eice ongituin motion moeing for noniner contro Ki E joub ou Giri Hmi Oui Luc Dugr tim Zr Coui To cite tis ersion: Ki E joub ou Giri Hmi Oui Luc Dugr tim Zr Coui. eice ongituin motion moeing for noniner contro. Contro Engineering Prctice Eseier 0 0 pp.69-8. <0.06/j.conengprc.0.09.005>. <-006889> HAL I: -006889 ttps://.rcies-ouertes.fr/-006889 Submitte on 7 No 0 HAL is muti-iscipinry open ccess rcie for te eposit n issemintion of scientific reserc ocuments eter tey re pubise or not. Te ocuments my come from tecing n reserc institutions in rnce or bro or from pubic or prite reserc centers. L rcie ouerte puriiscipinire HAL est estinée u épôt et à iffusion e ocuments scientifiques e nieu recerce pubiés ou non émnnt es étbissements enseignement et e recerce frnçis ou étrngers es bortoires pubics ou priés.
eice Longituin otion oeing for noniner contro K. E joub c. Giri H. Oui c L. Dugr b.z. Coui c GREYC Lb UR CNRS Uniersity of Cen Bsse-Normnie Cen rnce b GIPSA Lb UR CNRS INPG Grenobe rnce c ENSET Uniersite e Rbt-Ag Rbt rocco Abstrct Te probem of moeing n controing eice ongituin motion is resse for front ee propee eices. Te cssis ynmics re moee using reent funment s tking into ccount eroynmic effects n ro sop rition. Te ongituin sip resuting from tire eformtion is cpture troug Kiencke s moe. A igy noniner moe is tus obtine n bse upon in eice ongituin motion simution. A simper but neerteess ccurte ersion of tt moe proes to be usefu in eice ongituin contro. or security n comfort purpose te eice spee must be tigty regute bot in cceertion n eceertion moes espite unpreictbe cnges in eroynmics efforts n ro sop. To tis en noniner controer is eeope using te Lypuno esign tecnique n formy son to meet its objecties i.e. perfect cssis n ee spee regution. Keyors eice ongituin contro ongituin sip tire Kiencke s moe spee contro Lypuno stbiity.. INTRODUCTION eice ongituin motion contro ims t ensuring pssenger sfety n comfort. It is n importnt spect in ynmic cobortie riing i.e. en mutipe eices sou coorinte to sre ro efficienty ie mintining sfety. In tis respect seer orks e been eote to t is commony referre to ptie cruise contro tt consists in mintining specifie ey beteen eices Ionnou n Cien 99; oon et. 009. Different contro tecniques e been use in tese orks incuing iner n ptie contro You et. 009 genetic fuy contro Poursm n onteri 008 siing moe contro Ling et. 00; Noueiere n mmr 007 n sceuing gin contro inoing PIDs Ren et. 008. Hoeer most preious orks on ongituin contro ere bse on simpe moes negecting importnt noniner spects of te eice suc s roing resistnce eroynmics effects n ro o. In some stuies te controer performnces ere not formy nye Ren et. 008. In Ymk et. 007 ongituin eice contro s been stuie focusing on torque mngement for inepenent ee rie. It is ort noticing tt in preious stuies on ongituin eice contro te contro esign s been bse on simpe moes not ccounting for tire-ro interction.
In te present stuy te probem of ongituin eice contro is reisite for front ee propee eices focusing on spee regution. Te im is to esign controer tt is be to tigty regute te cssis n ee eocities in bot cceertion n eceertion riing moes espite cnging n uncertin riing conitions. Tis probem s not been et it preiousy. A furter originity of te present pper is tt te contro esign reies upon more compete moe tt ccounts for most eice noniner ynmics incuing tire-ro interction. Tt is te stuy incues to mjor contributions. irst suitbe contro moe is eeope for te eice ongituin beior. In tis respect rec tt conenient moe is one tt is sufficienty ccurte but remins simpe enoug to be utiibe in contro esign. To meet te ccurcy requirement te moe must ccount not ony for eroynmic penomen but so n especiy for tire-ro friction. oeing te tire/ro contct is quite compe issue inoing mutipe spects reent to tire crcteristics e.g. structure pressure n to enironment fctors e.g. ro o temperture. Seer tire moes e been propose in te iterture e.g. Guo s moe Guo n Ren 000 Pcejk s moe Pcejk n Besseink 997 Dugoff s moe Dugoff n Sege 970 Gim s moe Gim n Nikres 990 Kiencke s moe Kiencke n Niesen 00. In te present ork Kiencke s moe is retine becuse it proes to be goo compromise beteen ccurcy n simpicity. Te oer eice moeing is crrie out ccoring to te bicyce moe principe. In ition to tire equtions te moe incues cssis ynmics equtions obtine from funment ynmics n eroynmics s n incorportes reent prctic prior knoege e.g. te tire ongituin sip is pysicy imite. Te oer eice moe turns out to be combintion of to noniner stte-spce representtions escribing respectiey te cceertion n eceertion ongituin riing moes. Its ig compeity mkes it ry utiibe for contro esign but ue to its ig ccurcy it proes to be quite suitbe for simutor buiing. Tis moe eeopment is one mjor cieement of te present stuy. Te secon contribution is te esign of noniner controer tt ensures gob stbiition n ongituin spee regution uring cceertion/eceertion riing moes. Tis is crrie out using te Lypuno esign tecnique Ki 00 bse on simper but sti ccurte ersion of te boe simution-oriente moe. It is formy proe tt te eeope controer ctuy ciees te stbiity n regution objecties it s esigne to. urtermore it is obsere troug numeric simutions tt te controer is quite robust it respect to uncertinties on enironment crcteristics. Te pper is orgnie s foos: Section II is eote to moeing te cceertion/eceertion eice ongituin beior; te obtine moe is use in Section III to esign controer n to nye te resuting cose-oop system; te controer performnces re iustrte in Section I by numeric simutions. A concusion n reference ist en te pper.
. ODELLING O CHASSIS LONGITUDINAL OTION Ecept for eroynmic forces etern efforts cting on eice re generte t te ee-ro contct. Te unerstning n moeing of te forces n torques eeope t ee-ro contct is essenti for stuying propery te eice ynmics. Tese re iscusse in te fortcoming subsections. In tis respect rec tt te eice motion is compose of to types of ispcements: trnstions ong te y es n rottions roun tese sme es ig.... Kiencke s Tire oeing Te tire is min component of te ee-ro contct s it ensures tree importnt functions Kiencke n Niesen 00: i bering te ertic o n bsorbing ro eformtions; ii proucing ongituin cceertion efforts n contributing to eice brking; iii proucing te require trnsers efforts tt ep te eice turning. Te efforts generte t te ee-ro contct incue ongituin cceertion/eceertion forces ter guiing forces n sef ignment torque. Te effect of tese efforts on te eice beiour is etermine by te tire-ro esion. or sm o ritions te ongituin coefficient of friction is crcterie by te fooing rtio: ere t t enotes te ongituin effort n te ertic o. Te rtio is ce ongituin esion or friction coefficient. Te ue of tis coefficient epens on te tire sip resuting from te eformtion of te tire in contct it te ro Kiencke n Niesen 00. Te ongituin sip is crcterie by te coefficient efine s foos:. in cceertion moe i.e. W one s: r W eff W ψ ertic motion Longituin motion Ro φ Pitc Trnsers y motion ig.. Degrees of freeom of eice
. in eceertion moe i.e. W one s: W r eff W ere r eff enotes te effectie ee rius esigntes te ee ngur eocity W spee of te tire-ro contct eformtion occurs en te ee presents sip nge ter force ty. is te is te iner eocity of te ee centre ig.. A simir i.e. te resuting ter sip prouces oeing te efforts t te ee-ro contct s been gien gret e of interest oer te st yers. In tis respect seer tire moes e been eeope it quite ifferent properties e.g. Guo n Ren 000; Pcejk n Besseink 997; Dugoff n Sege 990; Gim n Nikres 990; Kiencke n Niesen 00. or contro esign use te most suitbe tire moe is one tt presents te best ccurcy/simpicity compromise. rom tis iepoint Kiencke s moe turns out to be quite stisfctory coice Kiencke n Niesen 00. Inee tis moe is sufficienty ccurte s it ccounts for te min fetures suc s te ertic o sip nge sip coefficient. On te oter n it s rey proe to be usefu in esigning simpe estimtors for stte ribes ike sip nge n ter efforts You et. 009. In te present pper tis moe i proe to be usefu in contro esign... Kiencke s oe Tis s eeope in Kiencke n Niesen 00 using te Burckrt s etene moe to compute te friction coefficient ongituin/ter sip coefficient representtion of Kiencke s ee moe ere:. Accoringy te tter is function of te combine n te forces cting on te tire. ig. gies scemtic C ep C C ep C C G 5 y 5 y ty t y r eff W ty t ig.. orces ppie on te ee t t W
t ty n tn W W y Acceertion 6 n y tn Deceertion 7 Te oter prmeters n ribes contine in equtions to 7 e te fooing menings: C C : prmeters epening on te ro stte. C C : coefficient epening on te mim riing spee. C 5 : coefficient epening on te ee oe mim o. : gob ongituin/ter sip. y : ter sip. G : spee of te eice centre of grity. : ertic o. In Kiencke s moe te ongituin n ter efforts re respectiey escribe by 8-9: cos C sin 8 t t y C cos sin 9 ty t y ere C t enotes eigting coefficient tking ues in te inter 0.9 0. 95. In ongituin motion te fooing simpifictions re use: i Te tire beiour is inepenent of te eice mim spee. Tt is te term in C is negecte in eqution. ig.. Kiencke s ee moe ii Te tire moe epens inery on te mim o. Ten te term in C5 is negecte in. iii Te eice moes ong strigt ine. Tt is te sip nge Using tese remrks equtions to 9 simpify to: is nu. C C y 0 0 ep C n 0 y t 0 ty 5
.. Roing Resistnce n Aeroynmic Resistnce It is obious tt tire eformtion cuses mecnic osses. Ri eformtions re cuse by te ertic o. ig. sos tt te ertic force istribution ong te eformtion re is not uniform. Te resuting force moes from te centr point I to point octe t istnce. Tis is ce ertic o mim tri n is efine s foos: rr m ig rr m rr Wen te ee strts moing torque is generte proie tt rrm 0. To bring bck te ertic o strengt to te centr point I compensting torque riing motor on te ee. Tis is ce roing resistnce torque n is gien by: rr must be ppie by te rr rr m rr 5.. Aeroynmic Resistnce Aeroynmic resistnce s ntury n impct on energy consumption onbor. ui mecnics s re resorte to epin ir fo roun moing eice. Accoringy eice forms re continuousy reinente to improe eroynmic performnces Poer n Nicstri 000. Aeroynmic efforts come in irect interction it te eice proucing rious forces rg ift n ter ic in turn generte torques y ro n pitc. Te eroynmic rg force is significnt en te eice moes t grn spee. Te eroynmic ift effort increses te roing resistnce becuse te surfce of tire-ro contct gros up improing tire steerbiity. On te oter n te eroynmic ift force ecreses te eice esion ic reuces its stbiity n security iiken n iiken 995. In presence of front-in te eroynmic resistnce is represente by to forces: te eroynmic rg force foos: e n te eroynmic ift force e. Tese re efine s e e C S C S 6 W r eff I rr ig.. Roing resistnce 6
it: : ir ensity epening on tmospere pressure n mbient temperture. C : eroynmic rg coefficient. C : eroynmic ift coefficient. S : front projection re eice. : eice spee. : in spee positie in presence of front-in negtie in presence of rer-in..5. oeing of Wee Submitte to Driing Coupe ig. 5 sos one-ee eice it mss. Te ee is rien by coupe m. Let enotes te inerti resuting from te ee te trnsmission sft n te riing motor. Inoking te ynmic funment principe one gets te fooing eqution: t W m t 8 rr ere eqution 5 s been use to ccount for te roing resistnce. Using te retion one gets: r W r eff eff m t rr 9 Eqution 9 togeter it n 0 escribe te one-ee eice beior. Tis is furter iustrte by te scemtic representtion of ig. 6..6. oe of To-Wee eice it One Driing Wee Longituin n trnsers beiors cn be ssume to be ecoupe en te steering nge is sm. Ten e mke use of te eice symmetry to perform projection of forces on te ongituin is reucing tus te four-ee moe into to-ee. ig. 7 iustrtes te forces inoe in bicyce moe. Te inoe nottions re escribe in Tbe I. No te focus i be me on te bicyce moe consiering tt ony one ee is submitte to motor torque W m. or eice stbiity purpose te front ee is riing. Te istribution of te eice o oer te tires is compute ppying te to ynmic funment s. Doing so one gets te fooing equtions: tf g sin 0 e g cos 0 f r e f f rr tf tr 0 Soing equtions n for f n r yies: 7
Boy of mss m W t W t ig. 5. One-ee eice moe ig. 6. Scemtic representtion of one-ee eice submitte to riing coupe e CoG e f tf f g g r r tr ig. 7. Te forces cting on bicyce type eice f g cos sin g e g e g r g cos sin g e g e g ere f r n f f r. rom eqution 0 it is esiy seen tt te eice spee unergoes te fooing eqution: tf g sin e 5 TABLE I. NOTATIONS O EHICLE LONGITUDINAL ODEL. f : Distnce beteen CoG n te front ee bse m r : Distnce beteen CoG n te rer ee bse m : Distnce beteen te bses of te to ees m : Heigt of te grity centre m e : Aeroynmic rg force N e : Aeroynmic crrying force N g : Grity cceertion m.s - : eice mss kg tf tr : ront n rer ee rie force N f r : Lo on te front n rer ee N θ : Ro sop r 8
9.7. Stte-Spce Representtion of eice Longituin Beior Te equtions obtine so fr re no combine togeter to bui-up stte-spce representtion of te eice ongituin cceertion/eceertion beior. Te eice ongituin ynmics re crcterie by to stte ribes i.e. eice cssis spee n front-ee spee. As te sip coefficient epens on te current riing moe cceertion or eceertion te eice is crcterie by to stte-spce representtions. Ec representtion escribes te eice in te corresponing opertion moe..7.. Stte-Spce Representtion in Deceertion oe Combining 0 n 5-8 one obtins te fooing stte-spce representtion: 7 6 5 ep m ep ep 0 9 8 6 5 ep ep ep 8 7 6 6b ere te rious prmeters re efine in Tbe II. A more compct representtion of 6-b is obtine introucing te nottions: m u n: 7 6 5 ep f ep ep 0 9 8 7 5 ep f ep ep 8 7 6 7b Wit te boe nottions te stte-spce representtion 6-b is gien te fooing usu compct form: f f u 8
: TABLE II. DEINITION O THE ODEL PARAETERS eice prmeters in eceertion : r eff eice prmeters in cceertion C : C : r eff C C : C C k : k : k C : k C : ep C C : k C ep C k 5 : g cos rr k C C 5 : 6 : g cos kc 6 : 7 : SC rr k C C 7 : g cos rr k C C g cos kc SC rr k C C 8 : SC k C 8 : SCkC 9 : g cos kc ep C 9 : g cos kc ep C 0 : r eff SCkC ep C 0 : SC k C ep C sin : g sin g : SC : SC : cos k C C g : g cos k C 5 : SCk C C 6 : SCkC g 7 7 : cos k C ep C : cos k C C g : g cos k C 5 : SCk C C 6 : SCkC : cos k C ep C g 8 : SCkC ep C : 8 SC C ep k C.7.. Stte-Spce Representtion in Acceertion oe Using 0 n 5-8 one gets te fooing stte-spce representtion: m ep 5 6 7 8 9 ep 0 ep 9 0
5 ep ep ep 8 7 6 9b ere te rious prmeters re efine in Tbe II. Let us introuce te nottions: 8 7 6 5 ep f ep ep 0 9 0 5 ep f ep ep 8 7 6 0b ere u n re s in 7-b. Wit tese nottions te stte-spce representtion 0-b is gien te fooing more compct form: f f u.8 Contro- n Simution-Oriente oes for To-Wee eice Wit Singe Driing Wee.8. Contro-Oriente oes.8.. Contro-Oriente oe Accounting or Tire Dynmics Combining te moe-epenent stte spce representtions 8 n one gets singe moe representing te eice in opertion moes i.e. g g u it: f f g f f g b c sign
In ition to eice eroynmics tis moe oes ccount for tire-ro contct effect. urtermore it i proe to be usefu in contro esign Section III..8.. Contro-Oriente oe Ignoring Tire Dynmics To better pprecite te benefit of ccounting of tire-ro contct comprison i be performe in Section I beteen te controer obtine from - n te one obtine from simpifie moe negecting tire-ro contct. Specificy te simpifie moe is obtine etting siing negigence ic immeitey impies tt resistnce ere r 0 tire i.e.. Ignoring so te roing rr one gets inoking te ynmic funments principes for trnstion n rottion: eff m tf tf g sin e 5 e fooing eqution: is efine by 6. It is reiy seen from -5 tt te eice spee unergoes te r r 6 r eff eff sin m g SC eff ere te ifferent prmeters re efine in Tbe. Introuce te fooing nottions: 7 r eff r f 7b eff g sin SC Ten eqution 6 cn be gien by te fooing more compct form ere u m n : u f 8 Te first-orer eqution 8 is simpifie ersion of te secon-orer moe -. As rey mentione bot moes i be bse upon in contro esign n te obtine controers i be compre ter in Section I..8. Simution oe Obious pysic consiertions so tt in re-ife ongituin motions te eice n ee spees re ys quite cose to ec oter. On te oter n tere is no gurntee tt te moe - ensures ys tt. In te net ines te moe i be sigty moifie so tt becomes structur property of it. Doing so one so i iscr ny risk of singurity in 7-b or 0-b by ensuring tt 0 n 0 it:
ep ep 9 Inee it is reiy seen tt te functions n re continuous n: ep ep 0 0 if Te continuity of n ten gurntees te eistence of 0 n 0 suc tt: inf inf 0 0 0 As mtter of fct te sie of n epens on te prmeters i i i. Te boe resut sos tt equtions - re representtie of te eice ongituin beior s ong s te stte ector stys in te fooing iity omin: D IR : A more reistic eice ongituin moe is one tt structury enforces te stte ribes to sty ys in te boe omin. To tis en introuce te ne ribe n its time- eritie. Ten eqution cn be reritten in term of te coupe s foos: u g it: u u g g It is reiy seen from tt: t t 0 u 0 Ten te prior knoege is ccounte for tking te sturte ersion of i.e.: t st t 5 it
t ef t 0 u 0 ere st. enotes te sturtion function efine by: 6 if if st if 7 It is reiy seen from 6 tt te uiiry ribe t unergoes te ifferenti eqution: t u n 0 0 8 Ten it foos from 5 tt see ig. 8. ere r u 9 r st 50 It is esiy cecke using 7 tt r. is te unit rectngur function efine by: 0 if if r if 5 0 Putting te secon eqution in togeter it 8-9 one gets te ne moe: t r u g u it 0 0 n 5 As it ccounts for te prior knoege 7 te ne moe 5 turns out to be more ccurte tn te moes - n 8. Hoeer moe 5 is too compe to be use for contro esign ue e.g. to te nonsmoot nture of te function r efine by 5. Terefore tt moe cn n presenty i ony be use to bui up simutors of te eice ongituin motion. oe - is quite stisfctory compromise beteen moe 8 simper but ess ccurte n moe 5 more ccurte but more compe. Empe. To iustrte preious n fortcoming resuts te empe of Citroën-C cr it te crcteristics of Tbe I i be consiere trougout in te rest of te pper. Using Tbe II n Tbe I one gets te numeric ues of te prmeters in te functions n see Tbe III. efine by 9 It is reiy seen by simpe cecking tt conitions 0 re fufie. urtermore it is seen from ig. 9 tt 0 if 0.9. On te oter n ig. 9b sos tt 0
0 u u. st. g. ig. 8. Synoptic sceme of te simution eice moe 5 b igs. 9.. s / ig. 9.b. s / TABLE III. ALUES O THE PARAETERS IN OR THE CITROËN-C CAR Deceertion Acceertion : -.99 :.99 :.600 :.50 : -0.0 : 0.0 : -6765556.987 : -9.7657 0 - eneer 0 i.e.. In te igt of tese obsertions it is seen tt te iity omin is 0.9. Tt is 0. 07 n. On te oter n te tire cnnot ssume in cceertion s e s in eceertion moe siing rger tn m 0% i.e. m. Ten one gets from n tt 0.9. ic gies 0. n 0.. Letting min 0. 07 n min 0. one gets tt 0.9.. Te 5
resuting bouning on te siing turn out to be it 0. 07 n 0.. EHICLE LONGITUDINAL REGULATORS DESIGN.. Reference Trjectory Genertion We seek spee controer for te eice moing in ongituin cceertion/eceertion moes. Te regutor esign is bse on te eice moe n te contro objectie is to enforce te eice cssis n ee spees n n to trck teir reference trjectories enote respectiey. Te tter re require to be time-ifferentibe. Tis requirement tt cn ys be compie it by pre-fitering gien non-ifferentibe spee setpoints: Tr s T s r it 0 0 0 ere te fiter time constnt T r is freey cosen by te user n 5 re positie spee setpoints. Te boe references cnnot bot be freey cosen becuse tey re inke by. As mtter of fct te esire reference of eice spee te ee reference is et to be of te form: is first cosen by te user. Ten 5b ere is constnt representing te siing tt is uniquey obtine from etting tere Ten b yies: n setting 0. Doing so one gets g 0 ue to. f f 0 5c Tis is n gebric eqution tt must be soe to get te equte ue of for ny fie. Rec tt. Ten it is reiy cecke tt te esire spees i be in te iity omin D if beongs to te inter. Empe. Tbe I sos te ues of te prmeter obtine by soing 5c for ifferent ues of in bot cceertion n eceertion moes. It is reiy obsere tt beongs to te 6
TABLE I. ALUE O PARAETER % IN DIERENT OPERATING CONDITIONS ALL SPEEDS ARE IN km/ = 0 θ = 0 = 0 θ =0 = 0 θ = 5 I 65 0.00 0.0980 0.780 II 60-6.6500-6.650-6.655 I: Acceertion II: Deceertion inter. In Empe e got 0. 07 n 0.. Hence te boe inter for turns out to be 7 % 0%... Spee Contro L Design As te contro objectie is to enforce te eice spees to trck teir reference trjectories et us introuce te fooing trcking errors: ef ef rom it reiy foos tt te errors unergo te fooing equtions: g u g Let us consier te fooing positie efinite Lypuno function cnite: 56 c c ere c c re ny positie esign prmeters. Derie t t it respect to time yies: c sign c sign c sign u g c sign g sign c u c g c sign sign g 5 55 57 ere equtions e been use in te secon equity. Eqution 6 suggests te fooing contro : u c c g c sign c sign sign g 58 it c 0 is ne esign prmeter. Inee substituting te rigt sie of 58 to u in 57 yies c ic in turn impies: 7
ct t 0 e 59 Eqution 8 os for ny t 0 proie tt 58 s been ppie oer te inter [ 0 t. On te oter n substituting 58 in yies te cose-oop system representtion in te coorintes: c g sign c Note tt te erities sign c g c n in 58 cn be obtine using 5. Specificy: 60 Tr T r 60b Teorem in resut. Consier te contro system iustrte by ig. 0 consisting of te sttespce moe in cose-oop it te regutor efine by te contro 58. Let te regutor prmeters be cosen s foos: 0 0 / c c 6 0 0 / c c 6b for some 0 rbitrriy cosen. Ten one s te fooing properties: Te omin D efine by is n inrint set of te cose-oop system 60-b. Wteer te initi conition 0 0 D one s: signs of te cose-oop system remin boune. b te trcking errors t nis symptoticy. t Proof. Prt. irst note tt te contro 58 is e efine i.e. inoes no singurity s ong s t t D becuse by efinition of D no singurity of g n g cn occurs terein. urtermore it is reiy seen from c- tt i.e. Regutor 58 ig. 0. Teoretic contro system nye in Teorem. It incue te eice ongituin moe in cose-oop it te regutor 58 u eice 8
neer nises. No et us so tt en suppose tt D is n inrint set of te cose-oop system 9-b. To tis 0 0 D. As te rigt sies of 60-b re pieceise continuous functions of tere eist 0 suc tt for t 0 one s t t D ic in ie of mens tt: for t [0 6 t im 6 t t On te oter n one gets from 59 tt t 0 for t [0 ic togeter it 56 yies: 0 c 0 c t t 0 c 0 c for t [0 Tis in turn gies: 0 c 0 c or equienty: 0 c 0 c t t t t 0 c 0 c 0 c 0 c for t [0 6 Using 5b it foos from 6 tt for t [0 : 0 c 0 c t t 0 c 0 c ic togeter it 6-b gies t t for t [0. But tis cery contricts 6 becuse 0. So D is ctuy n inrint set of te system 60-b. Prt. rom Prt it foos tt if 0 0 D ten t t D for t 0. Ten eqution 58-59 o for t 0. Eqution 59 impies tt t is boune n consequenty so re n ue to 5 n 56. urtermore since te rigt sie of 58 is pieceise t t 9
function of n inoes no singurity becuse t t D it foos tt te contro sign u t is so boune proing Prt. Prt b is irect consequence of 59. Tis competes te proof of Teorem. Remrks. Teorem. ensures tt te cose-oop system is symptoticy stbe in te -coorintes it n ttrction region contining te oe iity omin D. A cruci step in te regutor eeopment s to fin out suitbe Lypuno function. In prticur qurtic ike functions turne out to be epess ue to te prticur structure of te controe system. Inequities in 6-b cn esiy be fufie etting rge. n c c be sufficienty Teorem sos tt te regutor 58 performs e en ppie to te contro esign moe. Te question is eter suc goo performnce is presere en te regutor is ppie to te more ccurte moe 5 see igs.. Tis question is inestigte by simution in Section I... Contro L Design Negecting Tire Dynmics Presenty te simpifie contro esign moe 8 is bse upon. Te contro objectie is to enforce te eice spee to foo te reference trjectory. Introuce te contro error: ef 65 rom it is foos tt tis error unergoes te fooing eqution: u f 66 Consier te Lypuno function cnite: 67 Using 66-67 one gets te time-eritie of it respect to time: u f 68 Tis suggests te fooing contro : u f c 69 it c 0 is esign prmeter. Inee substituting 67 in 68 gies c ic sos tt is eponentiy nising. Hence te contro objectie i.e. 0 is ensure if te contro 0
69 is ppie to te simpifie moe 8. Te question is eter suc goo performnce is presere en te contro 69 is ppie to te more reistic moe 5 ig. b. Tis question is inestigte in Section I.. urter Discussion on Prctic Impementtion irst notice tt te contro ction generte by eice ongituin controer ike 58 constitutes te torque reference sign for te engine torque regutor not iscusse in tis stuy. Accoringy te ongituin spee regutor n te engine torque regutor constitute togeter more gob csce controer ig.. In csce contro jrgon te torque regutor is referre to inner or se ie te spee regutor is ce outer or mster. or te gob csce controer to perform e te torque regutor must be esigne so tt te inner oop is muc more rpi tn te outer oop. Te preious remrks re inepenent of te engine tecnoogic nture term eectric or ybri. Rec so tt te nytic esign of te outer ongituin spee regutor 58 eeope in Section III reies on te ongituin eice moe -. Te numeric ues of te inoe prmeters epen on te prticur eice uner stuy for instnce Tbe I sos tose of Citroën-C. Tey sou normy be proie by te eice mnufcturer; oterise tey must be etermine using moe ientifiction metos. Te controer esign prmeters c c c T must be gien suitbe numeric ues before onine r running of te contro goritm. As son by simution Section I suitbe ues cn simpy be seecte fooing te usu try-n-error serc meto. Te prctic ppiction of te ongituin spee regutor necessittes te numeric impementtion of te contro 58 n spee mesurements n. As te controe system e retiey so ynmics ue to its mecnic nture smping frequency of 00 H ou be conenient for t cquisition. ooing te usu prctice te conitioning of t cquisition is me better if te signs re propery processe before smping. Onine pre-processing opertions incue sign fitering mpifiction moution emoution etc. Gien te simpicity of inoe onine gebric opertions in contro n sign processing te sm number of onine input/output mesurements n te retiey o t-cquisition frequency te require computtion resources re retiey moest. A o-cost DSP n ny one of toy s microprocessors or progrmmbe ogic controers PLC ou be sufficient for re-time impementtion of te contro goritm. Te mjor numeric impementtion tsks re escribe by Tbe.
Torque reference u Engine torque regutor Engine & Ger Bo Torque Driesft & Tires Longituin spee regutor 58 Cssis & ee spee mesurements ig.. Gob csce controer for eice ongituin spee contro. Te grey prt refers to te outer spee contro oop et it in te present stuy. Step #0# TABLE. CONTROLLER IPLEENTATION ALGORITH Coice of esign prmeter n smping time Step ## Acquisition of prmeters n sttes ribes k n k k Step ## Generte te reference trjectories Step ## Compute te contro u k Step ## Appy te contro ue to te engine Step #5# Set k k n Go to step ## k n. SIULATION Te performnces of te sopisticte controer 58 obtine from moe i be compre using numeric simutions to tose of te simper controer 69 obtine from te simper n ess ccurte moe 8. Bot controers re ppie to te most ccurte simution-oriente eice ongituin moe 5 tt not ony ccount for tire ynmics but so for te iity omin so tt te moe opertion is singurity-free. Tt is te simution stuy i be performe it tb-simuink ccoring to te eperiment setups iustrte by igs n b respectiey. Te crcteristics of te eice re tose of Citroën-C of Tbe I. Regutor 58 u eice moe 5 Regutor 69 u eice moe 5 ig.. eice ongituin contro inoing te simution-oriente moe 5 in cose-oop it te sopisticte spee regutor 58 ig.. eice ongituin contro inoing te simution-oriente moe 5 in cose-oop it te simper spee regutor 69
TABLE I. NUERICAL CHARACTERISTICS O THE CITROËN-C CAR RO CHAIBET 006 AD : Aspt Dry CW : Cobbestone Wet AD CW C : Primry tire prmeter.80 0.5 C : Primry tire prmeter.99 0 C : Primry tire prmeter 0.5 0. : Cssis mss 560 kg : Wee inerti 000 kg m r eff : Effectie ee rius 0.8 m rr : Roing resistnce coefficient 0.05 : Rete eigt of te center of grity 0. : Rete position of center of grity 0. K : Lo correction fctor 0.55 : Density of ir.0 kg/m C : Aeroynmic rg coefficient 0.5 C : Aeroynmic ift coefficient 0.59 S : ront re eice 0.8 m As mentione preiousy tere is no simpe y to fin te best coice for te esign prmeters i.e. c c c Tr. ooing te usu prctice suitbe numeric ues re seecte using te euristic try-n-error serc meto. Accoringy te net ues e been obtine:. te moe reference time constnt in 5 is set to T r s. te esign prmeters c c c for te controer 58 re gien te ues c 0. c 60 c. te best coice of te esign prmeter c in te controer 69 turne out to be c... Contro Performnces in Esy Driing Conitions Presenty te contro performnces of bot controers re iustrte in ie riing conitions crcterie by ry n ft ro n ek front in. Specificy te riing conitions re efine by te moe prmeters 0 0 km/ C C.80.89 0.5. C urtermore it is suppose tt boe ues re perfecty knon to te esigner n use in te controer esign. igs. -e so te contro system responses obtine it bot controers in tese esy riing conitions. or bot te spee responses n conerge to teir respectie references it setting time ess tn.75s n present no oersoot in te cceertion moe igs. -b. But s epecte te performnces of te sopisticte controer 58 re cery better tn
tose of 69. Inee te former is muc more speeier eeops smer contro effort ig. c- n ensures eker siing ig. e... Contro Performnces in Hr Driing Conitions Presenty te riing conitions re muc rer compre to te preious subsection. Tt is te ro is gien te profie of ig. soing n scennt stge fooe immeitey by escent stge. Anyticy te rying ro sop is efine s foos: 0 for 0 t t t m cos for t t t 70 t t for t t 0 it m 0 t 8s t s. urtermore te ro is Aspt Dry crcterie by: C C C.80.990.5 t ig. 5-b so te eice n ee eocities obtine it bot controers. It is seen tt te eition beteen ee n cssis eocities gets rger en te eice psses oer te ro bump beteen t t 8s n t s. Hoeer te eition is muc rger it te simper controer 69 tn it te sopisticte one 58. Tis fct is better iustrte by ig. 6 tt so te beior of siing is better it controer 58. ig. 7 sos te ee torque eeope by bot controers. Cery te regutor 69 eeops uge torque on te ee uring te scennt stge. In te escennt stge te eeope torque ecys rsticy tking sm ues. As mtter of fct suc beior is prcticy uncceptbe becuse it is rmfu for person n eice sfety n is costy from n energetic iepoint. Cery te regutor 58 is better. Te supremcy of 58 oer 69 is no quntifie consiering te erge torque n friction ork one by te front tires. Tese re respectiey efine s foos: T A 0 ut 7 DA tf t 7 0 ere u is te torque ppie to te ees tf is te component of te friction force projecte onto te tngenti irection of te contct surfce is te spee of contct tire-ro n is te entire simution time inter. ig. 8 sos te rition of te erge torque n friction ork it te mimum sop ro sttes i.e.: m. Te comprison beteen te to controers is et it consiering to - Aspt Dry C C.80.990.5 C
- Cobbestone Wet C C 0.5 0 0.. C Te figure inictes tt in riing conitions te erge torque of te eice being rien by te regutor 58 is muc smer tn it te regutor 69. Tis confirms tt energy consumption is oer it 58 n sfety is better. rom ig.8b it is seen tt tire friction ctiity is muc eker en te eice is rien it te regutor 58 especiy en te eice is going troug bump by cobbestone et ro conition. Ten ig tire friction ctiity is inoe it te controer 69 ic mens tt te tires re ten subject ig pressure. ig. Cose-oop cssis spee responses. Soi: reference spee. Dse: eice spee response obtine it regutor 58. Dotte: eice spee response obtine it regutor 69 ig b. Wee spee responses. Soi: reference. Dse: spee response it regutor 58. Dotte: spee response it regutor 69 5
ig c. Contro sign u. Soi: contro sign it regutor 58. Dse: contro sign it regutor 69 ig. Trction force tf. Soi: tf it regutor 58. Dse: tf it regutor 69 ig e. Siing ; soi: Siing 58; se: Siing 69 6
m t t ig.. Ro profie consiere in Subsection.. ig. 5. eice spee cose-oop response. Soi: true eice spee response obtine it regutor 58. Dotte: eice spee response obtine it regutor 69 ig. 5b. Wee spee cose-oop responses. Soi: true ee spee obtine it regutor 58. Dotte: ee spee obtine it regutor 69 ig. 6. Siing responses. Soi: siing obtine it 58. Dse: siing obtine it regutor 69 7
ig. 7. Contro sign u. Soi: contro sign it regutor 58. Dse: contro sign it regutor 69 Regutor 69 Cobbestone Wet Regutor 69 Aspt Dry Regutor 58 Cobbestone Wet Regutor 58 Aspt Dry ig. 8. Aerge torque en going troug te bump Regutor 69 Cobbestone Wet Regutor 69 Aspt Dry Regutor 58 Cobbestone Wet Regutor 58 Aspt Dry ig. 8b. riction effort one by te tire 5. CONCLUSION Te probem of eice ongituin contro is resse bot in cceertion n eceertion moes bse on te to ne moes efine by n 5. Te originity of tese moes ies in te fct tt tey epicity ccounts for te ongituin sip resuting from tire eformtion using Kiencke s 8
moe. urtermore te moe 5 so ccounts for re-ife consiertions tt pysicy imit te ongituin sip ue. Consequenty 5 turns out to be quite suitbe for simuting te eice ongituin beior. Te contro esign is bse on te sigty simper moe ic ignores te sip imittion but sti ccounts for te tire ongituin sip. It is son tt spee controer cn ctuy be obtine from tt moe using Lypuno type esign tecnique. It is formy son tt te obtine controer efine by 58 oes meet its performnces i.e. stbiity n perfect spee reference. It is so cecke by simution tt te controer 58 is muc better tn te simper one efine by 69 n ignoring ongituin sip. Te supremcy of 58 is especiy pprecite in r riing conitions i.e. crossing rmpnt n et ros. In simute eperiments te eice ongituin motion is represente by te most ccurte n igy noniner moe 5. REERENCES Cibet A. 006. Contrôe Ltér et Longituin pour e Suii e éicue PD Uniersité Ery Essonne rnce. Dugoff P..H. n L. Sege 970. An nysis of tire trction properties n teir infuence on eice ynmic performnce SAE Tecnic Pper 70077 o pp. 9- oi:0.7/70077. Gim G. n P. Nikres 990. An nytic moe of pneumtic tyres for eice ynmic simutions prt: Pure sips Interntion ourn of eice Design o. No. 6 pp. 589-68. Guo K. n L. Ren 000. A non-stey n non-iner tire moe uner rge ter sip conition SAE Tecnic Pper 000-0-058 000 oi:0.7/000-0-058. Ionnou P.A. n C.C. Cien 99. Autonomous Inteigent Cruise Contro. IEEE Trnsctions on eicur Tecnoogy o. No. pp. 657-67. Ki H.K. 00. Noniner Systems Prentice-H r eition. Kiencke U. n L. Niesen 005. Automotie contro system for engine rieine n eice n eition Springer. Ling H. K.T. Cong n T.S. No S.Y. Yi 00. eice ongituin brke contro using ribe prmeter siing contro. Contro Engineering Prctice o. No. pp. 0- iiken.w. n L.D. iiken Rce cr eice ynmics SAE Interntion 995. oon S. I. oon n K. Yi 009. Design tuning n eution of fu-rnge ptie cruise contro system it coision oince. Contro Engineering Prctice o. 7 No. pp. -55. Noueière L. n S. mmr 007. Eperiment eice ongituin contro using secon 9
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