Adaptive throttle controller design based on a nonlinear vehicle model

Transcription

1 Adatv throttl controllr dsgn basd on a nonlnar vhcl odl Fng Gao, Kqang L, Janqang Wang, Xaon Lan Stat Ky Laboratory of Autootv Safty and Enrgy snghua Unvrsty Bjng, 84, P.. Chna Abstract Basd on study of nonlnarts of vhcl longtudnal odl and ts slfcaton, a odl rfrnc adatv controllr for throttl control s dsgnd usng a slfd nonlnar vhcl odl and ts stablty n th rsnc of unodld dynacs s rovd usng Lyaunov stablty thory n ths ar. Snc th slfd nonlnar odl s t nvarant whn gar s fxd and t ts rqurnt of t nvarant for dsgnng adatv control syst, th controllr basd on th slfd nonlnar odl has bttr rforanc of convrgnc than that basd on th slfd lnar odl. Sulaton rsults on a full ordr nonlnar vhcl longtudnal odl show that th adatv controllr basd on slfd nonlnar odl can rjct dsturbancs that ars fro aratr rrors and s robust to unodld dynacs. Furthror t has bttr rforanc of convrgnc than controllr basd on th slfd lnar odl. I. INODUCION Sd tracng s an ortant art of Adatv Crus Control (ACC). Many thods, such as PID and LQ, hav bn usd and so achvnts hav bn obtand []-[3]. But odlng rrors such as unodld dynacs, aratr rrors, tc., hav not bn fully tan nto account whn dsgnng controllrs. In fact varatons of vhcl and nvronnt aratrs such as vhcl ass, slo, arodynac drag forc and rollng rsstanc [4] sgnfcantly affct controllr rforancs and vn a syst unstabl. In ordr to achv good rforanc n th rsnc of odlng rrors, H thory and robust adatv thory ar usd [5]-[8]. But thr ar so ractcal dffcults whn usng ths nds of thods. h lnar odl and bounds of odl rror for dsgnng H controllr ar dffcult to dtrn. In [8] a frst ordr odl s obtand by aylor lnarzaton thod. Although th odl s vry sl, ts aratrs chang wth oratng onts. Actually ths sl lnar odl s t varyng, whch s n conflct wth rqurnt of t nvarant for dsgnng adatv control syst. In ths ar, a slfd nonlnar odl s usd to dscrb vhcl longtudnal dynacs. It s t nvarant whn gar s fxd and t ts rqurnt of t nvarant for dsgnng adatv control syst. Usng ths slfd nonlnar odl, a odl rfrnc adatv controllr for throttl control s dsgnd. hortcal analyss and sulaton rsults show that th adatv controllr basd on th slfd nonlnar odl can rjct dsturbancs that ars du to aratr rrors, unodld dynacs, and has bttr rforanc of convrgnc than that basd on th slfd lnar odl. II. LONGIUDINAL VEHICLE MODEL Fgur shows th basc blocs and nuts, oututs of a longtudnal vhcl odl [9]. h outut of th ngn subsyst s ngn torqu that s a nonlnar functon of throttl angl and ngn sd. A frst ordr syst s usd to dscrb th dynacs of ngn. h transsson subsyst s rsonsbl for transfrrng ngn torqu to drvtran. It s an autoatc transsson wth hydraulc torqu coulng and four forward transsson gars. h gar stat s a nonlnar functon of throttl angl and vhcl sd. h nut of th drvtran subsyst s drv torqu and ts oututs ar vhcl sd, acclraton or dclraton whch ar affctd by road condton, arodynac drag and vhcl ass. In ths ar only longtudnal control usng throttl control s consdrd so brang torqu s st to zro. Manuscrt rcvd Stbr, 3. Fng Gao s wth th Stat Ky Laboratory of Autootv Safty and Enrgy, snghua Unvrsty, Bjng, 84, P.. Chna (al: gao-f@als.tsnghua.du.cn) Fg.. Vhcl longtudnal odl

2 Snc th abov nonlnar odl s vry colcatd, t s dffcult to us ths nonlnar odl to drctly dsgn a throttl controllr. aylor lnarzaton thod s usd to slfy th abov nonlnar vhcl longtudnal odl [8] and hgh ordr rror s nglctd. h followng lnar odl s obtand. b ( s) ( s )( s )( s ) V s = V = V V () = V s vhcl sd s throttl angl V, ) s oratng onts ( b,, =,, 3 ar functons of th oratng onts For all oratng onts, th followng s obtand ( ) < 3 <.4 and > 7, =, 3 n So 3 s th donant ol and th fast ods can b nglctd, ladng to a sl odl of V s b = () s + 3 ( s) hs frst ordr odl s vry sl, but ts aratrs chang wth th oratng onts. For ACC syst, vhcl sd s always odfd accordng to surroundngs. hs odl s actually t varyng. Furthror th hgh ordr rror s nglctd durng lnarzaton rocdur. Modl rrors of ths sl lnar odl nclud not only unodld dynacs and aratr rrors but also hgh ordr rror. Norally, th bggr th odl rrors ar, th or srous th ngatv ffcts to th control syst ar. Consdrng th abov robls, a slfd nonlnar odl s usd to dscrb th vhcl longtudnal dynacs n ths ar: M V = CaV F f (3) r M s total vhcl nrta s torqu rato of transsson r s radus of whl C s arodynac drag coffcnt a F f s th rsstanc forc of road s ngn torqu. Coard wth th slfd lnar odl, t has th followng advantags: ) Whn gar s fxd, t s t nvarant. Although vhcl sd s odfd constantly accordng to surroundngs for ACC syst, gar shftng dos not han oftn. So aratrs of ths slfd nonlnar odl ar constant for a long t and ths odl acts as a t nvarant syst. Evn f gar shftng occurs, th aratr rrors du to gar shftng can b tratd as ntal aratr rrors and wll b rjctd du to tunng ablty of th adatv controllr. ) h dynacs of owrtran s nglctd whn sttng u th slfd nonlnar odl. Man odl rrors of ths nonlnar odl ar unodld dynacs and aratr rrors. It s wll nown that th dsgn tchnqus for adatv systs ar basd on thory for t nvarant lants, so th slfd nonlnar odl s or sutabl for dsgnng adatv controllr. In th nxt scton, ths two slfd odls ar usd to dsgn adatv controllrs for throttl control rsctvly. III. ADAPIVE CONOLLE DESIGN In ths scton, th two slfd odl ar usd to dsgn th adatv controllrs rsctvly. In ordr to dstngush btwn ths two controllrs, adatv controllr basd on th slfd lnar odl s calld controllr A and th othr s calld controllr B. Basd on th lnar odl, controllr A s dscrbd as follows [8]: = f V V V + d d, u, ε X > =, l, ε X <, =, γε X, othrws V s th dsrd vhcl sd d s statd aratr = f V dscrbs th rlatonsh of u (4) and V and ar rsctvly ur bound and lowr l bound of ε s th noralzd rror sgnal X = V V and X = ar asurd sgnals d

3 Usng th frst ordr lnar odl an adatv controllr can b obtand, but t s not ror, snc ths lnar odl s t varyng whch s n conflct wth rqurnt of t nvarant for dsgnng adatv control syst. o ovrco ths shortcong, th slfd nonlnar odl s usd to dsgn adatv controllr for throttl control n th followng. h objctv of throttl control s to a vhcl trac th rfrnc sd V dscrbd by V + a V = a Vd (5) a > s chosn basd on th rqurnts of syst rsons sd, rdng qualty, tc. h structur of th control syst s shown n Fg.. Fg.. Structur of th control syst h dsrd ngn torqu s = W (6) =,, s statd aratr W [ ] 3 = V d V, V, s asurd sgnal. Actual odl aratr s * a rc f =,, a Fr Usng th nvrs of ngn torqu a, th throttl angl s: = MAP (, ω ) (7) ω s ngn sd MAP s th nvrs of ngn torqu a A nw rror sgnal s dfnd as ε ε = (8) s + a = s s = s + = s s dtrnd by th ols of th transfr functon of owrtran. = V V s th tracng rror h adatv law s =Γ ε W (9) Γ γ = γ γ 3 γ >, =,,3 s adatv gan For robustnss, th adatv laws can b odfd to, > u, > =, < l, < () γεw, othrws and ar ur bound and lowr bound of u l W s th lnt of W. IV. OBUSNESS ANALYSIS OF CONOLLE B In th last scton, odl rfrnc adatv controllrs for throttl control ar dsgnd basd on slfd lnar and nonlnar odl rsctvly. h slfd nonlnar odl s t nvarant whn gar s fxd, but th dynacs of owrtran ar nglctd. In ths scton, Lyaunov stablty thory s usd to rov stablty of th control syst n th rsnc of unodld dynacs. ang th dynacs of owrtran nto account, whn gar s fxd, th nonlnar odl s M V = G ( s) CaV Ff () r M ( s) G( s) = s th transfr functon of owrtran N s h odl rror du to unodld dynacs s dfnd as

4 ( s) N( s) N( s) M s = () Substtutng () nto (), so that M V = ( + ( s) ) CaV F f (3) r Substtutng (5) nto (3), so that + a = Φ W +η (4) Φ = * s th aratr rror h dsturbanc η s dfnd as η = ( s ) (5) h followng Lyaunov functon s chosn. V = ε + Φ Γ Φ (6) h t drvatv of V s V = εε+ Φ Γ Φ (7) By th dfnton of th nw rror sgnal dscrbd by (8) thrfor a ε a ε = + ε (8) h aratrs ar t nvarant whn gar s fxd, so that Φ = (9) Substtutng (8), (4) and (9) nto (7), so that V =ε aε + ε Φ W + η () + Φ Γ Substtutng th adatv law (9) nto (), so that V = ε a ε + ε η () hn th followng s obtand V ε a ε + ε η () Usng.5 η ε ε ε η = + η 4 th followng s obtand 3 V ε ε η η + 4 Fro [] th followng thor s nown. Lt y = H s u [ ] H( s ) s a strctly ror transfr functon. If H( s ) s analytc n [ ].5 thn yt s+ H s (3) s for so > t u s an arbtrary constant. It s dffcult to dtrn th odl rrors, ( s), du to th unodld dynacs of owrtran, but sly by s can b xrnc th rang of th ols of ( s) dtrnd. Choosng that as [ s].5 analytc n thn accordng to th abov thor, th followng s obtand ηt s + s (4) And thn t η s + s = C (5) C s a constant. Substtutng (5) nto (3), so that 3ε ε η V + C 4 (6) Fro (6) w now that f V s gratr than a crtan constant, thn V <. hs rorty of V ls that th control syst s stabl and, ε, Φ L. V. SIMULAION ESULS In th last scton, stablty of controllr B n th rsnc of unodld dynacs s rovd usng Lyaunov stablty

5 thory. h analyss of scton shows that th slfd nonlnar odl s or sutabl for dsgnng adatv controllr. In ordr to tst th rforanc of controllr A and controllr B, thy ar both ald n th full ordr nonlnar vhcl longtudnal odl [9] n ths scton. h ntal control aratrs can b chosn arbtrarly btwn th ur and lowr bounds. h aratrs of th controllrs ar chosn as follows n th sulaton. ) Adatv controllr A a =.5 Γ = dag[ 3.5] =, = u 4 =, =6 u 6 l l ) Adatv controllr B a =.5 Γ = dag[.5.5 5] u = 93, l = 35 u =.43, l =.94 u3 = 34.3, l3 =3 (a) Sd tracng rrors of both controllrs In ordr to tst th rforanc of th controllrs, two acclraton scnaros and a dclraton scnaro ar usd n sulaton. On acclraton scnaro s at low sd and th othr s at hgh sd. h dsrd and rfrnc vhcl sd ar shown n Fg. 3. (b) Gar stat Fg. 4. Sulaton rsults Fg. 3. Dsrd and rfrnc sd And th rsults ar shown n Fg. 4. Fro t= s to t= s, th dsrd sd ncrass to /s wth an acclraton of.5 /s. Fro th sulaton rsults, tracng rrors ar larg. On rason s that ntal aratrs of controllrs ar chosn arbtrarly, whch rsults n larg aratr rrors. Anothr rason s that gar shftng s frqunt durng ths rod, whch rsults n shar chang of ral aratrs. Whn th dsrd sd s stabl, tracng rror of controllr B convrgs to or radly than that of controllr A. Fro t=9 s to t= s, th dsrd sd ncrass to 9.5 /s wth an acclraton of.3 /s. In ths rod, only on shft hans, and du to th accoodaton ablty, th aratr rrors bco sallr, so th tracng rrors bco sallr too. It s th sa as th last acclratng scnaro. racng rror of controllr B convrgs to or radly than that of controllr A whn th dsrd sd s stabl.

6 Fro t=35 s to t=55 s, th dsrd sd dcrass to 6.5 /s wth a dclraton of -.3 /s. No shft occurs n ths rod. Fro th sulaton rsults, rforanc of convrgnc and tracng ablty of controllr B s bttr than that of controllr A. h tracng rror bounds of controllr A of thr scnaros ar /s,. /s and. /s rsctvly and that of controllr B ar.45 /s,. /s and. /s rsctvly. Fro th sulaton rsults, It s concludd that throttl control syst usng controllr B convrgs to dsrd sd or qucly and has sallr tracng rrors, whch concds wth th concluson n scton that th slfd nonlnar odl s or sutabl for dsgnng adatv controllr. VI. CONCLUSION In ths ar, odl rfrnc adatv controllrs ar dsgnd for throttl control. Bfor dsgnng th controllr, two sl vhcl odls, on s a frst ordr syst and th othr s a nonlnar on, ar analyzd and th followng charactrstcs ar obtand. ) Paratrs of th slfd lnar odl ar functons of oratng onts, so n fact t s t varyng. h an odl rrors ar unodld dynacs, aratr rrors and hgh ordr rror, whch s nglctd n aylor lnarzaton. ) Slfd nonlnar odl s t nvarant whn gar s fxd. Its an odl rrors ar unodld dynacs and aratr rrors. Snc adatv control s anly usd to t nvarant or slowly t-varyng lants, and for ACC syst, vhcl sd should b odfd frquntly to saf dstanc to front vhcl, t s concludd that th slfd nonlnar odl s or sutabl for dsgnng adatv controllr. Basd on th slfd nonlnar odl, a odl rfrnc adatv controllr for throttl control s dsgnd. hortcal analyss and sulaton rsults show that th adatv throttl controllr has th followng advantags: ) h controllr can rjct dsturbanc that ars du to aratr rrors bcaus of th larnng caacty. ) h control syst s stabl n th rsnc of unodld dynacs. 3) Snc th slfd nonlnar odl s or sutabl for dsgnng adatv controllr, controllr B has bttr rforanc of convrgnc and tracng ablty than controllr A. [] Kyongsu Y, Youngjoo Cho, A throttl/bra control law for vhcl ntllgnt crus control, n Proc. World Autootv Control, Kora, Jun. [3] J.K. Hdrc, M. ozua, Control ssus n autoatd hghway systs, IEEE Control Systs, 994 Dcbr,.-3. [4] J.C. Grds, J.K. Hdrc, Vhcl sd and sacng control va coordnatd throttl and bra actuaton, Control Eng. Practc, Vol.5, No., 997. [5] Hong S.Ba, J.Chrstan Grds, Paratr staton and coand odfcaton for longtudnal control of havy vhcls, n Proc. 5th Int Syosu on Advancd Vhcl Control, Mchgan, August -4,. [6] Yoshnor Yaaura, Masaho ab, Dvlont of an adatv crus control syst wth sto and go caacty, SAE,. [7] Ara Hgashata, Kazutaa Adach, anor Hashzu, Dsgn of a hadway dstanc control syst for ACC, JSAE vw,,,.5-. [8] Z. Xu, P. Ioannou, Adatv throttl control for sd tracng, Vhcl Syst Dynacs, 994, [9] Dzao Hou, Fng Gao, Kqang L, Vhcl longtudnal odl for collson avodanc syst, Journal of snghua Unvrsty, Vol. 44, No., 4. [] Ptros A. Ioannou, Anruddha Datta, obust adatv control: a unfd aroach, Procdngs of th IEEE, Vol. 79, No., Dcbr 99. EFEENCES [] Grg Marsdn, M Mcdonald, Mar Bracston, owards an undrstandng of adatv crus control, Elsvr Scnc, ransortaton sarch, Part C, Stbr.