RESISTANCE ESTIMATION OF A PWM-DRIVEN SOLENOID

Transcription

1 Inernaional Journal of Auomoive echnology, Vol. 8, No. 2, pp (2007) Copyrigh 2007 KSAE /2007/ RESISANCE ESIMAION OF A PWM-DRIVEN SOLENOID H. G. JUNG 1),2)*, J. Y. HWANG 1), P. J. YOON 1) and J. H. KIM 2) 1) Mando Cenral R&D Cener, Gomae-dong, Giheung-gu, Yongin-si, Gyeonggi , Korea 2) School of Elecrical and Elecronic Engineering, Yonsei Universiy, Seoul , Korea (Received 26 Sepember 2006; Revised 26 February 2007) ABSRAC his paper proposes a mehod ha can be used for he resisance esimaion of a PWM (Pulse Widh Modulaion)-driven solenoid. By using esimaed solenoid resisance, he PWM duy raio was compensaed o be proporional o he solenoid curren. he proposed mehod was developed for use wih EHB (Elecro-Hydraulic Braking) sysems, which are essenial feaures of he regeneraive braking sysem of many elecric vehicles. Because he HU (Hydraulic Uni) of mos EHB sysems performs no only ABS/CS/ESP (Elecronic Sabiliy Program) funcions bu also service braking funcion, he possible duraion of coninuous solenoid driving is so long ha he generaed hea can drasically change he level of solenoid resisance. he curren model of he PWM-driven solenoid is furher developed in his paper; from his a new resisance equaion is derived. his resisance equaion is solved by using an ieraive mehod known as he FP (fixed poin heorem). Furhermore, by aking he average of he resisance esimaes, i was possible o successfully eliminae he effec of measuremen noise facors. Simulaion resuls showed ha he proposed mehod conained a sufficien pass-band in he frequency response. Experimenal resuls also showed ha adapive solenoid driving which incorporaes resisance esimaions is able o mainain a linear relaionship beween he PWM duy raio and he solenoid curren in spie of a wide variey of ambien emperaures and coninuous driving. KEY WORDS : Esimaion algorihm, Elecrical driving, Elecro-hydraulic sysem, Auomoive braking conrol 1. INRODUCION he EHB (Elecro-Hydraulic Braking) sysem is a weype BBW (Brake By Wire) sysem and is expeced o make up he nex generaion of braking sysems (Reuer e al., 2003; Peruccelli e al., 2003). EHB sysems are generally composed of a BPU (Brake Pedal Uni), a HU and an ECU (Elecronic Conrol Uni). Figure 1(a) depics he hydraulic circui as found in EHB sysems. During normal operaions, an isolaion valve disconnecs he hydraulic circui beween he BPU and he HU and he HU conrols wheel pressures according o he driver s inenions. he BPU absorbs brake oil obained by brake pedal operaions and makes drivers feel like hey are using a convenional braking sysem. When here are sysem failures or siuaions when he sysems are urned off, he hydraulic circui beween he BPU and he HU can provide a mechanical backup line o ensure emergency braking. Alhough EHB sysems canno compleely eliminae hydraulic componens, hey can separae wheels from brake pedals and provide perfec acive braking (Reuer e al., 2003). Acive braking, which can be defined as a braking operaion ha does no require pedal operaions *Corresponding auhor. hgjung@mando.com by drivers, is an imporan requiremen of new sysem funcions such as ACC (Adapive Cruise Conrol), CA (Collision Avoidance) and auonomous vehicles. Addiionally, because he regeneraive braking sysems of EV (Elecric Vehicles)/HEV (Hybrid Elecric Vehicles) require complee separaion beween brake pedals and wheels, EHB are hough o provide a unique pracical soluion. Almos every HEV has already adoped an EHB sysem o implemen regeneraive braking, an essenial way of achieving high fuel efficiency (Sakai, 2005; Nakamura e al., 2002; Gao and Ehsani, 2002). Consequenly, he growh of he HEV marke means an increase in he populariy of EHB sysems. he Prius, which was launched successfully in he Japanese marke as he firs massproduced hybrid vehicle in 1997, also adoped an EHB sysem (Sakai, 2005). he Prius was an immediae success and more han 127,000 unis were sold in six years (Yaegashi, 2005). EHB sysems generally require a robus mehod for solenoid curren conrol. In general, EHB HU sysems use poppe-ype valves acuaed by solenoids, and he solenoid curren is conrolled by a PWM-driving mehod. Figure 1(c) shows a ypical model of PWM driving. he poppe-ype valve acuaed by a PWM-driven solenoid can reduce he price of he acuaor and he oal sysem 249

2 250 H. G. JUNG, J. Y. HWANG, P. J. YOON and J. H. KIM Figure 1. EHB sysem valve conrol. (Verseveld and Bone, 1997). Some forces relaed o he poppe-ype valve include spring forces, hydraulic forces and magneic forces, as shown in Figure 1(b). By conrolling he solenoid curren, he ECU can conrol he magneic force and ne force imposed on he valve. Because he solenoid resisance changes over a wide range, e.g. ~70% ~ 100%, he relaionship beween he PWM duy raio and he solenoid curren varies as solenoid resisance changes. herefore, an esimaion of solenoid resisance is ineviable for successful solenoid curren conrol and pressure conrol o compensae for he PWM duy raio. Because EHB HU sysems implemen a CBS (Convenional Braking Sysem) funcion, he duraion of coninuous solenoid driving can be exended drasically and he compensaion mehod used by he ESP (Elecronic Sabiliy Program) canno be applied o he EHB sysem. In erms of regeneraive braking, he duraion of coninuous driving becomes longer and he acuaion srengh becomes sronger because he necessary wheel pressures have o change coninuously, even if he driver mainains heir brake pedal posiion (Nakamura e al., 2002). his paper inroduces a model for he average solenoid curren driven by he PWM mehod. If he frequency of he PWM signal is sufficienly high wih respec o he ime consan of he solenoid, he average curren of he solenoid can be modeled as if he solenoid was driven by a DC volage source proporional o he PWM duy raio. Wih his simple model, an equaion can be derived ha explains he solenoid curren wih he given iniial curren, he volage of he power source and he PWM duy raio. Because he solenoid curren is measured wih an embedded circui, he solenoid curren equaion refers o solenoid resisance. Unforunaely his equaion canno be solved direcly. By using an ieraive mehod, i.e. a FP (Fixed Poin heorem), he solenoid resisance equaion can be solved economically. Because he equaion is simple and can be calculaed in erms of ineger operaions, he consumpion of compuaional power is limied o a very small amoun. aking he average of he calculaed solenoid resisances over a fixed duraion eliminaes he effec of modeling errors and measuremen noise because averaging can generally eliminae Gaussian noise. Wih he esimaed solenoid resisance, he PWM duy raio can be compensaed o make he relaionship beween he PWM duy raio and he solenoid curren linear and saic. he period of he curren conrol of he EHB sysem is shor wih respec o changes in solenoid resisance. herefore, an adapive conrol mehod ha incorporaes he esimaion of slow varying parameers, i.e. solenoid resisance, is more efficien and effecive han direc conrol of a fas varying curren. Wih simple ineger calculaions and averaging, he proposed mehod economically achieves a robus curren conroller. Simulaion refers o he frequency response of he proposed mehod which shows ha he proposed mehod has a pass-band wide enough o immediaely follow changing solenoid resisances even in he wors siuaions. Experimenal resuls confirmed ha he suggesed mehod is very robus o changes of solenoid resisance and measuremen noise facors. 2. NEW REQUIREMENS EHB sysems mus be able o esimae solenoid resisance while driving solenoids because he duraion of coninuous driving can be long and he solenoid resisance variaions caused by generaed hea are oo large o be ignored Previous Mehod Used by he ESP Alhough he ESP uses he same solenoid driving mechanism, i can ignore he variaions of solenoid resisance because he duraion of solenoid driving is shor. Before he ESP sars o drive he solenoid, he ESP esimaes he solenoid resisance and assumes ha his resisance will

3 RESISANCE ESIMAION OF A PWM-DRIVEN SOLENOID 251 erms o adjus PWM duy raios need complicaed hardware circuis. Also, he frequency characerisics of arge currens have o be limied o low frequency ranges because of heavy low-pass filers used for curren measuremen. Figure 2. he ESP mehod. remain consan during solenoid driving. Figure 2 shows he circui and formula used in he ESP mehod. Before he sar of solenoid driving, he ESP urns off he power source swich (FSR). When he ESP urns on he solenoid driving FE (Field Effec ransisor), curren flows from he regulaed DC power V CC o he ground. he ESP measures wo volages, i.e. V S1 and V S2, in order o esimae he forward volage of he diode. he flowing curren can be calculaed wih he V CC -V S1 and he R S. Wih he curren and he V S2, he solenoid resisance R C can be calculaed (Choi e al., 2003). However, because his mehod can be used only when he power source swich is urned off, his mehod canno be applied o EHB sysems Some Problems of Alernaive Soluions he firs soluion is a able-based mehod; his is he usual procedure in auomoive conrol o compensae for parameer variance. However, consrucing a able for he esimaion of solenoid resisance requires hree indexes: he curren difference, he power source volage, and he PWM duy raio. A able like his would be oo large o be embedded ino a pracical ECU. A sofware-based curren feedback conroller is he second soluion. However, because his mehod requires a shorer conrol period and a larger amoun of compuaional power, i seems o go beyond he scope of a 16- bi micro-conroller. A curren feedback circui is he hird soluion because his kind of circui is widely used in solenoid curren conrol sysems. However, his mehod requires a DAC (Digial o Analog Conversion) pors: hese are no commonly used in auomoive ECU sysems o esablish arge currens. Furhermore, curren feedback circuis which compare arge currens wih measured currens hen urn he driving FE sysems on or off are supposed o generae random driving signals a a low frequency range; hese signals can cause EMC (Elecro-Magneic Compaibiliy) problems. PI conroller circuis which use error 2.3. New Requiremens he new requiremens of solenoid resisance esimaion for EHB sysems can be summarized as follows: New mehods mus be able o esimae solenoid resisance while driving solenoids. New mehods mus be able o cope wih noise facors in he measuremen of solenoid currens. Addiional compuaional loads mus be small. Addiional memory consumpion needs mus be small. 3. MODEL FOR AN AVERAGE SOLENOID CURREN In his secion, a model for an average solenoid curren is derived. Because a solenoid shows inducance properies as well as resisance properies, a solenoid curren can be modeled by firs order dynamics Differenial Equaion of he Solenoid Curren Figure 3 shows a solenoid driving circui. he solenoid is replaced wih a series connecion of he resisance propery R and he inducance propery L. he FE is modeled as an ideal swich driven by a PWM signal and he diode is modeled as a unidirecional swich. During he PWM-ON sae, he solenoid is driven by a DC volage source V BA as shown in Figure 3(a); Equaion (1) is se up by using he KVL (Kirchhoff Volage Law) (Vaughan and Gamble, 1996). Here, i() denoes he solenoid curren and is he ime consan of he solenoid. Equaion (2) is he soluion of Equaion (1) and I S refers o a sauraed curren. he solenoid curren exponenially increased from he iniial curren I O o he sauraed curren I S. Figure 3. Equivalen circuis for he PWM-ON/OFF saes.

4 252 H. G. JUNG, J. Y. HWANG, P. J. YOON and J. H. KIM Ri+L di ----=V d BA - i()=is +Io e =Io+ ( Is Io) 1 e where, Is= V BA , = L R R -- During he PWM-OFF sae, he freewheeling diode, or he flyback diode, was urned on and i made a loop wihou a volage source, as shown in Figure 3(b). Ignoring he forward volage drop of he freewheeling diode, Equaion (3) was se up by he KVL. Equaion (4) is he soluion of Equaion (3). he solenoid curren decreased exponenially from he iniial curren I O o zero. Ri+L di ----=0 d i()=io e (1) (2) (3) (4) 3.2. Sequence of he Iniial Curren Generally, when PWM signal drives he solenoid FE, he PWM-ON sae and he PWM-OFF sae are repeaed in urn. he final solenoid curren of he previous sae becomes he iniial curren of he ongoing sae. Equaions (2) and (4) are generalized as a summaion of he geomeric series. I O+ (n) denoes he iniial curren of he n h PWM-ON sae and I O (n) denoes he iniial curren of he n h PWM- OFF sae as depiced in Figure 4. δ denoes he PWM duy raio and has a value from 0 o 1. is he period of he PWM driving signal. he iniial curren was calculaed recursively from he firs period, as shown below: I 0+ ( 1)=0 I 0 ( 1)=Is Ise I 0+ ( 2)=Ise ( 1 δ) I 0 ( 2)=Is Ise I 0+ ( 3)=Ise ( 1 δ) δ δ -- +Ise -- δ δ +Ise ( ) Figure 4. Sequence of he iniial curren. 2 I 0 ( 3)=Is Ise I 0+ ( 4)=Ise +Ise ( 1 δ) δ +Ise -- δ Ise -- + ( 1 δ) Ise 2 + ( 1 δ) δ Careful observaion shows ha he n h iniial curren is he summaion of he geomeric series. Equaion (5) defines he iniial curren of he n h PWM-ON sae and Equaion (6) defines he iniial curren of he n h PWM- OFF sae. I 0+ I 0 ( n 1) ( 1 δ) ( n)=is e -- e e -- 1 e δ ( n)=is 1 e e -- 1 e n (5) (6) 3.3. Average Solenoid Curren he inegraion of he solenoid curren during he n h PWM-ON sae was calculaed by inegraing Equaion (1) and Equaion (5) from 0 o δ. I ON = δ V BA R V BA e R Similarly, he inegraion of he solenoid curren during he n h PWM-OFF sae was calculaed by inegraing Equaion (4) and Equaion (6) from 0 o (1 δ). I OFF = ( 1 δ) 0 By dividing he summaion of I ON and I OFF by he PWM period, he equaion of he average solenoid curren a ime index n was derived as in Equaion (7). By replacing he discree index n wih he coninuous ime, he equaion of he average solenoid curren a ime was derived as in Equaion (8). If he iniial curren of he firs sae was no zero, a erm reflecing he iniial curren which decreased exponenially, had o be added. + V ( 1 δ) BA e -- e 1 e R δ 1 e V BA 1 e R i( n)= V BA δ 1 + R δ e i()= V BA δ e R δ δ δ ( n 1) e -- 1 e n e -- 1 e n e e d d (7) (8)

5 RESISANCE ESIMAION OF A PWM-DRIVEN SOLENOID 253 Figure 6. Behavior of an average solenoid curren. Figure 5. Coefficien funcion Approximaion of he Average Solenoid Curren By applying realisic values o he parameers, Equaion (8) was approximaed in a very simple form. he coefficien of he exponenial erm of Equaion (8) is a funcion of he ime consan, he PWM period and he duy raio δ. he inroducion of a new variable x made he coefficien become a funcion of x as shown below: x= δ -----, e δ δ ( 1 e x ) x Figure 5(a) shows a graph of he coefficien funcion when x ranged from 6 o 6. If he range of x was 0~0.1, he coefficien funcion had a value of 1, as shown in Figure 5(b). Consequenly, if he PWM period was sufficienly smaller han he solenoid ime consan, x had a very small value and he coefficien funcion was approximaed o he consan 1. In our case, he solenoid ime consan was and he PWM frequency is 20 khz (= ). herefore, x < and coefficiens were reaed as 1. By replacing he coefficien in Equaion (8) wih he approximaion value 1, he approximaed equaion of average solenoid curren was derived as shown in Equaion (9). his equaion shows ha he average solenoid curren had he same ime consan as he insan solenoid curren. Furhermore, i shows ha he solenoid curren exponenially changed and converged o a sauraed curren which was proporional o he PWM duy raio. If he iniial curren of he firs sae was no zero, Equaion (9) was modified as shown in Equaion (10). Hereafer, Equaion (10) is used o esimae he solenoid resisance. V BA δ R e δ Io 1 e i()= V BA δ 1 e R i()= V BA δ + Io R =Io+ V BA R (9) (10) In Figure 6, he upper graph corresponds o he solenoid curren ha was driven by a DC volage source. he lower graph shows he solenoid curren driven by a PWM signal wih a 30% duy raio. Alhough he lower graph exponenially increased and decreased according o he PWM-ON or OFF saes, he envelope of he lower graph was he same as ha of he upper graph. his shows ha he lower graph converged o a sauraed curren proporional o he PWM duy raio. 4. ESIMAION OF SOLENOID RESISANCE Because he equaion of solenoid resisance canno be solved in a closed form, an FP (Fixed Poin heorem)- based ieraive mehod was inroduced (Burden, 2001). his secion shows ha his mehod proved robus wih respec o emperaure changes, measuremen noise facors and modeling errors Approximaion of Solenoid Resisance Equaion Solenoid currens measured a every conrol period C were defined as in Equaion (11), which was acquired by replacing in Equaion (10) wih C. I O denoed he iniial curren of he ongoing conrol period. Solenoid resisance R saisfied he equaion, which unforunaely could no be solved in a closed form. i( c)= V c BA δ + V Io BA δ (11) R R e Because an exponenial funcion would have been unsuiable for implemenaion in an embedded environmen, i was approximaed wih a polynomial expression. Also, because he resisance of he solenoid used in he EHB sysem ranged beween 4~8 Ω, he aylor series (cenered a R=6) was seleced. he firs order polynomial expression yields many errors. he second order polynomial expression successfully approximaed he exponenial erm. his expression is depiced in Figure 7 ( C =1 msec, L=15 mh). Neverheless, because Equaion (12) was he hird order polynomial expression in erms of R and replaced he exponenial erm wih he second order aylor series, i could no be solved in a closed form, eiher. he resisance R erm conained in he ime consan was explicily expressed.

6 254 H. G. JUNG, J. Y. HWANG, P. J. YOON and J. H. KIM Figure 7. aylor approximaion of he exponenial erm. i( c)= V BA δ + Io V BA δ (12) R R 4.2. he Ieraive Mehod In general, he ieraive mehod requires a large compuaional load. However, because in his case he ieraion period was he same as ha of he conrol period C, he required compuaional load was smaller. R= V BA δ (13) ic ( ) + Io R V BA δ ic ( ) ic ( ) ( R R ) R[ n]= V BA[ n] δ[ n] in [ 1] Rn [ 1] (14) in [ ] in [ ] V BA [ n] δ[ n] in [ ] ( p 1 R 2 [ n 1] p 2 R[ n 1]+p 3 ) where, p 1 = , p 2 = , p 3 = o apply he FP, Equaion (12) was ransformed ino an ieraive equaion form, Equaion (13). Equaion (14) characerized he corresponding difference equaion, represening a clear ime sequence of relaed variables. δ[n] denoed he PWM duy raio of he n h conrol period and V BA [n] denoed he measured volage of he power source a he beginning of he n h conrol period. i[n 1] represened he iniial curren of he ongoing conrol period and was measured a he beginning of he n h conrol period. i[n] denoed he measured curren a he end of he n h conrol period. Consequenly, he n h resisance R[n] was calculaed wih he n-1 h resisance R[n 1], he measured currens, he measured power source volage, and he applied PWM duy raio. Equaion (14) was calculaed only once per each conrol period. A fixed poin refers o he cross poin beween he LHS (Lef Hand Side) graph and he RHS (Righ Hand Side) graph of Equaion (13). As R was calculaed repeaedly according o Equaion (14), R approached a rue value as depiced in Figure 8. In he simulaion, he rue value of Figure 8. Calculaed R approaching he fixed poin. Figure 9. Fixed poins of various iniial currens and duy raio. solenoid resisance was 5.2 Ω; i was assumed ha he iniial curren was 1.2 A and he PWM duy raio was 0.7. In pracical siuaions, a pressure conroller generally ses he PWM duy raio a every conrol period. According o he PWM duy raio, he iniial curren and he final curren varied. Furhermore, he volage of he power source was no consan. In fac, he graphical shape of Equaion (8) RHS changed a every conrol period. However, because he fixed poin was he soluion of he resisance equaion, in any siuaion he poin was fixed and ensured he convergence of solenoid resisance esimaion. Figure 9 shows ha in spie of various iniial currens and PWM duy raios, he fixed poin was mainained. Figure 10 is he resul of simulaion in which he PWM duy raio represens a random number beween 0.1 and 1. Figure 10 shows ha he calculaed solenoid resisance rapidly approached he rue value. In pracical siuaions, he ESP mehod can be used o esimae he iniial value and he errors can be confined o a low level.

7 RESISANCE ESIMAION OF A PWM-DRIVEN SOLENOID 255 Figure 11. Relaionship beween he resisance PDF and he curren PDF. Figure 10. Simulaion resuls wihou noise facors Consideraion of Noise Facors and Modeling Errors aking he sample mean of he calculaed solenoid resisance made he proposed mehod robus wih respec o measuremen noise facors and modeling errors. Curren measuremens were supposed o produce addiional noise facors. Addiionally, modeling of he average solenoid curren conained very few errors which were confined by a sufficienly high PWM frequency. In general, i is possible o assume ha noise facors and modeling errors are WGN (Whie Gaussian Noise). A graphical inerpreaion of Equaion (11) reveals he effecs of noise facors and modeling errors. he LHS graph of Equaion (11) is a line parallel wih he x-axis; he RHS graph of Equaion (11) is a smoohly decreasing curve as depiced in Figure 11. Alhough i was impossible o calculae he resisance PDF (Probabiliy Disribuion Funcion) from he curren measuremen PDF, we assumed ha he mapping from he curren measuremen PDF o he resisance PDF was approximaely linear. herefore, he resisance PDF was supposed o be WGN and he sample mean of he calculaed solenoid resisance was supposed o be a good esimaion of solenoid resisance. Simulaion resuls showed ha if he WGN was added Figure 12. Simulaion resuls wih noise facors. o he curren measuremen as shown in Figure 12(a), he calculaed solenoid resisance also conained a WGN as shown in Figure 12(b). Even if he magniude of he curren noise facor was large ( 50 ma~50 ma), he calculaed solenoid resisance rapidly approached he rue value and he sample mean was a good esimaion of solenoid resisance, as shown in Figure 12(c) Frequency Response he simulaion resuls showed ha he proposed mehod conained sufficien pass-band in he frequency responses. o verify wheher he proposed mehod was able o immediaely esimae he changing solenoid resisance, he frequency response of he proposed mehod was

8 256 H. G. JUNG, J. Y. HWANG, P. J. YOON and J. H. KIM Figure 14. Frequency response of he proposed mehod. characerisics as a ypical low-pass filer. he cuoff frequency, or 3 db poin, was locaed a abou 60 Hz. herefore, he response of he proposed mehod would be sufficienly fas o cover he whole range of realisic siuaions. 5. COMPENSAION OF HE PWM DUY RAIO Figure 13. Experimenal resuls for frequency response measuremen. measured by simulaion. While changing he solenoid resisance in a sinusoidal waveform, he waveform of he esimaion resul was recorded. Such measuremens were repeaed while changing he sinusoidal frequency from 1 Hz o 500 Hz by 1 Hz seps, considering he conrol period as 1 msec. he sinusoidal offse was se o he resisance of normal emperaure, i.e. 5.2 Ω, and he sinusoidal ampliude of inpu resisance was se o a sufficienly large value, i.e. 2 Ω, o cover he operaional emperaure range of our EHB sysem. Because he waveform of esimaed solenoid resisance was supposed o have he same frequency as he inpu resisance sinusoid, he ML (Maximum Likelihood) esimaion of he sinusoidal ampliude was able o measure he ampliude of he esimaion resuls (Key, 1993). Figure 13(a) shows he waveform of esimaed resisance when he inpu resisance was 60 Hz. Figure 13(b) shows he reconsruced sinusoid in which he parameer was esimaed by he ML esimaion. Wih he sinusoidal ampliude of esimaed resisance, he frequency response was consruced as shown in Figure 14. he acquired frequency response had he same Wih he esimaion of solenoid resisance, he PWM duy raio was compensaed o achieve a linear relaionship beween he PWM duy raio and he solenoid curren. he compensaion mehod also used he measured volage of he power source. he wheel pressure conroller sen a PWM duy raio o a hardware conroller; he hardware conroller hen compensaed for he PWM duy raio by considering he variaion of solenoid resisance and he power source volage, as shown in Equaion (15). Here, he reference volage was 12 V and he reference resisance was he solenoid resisance of normal emperaure. δ modified [ n]= Ṽ BA V Rˆ [ n ] δ[ n] BA [ n] R (15) where, Ṽ BA : reference volage, R : reference resisance A he beginning of a given conrol period, he hardware conroller calculaed he solenoid resisance R[n] by using Equaion (14). I accumulaed he calculaed solenoid resisance over a fixed duraion, e.g. one second, and updaed he esimae of solenoid resisance Rˆ [ n] once in a fixed duraion, as shown by Equaion (16). Because he accumulaed resisance was sufficien o calculae he solenoid esimae Rˆ [ n], i was no necessary o mainain he calculaed solenoid resisance sequence. herefore, required memory was only he accumulaion of resisance esimaes. Simulaneously, using he sample mean of

9 RESISANCE ESIMAION OF A PWM-DRIVEN SOLENOID 257 Figure 15. Solenoids insalled on he HECU. calculaed solenoid resisances eliminaed he effec of measuremen noise facors. Rˆ [ n]= 1 N --- n N Rk [ ] k = n (16) herefore, he proposed mehod saisfied he aforemenioned new requiremens: Wih he model of an average solenoid curren, solenoid resisance was calculaed in spie of coninuous driving. By updaing he esimae wih he sample mean of he calculaed solenoid resisances, he effecs of measuremen noise facors and modeling errors were eliminaed. he equaion of solenoid resisance was a simple polynomial expression which was implemened in ineger operaions. Furhermore, because he equaion was calculaed only once every conrol period, addiional compuaional power requiremens were very small. Because he proposed mehod mainained only an accumulaion of he calculaed solenoid resisances, addiional memory requiremens were very small. 6. EXPERIMENAL RESULS hrough several ess, i was confirmed ha he proposed mehod represens a useful way o make hardware conrollers robus wih respec o emperaure changes and noise facors Seing of Experimens he micro-conroller used in he experimens was he Freescale 6812DP256, a fixed-poin 16-bi micro-conroller which operaed a 25 MHz. he PWM frequency was 20 khz and he conrol period was 1msec. he circui for curren measuremen consised of a shun resisor and an insrumenal OP-AMP wih an 80 db CMRR (Common Mode Rejecion Raio). Figure 15 shows he solenoids insalled on he EHB HECU (Hydraulic Elecronic Conrol Uni). Figure 16. Invariance o ambien emperaures Invariance o Ambien emperaures In consan emperaure chambers, solenoids are driven by PWM duy raio es paerns under hree differen emperaure condiions: 40ºC, 25ºC, 110ºC. Figure 16(a) shows he resuls of an uncompensaed case. I can be observed ha he paerns of he solenoid curren differed from each oher. Figure 16(b) shows he resuls of he compensaed case. he paerns of he solenoid curren were all he same in spie of differen ambien emperaures Invariance o Coninuous Driving While driving a solenoid wih a riangular PWM duy raio paern, he effec of resisance change caused by generaed hea was invesigaed. Figure 17(a) shows he resul of he uncompensaed case. As he ime passed by, he solenoid curren decreased smoohly. Figure 17(b) shows he resuls of he compensaed case. he paern of

10 258 H. G. JUNG, J. Y. HWANG, P. J. YOON and J. H. KIM and he solenoid curren in spie of various ambien emperaures and resisance changes caused by generaed hea. herefore, his paper has presened hree conribuions. Firs, a simplified curren model of he PWMdriven solenoid and is condiions, i.e. PWM frequency requiremens, were derived. Second, he resisance of he PWM-driven solenoid was esimaed based on he fixed poin heorem. hird, he adapive conrol mehod incorporaing he esimaion of a slow varying parameer, i.e. solenoid resisance, was more efficien and effecive han direc conrol of a fas varying curren. REFERENCES Figure 17. Invariance o coninuous driving. he solenoid curren did no change in spie of coninuous driving. 7. CONCLUSION his paper proposed a novel resisance esimaion mehod of a PWM-driven solenoid. Simulaion resuls showed ha he proposed mehod was robus o noise and sufficienly fas o follow changing resisance. Experimenal resuls showed ha he proposed mehod was able o ensure a linear relaionship beween he PWM duy raio Burden, R. L. and Faires, J. D. (2001). Numerical Analysis. Brooks/Cole. CA. USA. Choi, S. H., Lee, J. G. and Hwang, I. Y. (2003). New generaion ABS using linear flow conrol and moor speed conrol. SAE Paper No Gao, Y. and Ehsani, M. (2002). Elecronic braking sysem of EV and HEV-Inegraion of regeneraive braking, auomaic braking force conrol and ABS. SAE Paper No Key, S. M. (1993). Fundamenals of Saisical Signal Processing, I: Esimaion heory. Prenice Hall. New Jersey Nakamura, E., Soga, M., Sakai, A., Oomo, A. and Kobayashi,. (2002). Developmen of elecronically conrolled brake sysem for hybrid vehicle. SAE Paper No Peruccelli, L., Velardocchia, M. and Sornioi, A. (2003). Elecro-hydraulic braking sysem modeling and simulaion. SAE Paper No Rueer, D. F., Lloyd, E. W., Zehnder, J. W. and Ellio, J. A. (2003). Hydraulic design consideraions for EHB sysems. SAE Paper No Sakai, A. (2005). he regeneraive braking sysem. Auo- echnology, 5, February, Vaughan, N. D. and Gamble, J. B. (1996). he modeling and simulaion of a proporional solenoid valve. J. Dynamic Sysems, Measuremen and Conrol, rans. ASME, 118, March, Verseveld, R. B. and Bone, G. M. (1997). Accurae posiion conrol of pneumaic acuaor using on/off solenoid valves. IEEE/ASME rans. Mecharonics 2, 3, Yaegashi,. (2005). he hisory of hybrid echnology. Auoechnology, 5, February, 8 12.