 118.99 19.81 140.6 Pro. IMehE Vol. 3 Part G: J. Aerospae Engineering
664 T L Grigorie andrmbotez Table Parameters of the FIS input s membership funtion after training ElongationFis ElongationFis delongationfis d0elongationfis Fore (N) Current (A) Fore (N) Time (s) Fore (N) Current (A) Fore (N) Time (s) a b a b a b a b σ / σ / a b a b mf1 9.7.1 11.4 6.51 3.088 0.47 9.31.04 119.5 0.9.0 0. 16.39 14. 0.78 4.75 6.8 0.8 83.3 5.7.5 0.5 mf 9.9.4 19.7 5.859 3.851 1.5 8.9 1.14 137.4 1.3.3.4 13.5 139. 0.43 1.03 6.9 0.5 96.8 6.0.1 11. mf3 9.45.8 148.7 7.036 1.078 5. 9.79.76 155.0 1. 1.7 4.5 16.38 03.8 1.0 5.35 7.1.0 110.7 3.9 3.1 19.9 mf4 9.46 1.0 166.6 6.391.005 38.3 9.57 4.3 176.4 1.8 1.9 8.0 16.7 14.1 0.97 1.3e-7 7.6 1.5 13.7 4.8 31.5 mf5 6.6.9 18 6.39.0069 51.1 7.90 1.64 193.4 1.4 3.3 9.9 1.05 177.9 0.97 1.e-6 6.1 3.9 137.8 5.8 3.6 43.6 mf6 9.51 4. 00 6.3914.0036 63.9 10.40 09.5 1.1 3. 1.6 16.48 186.6 1.04 5. 6.3 1.3 151.4 4.8 0.3 54.0 mf7 6.3917.00 76.7 0.6 1.4 15.1 7.5.8 163.5 4.5 4.1 64.6 mf8 6.3918.001 89.4 1.0.6 16.6 3.8 4. 177.6 6.7 1.7 75.0 mf9 6.3918.0007 10. 1.6.5 19. 5. 8.4 191.5 5.9 1.7 86. mf10 6.3919.0005 115.0 1.3.0.0 7..6 04.8 5.6 1.9 97.1 mf11 6.3919.0003 17.8 1.5 1.8 4.3 7.9 1.7 19.0 5.6 1.6 108. mf1 6.39.00 140.6 1.4 0.6 7.0 6.8 0.7 3.7 5.5 0.7 119.0 mf13 5.7 1.1 19.7 mf14 5.5 1.5 140.5 genfis1 funtion, the parameters are the membership funtion entre () defining their position, and a, b that define their shape. For the Gaussian-type membership funtions, generated with the genfis funtion, the parameters are one-half of the dispersion (σ /) and the entre of the membership funtion (). For our system, a set of 7 rules for ElongationFis and another 7 for ElongationFis, 6 rules for delongationfis and 168 rules for d0elongationfis are generated. Comparison of the FISs harateristis and membership funtions parameters before and after training, from Tables 1 and, indiates a redistribution of the membership funtions in the working domain and a hange in their shapes, by modifiation of the a, b, and σ parameters. Aording to the parameter values from Table 1, generating FISs with the genfis1 and genfis funtions primarily results in the same values for the a, b, and σ / parameters for all of the membership funtions that haraterize an input. A seondary result is the separation of the working spae for the respetive input using a grid partition on the data (no lustering) if the genfis1 funtion is used, or using the fuzzy subtrative lustering method if generating with the genfis funtion. For the delongationfis FIS (initially generated by using the genfis funtion) the rules are of the type: if (in1 is in1luster k ) and (in is inluster k ) then (out1 is out1luster k ). For both of the inputs of this FIS, six Gaussian-type mf were generated; within the set of rules they are noted by in j luster k ; j is the input number (1/), and k is the number of the membership funtion (1/6). The delongationfis FIS has the struture shown in Fig. 1, whereas the orresponding ontroller (Controller 3) has the struture presented in Fig. 13. For the other three FISs (initially generated by using the genfis1 funtion) the rules are of the type: if (in1 is in1mf k ) and (in is inmf 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 these three FISs, generalized Bell-type membership funtions were generated; within the sets of rules they are noted by in j mf n ; j is the input number (1/) and n is the number of the membership funtions. For ElongationFis and ElongationFis, six membership funtions for the first input (k = 6) and 1 membership funtions for the seond input (p = 1 r = 7) are produed. The d0elongationfis results in 1 membership funtions for the first input (k = 1) and 14 for the seond input (p = 14 r = 168). For example, the ElongationFis FIS has the struture shown in Fig. 14, whereas the orresponding ontroller (Controller 1) has the same struture as Controller 3 (see Fig. 13). Eah of the four FISs is imported at the fuzzy ontroller level, resulting in four ontrollers: Controller 1 ( ElongationFis ), Controller ( ElongationFis ), Controller 3 ( delongationfis ), and Controller 4 Pro. IMehE Vol. 3 Part G: J. Aerospae Engineering JAERO5
Adaptive neuro-fuzzy inferene system-based ontrollers 665 Fig. 1 Struture of the delongationfis FIS Fig. 13 The struture of Controller 3 ( d0elongationfis ). These four ontrollers are integrated using the logial sheme given in Fig. 3; the Matlab/Simulink model in Fig. 15 is the result. In the Matlab/Simulink model shown in Fig. 15, the seond input of Controller and that of Controller 4 (time) are generated by using integrators, starting from the moment that these inputs are used in Controller or Controller 4 (the input of the Gain blok is 0 if the shema deides not to work with one of the Controllers or 4). It is possible that the simulation sample time may be different from the sample time used in the experimental data aquisition proess, and therefore the Gain blok that gives their ratio is used; Te is the sample time in the experimental data and T is the simulation sample time. In the shema, the onstant C represents the maximum time onsidered for the atuator to reover its initial temperature ( 4 C) when the urrent beomes 0 A. Evaluating the integrated ontroller model (see Fig. 15) for all five experimental data ases produes the results shown in Figs 16 and 17. These graphis show the elongations versus the number of experimental data points and versus the applied eletrial urrent, respetively, using the experimental data and the integrated neuro-fuzzy ontroller model for the SMA. A good overlapping of the outputs of the integrated neuro-fuzzy ontroller with the experimental data an be easily observed. Fig. 14 Struture of the ElongationFis FIS JAERO5 Pro. IMehE Vol. 3 Part G: J. Aerospae Engineering
666 T L Grigorie andrmbotez Fig. 15 The integration model shema in Matlab/Simulink Fig. 16 Elongations versus the number of experimental data points Pro. IMehE Vol. 3 Part G: J. Aerospae Engineering JAERO5
Adaptive neuro-fuzzy inferene system-based ontrollers 667 Fig. 17 Elongations versus the applied eletrial urrent Fig. 18 Three-dimensional evaluation of the integrated neuro-fuzzy ontroller The same observation an be made from the threedimensional harateristis of the experimental data and the neuro-fuzzy modelled data in terms of temperature, elongation, and fore, depited in Fig. 18(a), and in terms of urrent, elongation, and fore, depited in Fig. 18(b). The mean values of the relative absolute errors of the integrated ontroller for the five load ases of the SMA, based on adaptive neuro-fiss, are 0.459 97 per ent for 10 N, 0.50 95 per ent for 140 N, 0.513 19 per ent for 150 N, 0.716 09 per ent for 180 N, and 0.507 75 per ent for 190 N. The mean value of the relative absolute error between the experimental data and the outputs of the integrated ontroller is 0.54 per ent. 5 CONCLUSIONS In this artile, an integrated ontroller based on adaptive neuro-fiss for modelling smart material atuators was obtained. The diret appliation of this ontroller is in a morphing wing system. The general aim of the smart material atuators desired model is to alulate the elongation of the atuator under the appliation of a thermo-eletro-mehanial load for a ertain time. Therefore, the smart material atuators were experimentally tested in onditions lose to those in whih they will be used. Testing was performed for five load ases, with fores of 10, 140, 150, 180, and 190 N. Using the experimental data, JAERO5 Pro. IMehE Vol. 3 Part G: J. Aerospae Engineering
668 T L Grigorie andrmbotez four FISs were generated and trained to obtain four neuro-fuzzy ontrollers: one ontroller for the urrent inrease ( ElongationFis ), one for a onstant urrent ( ElongationFis ), one for the urrent derease ( delongationfis ), and one ontroller for the null urrent, after its derease ( d0elongationfis ). The genfis1 and genfis Matlab funtions were used to generate the initial FISs, and the adaptive neuro-ifs tehnique was then used to train them.the final values of the relative absolute errors for the four generated and trained FISs were 0.69 09 per ent for ElongationFis, 0.65 95 per ent for ElongationFis, 0.003 47 per ent for delongationfis, and 0.607 46 per ent for d0elongationfis. Eah of the four obtained and trained FISs were imported at the fuzzy ontroller level, resulting in four ontrollers. Finally, these four ontrollers were integrated by using the logial sheme given in Fig. 3; resulting in the Matlab/Simulink model for the integrated ontroller shown in Fig. 15. The integrated ontroller performanes were evaluated for all five load ases; the values obtained for the mean relative absolute errors were 0.459 97 per ent for 10 N, 0.50 95 per ent for 140 N, 0.513 19 per ent for 150 N, 0.716 09 per ent for 180 N, and 0.507 75 per ent for 190 N. Thus, the mean value of the relative absolute error between the experimental data and the outputs of the integrated ontroller was 0.54 per ent. A partiular advantage of this new model is its rapid generation, thanks to the genfis1, genfis, and ANFIS funtions already implemented in Matlab. The user need only assume the four FIS s training performanes using the anfisedit interfae generated with Matlab. Authors 009 REFERENCES 1 Sivanandam, S. N., Sumathi, S., and Deepa, S. N. Introdution to fuzzy logi using MATLAB, 007 (Springer, Berlin, Heidelberg). Kosko, B. Neural networks and fuzzy systems a dynamial systems approah to mahine intelligene, 199 (Prentie Hall, New Jersey, NJ). 3 Matlab fuzzy logi and neural network toolboxes, available from http://www.mathtools.net/matlab/books/ Neural_Network_and_Fuzzy_Logi/. 4 Khezri, M. and Jahed, M. Real-time intelligent pattern reognition algorithm for surfae EMG signals. BioMed. Eng. OnLine, 007, 6, 45. DOI: 10.1186/1475-95X-6-45. 5 Kung, C. C. and Su, J.Y. Affine TakagiSugeno fuzzy modelling algorithm by fuzzy -regression models lustering with a novel luster validity riterion. IET Control Theory Appl., 007, 1(5), 155165. 6 Mahfouf, M., Linkens, D. A., and Kandiah, S. Fuzzy TakagiSugeno Kang model preditive ontrol for proess engineering, 1999, p. 4 (IEE, Savoy plae, London WCPR OBL, UK). APPENDIX Notation a, b parameters of the generalized bell membership funtion a i k parameters of the linear funtion (k = 1,, i = 1, N) Aq i assoiated individual anteedent fuzzy sets of eah input variable (i = 1, N) b0 i salar offset (i = 1, N) luster entre q i luster entre (q = 1, ) C p pressure oeffiient F fore i eletrial urrent k variable for neuro-fuzzy ontroller seletion k y step size l dimension of the data vetors M number of data points N number of rules r m radius within the fuzzy neighbourhood, ontributes to the density measure Re Reynolds number t time T temperature of the smart material atuator T amb ambient temperature u j entre of the jth luster u k data vetors V speed w i degree of fulfilment of the anteedent, i.e. the level of firing of the ith rule x input vetor x q individual input variables (q = 1, ) y output of the fuzzy model y i first-order polynomial funtion in the onsequent (i = 1, N) α δ t ε ρ σ σ i q angle of attak atuator elongation time variation ost funtion density measure dispersion luster dispersion Pro. IMehE Vol. 3 Part G: J. Aerospae Engineering JAERO5