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2 4 New Applcatons of Fuzzy Logc Methodologes n Aerospace Feld Teodor Lucan Grgore and Ruxandra Mhaela Botez École de Technologe Supéreure Canada. Introducton Automatc control can be defned as a way of analyzng and desgnng a system that can self-regulate wth mnmal human nterventon. It s based on control theory, vewed as an nterdscplnary branch of engneerng and mathematcs. The devce that montors and modfes the operatonal condtons of a dynamc system s called a controller. The global technology evoluton has trggered an ever-ncreasng complexty of applcatons, both n ndustry and n the scentfc research felds. Many researchers have concentrated ther efforts on provdng smple control algorthms to cope wth the ncreasng complexty of the controlled systems (Al-Odenat & Al-Lawama, 8). The man challenge of a control desgner s to fnd a formal way to convert the knowledge and experence of a system operator nto a well-desgned control algorthm (Kovacc & Bogdan, ). From another pont of vew, a control desgn method should allow full flexblty n the adjustment of the control surface, as the systems nvolved n practce are, generally, complex, strongly nonlnear and often wth poorly defned dynamcs (Al-Odenat & Al- Lawama, 8). If a conventonal control methodology, based on lnear system theory, s to be used, a lnearzed model of the nonlnear system should have been developed beforehand. Because the valdty of a lnearzed model s lmted to a range around the operatng pont, no guarantee of good performance can be provded by the obtaned controller. Therefore, to acheve satsfactory control of a complex nonlnear system, a nonlnear controller should be developed (Al-Odenat & Al-Lawama, 8; Hampel et al., ; Kovacc & Bogdan, ; Verbruggen & Brujn, 997). From another perspectve, f t would be dffcult to precsely descrbe the controlled system by conventonal mathematcal relatons, the desgn of a controller usng classcal analytcal methods would be totally mpractcal (Hampel et al., ; Kovacc & Bogdan, ). Such systems have been the motvaton for developng a control system desgned by a sklled operator, based on ther mult-year experence and knowledge of the statc and dynamc characterstcs of a system; known as a Fuzzy Logc Controller (FLC) (Hampel et al., ). FLCs are based on fuzzy logc theory, developed by L. Zadeh (Zadeh, 9). By usng multvalent fuzzy logc, lngustc expressons n antecedent and consequent parts of IF-THEN rules descrbng the operator s actons can be effcently converted nto a fully-structured control algorthm sutable for mcrocomputer mplementaton or mplementaton wth specally-desgned fuzzy processors (Kovacc & Bogdan, ). In contrast to tradtonal lnear and nonlnear control theory, an FLC s not based on a mathematcal model, and t does provde a certan

3 4 Fuzzy Controllers, Theory and Applcatons level of artfcal ntellgence compared to conventonal PID controllers (Al-Odenat & Al- Lawama, 8). The objectve of the research presented here s to develop a new morphng mechansm usng smart materals such as Shape Memory Alloy (SMA) as actuators and fuzzy logc technques. These smart actuators deform the upper wng surface, made of a flexble skn, so that the lamnar-to-turbulent transton pont moves closer to the wng tralng edge. The ultmate goal of ths research project s to acheve drag reducton as a functon of flow condton by changng the wng shape. The transton locaton detecton s based on pressure sgnals measured by optcal and Kulte sensors nstalled on the upper wng flexble surface. Dependng on the project evoluton phase, two archtectures are consdered for the morphng system: open loop and closed loop. The dfference between these two archtectures s ther use of the transton pont as a feedback sgnal. Ths research work was a part of a morphng wng project developed by the Ecole de Technologe Supéreure n Montréal, Canada, n collaboraton wth the Ecole Polytechnque n Montréal and the Insttute for Aerospace Research at the Natonal Research Councl Canada (IAR-NRC) (Bralovsk et al., 8; Coutu et al., 7; Coutu et al., 9; Georges et al., 9; Grgore & Botez, 9; Grgore & Botez, ; Grgore et al., a; Grgore et al., b; Grgore et al., c; Popov et al., 8 a; Popov et al., 8 b; Popov et al., 9 a; Popov et al., 9 b; Popov et al., a; Popov et al., b; Popov et al., c; Sanmont et. al., 9), ntated and fnancally supported by the followng government and ndustry assocatons: the Consortum for Research and Innovaton n Aerospace n Quebec (CRIAQ), the Natonal Scences and Engneerng Research Councl of Canada (NSERC), Bombarder Aerospace, Thales Avoncs, and the Natonal Research Councl Canada Insttute for Aerospace Research (NRC-IAR). Recently, morphng wng system studes have branched out nto new research drectons. Extremely complex and catalogued as nter- and multdscplnary studes, morphng wng studes contnue to push the scence to the extreme boundares of mathematcs and physcs. These multdscplnary studes therefore requre knowledge of the followng dscplnes: aerodynamcs and computatonal flud dynamcs, aeroelastcty, automatc control, ntellgent materals, sgnal detecton usng the latest mnaturzed sensors, hgh computer-tme calculatons, wnd tunnel and flght testng, nstruments, and sgnal acquston -- these sgnals have such speed that they are rasng serous problems for the exstng calculus technology. Consequently, real-tme system functonng s condtoned (n addton to other factors) by beng able to obtan the best processng algorthms and employng easy-to-mplement software for the command and control unt. Fuzzy logc theores, whch offer remarkable facltes, may therefore be used n these algorthms. They facltate sgnal processng by allowng emprcal models to be desgned based on expermental ; thus avodng the complex mathematcal calculus currently n use. In addton, fuzzy logc can be used to model hghly non-lnear, multdmensonal systems, ncludng those wth parameter varatons, or where the sensors sgnals are not accurate enough for other models. Ths research project ncluded the followng: optcal sensor selecton and testng for lamnar-to-turbulent flow transton valdaton (by use of XFol code and Matlab), smart materal actuator modelng, aeroelastcty wng studes usng MSC/Nastran, open loop and closed loop transton delay controller desgn, and ntegraton and valdaton on a wng equpped wth SMAs and optcal sensors (smulaton versus expermental test results) (Fg. (Grgore et al., b)).

4 New Applcatons of Fuzzy Logc Methodologes n Aerospace Feld A frst phase of ths project nvolved the determnaton of optmzed arfols avalable for 3 dfferent flow condtons expressed n terms of fve Mach numbers (.,.,.,.7,.3) and seven angles of attack (-, -.,,.,,., ) combnatons. The optmzed arfols, derved from a lamnar WTEA-TE reference arfol, were calculated and used as a startng pont n the actuaton system desgn. Three steps were completed n the actuaton system desgn phase: optmzaton of the number and postons of flexble skn actuaton ponts, establshment of each actuaton lne s archtecture, and modelng of the smart materals actuators used n ths applcaton wth fuzzy logc technques. The next phase of the project was about the desgn of the actuaton control, for whch a fuzzy PD archtecture was chosen. In ths desgn, numercal smulatons of the open loop morphng wng ntegrated system, based on an SMA non-lnear model, were performed. As subsequent valdaton methods, a bench test and a wnd tunnel test were conducted. Rgd part of the extrados Tralng edge Flexble skn (morphed extrados) Actuaton lnes Leadng edge Support plate for actuaton system Rgd ntrados Cavtes for nstrumentaton Fg.. General archtecture of the mechancal model The shape memory actuator wres were made of nckel-ttanum, known as Ntnol, and they contract as muscles do when electrcally drven. Ths ablty to flex or shorten s a characterstc of certan alloys that dynamcally change ther nternal structure at certan temperatures. These alloys have the propertes of exhbtng martenstc transformaton when they deform at a low temperature phase, and may recover ther orgnal shape after heatng (Popov et al., 8 a). Ths phase change, from martenste to austente, s shown n Fg. (Baron et al., 3; Thll et al., 8). The load changes the nternal forces between the atoms, forcng them to change ther postons n the crystals and consequently forcng the wres to lengthen, whch s called the SMA actvaton or the ntal phase. When the wre s heated usng a current, the heat generated by the current resstvty causes the atoms n the crystallne structure to realgn and force the alloy to recover ts orgnal shape. Therefore, any change n the alloy s nternal temperature would modfy the crystallne structure accordngly and thus the wre s exteror shape. Ths property of changng the wre length as a functon of the electrcal current passng through the wre s used for actuaton purposes (Popov et al., 8 a). Another major reason for usng Ntnol s that t s the most effectve materal at wthstandng repeated cycles of heatng and coolng wthout exhbtng a fatgue phenomenon (Gonzalez, ). SMA wres can process the deflectons obtaned usng the appled forces and they provde a varety of shapes and szes that are extremely useful to acheve actuaton system goals. For example, SMA wres can provde hgh forces correspondng to small strans to acheve the correct balance between the forces and the deformatons, as requred by the actuaton system. To ensure a stable system, a compromse or balance must be establshed and mantaned. The structural components of the actuaton system should be desgned to respect the actuators capabltes to accommodate the requred deflectons and forces.

5 Fuzzy Controllers, Theory and Applcatons Each of our actuaton lnes uses three shape memory alloys wres as actuators, and contans a cam, whch moves n translaton relatve to the structure (on the x-axs n Fg. 3 (Georges et al., 9). The cam causes the movement of a rod related on the roller and on the skn (on the z-axs). The recall employed here s a gas sprng. So, when the SMA s heatng the actuator contracts and the cam moves to the rght, resultng n the rse of the roller and the dsplacement of the skn upwards. In contrast, the coolng of the SMA results n a movement of the cam to the left, and thus a movement of the skn downwards. The horzontal dsplacement of each actuator s converted nto a vertcal dsplacement at a fxed rate. Austente Temperature Coolng Loadng Heatng Fg.. SMA phase change Twnned martenste Deformed martenste Load SMA wres can execute the deflectons resultng from contractng or expandng forces and can provde a varety of shapes and szes that are extremely useful to acheve actuaton system goals. To ensure a stable system, a compromse or balance must be establshed and mantaned. The structural components of the actuaton system should be desgned to respect the actuators capabltes to accommodate the requred deflectons and forces. Flexble skn z x Roller Rod Cam Coolng Heatng Fg. 3. The actuaton mechansm concept Compresson sprng Support plate for actuaton system Three SMA wres The SMA actuator control can be acheved usng any method for poston control. However, the specfc propertes of SMA actuators such as hysteress, the frst cycle effect and the mpact of long-term changes must be consdered. The operatng scheme of our open loop controller can be developed as llustrated n Fg. 4 (Grgore et al., b; Grgore et al., c). Based on the 3 studed flght condtons, a base of the 3 optmzed arfols was bult. For each flght condton, a par of optmal vertcal deflectons (dy opt, dy opt ) for the two actuaton lnes s apparent (Fg. ). The SMA actuators morphed the arfol untl the vertcal deflectons of the two actuaton lnes (dy real, dy real ) became equal to the requred

6 New Applcatons of Fuzzy Logc Methodologes n Aerospace Feld 7 Plot Control Flght condtons α, M, Re Optmsed arfols base dy opt dy opt e=dy opt - dy real dy real dy real Controller Current SMA actuators Arflow perturbatons Real arfol Poston transducers dy real dy real Fg. 4. Operatng scheme of the SMA actuators control deflectons (dy opt, dy opt ). The vertcal deflectons of the real arfol at the actuaton ponts were measured usng two poston transducers. The controller s role s to send a command to supply an electrcal current sgnal to the SMA actuators, based on the error sgnals (e) between the requred vertcal dsplacements and the obtaned dsplacements. The desgned controller was vald for both actuaton lnes, whch are practcally dentcal. dy [mm] alpha alpha alpha3 alpha4 alpha alpha alpha7 dy [mm] Mach number Mach number Fg.. dy opt and dy opt, dy opt as functons of M for varous angles of attack Durng the frst phase of the controller desgn, numercal smulaton of the controlled actuaton system was performed; a step whch requred an SMA actuator model. In the lterature, the modelng and control of smart materal actuators can be categorzed as recent research felds. Techncal lterature s avalable n three ndependent domans: modelng, control and smart materals. A smart actuator s formulated for a large range of smart materals and devces, and can be found n a varety of dfferent confguratons. It s common knowledge that all physcal systems, ncludng smart actuators, contan nonlneartes. As a consequence, lnear modelng of smart materal actuators may contan errors, whle non-lnear modelng remans possble. In order to conceve such a model, a fuzzy set must be desgned, whch may be gven by the orgnal fuzzy logc theory conceved by Lotf A. Zadeh (Zadeh, 9). The most serous problem arses from the determnaton of a complete set of rules and the membershp functons correspondng to each nput. The multple attempts requred to reduce errors and to optmze the model are tme-consumng and, very often, the results are far from what was expected. A modern desgn method allows fuzzy model desgn to be completed n a relatvely short tme nterval. The Adaptve Neuro-Fuzzy Inference System (ANFIS) desgn technque allows the generaton and the optmzaton of the set of rules and the membershp functons parameters by use of Neural Networks. Moreover, the ANFIS desgn technque already mplemented n Matlab s Neuro-Fuzzy software tools should be relatvely easy to use.

7 8 Fuzzy Controllers, Theory and Applcatons Consderng the numercal values resultng from the SMA expermental testng (forces, currents, temperatures and elongatons), an emprcal model can be developed, based on a neuro-fuzzy network. The model can learn the process behavor based on the nput-output process by usng a Fuzzy Inference System (FIS), whch should model the expermental.. SMA actuator fuzzy model The general am of the SMA model s to calculate the elongaton of the actuator (Δδ) under the applcaton of a thermo-electro-mechancal load for some tme (Δt). The load s soqualfed because the actuator can be operated by varyng temperature (T amb ), by njecton of electrc current () or by applyng a force (F). The geometry of the actuator s an SMA wre wth constant secton and permeter over the length of the actuator. For these specfc model objectves, n the frst phase, the SMA actuators were expermentally tested n condtons close to those n whch they wll be used. The SMA testng was performed usng at T amb =4 C, for sx load cases wth the forces of 7 N, 8 N, N, N, N and N. The electrcal currents followng the ncreasngconstant-decreasng-zero values evoluton were appled to the SMA actuator for each of the sx load cases. In each case, the followng parameters were regstered: tme, the electrcal current suppled to the SMA, the load force, the materal temperature and the actuator elongaton. To model the SMA we wll bult an ntegrated controller based on Adaptve Neuro-Fuzzy Inference Systems. The expermental elongaton-current curves obtaned from the sx load cases are ndcated n Fg.. One can observe that all sx of the curves are characterzed by four dstnct zones: electrcal current ncrease, constant electrcal current, electrcal current decrease and null electrcal current n the coolng phase of the actuator. Therefore, four Fuzzy Inference Systems (FIS s) are used to obtan four neuro-fuzzy controllers: one controller for the current ncrease, one for a constant current, one for the current decrease, and one controller for the null current (after ts decrease). For the frst and the thrd controllers, nputs such as the force and the current are used, whle for the second and the fourth controller, nputs such as the force and the tme values reflectng the SMA thermal nerta are used (for the four controllers the tme values used are those requred for the SMA to recover ts ntal temperature value (approxmately 4 C)). Fnally, the four obtaned controllers must be ntegrated nto a sngle controller. The reasonng behnd the desgn of the frst and the thrd controllers s that from the avalable expermental, two elongatons for the same values of forces and currents are used (see Fg. ). Due to the expermental values, ths cannot be represented as algebrac functons, and therefore t s mpossble to use the same FIS representaton. An nterpolaton between the two elongaton values obtaned for the same values of forces and currents can be performed n Matlab, but t s not vald for our applcaton. Also, the constant values, respectvely the null values of the current before, respectvely after the current decrease phase are not suggestve to be consdered lke nputs n the second and n the four controllers. Practcally, wth these phases the values of the actuator temperature could be used. The tme values for these phases do prove very useful, because these values represent a measure of the thermal nerta of the actuator. We use the tme value as the second nput of the thrd controller, and therefore, as the second nput of the second and of the fourth controllers snce force was consdered as the frst nput (the tme values must be consdered from the moment when the current becomes constant, or null).

8 New Applcatons of Fuzzy Logc Methodologes n Aerospace Feld 9 3 N N N N 8 N N Current [A] Fg.. Elongaton versus the current values for dfferent forces values for sx load cases. SMA model archtecture based on fuzzy logc controllers A fuzzy nference system (FIS) can be easly generated usng Matlab s genfs or genfs functons. The genfs functon generates a sngle-output Sugeno-type fuzzy nference system (FIS) usng a grd partton on the (no clusterng). The FIS thus obtaned s used to provde the ntal condtons for ANFIS tranng. The genfs functon uses generalzed bell-type membershp functons for each nput. Each rule generated by a genfs functon has one output membershp functon, whch s a lnear type by default. It s also possble to create the FIS usng the Matlab genfs functon, whch frst generates an ntal Sugenotype FIS by decomposton of the operaton doman nto dfferent regons usng the fuzzy subtractve clusterng method. For each regon, a low order lnear model can descrbe the local process parameters. The non-lnear process can then be locally lnearzed around a functonng pont by usng the Least Squares method. The obtaned model s consdered vald n the entre regon around ths pont. To lmt the operatng regons mples the exstence of overlappng among these dfferent regons, whose defnton s gven n a fuzzy manner. Thus, for each model nput, several fuzzy sets are assocated wth ther correspondng defntons of ther membershp functons. By combnng these fuzzy nputs, the nput space s dvded nto fuzzy regons. For each such regon, a local lnear model s used, whle the global model s obtaned by defuzzfcaton wth the center-of-gravty method (Sugeno), whch nterpolates the local models outputs (Svanandam et al., 7; MathWorks Inc., 8). Based on the concept of fndng regons wth a hgh densty of ponts n the feature space, the subtractve clusterng method dvdes space nto a number of clusters. Centers of clusters are selected, startng wth the ponts wth the hghest number of neghbours. The clusters are dentfed one by one; for each cluster the ponts wthn a prespecfed fuzzy radus are removed (subtracted). After each cluster dentfcaton, the algorthm looks for a new one untl all of the ponts have been examned. If a collecton of M ponts, specfed by l-dmensonal vectors u k, k =,..., M, s consdered, a densty measure at pont u k can be defned as follows: ρ k M = exp u u k j. = j ( r /) () m

9 Fuzzy Controllers, Theory and Applcatons where r m s a postve constant that defnes the radus wthn the fuzzy neghborhood and contrbutes to the densty measure. The pont wth the hghest densty s selected as the frst cluster center. Let u c be the pont selected and c ts densty measure. Next, the densty measure for each pont u k s revsed by the formula: exp u u. ( /) k c ρ = ρ ρ k k c rn n whch r n s a postve constant, larger than r m, and defnes a neghborhood to be reduced n ts densty measure to prevent closely-spaced cluster centers. In ths way, the ponts near the frst cluster center u c wll have sgnfcantly reduced densty measures, and these ponts cannot be selected as centers for the next clusters. After the densty measure for each pont has been revsed, the next cluster center u c s selected and all the densty measures are revsed agan. The process s repeated untl all of the ponts have been examned and a suffcent number of cluster centers generated. When the subtractve clusterng method s appled to an nput-output set, each of the cluster centers are used as the centers for the premse sets n a sngleton type of rule base (Khezr & Jahed, 7). The Matlab genfs functon generates membershp functons of a generalzed bell type, defned as follows (Kosko, 99; Kung & Su, 7): b A ( x) = (+ ( x c )/ a ), (3) q where c s the cluster center defnng the poston of the membershp functon, a, b are two q parameters whch defne the shape of the membershp functon, and A ( =, N) are q assocated ndvdual antecedent fuzzy sets of each nput varable (N - number of rules). The Matlab genfs functon generates membershp functons of the Gaussan type, descrbed by the followng expresson (Kosko, 99; Kung & Su, 7): q A ( x) = exp{.(( x c )/ σ ) }, (4) q where c s the cluster center, and σ s the dsperson of the cluster. q q The Sugeno fuzzy model was proposed by Takag, Sugeno and Kang to generate the fuzzy rules from a gven nput-output set (Mahfouf et al., 999). For our system, for all four of the FIS s (two nputs and one output) a frst-order model s consdered, and for N rules s gven by (Kung & Su, 7; Mahfouf et al., 999): q q () Rule : If x Rule : If x Rule N : If x s A s A s A N and x and x and x s A s A s A N, then y ( x, x ) = b + a x + a x,, then y ( x, x ) = b + a x + a x, N N N N, then y ( x, x ) = b + a x + a x, () where x q ( q =,) are ndvdual nput varables, and y ( =, N) s the frst-order polynomal functon n the consequent. a ( k =,, =, N) are the parameters of the lnear k functon and b ( =, N) denotes a scalar offset. The parameters a, b ( k =,, =, N) are k optmzed by Least Square method.

10 New Applcatons of Fuzzy Logc Methodologes n Aerospace Feld For any nput vector, T x = x, x ], f the sngleton fuzzfer, the product fuzzy nference and [ the center-average defuzzfer are appled, the output of the fuzzy model y s nferred as follows (weghted average): where N N y = w ( x ) y / w ( x), () = = w ( x x ) = A ( x ) A ( ). (7) w (x) represents the degree of fulfllment of the antecedent, that s, the level of frng of the th rule. The adaptve neuro-fuzzy nference system adapts the parameters of Sugeno-type fuzzy nference systems usng the neural networks. A very smple way to realze the FIS s tranng s by usng the Matlab ANFIS functon, whch use a learnng algorthm for the dentfcaton of the membershp functons parameters of a Sugeno-type fuzzy nference system wth two outputs and one nput. As a startng pont, the nput-output and the s generated wth the genfs or genfs functons are consdered. The ANFIS optmzes the membershp functons parameters for a number of tranng epochs; ths number s set by the user. The optmzaton s realzed for a better process approxmaton performed by the neuro-fuzzy model by means of a qualty parameter present n the tranng algorthm (MathWorks Inc., 8). Followng the tranng phase, the models may be used for elongaton value generaton correspondng to the parameters at the nput. For tranng the fuzzy system, ANFIS employs a back-propagaton algorthm for the parameters assocated wth the nput membershp functons, and a least mean square estmaton for the parameters assocated wth the output membershp functons. For the FISs generated usng the genfs or genfs functons, the membershp functons are of the generalzed bell type and gaussan type, respectvely. In accordance wth equatons (3) and (4), n these knds of membershp functons, a, b and c, and σ and c, respectvely, are consdered varables and must be adjusted. Therefore, the back-propagaton algorthm may be used to tran these parameters. In ths way, we can acheve our goal to mnmze a cost functon of the form ( y ) des y /, ε = (8) where y des s desred output. The output of each rule y x, x ) ( s defned by: n whch k y s the step sze. Startng from the Sugeno system s output (eq. ()), we fnd: y ( t + ) = y ( t) k ( ε/ y ), (9) y ε y ε y = y y, () wth

11 Fuzzy Controllers, Theory and Applcatons ε = y y des y, y y N = w ( x )/ w ( x). () = Therefore, the followng equaton for the output of each rule s y ( t + ) = y ( t) k y ( y des N y) w ( x )/ w ( x). () If a generalzed bell-type membershp functon s used, for the j th membershp functon of the th fuzzy rule the parameters are determned wth the relatons: ε ε ε a ( t + ) = a ( t) k, b ( t + ) = b ( t) k, c ( t + ) = c ( t) k. (3) j j a j j b j j c a b c j For a Gaussan-type membershp functon, the parameters of the j th membershp functon of the th fuzzy rule are calculated wth the relatons: σ ( t + ) = σ ( t) k ( ε/ σ ), c ( t + ) = c ( t) k ( ε/ c ). (4) σ j j j After the four controllers (Controller for ncreasng current, Controller for constant current, Controller 3 for decreasng current and Controller 4 for null current) have been determned, they must be ntegrated, resultng n the logcal scheme n Fg. 7. The decson to use one of the four controllers depends on the current vector type (ncreasng, decreasng, constant or zero) and on the value of varable k. Dependng on the k varable value, we may decde f a constant current value s a part of an ncreasng vector or a part of a decreasng vector. The ntal k value s equal to when Controller s used, and s equal to when Controllers, 3 or 4 are used. j j j = c j j START k= not I(j)>I(j-) yes go to Controller and k(j)= I(j)=I(j-) not I(j)<I(j-) yes go to Controller 3 and k(j)= not I(j)= yes go to Controller and k(j)= k(j-)= not k(j-)= yes go to Controller 4 and k(j)= go to Controller and k(j)= Fg. 7. The logcal scheme for the four controller s ntegraton

12 New Applcatons of Fuzzy Logc Methodologes n Aerospace Feld 3. The SMA model desgn and evaluaton In a frst phase, the genfs Matlab functon (MathWorks Inc., 8) was used to generate and tran the FISs assocated wth the four controllers n Fg. 7: ControllerFs (for the ncreasng current phase), ControllerFs (for the constant current phase), Controller3Fs (for the decreasng current phase) and Controller4Fs (for the null values of the current obtaned after the decreasng phase). The frst FIS, wth force and electrcal current as ts nputs, was traned for epochs usng the ANFIS Matlab functon. The rules were of the type: f (n s ncluster k ) and (n s ncluster k ) then (out s outcluster k ). For both of these nputs, nne Gaussantype membershp functons (mf) were generated; wthn the set of rules they are noted by: n j cluster k ; where j s the nput number ( ), and k s the number of the membershp functon (-9). ControllerFs fuzzy nference system thus has the structure shown n Fg. 8, whle Controller has the structure ndcated n Fg. 9. The rules of ControllerFs fuzzy nference system, before and after tranng, are presented n Fg., and Fg. dsplays the devaton between the neuro-fuzzy models and the expermentally obtaned, defnng the qualty parameter from the tranng algorthm, for dfferent tranng epochs. Fgure shows a rapd decrease n the devaton between the expermental and the neuro-fuzzy model for the qualty parameter wthn the tranng algorthm over the frst tranng epochs, from a value of. to.3. Evaluatng the FIS before and after tranng for the expermental, usng the evalfs command, the characterstcs n Fg. were obtaned. The mean of the relatve absolute values of the errors decreased from.33% before tranng to.9% after tranng, whle ts maxmum value decreased from.9339% to.434%. Snce the error determned for ControllerFs was very small, ths FIS was selected to be mplemented n the Smulnk ntegrated controller. Fg. 8. Structure of the ControllerFs fuzzy nference system

13 4 Fuzzy Controllers, Theory and Applcatons n n Force Current Sugeno FIS out Elongaton Fg. 9. The structure of Controller Fg.. The ControllerFs rules, before and after tranng From Fg. one observes a good overlappng of the wth the elongaton expermental. Ths superposton s dependent upon the tranng epochs number, and mproves as the number of tranng epochs ncreases. Because the tranng errors take constant values, an mproved approxmaton of the real model can be acheved wth neurofuzzy methods only n the case when a larger amount of expermental s used.. "ControllerFs".. Devaton..4.4 Fg.. The tranng error for ControllerFs Number of tranng epochs

14 New Applcatons of Fuzzy Logc Methodologes n Aerospace Feld 4 "ControllerFs" before tranng F=N F=N 4 "ControllerFs" before tranng F=N F=N 8 F=N F=N 8 F=N F=N 4 F=8N 4 F=8N 3 4 Number of expermental ponts 4 F=7N "ControllerFs" after tranng F=N F=N 4 F=7N 4 8 Current [A] "ControllerFs" after tranng F=N F=N 8 F=N F=N 8 F=N F=N 4 F=8N 4 F=8N F=7N 3 4 Number of expermental ponts Fg.. ControllerFs evaluaton, before and after tranng F=7N 4 8 Current [A] The parameters of the nput s membershp functons for ControllerFs, before and after tranng, are shown n Table, whle the membershp functons shapes are depcted n Fg. 3. For the Gaussan-type membershp functons generated wth genfs, the parameters are half of the dsperson (σ/) and the center for the membershp functon (c). Status Input Param. mf mf mf3 mf4 mf mf mf7 mf8 mf9 Force σ/ Before [N] c tranng Current σ/ [A] c Force σ/ After [N] c tranng Current σ/ [A] c Table. Parameters of the ControllerFIS nput s mf, before and after tranng

15 Fuzzy Controllers, Theory and Applcatons "ControllerFs" before tranng "ControllerFs" before tranng mf mf4 mf8 mf mfmf7 mf3 mf9 mf mf8 mf mf mf mf mf3 mf9mf4 mf7 Degree of membershp Degree of membershp Force [N] "ControllerFs" after tranng Current [A] "ControllerFs" after tranng mf mf4 mf8 mf mfmf7 mf3 mf9 mf mf mf8 mf mf mf3mf mf4 mf9mf7 Degree of membershp Degree of membershp Force [N] Current [A] Fg. 3. Membershp functons of ControllerFs, before and after tranng Comparson of the FIS characterstcs and the membershp functon parameters n Table, before and after tranng, ndcates a redstrbuton of the membershp functons n the workng doman (modfcaton of the c parameter) and a change n ther shapes by the modfcaton of the σ parameter. Accordng to the parameter values from Table, the FIS s generated wth the genfs functon gve, as a frst result, the choce of the same values for the σ/ parameter, for all membershp functons whch characterze an nput. A second result s the separaton of the workng space for the respectve nput, usng the fuzzy subtractve clusterng method. Surfaces that reproduce the expermental before and after the ControllerFs tranng are presented n Fg. 4. The second FIS, ControllerFs, wth nputs of force and tme, was traned for the epochs usng the ANFIS Matlab functon. The rules here were also of the type: f (n s ncluster k ) and (n s ncluster k ) then (out s outcluster k ). For both of ths FIS s nputs, eght Gaussan-type membershp functons (mf) were generated. Therefore, ControllerFs fuzzy nference system has the structure shown n Fg., whle Controller has the structure gven n Fg..

16 New Applcatons of Fuzzy Logc Methodologes n Aerospace Feld 7 "ControllerFs" before tranng "ControllerFs" after tranng Current [A] 4 Force [N] Current [A] Fg. 4. Control surfaces resulted for ControllerFs, before and after tranng 4 Force [N] Fg.. Structure of the ControllerFs fuzzy nference system n n Force Tme Fg.. The structure of Controller Sugeno FIS out Elongaton The rules of the ControllerFs fuzzy nference system, before and after tranng, are presented n Fg. 7, whle Fg. 8 dsplays the devaton between the neuro-fuzzy models and the expermentally obtaned, defnng the qualty parameter from the tranng algorthm, for dfferent tranng epochs.

17 8 Fuzzy Controllers, Theory and Applcatons Fg. 7. The ControllerFs rules, before and after tranng.4 "ControllerFs".3.3 Devaton... Fg. 8. The tranng error for ControllerFs Number of tranng epochs x 4 Fgure 8 shows a rapd decrease n the devaton between the expermental and the neuro-fuzzy model for the qualty parameter wthn the tranng algorthm over the frst tranng epochs, from.3 untl a value of.9. Evaluatng the FIS before and after tranng for the expermental, the characterstcs n Fg. 9 were obtaned. The mean of the relatve absolute values of the errors decreased by 3.7 tmes -- from 3.33% before tranng to.89% after tranng. Consderng that the error for the ControllerFs s n the desred lmts after tranng epochs, ths FIS was selected to be mplemented n the Smulnk ntegrated controller. In Fg. 9, a good overlappng of the s wth the elongaton expermental s clearly vsble. As n the prevous FIS case, ths superposton s dependent on the tranng epochs number, and mproves as the number of tranng epochs ncreases. The parameters of the nput s membershp functons for the ControllerFs, before and after tranng, are shown n Table, whle the membershp functons shapes are depcted n Fg.. Comparson of the FIS characterstcs and the membershp functons parameters, before and after tranng, ndcates a redstrbuton of the membershp functons n the workng doman

18 New Applcatons of Fuzzy Logc Methodologes n Aerospace Feld 9 (modfcaton of the c parameter) and a change n ther shapes by modfcaton of the σ parameter (Table ). 3 "ControllerFs" before tranng 3 "ControllerFs" before tranng F=8N F=N F=N "ControllerFs" after tranng F=N F=N F=7N 3 Number of expermental ponts 3 Tme [s] F=N F=N F=8N F=7N F=N F=N "ControllerFs" after tranng F=N F=N F=N F=N F=8N F=7N 3 Number of expermental ponts Fg. 9. ControllerFs evaluaton, before and after tranng F=N F=N F=N F=N F=8N F=7N 3 Tme [s] Status Input Param. mf mf mf3 mf4 mf mf mf7 mf8 Before tranng After tranng Force [N] Tme [s] Force [N] Tme [s] σ/ c σ/ c σ/ c σ/ c Table. Parameters of the ControllerFIS nput s mf before and after tranng

19 7 Fuzzy Controllers, Theory and Applcatons "ControllerFs" before tranng "ControllerFs" before tranng mf8mf3 mfmf mf4 mf mf7mf mfmf8mf7 mf mf3 mf mf mf4 Degree of membershp Degree of membershp Force [N] "ControllerFs" after tranng Tme [s] "ControllerFs" after tranng mf8mf3 mf mf mf4 mf mf7mf mfmf8 mf mf7 mf mfmf3 mf4 Degree of membershp Degree of membershp Force [N] Tme [s] Fg.. Membershp functons of ControllerFs, before and after tranng Surfaces whch reproduce the expermental, before and after the ControllerFs tranng, are represented n Fg.. "ControllerFs" before tranng "ControllerFs" after tranng 4 Force [N] Tme [s] 8 4 Force [N] 8 Tme [s] Fg.. Control surface resulted for ControllerFs, before and after tranng

20 New Applcatons of Fuzzy Logc Methodologes n Aerospace Feld 7 The thrd FIS, Controller3Fs, whch has the force and the current as ts nputs, was traned for. epochs. The rules were also of the type: f (n s ncluster k ) and (n s ncluster k ) then (out s outcluster k ). For both of ths FIS s nputs, seven Gaussantype membershp functons (mf) were generated. Therefore, Controller3Fs fuzzy nference system has the structure presented n Fg., whle Controller 3 has the same structure as Controller, represented n Fg. 9. The rules of the Controller3Fs fuzzy nference system, before and after tranng, are presented n Fg. 3, and Fg. 4 dsplays the devaton between the neuro-fuzzy models and the expermentally obtaned for dfferent tranng epochs, defnng the qualty parameter from the tranng algorthm. Fg.. Structure of the Controller3Fs fuzzy nference system Fg. 3. The Controller3Fs rules, before and after tranng

21 7 Fuzzy Controllers, Theory and Applcatons Fgure 4 shows a decrease n the devaton between the expermental and the neurofuzzy model for the qualty parameter (wth some oscllatons) wthn the tranng algorthm over the frst 3 tranng epochs, from the value of. -4 to that of. -4. Evaluatng the FIS before and after tranng for the expermental, the characterstcs n Fg. were obtaned. The mean of the relatve absolute values of the errors decreased from.4-3 % before tranng, to.3-3 % after tranng. Controller3Fs was selected to be mplemented n the Smulnk ntegrated controller because ts obtaned error was wthn the desred lmts after tranng epochs. From Fg. one observes a good overlappng of the s wth the elongaton expermental. As n the prevous FISs cases, ths superposton s dependent upon the tranng epochs number, and s better as the number of tranng epochs s hgher. x -4. "Controller3Fs"..4.3 Devaton.. Fg. 4. The tranng error for Controller3Fs Number of tranng epochs x 4 The parameters of the nput s membershp functons for Controller3Fs, before and after tranng, are shown n Table 3, whle the membershp functons shapes are depcted n Fg.. Status Input Param. mf mf mf3 mf4 mf mf mf7 Before tranng Force [N] σ/ c Current σ/ [A] c..44. After tranng Force [N] σ/ c Current σ/ [A] c Table 3. Parameters of the Controller3FIS nput s mf before and after tranng

22 New Applcatons of Fuzzy Logc Methodologes n Aerospace Feld 73 4 "Controller3Fs" before tranng F=N F=N 4 F=N F=N "Controller3Fs" before tranng 8 4 F=8N F=N F=N 8 4 F=N F=N F=8N F=7N F=7N 4 8 Number of expermental ponts "Controller3Fs" after tranng F=8N F=N F=N F=N F=N 4 8 Current [A] F=N F=N F=N F=N F=8N "Controller3Fs" after tranng F=7N F=7N 4 8 Number of expermental ponts Fg.. Controller3Fs evaluaton, before and after tranng 4 8 Current [A] Comparson of the FIS characterstcs and the membershp functons parameters, before and after tranng, ndcates a redstrbuton of the membershp functons n the workng doman (modfcaton of the c parameter) and a change n ther shapes by modfcaton of the σ parameter (Table 3). The surfaces reproducng the expermental, before and after tranng of the Controller3Fs, are presented n Fg. 7. The fourth and last controller FIS, Controller4Fs, wth nputs of force and tme, was traned for epochs. As wth the others, the rules were of the type: f (n s ncluster k ) and (n s ncluster k ) then (out s outcluster k ). Seven Gaussan-type membershp functons (mf) were generated for each of the two nputs. Therefore, the Controller4Fs fuzzy nference system has the structure gven n Fg. 8, whle Controller 4 has the same structure as Controller, shown n Fg.. The rules of the Controller4Fs fuzzy nference system, before and after tranng, are presented n Fg. 9, whle Fg. 3 dsplays the devaton between the neuro-fuzzy models and the expermentally obtaned, defnng the qualty parameter from the tranng algorthm, for dfferent tranng epochs. Fgure 3 shows a rapd decrease n the devaton between the expermental and the neuro-fuzzy model for the qualty parameter wthn the tranng algorthm over the frst tranng epochs, from the value of.7 to that of.3. By evaluatng the FIS before and after tranng for the expermental, the characterstcs shown n Fg. 3 were obtaned.

23 74 Fuzzy Controllers, Theory and Applcatons The mean of the relatve absolute values of the errors decreased from.8% before tranng, to.3% after tranng. Snce the error found for the Controller4Fs was wthn the desred lmts after tranng epochs, ths FIS was chosen to be mplemented n the Smulnk ntegrated controller. "Controller3Fs" before tranng "Controller3Fs" before tranng Degree of membershp mf4 mf mf mf3 mf mf mf7 Degree of membershp mf mf4 mf mf7 mf mf3 mf Force [N] "Controller3Fs" after tranng 4 8 Current [A] "Controller3Fs" after tranng Degree of membershp mf4 mf mf mf3 mf mf mf7 Degree of membershp mf mf4 mf mf7 mf mf mf Force [N] Current [A] Fg.. Membershp functons of Controller3Fs, before and after tranng "Controller3Fs" before tranng "Controller3Fs" after tranng Current [A] Current [A] 8 Force [N] 8 Force [N] Fg. 7. Control surface resulted for Controller3Fs, before and after tranng

24 New Applcatons of Fuzzy Logc Methodologes n Aerospace Feld 7 Fg. 8. Structure of the Controller4Fs fuzzy nference system Fg. 9. Rules of the Controller4Fs before and after tranng.8 "Controller4Fs".7. Devaton..4 Fg. 3. The Controller4Fs tranng error x Number of tranng epochs

25 7 Fuzzy Controllers, Theory and Applcatons 3 "Controller4Fs" before tranng F=N 3 "Controller4Fs" before tranng F=N F=7N F=8N F=N F=N F=N F=N F=N F=N F=7N F=8N Number of expermental ponts 3 F=7N "Controller4Fs" after tranng F=8N F=N F=N F=N F=N Tme [s] 3 "Controller4Fs" after tranng F=N F=N F=N F=8N F=N F=7N Number of expermental ponts Fg. 3. Controller4Fs evaluaton, before and after tranng Tme [s] From Fg. 3, a good overlappng of the s output wth the elongaton expermental can be observed. As n the prevous FIS cases, ths superposton s dependent upon the tranng epochs number, and s better as the number of tranng epochs s hgher. The parameters of the nput s membershp functons for the Controller4Fs, before and after tranng, are shown n Table 4, whle the membershp functons shapes are depcted n Fg. 3. Status Input Param. mf mf mf3 mf4 mf mf mf7 Before tranng After tranng Force [N] Tme [s] Force [N] Tme [s] σ/ c σ/ c σ/ c σ/ c Table 4. Parameters of the Controller4FIS nput s mf before and after tranng

26 New Applcatons of Fuzzy Logc Methodologes n Aerospace Feld 77 "Controller4Fs" before tranng "Controller4Fs" before tranng mf4 mf mf mf mf3 mf7 mf mf7 mf mf4 mf3mf mf mf Degree of membershp Degree of membershp Force [N] Tme [s] "Controller4Fs" after tranng "Controller4Fs" after tranng mf4 mf mf mf mf3 mf7mf mf mf7 mf3 mf4 mfmf mf Degree of membershp Degree of membershp Force [N] Tme [s] Fg. 3. Membershp functons of Controller4Fs, before and after tranng Comparson of the FIS characterstcs and the membershp functons parameters, before and after tranng, ndcates a redstrbuton of the membershp functons n the workng doman (modfcaton of the c parameter) and a change n ther shapes by the modfcaton of the σ parameter (Table 4). The surfaces reproducng the expermental, before and after tranng of the Controller4Fs, are presented n Fg. 33. "Controller4Fs" before tranng "Controller4Fs" after tranng Tme [s] 8 Force [N] 4 Tme [s] Fg. 33. Control surface resulted for Controller4Fs, before and after tranng 8 Force [N] 4

27 78 Fuzzy Controllers, Theory and Applcatons Each of the four obtaned FISs was mported at the fuzzy controller level, resultng n four controllers: Controller ( ControllerFs ), Controller ( ControllerFs ), Controller 3 ( Controller3Fs ), and Controller 4 ( Controller4Fs ). The ntegraton of these four controllers s carred out usng the logcal scheme gven n Fg. 7; resultng n the Matlab/Smulnk model below, n Fg. 34. Force Force Fuzzy Logc Controller Constant Constant varable " k " z Swtch Sgnal From Workspace Current Current Detect Increase U > U/z Elongaton El Sgnal From Workspace Fuzzy Logc Controller 3 Swtch4 To Workspace Constant AND Logcal Operator Constant3 NOT Logcal Operator Te s the sample tme n the expermental T s the value of the sample tme for smulaton Swtch7 Constant C s the maxmum tme for the actuator to recover ts ntal temperature f the current becomes null Constant AND Logcal Operator3 Constant4 T/Te Gan Swtch s Integrator T/Te Gan C s Integrator Swtch Detect Decrease U < U/z == Compare To Zero Fuzzy Logc Controller 4 Compare To Zero Fuzzy Logc Controller Fg. 34. The ntegraton model schema n Matlab/Smulnk In the Matlab/Smulnk model shown n Fg. 34, the second nput for Controller and for Controller 4 (Tme) s generated by usng an ntegrator, and starts from the moment that ether of these controllers s used (the nput of the Gan block s f the schema decdes not to work wth ether Controller or 4). Because s possble that the smulaton sample tme may be dfferent than the sample tme used n the expermental acquston process, we use the Gan block that gves ther rapport; Te s the sample tme n the expermental and T s the smulaton sample tme. In the scheme, the constant C represents the maxmum tme that t takes for the actuator to recover ts ntal temperature (approxmately 4 C) when the current becomes null. Evaluatng the ntegrated model for controller (Fg. 34) n all sx cases of expermental, the results n Fg. 3 and Fg. 3 are obtaned. These results represent the elongatons versus the number of expermental ponts and versus the appled electrcal current, respectvely, usng the expermental and the ntegrated neuro-fuzzy controller model for the SMA. A good overlappng of the outputs of the ntegrated neurro-fuzzy controller wth the expermental can be easly observed. == Swtch Swtch Swtch3

28 New Applcatons of Fuzzy Logc Methodologes n Aerospace Feld 79 F=7N 4 F=8N Number of expermental ponts 3 3 Number of expermental ponts 8 F=N F=N Number of expermental ponts 3 3 Number of expermental ponts F=N F=N Number of expermental ponts Number of expermental ponts Fg. 3. Elongatons versus the number of expermental ponts The same concluson can be devolved from the 3D characterstcs for the expermental, and for neuro-fuzzy modeled n terms of temperature, elongaton and force, as depcted n Fg. 37 a., and n terms of current, elongaton and force, depcted n Fg. 37 b. The mean values of the relatve absolute errors of the obtaned model for the sx load cases of the SMA actuator, based on adaptve neuro-fuzzy nference systems, are:.738% for 7N,.738% for 8N,.94% for N,.8% for N,.779% for N and.3 for N. Therefore, the mean value of the relatve absolute error between the expermental and the outputs of the obtaned model s.933%. A very mportant advantage of ths new model s ts rapd generaton due to the genfs and ANFIS functons already mplemented n Matlab. The user only need assume the four FIS s tranng performances usng the anfsedt nterface generated wth Matlab.

29 8 Fuzzy Controllers, Theory and Applcatons F=7N 4 F=8N Current [A] 4 8 Current [A] 8 F=N F=N Current [A] 4 8 Current [A] F=N F=N Current [A] Current [A] Fg. 3. Elongatons versus the appled electrcal current Neuro-fuzzy controller Neuro-fuzzy controller 4 4 Force [N] 8 Force [N] 8 3 Temperature [ o C] 3 Current [A] Fg D evaluaton of the ntegrated neuro-fuzzy controller

30 New Applcatons of Fuzzy Logc Methodologes n Aerospace Feld 8 Another alternatve to desgn the SMA model, necesstatng some supplementary work, but also wth very good results, uses the genfs Matlab functon. In ths case, generalzed bell-type membershp functons are generated for the FISs; wthn the sets of rules they are noted by: n j mf n ; j s the nput number ( ), and n s the number of membershp functons. The rules are of the type: f (n s nmf k ) and (n s nmf p ) then (out s outmf r ). The number of the output membershp functons (mf) s k p (r= (k p)) and s equal to the number of rules; k and p are the number of mf of the two FIS s nputs. The genfs functon allows the membershp functon number to be chosen for each FIS nput (k and p), whle genfs automatcally generates the membershp functon s number. For example, f k= and p= are chosen, the structure of the ControllerFs generated wth the genfs functon can be organzed as n Fg. 38, whle Controller has the same structure as that presented n Fg. 9. Inputs Input mf Rules Output mf Output n Force n Current nmf nmfk nmf nmf nmfp nmf p r... 7 r = k p outmf outmfp outmf outmfr outmf7 Agregated Output out Elongaton Fg. 38. Structure of ControllerFs f the genfs functon s used for k= and p= By usng the genfs functon, generalzed bell-type membershp functons are generated; ther parameters are the membershp functon center (c) defnng ther poston, and a, b whch defne ther shape (see eq. (3)). Generatng FIS s wth the genfs functon has as a prmary result the choce of the same values for the a and b parameters for all of the membershp functons that characterze an nput, and as a secondary result the separaton of the workng space for the respectve nput usng a grd partton on the (no clusterng). FIS tranng wth the ANFIS functon produces an optmzed redstrbuton of the membershp functons n the workng doman (modfcaton of the c parameter) and a change n ther shapes by modfcaton of the a and b parameters. Usually, for an expermental set modelng, the genfs functon s frst used for FIS generaton, followed by FIS tranng wth the ANFIS functon over a dfferent number of tranng epochs. If the obtaned results are not the ones desred, the genfs functon wll be used to generate the FIS n order to mprove the accuracy of the obtaned model. 3. Actuaton lnes control 3. Controller desgn Startng from the developed SMA actuators model and based on the operatng scheme of the SMA actuators control n Fg. 4, a controller must be desgned n order to control the SMA actuators by means of the electrcal current supply, n order to cancel the devaton e between the requred values for vertcal dsplacements (correspondng to the optmzed

31 8 Fuzzy Controllers, Theory and Applcatons arfols) and the real values, obtaned from two poston transducers. The desgn of such a controller s dffcult due to the strong nonlneartes of the SMA actuators characterstcs. In these condtons, and consderng our research team experence n fuzzy logc control systems desgn, we decded that one varant of control would be developed wth fuzzy logc. The smplest fuzzy logc controller s the Fuzzy Proportonal (FP) controller, beng relevant for state or output feedback n a state space controller. Its nput s the error and the output s the control sgnal. From another perspectve, dervatve acton helps to predct the error, and the Proportonal-Dervatve (PD) controller uses further the dervatve acton to mprove closed-loop stablty (Jantzen, 998). The equaton of a PD controller can be expressed as follows: de( t) de( t) ( t) = K e( t) + K = K ( ), d e t + T t d () P D P D t where (t) s the command varable (electrcal current n our case), that s tme dependent; e s the operatng error (see Fg. 4), K P s the proportonal gan and K D s the dervatve gan. In dscrete form, the equaton () becomes (Kumar et al., 8): ( k) = K e( k) + K Δe( k), () P D [ e( k) e( k )] ( k) = K e( k) + K, (7) P D T where k s the dscrete step, T S s the sample perod, and Δ e(k) s the change n error. Therefore, the nputs to the Fuzzy Proportonal-Dervatve (FPD) controller are the error and ts dervatve (called change n error n fuzzy control language), whle the output s the control sgnal. We have chosen the structure shown n Fg. 39 for our FLC, where K D s the change n the output gan. S e Error Δe Change n error K P Proportonal gan K D Dervatve gan Fuzzy rule base K O Change n output gan FPD controller Command Fg. 39. Fuzzy PD controller archtecture To realze the nput-output mappng of the desgned controller, we must consder that n the SMA coolng phase the actuators would not be powered or the supplyng current would be very small. Ths coolng phase may occur not only when controllng a long-term phase, when a swtch between two values of the actuator dsplacements s ordered, but also n a short-lved phase, whch occurs when the real value of the deformaton exceeds ts desred value and the actuator wres need to be cooled. Each of the FLC nput or output sgnals have the real lne as the unverse of dscourse. In practce, the unverse of dscourse s restrcted to a comparatvely small nterval, many

32 New Applcatons of Fuzzy Logc Methodologes n Aerospace Feld 83 authors and several commercal controllers usng standard unverses such as [-, ], or [-, ] correspondng to percentages of full scale. For our system, the [-, ] nterval was chosen as the unverse for nputs sgnals, and [,.] nterval was chosen as the unverse for output sgnal. Also, followng numercal smulatons, we have chosen a number of three membershp functons for each of the two nputs, and three membershp functons for the output. The shapes chosen for nputs membershp functons were s-functons, -functons, and z-functons. Generally, an s-functon shaped membershp functon can be mplemented usng a cosne functon:, f x < xleft x x rght s ( x, x, x) = + cos π, f x x x, (8) left rght left rght x x rght left, f x > xrght a z-functon shaped membershp functon s a reflecton of a shaped s-functon:, f x < xleft x x left z ( x, x, x) = + cos π, f x x x, (9) left rght left rght x x rght left, f x > xrght and a -functon shaped membershp functon s a combnaton of both functons: π x, x, x, x, x) mn[ s( x, x, x), z( x, x, )], () ( = x left m m rght left m m rght wth the peak flat over the [x m, x m ] mddle nterval. x s the ndependent varable on the unverse, x left s the left breakpont, and x rght s the rght breakpont (Jantzen, 998). To defne the rules, a Sugeno fuzzy model was chosen, whch for a two nput - sngle output system wth N rules s gven by eq. (): Rule : If x Rule : If x Rule N : If x s A s A s A N and x and x and x s A, then y ( x, x ) = b + a x + a x, s A, then y ( x, x ) = b + a x + a x, N N N N N s A, then y ( x, x ) = b + a x + a x. () In the [-, ] unverse nterval, a three range partton, Negatve (N), Zero (Z) and Postve (P), were chosen for the nputs e and Δe whle n the [,.] unverse nterval three-range partton, Zero (Z), Postve-Small (PS) and Postve-Bg (PB) were used for the output. Accordng to the values n the Table, the membershp functons for the nputs are by the form depcted n Fg. 4, and are gven by the eq. (8), (9) or ():, f x <. A ( x) = z(.,, x) = [ cos( ) ], f., + x + π x (), f x >

33 84 Fuzzy Controllers, Theory and Applcatons, f x < A ( x) = z(,, x) = [ cos( ) ], f, + x + π x (3), f x >, f x < A ( x) = π(,,,, x) = mn[ s(,, x), z(,, x)] = [ cos( )], f, + πx x (4), f x >, f x < x + cos, f. + π 9 x A ( x) = π(,.,.,, x) =, f.., < x < () x cos, f. + π 9 x, f x >, f x < 3 3 A ( x) = A ( x) = s(,, x) = [ cos( ) ], f. + x π x (), f x > mf parameters Input mf mf type x left x m x m x rght mf ( A ) z - functon e Δ e mf ( A ) -functon - 3 mf3 ( A ) s - functon - - mf ( A ) z - functon mf ( A ) - functon mf3 ( A ) s - functon - - Table. Parameters of the nput s membershp functons For the output membershp functons constant values were chosen (Z=, PS=., PB=.), so the values of a ( k =,, =, N) parameters n eq. () were zero. Startng from the k nputs and output s membershp functons, a set of nference rules were obtaned (N=): Rule : Rule : Rule 3 : Rule 4 : Rule : If e s A If e s A If e s A If e s A If e s A 3 and Δe s A 3 and Δe s A, and Δe s A, and Δe s A and Δe s A 3,,, then y ( e, Δe) =., then y ( e, Δe) =, 3 then y ( e, Δe) =., 4 then y ( e, Δe) =, then y ( e, Δe) =. (7)

34 New Applcatons of Fuzzy Logc Methodologes n Aerospace Feld 8 Degree of membershp Degree of membershp nput (e) mf (N) mf (Z) mf3 (P) mf (N) mf (Z) mf3 (P) nput (Δe) Fg. 4. Membershp functons for the fuzzy logc controller nputs The rule-based nference chosen for each consequent s presented n Table. Wth the prevous consderatons, the fuzzy control surface results by the form presented n Fg. 4 (two vews for dfferent angles). Δe/e N Z P N - PS(.) - Z PB(.) - Z() P Z() Z() - Table. Rule-based nference for the fuzzy logc controller output nput (e) nput (Δe) output nput (Δe) nput (e). Fg. 4. The fuzzy control surface (two vews for dfferent angles) 3. Actuaton lnes controller mplementaton and numercal smulaton Implementng the operatng scheme of the SMA actuators control (Fg. 4) n Matlab- Smulnk, the model shown n Fg. 4 was obtaned. The nput varable of the scheme s the desred skn deflecton, whle the output s the real skn deflecton.

35 8 Fuzzy Controllers, Theory and Applcatons desred deflecton [mm] desred deflecton [mm] 3 cam factor FPD controller Dff error Current out Current [A] Current Dsplacement Force SMA Fuzzy Model SMA elongaton skn [m] deflecton [mm] Scope Memory.8 SMA Intal length F aero [N] F SMA [N] Aerodynamc force [N] x [m] Mechancal system y [mm] skn deflecton [mm] SMA elongaton [m] skn deflexon [mm] Fg. 4. The smulaton model for the controlled SMA actuator wth the neuro-fuzzy model The FPD controller block contans the mplementaton of the controller presented n Fg. 39; the detaled Smulnk scheme of ths block s shown n Fg. 43. The block has as nput the control error (the dfference between the desred and the obtaned dsplacements), and the controlled electrcal current appled on the SMA actuators as output. The SMA Fuzzy Model block has the schema presented n Fg. 34; ts nputs are the SMA loadng force and the electrcal current, whle ts output s the SMA elongaton. The Mechancal System block n Fg. 43 models the SMA loadng force startng from the aerodynamc force, skn elastc force, gas sprng elastc force and gas sprng pretenson force. Dff error Gan Gan z- den (z) Dervatve Fg. 43. FPD controller block n Smulnk Fuzzy Logc Controller -K- -K- -K- Gan Current out To obtan an automatc control system, the preloaded forces on the gas sprngs n the two actuaton lnes must be vald for all 3 studed cases. By estmatng the aerodynamc forces for all 3 studed flght condtons and optmzed arfols, a compromse should be done to balance the aerodynamc forces wth the preloaded forces of the gas sprng. Followng estmaton calculatons, the pretenson force of the gas sprngs n Mechancal system Smulnk block (see Fg. 4) was consdered wth the value of N. In ths stuaton, f the smulated model n Fg. 4 was loaded wth an aerodynamc force F aero = N, for a successve steps nput sgnal appled to the controlled actuator, the characterstcs shown n Fg. 44 are obtaned. The results shown n Fg. 44 confrm that the obtaned FPD controller works very well n both phases (heatng and coolng) of the SMA actuators. To see the manner n whch the controller works, screenshots were taken at dfferent tmes of the numercal smulaton presented n Fg. 44. The screenshots (Fg. 4) hghlghted the fuzzy model nput-output mappng of the eght analyzed ponts (P P8). The chosen tme values, shown on Fg. 44, are:.7 s (P), 7.3 s (P), 9.4 s (P3),.3 s (P4),.7 s (P), 8.4 s (P), 9. s (P7)

36 New Applcatons of Fuzzy Logc Methodologes n Aerospace Feld 87 Skn deflecton [mm] P3 P4 P 3 P P8 P Tme [s] P obtaned desred analyzed ponts P7 Fg. 44. Response for a successve steps nput when F aero = N and F pretenson = N 7.7 s (P8). Fg. 4 shows that the correspondence between the membershp functons of the nputs and the membershp functons of the output through the nference engne of the desgned fuzzy model was correctly establshed. The same observaton can be found by correlatng Fg. 4 wth the postons of the analyzed ponts n Fg. 44 and wth the error e and change n error Δe sgns and trends. 3.3 Bench test and wnd tunnel expermental valdaton From the SMA theory and based on the numercal smulatons of the morphng wng system, the lmts for the electrcal current used to drve the actuators, correlated wth the SMA temperature and SMA loadng force, were estmated. As a consequence, two Programmable Swtchng Power Supples AMREL SPS-33 (Bralovsk et al., 8; Coutu et al., 7; Coutu et al., 9), controlled by Matlab/Smulnk through a Quanser Q8 acquston card (Fg. 4) were chosen to mplement the controller model (Austerltz, ; Kranak et al., ; Park & Mackay, 3). The AMREL SPS-33 Power Supples have RS-3 and GPIB IEEE-488 as standard features, and ther techncal characterstcs nclude: Power 3.3kW, Voltage (dc) - V, Current (dc) -33 A. The Quanser acquston card has 8 sngle-ended analog nputs wth 4-bt resoluton, whch can be sampled smultaneously at khz, wth A/D converson tmes of.4 µs/channel, and s equpped wth 8 analog outputs, software programmable voltage ranges, that allow the control of the SMA actuators. The Q8 acquston card was connected to a PC and programmed va Matlab/Smulnk Rb and WnCon. (Fg. 47). As observed on Fg. 47, all acquston card sngle-ended analog nputs were used: two sgnals ndcatng the vertcal dsplacements dy and dy of the SMA actuators are provded by two Lnear Varable Dfferental Transformer (LVDT) potentometers, and sx sgnals are provded by sx thermocouples nstalled on each of the SMA wres components. Two of the card output channels were used to control each power supply through analog/external control by means of a DB- I/O connector, and other two card output channels were used to start the power supples wth a V analog sgnal. The SMA block had the scheme presented n Fg. 48.

37 88 Fuzzy Controllers, Theory and Applcatons P (.7 s) P (7.3 s) P3 (9.4 s) P4 (.3 s) P (.7 s) P (8.4 s) P7 (9. s) P8 (7.7 s) Fg. 4. Fuzzy model nput-output mappng of the analyzed ponts As seen n Fg. 48, the SMA block, controllng the frst actuaton lne, contans the mplementaton of the controller presented n Fg. 43 and the observatons related to the SMA actuators physcal lmtatons n terms of temperature and supplyng currents. The current suppled to the actuator was lmted at A, and the control sgnal was set to be -.V (maxmum voltage for the power supply s V for a 33A current supply). The upper lmt of the SMA wres temperature n the Temperature lmter block was establshed at

38 New Applcatons of Fuzzy Logc Methodologes n Aerospace Feld 89 3 C. The control scheme n Fg. 43 was mproved wth condtoners related to physcal model protecton. In ths way, a software protecton of the actuaton lnes was realzed. Reference arfol Optmum arfol dy dy... pressure sensors SMA# Poston transducers SMA# Desred dy, dy Morphng wng Sgnal from poston transducers Sgnal from sx thermocouples... Real dy, dy AMREL SPS-33 power supples Output sgnal for power supply control Quanser Q8 acquston card Fg. 4. Bench test physcal model operatng schema termocouple termocouple termocouple3 poston temp amps control V start V StartVolt SMA Q8 Quanser Analog Input Analog Input Q8 Quanser Analog Output Analog Output Temp termocouple termocouple termocouple3 poston temp amps control V start V Amps SMA Volts Fg. 47. Q8 acquston card usng to control the actuators After some tests wth the expermental model, the preloaded force of the gas sprngs that mantans the SMA wres n tenson was chosen to be N, snce n the laboratory the exstence of aerodynamc forces could not be consdered. The Matlab/Smulnk mplemented controller was used n the same way for both actuaton lnes of the morphng wng system, so the SMA block had a smlar scheme to the SMA block, wth the excepton of the numercal values of the thermocouples calbraton gans and constants.

39 9 Fuzzy Controllers, Theory and Applcatons poston sensor calbraton gan 9.43 dsplacement sensor calbraton constant FPD Controller current amps SMA desred dsplacement Manual Swtch for testng. current to control voltage SMA current (manually requred for testng) Saturaton temperature lmter max -K- 3 termocouple termocouple 3 termocouple MnMax Gan control V for hgh SMA temperature control V temp Termocouples calbraton gans and constants Fg. 48. SMA block, controllng the frst actuaton lne Sgnal START command for power supply 4 start V The valdaton of the desgned controller durng bench test runs conssted n two man steps, followed by a secondary one. Frstly, each of the two actuaton lnes of the morphng system were tested ndependently, the control prescrbed values (the desred dsplacements dy and dy ) beng presented under the form of successve steps. In ths way, the actuaton lnes responses were obtaned n Fg. 49; the characterstcs confrmed that the control works very well for both actuaton lnes. After ths frst step, the challenge was to test the actuaton lnes workng smultaneously (synchronzed commands), for desred dsplacements (dy and dy ) under the form of successve steps sgnals appled at ther nputs. A stuaton acqured durng ths test s presented n Fg., and valdates the good functonng of the desgned controller. The obtaned results presented n Fgs. 49 and show that the controllers, n the two actuaton lnes, work even at zero values of the desred sgnal because of the pre-tensoned gas sprngs. Small oscllatons of the obtaned dsplacements are observed around ther desred values. The ampltude of the oscllatons n ths phase s due to the LVDT potentometers mechancal lnk (were not fnally fxed because the model was not equpped wth the flexble skn n ths test Fg. ) and to the SMA wres thermal nerta; the smallest ampltude s less than. mm. In the secondary step of the bench test all pars of the desred dsplacements characterzng the 3 optmzed arfol cases were mposed smultaneously as nput sgnals on the two actuaton lnes, whle the skn was provsonally mounted on the model. In ths step, we could see f the skn supports both strans smultaneously; the recorded results for all 3 tested cases confrmed the good workng of the ntegrated morphng wng system.

40 New Applcatons of Fuzzy Logc Methodologes n Aerospace Feld 9 8 Dsplacement dy [mm] 4 3 act # desred act # obtaned Dsplacement dy [mm] act # desred act # obtaned Tme [s] Fg. 49. Actuaton lnes ndependent bench test Tme [s] Ths secondary step of the bench test was consdered n wnd tunnel, and get together wth the transton pont real tme poston detecton and vsualzaton, n order to valdate expermentally all of the 3 optmzed arfols theoretcally obtaned. A typcal test for one of the 3 flght condtons conssted n a wnd tunnel tare run, followed by a run for the reference (un-morphed) arfol, and fnally by a run for morphed arfol, reproducng the correspondng optmzed arfol. The morphng wng system durng wnd tunnel tests s shown n Fg act # desred act # obtaned 7 act # desred act # obtaned Dsplacement dy [mm] 4 3 Dsplacement dy [mm] Tme [s] Fg.. Actuaton lnes smultaneously bench test Tme [s] Fg.. Morphng wng system n the bench test

41 9 Fuzzy Controllers, Theory and Applcatons Fg.. Morphng wng system n the wnd tunnel test runs Because of the presence of the aerodynamc forces on the flexble skn of the wng for the wnd tunnel tests, the preloaded forces of the gas sprngs were reconsdered at N. The control results for test run 4, characterzed by the angle of attack α= and Mach number Mach=. (deflectons of both actuators are dy =.73 mm and dy =7.4 mm), are shown n Fg 3. 8 Dsplacement dy [mm] 4 3 dy=.73 mm act # desred act # obtaned Dsplacement dy [mm] dy =7.4 mm act # desred act # obtaned Dsplacements dy, dy [mm] Tme [s] act # act # Temperature [ o C] Tme [s] act # act # Temperature [ o C] Tme [s] Fg. 3. Wnd Tunnel Test for α=, Mach=. (dy =.73 mm, dy =7.4 mm) From the expermental results, t can stll be observed a hgh frequency nose nfluencng the LVDT sensors and thermocouple s nstrumentaton amplfers, but wth small ampltudes wth respect to those for the bench tested cases. A postve mpact on the nose ampltude

42 New Applcatons of Fuzzy Logc Methodologes n Aerospace Feld 93 reducton s the completon of mechancal model by fnal fxng of the skn on the model; ths tme, the nose sources are the wnd tunnel vbratons, the nstrumentaton electrcal felds and the wnd tunnel supplyng and montorng equpments electrcal felds. Fg. 3 and all the tested stuatons results confrmed that the desgned controller works very well n the wnd tunnel, beng postvely nfluenced by the aerodynamc forces presence. 4. Conclusons The approaches for the desgn and to the valdaton of a morphng wng fuzzy logc applcaton were presented. The developed morphng mechansm used smart materals such as Shape Memory Alloy (SMA) n the actuaton control concept. Two mportant applcatons of the fuzzy logc technque were hghlghted n ths work: the dentfcaton of a model for a system startng from some expermental nput-output, and the automatc control of a system. In ths way, n our morphng applcaton two drectons were developed: smart materal actuator modelng and actuaton lnes control. Based on a neuro-fuzzy network and usng numercal values resulted from the SMA expermental testng (forces, currents, temperatures and elongatons), an emprcal model was developed for the SMA actuators that could be used n the desgn phase of the actuaton lnes control. The SMA testng was performed at T amb =4 C for sx load cases wth the forces of 7 N, 8 N, N, N, N and N. The electrcal currents followng the ncreasng-constant-decreasng-zero values evoluton were appled on the SMA actuator n each of the sx cases consdered. Four Fuzzy Inference Systems (FIS s) were used to obtan four neuro-fuzzy controllers: one controller for the current ncrease, one for a constant current, one for the current decrease, and one controller for the null current (after ts decrease). For the frst and for the thrd controllers, nputs such as the force and the current were used, whle for the second and the fourth controllers, nputs such as the force and the tme values reflectng the SMA thermal nerta were used. Fnally, the four obtaned controllers were ntegrated nto a sngle controller. The genfs Matlab functon was used to generate and tran the fuzzy nference systems assocated wth the four controllers. The four ntally obtaned fuzzy nference systems were traned for,,, and tranng epochs. For the four FISs, the mean of the relatve absolute values of the errors decreased from.33%, 3.33%,.4-3 %, and.8%, respectvely, before tranng, to.9%,.89%,.3-3 %, and.3%, respectvely, after tranng. Evaluatng the model obtaned for the SMA actuators (the fnal, ntegrated controller) n all sx cases of expermental, the mean values of the relatve absolute errors were:.738% for 7N,.738% for 8N,.94% for N,.8% for N,.779% for N, and.3 for N. Therefore, the mean value of the relatve absolute error between the expermental and the outputs of the obtaned model was.933%. A very mportant advantage of ths new model s ts rapd generaton, snce the genfs and ANFIS functons are already mplemented n Matlab. The user only needs to assume the four FIS s tranng performances usng the anfsedt nterface generated wth Matlab. The second applcaton of fuzzy-logc technques n our project (actuaton lnes control) supposed the desgn of an SMA actuators controller startng from the developed SMA actuators model. The controller was desgned to control the SMA actuators by means of the electrcal current supply, n order to cancel the devaton e between the requred values for

43 94 Fuzzy Controllers, Theory and Applcatons vertcal dsplacements (correspondng to the optmzed arfols) and the real values, obtaned from two poston transducers. Fnally, a fuzzy PD archtecture was chosen for the controller. In ts desgn, numercal smulatons of the open loop morphng wng ntegrated system, based on a SMA neuro-fuzzy model, were performed. A bench test and a wnd tunnel test were conducted as subsequent valdaton methods. A [-, ] nterval was chosen as the unverse for the nputs sgnals, and a [,.] nterval was chosen as the unverse for the output sgnal. Also, followng numercal smulatons, three membershp functons for each of the two nputs, and three membershp functons for the output were chosen. The expermental valdaton tests (bench tests and wnd tunnel test) confrmed that the desgned controller works very well. The wnd tunnel tests were qute postve, wth ther transton pont real tme poston detecton and vsualzaton, whch expermentally valdated all of the 3 theoretcally-obtaned optmzed arfols. As a general concluson, the work presented here has proved the feasblty of usng fuzzy logc methodologes n multdscplnary research studes n the aerospace feld, especally for morphng wng or morphng arcraft studes.. References Al-Odenat, A.I. & Al-Lawama, A.A. (8). The Advantages of PID Fuzzy Controllers Over The Conventonal Types, Amercan Journal of Appled Scences, Vol., No., pp. 3-8, June 8, ISSN: Austerltz, H. (). Data acquston technques usng PCs, Elsever, ISBN: , USA Baron, A.; Benedct, B.; Branchaw, N.; Ostry, B.; Pearsall, J.; Perlman, G. & Selstrom, J. (3). Morphng Wng (MoW), ASEN 48 Senor Projects Techncal Report, Department of Aerospace Engneerng, Unversty of Colorado, December 3, Boulder, Colorado, USA, pages Bralovsk, V.; Terrault, P.; Coutu, D.; Georges, T.; Morellon, E.; Fscher, C. & Berube, S. (8). Morphng lamnar wng wth flexble extrados powered by shape memory alloy actuators, Proceedngs of ASME 8 Conference on Smart Materals, Adaptve Structures and Intellgent Systems (SMASIS8), pp. -3, ISBN: , Maryland, USA, October 8 3, 8, Publsher ASME, Ellcott Cty Coutu, D.; Bralovsk, V.; Terrault, P. & Fscher, C. (7). Expermental valdaton of the 3D numercal model for an adaptve lamnar wng wth flexble extrados, Proceedngs of the 8 th Internatonal Conference of Adaptve Structures and Technologes, pages, Ottawa, Ontaro, Canada, 3- October, 7 Coutu, D.; Bralovsk, V. & Terrault, P. (9). Promsng benefts of an actve-extrados morphng lamnar wng, AIAA Journal of Arcraft, Vol. 4, No., pp , March- Aprl 9, ISSN: -89 Georges, T.; Bralovsk, V.; Morellon, E.; Coutu, D. & Terrault, P. (9). Desgn of Shape Memory Alloy Actuators for Morphng Lamnar Wng Wth Flexble Extrados, Journal of Mechancal Desgn, Vol. 3, No. 9, 9 pages, 9, September 9, ISSN: -47 Gonzalez, L. (). Morphng Wng Usng Shape Memory Alloy: A Concept Proposal, Fnal research paper n Summer Research Experence for Undergraduates (REU) on Nanotechnology and Materals Systems, Texas Insttute of Intellgent Bo-Nano

44 New Applcatons of Fuzzy Logc Methodologes n Aerospace Feld 9 Materals and Structures for Aerospace Vehcles (TMS) NASA Research Unversty, Texas A&M Unversty, July, College Staton, Texas, USA Grgore, T.L. & Botez, R.M. (9). Adaptve neuro-fuzzy nference system based controllers for Smart Materal Actuator modelng, Proceedngs of the Insttuton of Mechancal Engneers, Part G: Journal of Aerospace Engneerng, Vol. 3, No., pp. -8, June 9, ISSN: 94-4 Grgore, T.L. & Botez, R.M. (). New adaptve controller method for SMA hysteress modelng of a morphng wng, The Aeronautcal Journal, Vol. 4, No., pp. -3, January, ISSN: -94 Grgore, T.L.; Popov, A.V.; Botez, R.M.; Mébark, Y. & Mamou, M. ( a). Modelng and testng of a morphng wng n open-loop archtecture, AIAA Journal of Arcraft, Vol. 47, No. 3, pp , May June, ISSN: -89 Grgore, T. L.; Popov, A.V.; Botez, R.M.; Mamou, M. & Mebark, Y. ( b). A morphng wng used shape memory alloy actuators new control technque wth b-postonal and PI laws optmum combnaton. Part : desgn phase, Proceedngs of the 7th Internatonal Conference on Informatcs n Control, Automaton and Robotcs ICINCO, pp. -, ISBN: , Madera, Portugal, -8 June,, ScTePress Scence and Technology Publcatons, Funchal Grgore, T. L.; Popov, A.V.; Botez, R.M.; Mamou, M. & Mebark, Y. ( c). A morphng wng used shape memory alloy actuators new control technque wth b-postonal and PI laws optmum combnaton. Part : expermental valdaton, Proceedngs of the 7th Internatonal Conference on Informatcs n Control, Automaton and Robotcs ICINCO, pp. 3-9, ISBN: , Madera, Portugal, -8 June,, ScTePress Scence and Technology Publcatons, Funchal Hampel, R.; Wagenknecht, M. & Chaker, N. (). Fuzzy Control Theory and Practce, Physca-Verlag, ISBN-3: , USA Jantzen, J. (998). Tunng of fuzzy PID controllers, Techncal Report 98-H87, Department of Automaton, Techncal Unversty of Denmark, September 998, pages Khezr, M. & Jahed, M. (7). Real-tme ntellgent pattern recognton algorthm for surface EMG sgnals, BoMedcal Engneerng OnLne, Vol., Paper 4, pages, December 7, ISSN: 47-9X Kranak, N.V.; Yursh, S.Y; Shpak, N.O. & Deynega, V.P. (). Data Acquston and Sgnal Processng for Smart Sensors. John Wley & Sons, ISBN: , UK Kosko, B. (99). Neural networks and fuzzy systems A dynamcal systems approach to machne ntellgence, Prentce Hall, ISBN: , New Jersey, USA Kovacc, Z. & Bogdan, S. (). Fuzzy Controller Desgn Theory and applcatons, Taylor and Francs Group, ISBN: , USA Kumar, V.; Rana, K.P.S. & Gupta, V. (8). Real-Tme Performance Evaluaton of a Fuzzy PI + Fuzzy PD Controller for Lqud-Level Process, Internatonal Journal of Intellgent Control and Systems, Vol. 3, No., pp. 89-9, June 8, ISSN: 8-79 Kung, C.C. & Su, J.Y. (7). Affne Takag-Sugeno fuzzy modellng algorthm by fuzzy c- regresson models clusterng wth a novel cluster valdty crteron, IET Control Theory and Applcatons, Vol., No., pp. -, September 7, ISSN: Mahfouf, M.; Lnkens, D.A. & Kandah, S. (999). Fuzzy Takag-Sugeno Kang model predctve control for process engneerng, IEE Workshop on Model Predctve Control:

45 9 Fuzzy Controllers, Theory and Applcatons Technques and Applcatons, 4 pages, 9 Aprl, 999, Prnted and publshed by the IEE, Savoy place, London WCPR OBL, UK MathWorks Inc. (8), Matlab Fuzzy Logc and Neural Network Toolboxes - Help. Park, J. & Mackay, S. (3). Practcal acquston for nstrumentaton and control systems, Elsever, ISBN: , UK Popov, A.V.; Botez, R.M. & Labb, M. (8 a). Transton pont detecton from the surface pressure dstrbuton for controller desgn, AIAA Journal of Arcraft, Vol. 4, No., pp. 3-8, January-February 8, ISSN: -89 Popov, A.V.; Labb, M.; Fays, J. & Botez, R.M. (8 b). Closed loop control smulatons on a morphng lamnar arfol usng shape memory alloys actuators, AIAA Journal of Arcraft, Vol. 4, No., pp , September-October 8, ISSN: -89 Popov, A.V.; Botez, R.M.; Mamou, M.; Mebark, Y.; Jahrhaus, B.; Khald. M. & Grgore, T.L. (9 a). Drag reducton by mprovng lamnar flows past morphng confguratons, AVT-8 NATO Symposum on the Morphng Vehcles, pages, -3 Aprl, 9, Publshed by NATO, Evora, Portugal Popov, A.V.; Botez, R. M.; Mamou, M. & Grgore, T.L. (9 b). Optcal sensor pressure measurements varatons wth temperature n wnd tunnel testng, AIAA Journal of Arcraft, Vol. 4, No. 4, pp , July-August 9, ISSN: -89 Popov, A.V.; Grgore, T. L.; Botez, R.M.; Mamou, M. & Mebark, Y. ( a). Morphng wng real tme optmzaton n wnd tunnel tests, Proceedngs of the 7th Internatonal Conference on Informatcs n Control, Automaton and Robotcs ICINCO, pp. 4-4, ISBN: , Madera, Portugal, -8 June,, ScTePress Scence and Technology Publcatons, Funchal Popov, A.V.; Grgore, T. L.; Botez, R.M.; Mamou, M. & Mebark, Y. ( b). Closed-Loop Control Valdaton of a Morphng Wng Usng Wnd Tunnel Tests, AIAA Journal of Arcraft, Vol. 47, No. 4, pp , July August, ISSN: -89 Popov, A.V.; Grgore, T. L.; Botez, R.M.; Mamou, M. & Mebark, Y. ( c). Real Tme Morphng Wng Optmzaton Valdaton Usng Wnd-Tunnel Tests, AIAA Journal of Arcraft, Vol. 47, No. 4, pp. 34-3, July August, ISSN: -89 Sanmont, C.; Paraschvou, I. & Coutu, D. (9). Multdscplnary Approach for the Optmzaton of a Lamnar Arfol Equpped wth a Morphng Upper Surface, AVT- 8 NATO Symposum on the Morphng Vehcles, -3 Aprl, 9, Publshed by NATO, Evora, Portugal Svanandam, S.N.; Sumath, S. & Deepa, S.N. (7). Introducton to Fuzzy Logc usng MATLAB, Sprnger, ISBN: , Berln, Hedelberg, New York Thll, C.; Etches, J.; Bond, I.; Potter, K. & Weaver, P. (8). Morphng skns, The Aeronautcal Journal, Vol., No. 9, pp. 7-39, March 8, ISSN: -94 Verbruggen, H.B. & Brujn, P.M. (997). Fuzzy control and conventonal control: What s (and can be) the real contrbuton of Fuzzy Systems?, Fuzzy Sets Systems, Vol. 9, No., pp., September 997, ISSN: -4 Zadeh, L.A. (9). Fuzzy sets, Informaton and Control, Vol. 8, No. 3, pp , June 9, ISSN: 9998

46 Fuzzy Controllers, Theory and Applcatons Edted by Dr. Lucan Grgore ISBN Hard cover, 38 pages Publsher InTech Publshed onlne 8, February, Publshed n prnt edton February, Tryng to meet the requrements n the feld, present book treats dfferent fuzzy control archtectures both n terms of the theoretcal desgn and n terms of comparatve valdaton studes n varous applcatons, numercally smulated or expermentally developed. Through the subject matter and through the nter and multdscplnary content, ths book s addressed manly to the researchers, doctoral students and students nterested n developng new applcatons of ntellgent control, but also to the people who want to become famlar wth the control concepts based on fuzzy technques. Bblographc resources used to perform the work ncludes books and artcles of present nterest n the feld, publshed n prestgous journals and publshng houses, and webstes dedcated to varous applcatons of fuzzy control. Its structure and the presented studes nclude the book n the category of those who make a drect connecton between theoretcal developments and practcal applcatons, thereby consttutng a real support for the specalsts n artfcal ntellgence, modellng and control felds. How to reference In order to correctly reference ths scholarly work, feel free to copy and paste the followng: Teodor Lucan Grgore and Ruxandra Mhaela Botez (). New Applcatons of Fuzzy Logc Methodologes n Aerospace Feld, Fuzzy Controllers, Theory and Applcatons, Dr. Lucan Grgore (Ed.), ISBN: , InTech, Avalable from: InTech Europe Unversty Campus STeP R Slavka Krautzeka 83/A Rjeka, Croata Phone: +38 () Fax: +38 () 8 InTech Chna Unt 4, Offce Block, Hotel Equatoral Shangha No., Yan An Road (West), Shangha, 4, Chna Phone: Fax: