Semi-active vibration control of an eleven degrees of freedom suspension system using neuro inverse model of magnetorheological dampers

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1 Jounal of Mechancal Scence and Technology () (0) 9~ DOI 0.00/s Sem-actve vbaton contol of an eleven degees of feedom suspenson system usng neuo nvese model of magnetoheologcal dampes Seyed Hamd Zaeh,*, Mesam Abbas, Had Mahdav and Kambz Ghaem Osgoue School of Scence and Engneeng, Shaf Unvesty of Technology, Ian Depatment of Mechancal Engneeng, Shahd Bahona Unvesty of Keman, Keman, Ian (Manuscpt Receved Septembe, 0; Revsed Mach, 0; Accepted Mach, 0) Abstact A sem-actve contolle-based neual netwok fo a suspenson system wth magnetoheologcal (MR) dampes s pesented and evaluated. An nvese neual netwok model (NIMR) s constucted to eplcate the nvese dynamcs of the MR dampe. The typcal contol stateges ae lnea quadatc egulato (LQR) and lnea quadatc gaussan (LQG) contolles wth a clpped optmal contol algothm, whle nheent tme-delay and non-lnea popetes of MR dampe le n these stateges. LQR pat of LQG contolle s also desgned to poduce the optmal contol foce. The LQG contolle and the NIMR models ae lnked to contol the system. The effectveness of the NIMR s llustated and vefed usng smulated esponses of a full-ca model. The esults demonstate that by usng the NIMR model, the MR dampe foce can be commanded to follow closely the desable optmal contol foce. The esults also show that the contol system s effectve and acheves bette pefomance and less contol effot than the optmal n mpovng the sevce lfe of the suspenson system and the de comfot of a ca. Keywods: Clpped optmal algothm; Invese model; MR dampe; Neual netwok; Sem-actve contol; Suspenson system Intoducton Suspenson systems have long been of geat concen fo ca manufactung ndustes. They pefom multple tasks such as mantanng contact between vehcle tes and the oad, addessng the stablty of the vehcle, and solatng the fame of the vehcle fom oad-nduced vbaton and shocks. In geneal, de comfot, oad handlng, and stablty ae the most mpotant factos n evaluatng suspenson pefomance. In the pesent study, the suspenson system of a passenge ca s modfed to educe the ampltude of the ca vbaton caused by appled oad pofle. In the passve suspenson system, the stffness and dampng paametes ae fxed and effectve ove a cetan ange of fequences. To ovecome ths poblem, the use of sem-actve suspenson systems whch have the capablty of adaptng themselves to changng oad condtons by usng an actuato has been consdeed; theefoe an MR dampe s added to an odnay suspenson system whle the othe pats of the suspenson system ae ntact. The sgnfcance of the MR dampe s that ts vscosty changes as the magnetc feld changes. A schematc model of the MR dampe s shown n Fg.. * Coespondng autho. Tel.: +990, Fax.: +99 E-mal addess: zaeh_hamd@alum.shaf.edu Recommended by Assocate Edto Hyoun Jn Km KSME & Spnge 0 Fg.. Schematc model of MR dampe. The consdeed suspenson model s contolled by the appled LQR and LQG contolles. By usng employed contolle esults, the amount of vscosty of the MR dampe ncopoated wth the clpped optmal stategy can be calculated. Unfotunately, due to the nheent nonlnea natue of the MR dampe to geneate foce, a model lke that fo ts nvese dynamcs s dffcult to mathematcally obtan. Fo ths eason, an nvese neual netwok model s constucted to copy the nvese dynamcs of the MR dampe. Inteest n a new class of computatonal ntellgence systems known as Atfcal Neual Netwoks (ANNs) has gown n the last few yeas. Ths type of netwok has been found to be a poweful computatonal tool fo oganzng and coelatng

2 0 S. H. Zaeh et al. / Jounal of Mechancal Scence and Technology () (0) 9~ nfomaton n ways that have been poven to be useful fo solvng cetan types of complex and pooly undestood poblems. The applcatons of ANNs to the aea of stuctual contol have apdly gown though system dentfcaton, system nvese dentfcaton o contolle eplcaton. The ntellgent neuo-fuzzy and fuzzy nvese model stateges fo sem-actve suspenson system ae utlzed by Zaeh et al. [, ]. Naayanan et al. [] appled a half ca model fo smulatng the sem-actve suspenson system. They modeled MR dampe paametes by the modfed Bouc Wen model and detemned that they ft the hysteetc behavo. Vbaton contol of a passenge ca utlzng a half ca model was dscussed by Yahaya et al. []. To mpove the accuacy of ca model Zaeh et al. [], modeled the vehcle suspenson system usng a full ca model n whch the accuacy of the model mpoved components to quate-ca and half-ca models. Sem-actve and actve contol methods have been developed usng dffeent actuatos such as Electo-Rheologcal (ER) and MR dampes. Salem et al. [] contolled an actve quate-ca suspenson system by fuzzy logc contolle. Sohn et al. [] utlzed an adaptve LQG contolle fo sem-actve suspenson systems fo a quate-ca model. Golnaagh et al. [] contolled a sem-actve quate ca suspenson system ntellgently. They utlzed an nvese mappng model to estmate the cuent based on an atfcal neual netwok and ncopoated t nto the fuzzy logc contolle. The pevous studes made full use of the advantages of the neual netwok and the optmal contolles and solved the dffeent poblems whch lay n the suspenson system. Few studes ncluded the combnaton of the two technques (LQG and neual netwok n that t s moe applcable) to solve the tme-delay and the nheent nonlnea natue of the MR dampe to geneate foce nstantaneously choosng contol paametes n sem-actve contol o actve contol fo full ca models wth hgh degees of feedom. In ths pape, fou MR dampes ae added to a suspenson system between body and wheels n paallel wth the passve dampes. Fo the ntellgent system, a thee-laye feed fowad neual netwok, taned on-lne unde the Levenbeg Maquadt (LM) algothm, s adopted. In ode to vefy the effectveness of the poposed NIMR contol stategy, the uncontolled system and the clpped optmal contolled suspenson system ae compaed wth the NIMR contolled system. Though a numecal example unde actual oad pofle exctaton, t can be concluded that the contol stategy s vey mpotant fo sem-actve contol, that the NIMR contol stategy can detemne cuents of the MR dampe quckly and accuately, and that the contol effect of the NIMR contol stategy s bette than the othes.. Full ca model In the full-ca model, -DOFs ae assumed, and all wheels and passenges ae dependent on each othe and on the ca body. It s assumed that each wheel has an effect on the spng Fg.. Full-ca model wth -DOFs []. and the dampe of othe wheels, and the two axes of the vehcle ae elevant. The MR actuato s utlzed to damp the effect of oad pofle on the passenges. Note that the MR shock absobe s added to the axel and the ca body. In the full-ca model, the effect of body otatons aound oll and the yaw axs s smulated. The suspenson system usng the full-ca model has -DOFs, fou of whch fo the fou wheels, thee fo the body dsplacement and ts otatons and the last fou fo passenges. The schematc of the full-ca model wth - DOFs and an addtonal MR dampe s shown n Fg.. The state space fom and the coespondng matces ae obseved n Eqs. ()-() []. E s the locaton matx of the actuatos. Matces K, S and T ae defned as stffness, dampng coeffcent and nput matx due to the wheels stffness, espectvely. In Eq. () x, u and w ae states of system, contol sgnal fom actuatos and nput oad pofle espectvely. x =Axu+Gw () 0 I A = M K M S 0 B = M T 0 G = M E M = dag(m, m, m, m, M b, m, m, m, m, I, I ) () T = [ kt,0,0,0; 0, kt,0,0; 0,0, kt,0; 0,0,0, kt; zeos(,)] E = [ eye(,);,,,; zeos(,);,,, ;,,, ] () () () () ()

3 S. H. Zaeh et al. / Jounal of Mechancal Scence and Technology () (0) 9~ + k t kt kt kt K= () S= (9) whee M b, m, m, m, m, m, m, m and m stand fo the mass of the ca body, masses of the fou wheels and mass of passenges, espectvely. I and I ae the moments of neta of the ca body aound the two axes, espectvely. The tems k, k, k, k, k, k, k and k ae the stffness of the spngs of the suspenson system and the stffness of the spngs of the passenge seat, espectvely. The tems k t, k t, k t and k t ae the stffness of the tes. The tems b, b, b, b, b, b, b and b ae the coeffcents of the ca and the passenge seat dampes, espectvely. Then, b, b, b and b ae the coeffcents of the MR dampes, espectvely. x, x, x, x, x, x, x, x, x 9, φ and θ ndcate the DOFs of the suspenson system model, espectvely. The tems x, x, x and x ndcate the load pofle dstubance, espectvely. The numecal values of the model dmensons fo obtanng the esponses ae shown n Table [].. NIMR contol stategy Fg. llustates the poposed NIMR contol stategy. Thee Fg.. NIMR contol stategy block dagam. s bascally no estcton on the type of contol algothm that should be used as long as t calculates a desable contol foce f c based on the esponse and/o exctaton. The desable contol foce and the esponse of the systems ae passed n to the nvese NN model. Ths NN model emulates nvese dynamcs of the MR dampe. The output of ths nvese NN model s the voltage equed to poduce the dese contol foce unde the cuent esponse condton. Ths voltage s an/the nput to the MR dampe whch then poduces foce f actng on the

4 S. H. Zaeh et al. / Jounal of Mechancal Scence and Technology () (0) 9~ Table. Numecal values of model. Symbol Quantty Value M b Mass of ca body (kg) 0 I Ineta aound yaw (kg/m ) 00 I Ineta aound oll (kg/m ) 00 m - Mass of wheels (kg) 0 m - Mass of passenges (kg) 0 k - Stffness of ca spngs (N/m) 00 k - Stffness of seats (N/m) 0 b - Vscosty of ca dampes (Ns/m) 0 b - Vscosty of seat dampe (Ns/m) 00 k t- Stffness of wheels (N/m) 00 Length dstance between the font left wheel and the mass cente (m).99 Wdth dstance between the font left wheel and the mass cente (m) 0.0 Length dstance between the font ght wheel and the mass cente (m).99 Wdth dstance between the font ght wheel and the mass cente (m) 0.0 Length dstance between the back ght wheel and the mass cente (m).99 Wdth dstance between the back ght wheel and the mass cente (m) 0.0 Length dstance between the back left wheel and the mass cente (m).99 Wdth dstance between the back left wheel and the mass cente (m) 0.0 Length dstance between the dve seat and the mass cente (m).99 Wdth dstance between the dve seat and the mass cente (m) 0.0 Length dstance between the font ght seat and the mass cente (m).99 Wdth dstance between the font ght seat and the mass cente (m) 0.0 Length dstance between the back left seat and the mass cente (m).99 Wdth dstance between the back left seat and the mass cente (m) 0.0 Length dstance between the back ght seat and the mass cente (m).99 Wdth dstance between the back ght seat and the mass cente (m) 0.0 suspenson system. Note that, the NN pat of contol stategy wll be taned by the esults of clpped optmal stategy.. Clpped optmal algothm The clpped optmal contol stategy fo an MR dampe usually nvolves two steps. The fst step s to assume an deal actvely contolled devce and constuct an optmal contolle fo ths actve devce. In the second step, a seconday contolle fnally detemnes the nput voltage of the MR dampe [9]. That s, the seconday contolle clps the optmal foce n a manne consstent wth the dsspatve natue of the devce. Fg.. Clpped optmal algothm block dagam. The block dagam of the clpped optmal algothm s shown n Fg.. The clpped optmal contol appoach s to append a foce feedback loop to nduce the MR dampe to poduce appoxmately a desed contol foce f c. It s known as a contol sgnal (u). The Lnea Quadatc Regulato (LQR) algothm has been employed both fo actve contol and fo sem-actve contol. Usng ths algothm, the optmal contol foce f c fo f, whch s foce geneated by an MR dampe, may be obtaned by mnmzng the followng scala pefomance ndex: t f J = ( T T x Qx+ F RF) dt (0) t 0 whee Q and R ae weghtng matces and the values ae selected based on the elatve mpotance gven to dffeent tems n the contbutons to the pefomance ndex J. Solvng the optmal contol poblem wth J defned by Eq. (0) esults n an optmal foce vecto f c egulated only by the state vecto x, such that: f ( T c = R B P) x = K x () lq whee matx K lq epesents the gan matx; and the matx P s the soluton of the Rccat equaton gven by Eq. (). PA + A T P PBR B T P + Q = 0 () Hee, Q * s an dentty matx (I * ) because all states ae mpotant and the best esponses ae obtaned by ths value. R * s *I *, whch s obtaned by tal and eo to eceve the desed esponse. The foce geneated by the MR dampe cannot be commanded. When the MR dampe s povdng the desed optmal foce (.e., f = f c ), the voltage appled to the dampe should eman at the pesent level. If the magntude of the foce poduced by the dampe s smalle than the magntude of the desed optmal foce and the two foces have the same sgn, the voltage appled to the cuent dve [9] vaes con-

5 S. H. Zaeh et al. / Jounal of Mechancal Scence and Technology () (0) 9~ Table. Numecal values of MR dampe model. α a (N cm - ) α b (N cm - V - ) γ (cm - ) β (cm - ) 0 9 Table. Numecal values of MR dampe model. c 0a (Nscm - ) c 0b (Nscm - V - ) n η (s - ) A m Fg.. Mechancal model of a shea mode type MR dampe []. tnuously n the ange of [0-V max ]. The seconday contolle fo contnuously vayng the command voltage can be stated as: Table. Typcal popetes of unpaved oad pofle. σ (m ) ω (ad/s ) V (m/s) α (ad/m) e- 00π. 0. { } v = V c H( f c f f ) () μ f c, fo f c f V max c = V max, fo f c f () > max whee V max s v and f max s the maxmum foce poduced by the dampe (=,000 N); μ s the coeffcent elatng the voltage to the foce (=V max /f max ); H (.) s the heavsde step functon expessed as 0 o [9]; and f s the foce poduced by th MR dampes and appled to the stuctue. In ths pape, a smple mechancal model consstng of a Bouc-Wen element shown n Fg. n paallel wth a vscous dampe s used. Ths model has been vefed to accuately pedct the behavo of a pototype shea-mode MR dampe ove a wde ange of nputs n a set of expements and s also expected to be appopate fo modelng a full-scale MR dampe. The equatons govenng the foce f exeted by ths model ae as follows: f = c x& +α z 0 () n n z& = γ x& z z& βx& z + Am x& () whee x s the dsplacement of the devce, and z s the evolutonay vaable that accounts fo the hstoy dependence of the esponse. The paametes γ, β, n and A m ae adjusted to detemne the lneaty n the unloadng and the smoothness of the tanston fom the pe-yeld egon to the post-yeld egon. The devce model paametes α and c 0 ae detemned by the dependency on the contol voltage u as follows: α = α ( u) = α a + α u b () c ( ) a 0b () Moeove, to account fo a tme-lag n the esponse of the devce to the changes n the command nput, the fst-ode flte dynamcs ae ntoduced nto the system as follows: Fg.. Hysteetc behavo of an MR dampe []. u& = η ( u υ) (9) whee v s a command voltage appled to the contol ccut, and η s the tme constant of the fst-ode flte. The numecal values of the paametes ae shown n Tables and [0]. The hysteetc behavo of the MR dampe model accodng to the nput voltage s shown n Fg... Road pofle smulaton The andom oad exctaton s geneated usng whte nose as a oad egulaty as a dstubance. The Powe Spectal Densty (PSD) functon of the oad egulaty s assumed to be n the fom of Eq. (0). σ α S V h = π ( ω + ( α V ) ) (0) whee σ s the vaance of the oad pofle, ω s the exctaton fequency of the oad nput V s the vehcle fowad constant velocty and α s a coeffcent dependng on the type of oad suface. The typcal popetes of the unpaved oad pofle ae shown n Table []. The PSD of the oad suface s obtaned

6 S. H. Zaeh et al. / Jounal of Mechancal Scence and Technology () (0) 9~ Fg.. Tanng of NIMR. Fg.. NN of nvese dynamcs of MR dampe (NIMR). usng MATLAB.. Neual netwok nvese dynamcs of MR dampe The MR dampe model dscussed eale n ths pape estmates dampe foces based on the nputs of the eactve velocty and the ssued voltage as descbed by Eqs. ()-(9). Thus, the voltage sgnal s the only paamete that can be modfed to contol the dampe foce to poduce the equed contol foce. The contol algothm, LQG, estmates the equed optmal contol foce but the MR dampe foce s contolled by voltage. In ths case, t s essental to develop an nvese dynamc model that pedcts the coespondng contol voltage to be sent to the dampe so that an appopate dampe foce can be geneated. Unfotunately, due to the nheent nonlnea natue of the MR dampe, a model lke that fo ts nvese dynamcs s dffcult to mathematcally obtan. Fo ths eason, a feed-fowad back-popagaton neual netwok that s shown n Fg. s constucted to copy the nvese dynamcs of the MR dampe. The pupose n a vehcle suspenson system s educton of tansmttance of vbatonal effects fom the oad to the vehcle s passenges, hence povdng de comfot. To accomplsh ths, one can fst desgn a clpped algothm fo the suspenson system, usng an optmal contol method and use t to tan a neuo contolle. Consde netwok can be taned usng the clpped contolle output eo on an onlne manne (clpped output selected as a dese tagets fo tanng). Once taned, the clpped contolle s automatcally emoved fom the contol loop and the neuo contolle takes on. In case of a change n the paametes of the system unde contol, the clpped optmal contolle entes the contol loop agan and the neual netwok gets taned agan fo the new exctaton []. An mpotant chaactestc of the poposed contolle s that no mathematcal model s needed fo the system components, such as the non-lnea actuato o shock absobes. At the ntal tme, the nputs of the netwok wll be taken to have the value of zeo n accodance wth the actual ntal ccumstance and smplcty n smulaton [, 9, ]. Befoe onlne tanng, the netwok taned off-lne to acheve update weghts nea to desed. Ths model s denoted as NIMR. Ths neual netwok model s taned usng nput-output data geneated analytcally by usng the smulated MR model based on Eqs. ()-(9). Ths NIMR calculates the voltage sgnal as an output based on the cuent and a few pevous hstoes of measued velocty and desable contol foce as nputs. Then the voltage sgnals ae sent to the MR dampe so that t can geneate the desable optmal contol foces. Tanng the NIMR eques the complaton of nput-output data. To completely dentfy the undelyng MR system model, the data must contan nfomaton about the ente opeatng ange of the system. The geneated foces ae the esults of the MR model descbed n Eqs. ()-(9). The samplng data fo 0-second peods esults n 0000 pattens fo tanng, testng and valdaton, Fg.. The next step s to select the netwok achtectue. To do so t s equed to detemne the numbe of nputs, outputs, hdden layes, and nodes n the hdden layes, and ths s usually done by tal and eo. The most sutable nput data fo ou case was found to be the cuent and the thee and two pevous hstoes fo the velocty (x k+, x k+, x k) and the foce ( f k+, f k ), espectvely. Also one hdden laye, whch t has sx nodes, wee adopted as one of the best sutable topologes fo the NIMR as can be seen n Fg.. Ths stuctue obtaned by tal and eo, because t equed mnmum tme fo onlne tanng wth acceptable accuacy. The logsgmod (angng fom 0 to ) actvaton functon s used fo the hdden laye s nodes and the lnea functon fo the output laye s nodes whch epesents the voltage pattens of the povded data wee chosen fo tanng whch equed 000 tanng epochs to acheve desed. The tanng s caed out upon the geneated data usng the Levenbeg Maquadt (LM) algothm, whch s encoded n neual netwoks toolbox n MATLAB unde tanlm outne. Neual netwok s a smplfed model of the bologcal stuctue found n human bans. Ths model conssts of elementay pocessng unts (also called neuons). It s the lage amount of nteconnectons between these neuons and the capabltes to lean fom data to enable neual netwok as a stong pedctng and classfcaton tool. In ths study, as mentoned, thee-laye feed fowad neual netwok, whch con-

7 S. H. Zaeh et al. / Jounal of Mechancal Scence and Technology () (0) 9~ ssts of an nput laye, one hdden laye, and an output laye s selected to pedct the esponses wth MR dampes. The net nput value net k of the neuon k n some laye and the output value O k of the same neuon can be calculated by the followng Eqs. ()-(): net = w O k jk j () O = f( net + θ ) k k k () whee w jk s the weght between the j th neuon n the pevous laye and the k th neuon n the cuent laye, O j s the output of the j th neuon n the pevous laye, f(.) s the neuon s actvaton functon whch can be a lnea functon, a adal bass functon, and a sgmod functon, and y k s the bas of the k th neuon. Feed fowad neual netwok often has one o moe hdden layes of sgmod neuons followed by an output laye of lnea neuons. Multple layes of neuons wth nonlnea tansfe functons allow the netwok to lean nonlnea and lnea elatonshps between nput and output vectos. In the neual netwok achtectue as shown n Fg., the logathmc sgmod tansfe functon s chosen as the actvaton functon of the hdden laye, Eq. (): O = f( net + θ ) =. k k k ( net + θ ) + e k k () The lnea tansfe functon s chosen as the actvaton functon of the output laye, Eq. (): O = f( net + θ ) = net + θ. () k k k k k We note that neual netwok needs to be taned befoe pedctng esponses. As the nputs ae appled to the neual netwok, the netwok outputs (.) ˆ ae compaed wth the tagets (.). The dffeence o eo between both s pocessed back though the netwok to update the weghts and bases of the neual netwok so that the netwok outputs match close wth the tagets. The nput and output data ae usually epesented by vectos called tanng pas (f k+, f k, x k+, x k+, x k, v^k+). The pocess as mentoned above s epeated fo all the tanng pas n the data set, untl the netwok eo conveged to a theshold mnmum defned by a coespondng pefomance functon. In ths eseach, the Mean Squae Eo (MSE) functon s adopted (desed MSE s e-). As mentoned, LM algothm s adapted to tan the neual netwok, whch can be wtten as Eq. (): E E w + = w + μi w w () Fg. 9. Dsplacement of ea-left seat fom font left wheel excted wth 0km/h. whee s the teaton ndex, E/ w s the gadent descent of the pefomance functon E wth espect to the paamete matx w, μ 0 s the leanng facto, and I s the unty matx. Dung the vbaton pocess, the neual netwok updates the weghts and bas of neuons eal tme n accodance wth samplng pas tll the objectve eo s satsfed,.e. the popety of the system s acqued. As we know, the man am of the neual netwok s to pedct the desed voltage esponses of the system and to povde the desed foce of MR dampes. Thus outputs of the neual netwok ae pedcted values of voltage v^k+. In ode to accuate pedcton, the most dect and mpotant facto whch affect the pedcted voltage s the feedback of pedcted voltage (v^k+).. Results The full-ca model wth the MR dampe s modeled by the dynamc equatons and state space matces. One of the desed ponts of ths study s to decease the ampltude of passenges dsplacements and spung mass acceleatons, when the suspenson system s excted by the oad pofle. Theefoe, the effect of LQR and LQG contolles and the NIMR stategy ae smulated fo oad exctaton wth calculatng the ampltude, and then compang them wth each othe. The andom oad suface s compatbly geneated wth the powe spectal densty gven n Eq. (0) usng a sum of bumpes cm hgh and cm wde. The dsplacement and acceleaton tajectoes fo ea-left passenge seat that s excted by the bumpe wth a heght cm and a wdth cm and wth 0 km/h constant velocty n the unpaved oad unde the font left wheel ae shown n Fgs. 9 and 0, espectvely (Consdeed system excted by dffeent veloctes (e.g. 0, 0 and km/h). The esults, to some extend have the same tend wth dffeent ampltude fo each velocty. Theefoe the esults coespond to 0 km/h ae shown.). Notce that, n all these gaphs, tme duaton s selected fo the best esoluton and ctcal esponses happen when the ca stkes the bumpe. The gaphs show that ths system can be contolled as well when the vehcle hts the bumpe. The pmay oscllatons ae due to the fewe numbe of netwok nputs to tan. On the othe hand, thee s not a stong hstoy avalable dung the tansent state; theefoe, the tansent pat of the

8 S. H. Zaeh et al. / Jounal of Mechancal Scence and Technology () (0) 9~ Fg. 0. Acceleaton of ea-left seat fom font left wheel excted wth 0km/h. Fg.. Geneated foce by ea-left MR dampe fom font left wheel excted wth 0 km/h. Fg.. Road holdng fo ea-left dampe excted wth 0 km/h. esponse s not as satsfactoy as the steady state pat. It s also shown that comfot dvng s attaned by deceasng (the ampltude of the) acceleaton. The oad holdng fo ea-left dampe that s excted by the -cm hgh and -cm wde bumpe wth 0 km/h constant velocty unde the font left wheel s shown n Fg.. The stablty of the automoble due to usng the NIMR stategy s bette than the othe two stateges because the oscllaton of the ea left wheel due to oad exctaton s less than the othes. The tajectoy fo the optmal foce whch poduces the desed dsplacement and acceleaton wth a velocty of 0 km/h velocty s shown n Fg.. One of the man advantages of NIMR s that contol effots of dampes ae less than the LQR and LQG esponses, leadng to less cost and enegy consumpton. The equement voltage to geneate the optmal foce s shown n Fg.. Accodng to the gaphs above, the tajectoes of the NIMR stategy can educe the ampltude of vbaton to some extent smla the optmal contolles wth less contol effot and oscllaton. Fg.. Requement voltages to ea-left MR dampe fom font left wheel excted wth 0km/h.. Conclusons Usual suspenson systems ae utlzed n vehcles and damp vbaton fom oad pofle. Howeve, passve suspenson systems have long settlng tme. When cas ae dven n bumpy oads, the passve suspenson system dve cannot eact effectvely. As a esult, the usual suspenson system cannot damp the exctaton wth small tme ntevals. In ode to emove ths poblem the popetes of the suspenson system should be vaable. Ths task s done by addng MR dampes as actuatos to the suspenson system. In ode to send commands to the actuato, the LQR contolle s utlzed. It can decease the ampltude of the vbaton of the passenge seat, but t cannot elmnate the effect of bumpy oads as a dstubance. Theefoe, the LQG contolle was desgned. Unfotunately, due to the nheent nonlnea natue of the MR dampe to geneate foce, a model lke that fo ts nvese dynamcs s dffcult to mathematcally obtan. Fo ths eason, a neual netwok s constucted to copy the nvese dynamcs of the MR dampe. Accodng to the gaphs above, the tajectoes of the NIMR stategy can educe the ampltude of vbaton to some extent smla the optmal con-

9 S. H. Zaeh et al. / Jounal of Mechancal Scence and Technology () (0) 9~ tolles wth less contol effot and oscllatons. Theefoe, t can ncease duablty of dampe and low cost of poducton. Acknowledgment The authos wsh to expess the gattude to the School of Scence and Engneeng of Shaf Unvesty of Technology fo suppotng ths eseach. Refeences [] S. H. Zaeh, A. Saafan, A. A. A. Khayyat and A. Zabhollah, Intellgent sem-actve vbaton contol of eleven degees of feedom suspenson system usng magnetoheologcal dampes, Jounal of Mechancal Scence and Technology, () (0) -. [] S. H. Zaeh and A. A. A. Khayyat, Fuzzy nvese model of magnetoheologcal dampes fo sem-actve vbaton contol of an eleven-degees of feedom suspenson system, Jounal of System Desgn and Dynamcs, () (0) -9. [] R. S. Pabaka, C. Sujatha and S. Naayanan, Optmal semactve pevew contol esponse of a half ca vehcle model wth magnetoheologcal dampe, Jounal of sound and vbaton, (009) [] M. S. Yahaya and H. S. Joha, Modelng and contol of the actve suspenson system usng popotonal ntegal sldng mode appoach, Asan Jounal of Contol, () (00) 9-9. [] S. H. Zaeh, A. Saafan and A. A. A. Khayyat, Clpped optmal contol of -DOFs of a passenge ca usng magnetoheologcal dampe, Intenatonal Confeence on Compute Contol and Automaton (0) -. [] M. M. M. Salem and A. A. Ayman, Fuzzy contol of a quate-ca suspenson system, Wold Academy of Scence Engneeng and Technology, (009) -. [] H. C. Sohn and K. T. Hong, An adaptve LQG contol fo sem-actve suspenson systems, Intenatonal Jounal Vehcle Desgn, () (00) 09-. [] M. Bglabegan, W. Melek and F. Golnaagh, Intellgent contol of vehcle sem-actve suspenson system fo mpoved de comfot and oad handlng, Fuzzy Infomaton Pocessng Socety (00) 9-. [9] O. Yoshda and S. J. Dyke, Sesmc contol of a nonlnea benchmak buldng usng smat dampes, Jounal of engneeng mechancs, 0 (00) -9. [0] Y. Km, R. Langa and S. Hulebaus, Sem-actve nonlnea contol of a buldng wth a magnetohelogcal dampe system, Mechancal Systems and Sgnal Pocessng, (009) 00-. [] M. Ahmadan and C. A. Pae, A quate ca expemental analyss of altenatve sem-actve contol methods, Jounal of Intellgent Mateal Systems and Stuctues, () (000) 0-. [] J. S. Chou and T. M. Lu, Usng fuzzy logc contolle and evolutonay genetc algothm fo automotve actve suspenson system, Intenatonal Jounal of Automotve Technology, 0 () (009) 0-0. [] J. Sun and Q. Yang, Modelng and ntellgent contol of vehcle actve suspenson system, IEEE Intenatonal confeence on RAM (00) 9-. [] S. Yldm and I. Esk, Vbaton analyss of an expemental suspenson system usng atfcal neual netwoks, Jounal of Scentfc & Industal Reseach, (009) -9. [] V. S. Atay and P. N. Roschke, Neuo-fuzzy contol of alca vbatons usng sem actve dampes, Compute- Aded Cvl and Infastuctue Engneeng, 9 (00) -9. [] S. H. Sadat, M. A. Shooedel and A. D. Panah, Desgnng a neuo-fuzzy contolle fo a vehcle suspenson system usng feedback eo leanng, Jounal of mechancs and aeospace, () (00) -. [] M. T. Hagan and M. B. Menhaj, Tanng feedfowad netwoks wth the Maquadt algothm, IEEE Tans. on Neual Netwoks, () (99) [] A. Khajekaamodn and H. Haj-Kazem, A. Rowhanmanesh and M. R. Akbazadeh, Sem-actve contol of stuctues usng neuo-nvese model of MR dampes, Poc. of Fst Jont Congess on Fuzzy and Intellgent Systems, Mashhad, Ian (00) 9-0. [9] A. Saafan, S. H. Zaeh, A. A. A. Khayyat and A. Zabhollah, Neuo-fuzzy contol stategy fo an offshoe steel jacket platfom subjected to wave-nduced foces usng magnetoheologcal dampes, Jounal of Mechancal Scence and Technology, () (0) -. [0] Y. W. Yun, S. M. Lee and M. Pak, A study on the effcency mpovement of a passve ol dampe usng an MR accumulato, Jounal of Mechancal Scence and Technology, () (00) 9-0. [] Z. D. Xu and Y. Q. Guo, Neuo-fuzzy contol stategy fo eathquake-excted nonlnea magnetoheologcal stuctues, Sol Dynamcs and Eathquake Engneeng, (00) -. Seyed Hamd Zaeh was bon n 9 and eceved hs B.Sc. degee fom the Techncal Faculty of Imam Hossen Unvesty, Tehan, Ian, n 00 and hs M.Sc. degee fom School of Scence and Engneeng of Shaf Unvesty of Technology, Ian, n 0, all n mechancal and mechatoncs engneeng. Ths autho became a membe of IEEE n 0. Hs eseach nteests ae mechatoncs system, smat mateals and stuctues, automatc contol, and ntellgent contol. He has one book n the feld of ol and gas wth the ttle of Stoage Tank Desgn, (00) and one book chapte (0). He also publshed fou jounal and twelve ntenatonal confeence papes. M. Zaeh s also a membe of Amecan Socety of Mechancal Engnees (ASME) and Ianan Socety of Mechancal Engnees (ISME). He eceved oal pesentaton awads at the nd, d and th symposum n eseach week n 009, 00, 0 and techncal pesentaton n Sngapoe 009, espectvely.