Stability Analysis of an Electric Parking Brake (EPB) System with a Nonlinear Proportional Controller

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1 Procdgs of th 17th World Cogrss h Itrtol Fdrto of utomtc Cotrol Stblty lyss of Elctrc Prkg Brk (EPB) Systm wth Nolr Proportol Cotrollr Youg O. L *, Choog W. L *, Chug C. Chug **, Yougsup So ***, Pljoo Yoo *** d Iyog Hwg*** *Dvso of Elctrcl d Computr Egrg, Hyg Uvrsty, Soul, 1-791, or (-ml: foryou5252@hotml.com, chugwoo.l@gml.com) **Dvso of Elctrcl d Bomdcl Egrg, Hyg Uvrsty, Soul, 1-791, or (-ml: cchug@hyg.c.kr) *** Ctrl R&D Ctr, MNDO Corporto, yogg-do, , or (-ml: ysso@mdo.com, pjyoo@mdo.com, yhwg@mdo.com) bstrct: I ths ppr, Elctrc Prkg Brk (EPB) systm s modlld s stt-dpdt swtchd systm. h modl volvs scrw frcto whch vrs dpdg o th oprto rgo. w olr proportol (P) cotrollr s proposd d ts stblty s lyzd v Lypuov d LSll s thory. It s show tht th qulbrum pot s loclly uform d ultmtly boudd. 1. INRODUCION Elctrc Prkg Brk (EPB) systm s kd of X-bywr systm. h systm rplcs th mul muvrg forc of th covtol lvr prkg systm wth motor torqu. O prformc rqurmt s tht th EPB systm mts brk forc wthout powr supply. o stsfy ths codto, th systm ds to us th frcto of th scrw, whch chgs ccordg to th oprto rgo. h cotrollr of th EPB systm should b dsgd to b robust d to provd uform prformc. Frst, t rqurs wd rg of oprto du to th vrous wghts of crs d th cltos of rods. Scod, t must tolrt lrg dsturbc forcs ctg upo th prkg cbl du to frcto o grs d scrws s th prssur o th prkg brk pds crss. For th EPB cotrollr, o-off, lr proportol (P) d olr P cotrollrs wr comprd (L, t l., 27). h olr P cotrollr provds good uformty prformc d robustss. s th EPB systm s oprto stt vrs, th scrw s frcto lso vrs. EPB systm c b modlld s stt-dpdt swtchd systm wth fv oprto rgos. It s put-ff olr systm whch c b pproxmtd s lr tm-vrt systm wth ch oprto rgo. h olr P cotrollr hs lr g for smll rror d xpotl g bout th motor s gl rror for lrg rror. Smlr olr PID cotrol lgorthms hv b ppld to clss of truck BS problms (Jg d Go, 21). h stblty of th olr P cotrollr hs ot b rportd yt. For swtchg systm, stblty c b show f ts dvdul systms shr commo Lypuov fucto wth gtv drvtv of th Lypuov fucto wth rspct to tm (Nrdr d Blkrsh, 1994). o prov Corrspodg uthor. -ml: cchug@hyg.c.kr th stblty of th swtchd EPB systm wth th proposd olr P cotrollr, commo Lypuov fucto s costructd. Bcus of frcto, thr s slf-lockg stt. It c b provd tht th qulbrum pot s loclly uform d ultmtly boudd usg LSll s thorm. hs ppr s orgzd s follows. I Scto 2, structur d chrctrstcs of th EPB systm r prstd. sttdpdt swtchd systm modl s proposd. Scto dscrbs th cotrollr s spcfctos d ts olr P cotrollr dsg s troducd. I Scto 4, th stblty lyss of th EPB systm s prstd. Scto 5 dmostrts th prformc of th cotrollrs by xprmt. 2. EPB SYSEM MODEL 2.1 EPB Systm Structur d Chrctrstcs. h EPB s systm whch cotrols th brk forc by pullg th prkg cbl s covtol xstg prkg brks. h structur of th EPB systm s show Fg. 1. It cluds DC motor, grbox, scrw, ut, currt ssor, Hll-ffct forc ssor, cclrto ssor d ECU. Grlly, f drvr or hgh lvl systm oprts th EPB systm, th cotrollr clcults trgt forc from th prkg cbl bsd o th cr mss s wll s th clto of th rod s msurd by th cclrto ssor. h EPB crss th brk forc by pullg th prkg cbl usg th DC motor utl th brk forc rchs th trgt forc. Brk forc s msurd by th Hll-ffct forc ssor. h EPB systm hs to mt th brk forc wthout powr. o stsfy ths rqurmt, t ds to us th frcto of th scrw. h drcto d vlu of th frcto my vry s th scrw s rottol drcto s chgd. h stt of th scrw s frcto dtrms th codto of th slf-lockg scrw /8/$2. 28 IFC / R-11.19

2 Cotrol Commd Cotrollr Forc Fdbck Motor Drvr PWM Motor Forc Ssor Gr Box Gr Box Scrw Jot Wr stt-dpdt swtchd systm. h grl frcto modl c b dscrbd s (1) (Olsso, t l., 1998). F dscrbs th xtrl forc. h mxmum sttc frcto forc F s s ssumd to b lrgr th zro. ( v, F) F ( v) Fr f v F f v d F < Fs Fs sg ( F) f v d F Fs (1) Fg. 1. Structur of th EPB systm Motor gl (rd) Wh th vlocty, v, s ot zro, th frcto modl cluds th Coulomb frcto F c, th mxmum sttc frcto F s, th coffct of vscous frcto F v d th Strbck frcto wth Strbck vlocty v s. δs v/ v s c s c v F v F + F F + Fv (2) Motor gulr vlocty (rd/s) rmtur currt () Fg. 2. Motor gl, gulr vlocty d rmtur currt stts. (pplyg brk forc mod) bl 1. Sg of th stt vrbls Stt vrbl pply Rls Motor gl θ θ Motor gulr vlocty θ θ rmtur currt Fg. 2 shows typcl smulto rsults for th EPB systm wth olr P cotrollr whch wll b dscussd scto. s th gulr vlocty pprochs zro, th rmtur currt dmostrts th Strbck ffct. Oc th scrw s slf-lockd, th gulr vlocty d th rmtur currt r mtd du to scrw frcto. bl 1 shows sgs of th stts t ch oprto mod. Notc tht th product of th motor gl d th rmtur currt s lwys gtv sm-dft rgrdlss of th oprto mod. 2.2 Stt Dpdt Swtchd Modl of EPB h EPB systm s hghly olr systm du to frcto. h frcto of th scrw vrs dpdg o th oprto rgo. hrfor, th EPB systm c b modlld s h scrw of th EPB systm hs lowr mgtud of vscous d Strbck frcto th sttc frcto d Coulomb frcto th rg of trst. h frcto modl c b smplfd for stblty lyss s follows: Fc f v Fr ( v, F) F f v d F < Fs Fs sg ( F) f v d F Fs Hr, xtrl forc F d Coulomb frcto F c c b dscrbd s (4) d (5). s th torqu whos drcto s prpdculr to th scrw pl, Q s th forc ppld th scrw xl drcto, d s th scrw dmtr, λ s th scrw ld gl d µ s s th frcto coffct of th scrw. () 2 F cos λ Q s λ d (4) 2 Fc µ s s λ+ Q cos λ d (5) α1 f v d F v > (6-) α 2 f v d F v < (6-b) Q α f v d F < Fs (6-c) α 4 f v d F Fs d F > (6-d) α 5 f v d F Fs d F < (6-) h rlto btw put torqu d output forc Q s modlld s lr ch scrw s oprto stt. Ech stt s s follows. (6-) d (6-b) show cclrto d dclrto mods, rspctvly. (6-c) s th cs of th slflockg stt, α s mtd. (6-d) s th cs whr th 14248

3 scrw bgs to rott forwrd to pply forc for pullg th prkg cbls. (6-) s th cs whr th scrw bgs to rott bckwrd to rls th prkg cbls. α s th scrw g corrspodg to ch cs. hs codtos rsult stt-dpdt swtchd EPB systm (7). h systm s put-ff olr systm. x x x+ Bu h systm mtrx (x) vrs dpdg o th systm swtch-stts d t c b rdfd s (8),, 1,,5 (7) x x Ω (8) whr th oprtg rgo, Ω, s dfd t ch swtchg codto. t ch Ω, (7) s lr tm-vrt systm. h st S cossts of 5 ordrd prs, systm mtrcs d oprtg rgos. It c b dfd s {( 1, 1),,( 5, 5) } S Ω Ω (9) whr d Ω stsfy th codtos of (1). dclrts (Ω 2 ). Dpdg o ts vlocty, th dclrto stt s swtchd to thr cclrto or slf-lockg stts. Stt spc qutos c b clssfd to two groups. Wh mchcl systm s movg or strtg to mov, (11-) s usd. Wh mchcl systm s lockd by frcto, (11-b) s usd. h stt vrbls r th motor gl,θ, th motor gulr vlocty, ω d th rmtur currt,. L s th rmtur ductc, R s th rmtur rsstor, J s th rt t th rmtur, b s th bck mf costt d t s th motor torqu costt. Cs 1,2,4,5 θ 1 θ β t ω ω+ V J J b R 1 (11-) L L L y [ 1 ] θ ω slf-lockg v, F Fs, F >? v, F < F? scp forwrd 4 4 s R, Ω R, 5,, 1 R m m Ω Ω Ω v, F < F? s v, F Fs, F <? dclrto 2 2 v,? Fg.. Stt dgrm of EPB systm v, Fv<? v, Fv<? v,? scp bckwrd 5 5 cclrto 1 1 (1) v,? v,? h stt swtchs occur rstrctd drctos. hs rltos c b dscrbd Fg.. Wh th scrw s slf-lockg stt (Ω ), f th mgtud of th xtrl forc s lrgr th th mxmum sttc frcto forc of th scrw, t bgs to rott thr bckwrd (Ω 5 ) or forwrd (Ω 4 ). It th cclrts (Ω 1 ). Grlly, t mts ts cclrto stt (Ω 1 ) for som tm. ftr som prod of tm, t Cs B θ θ ω ω+ V R 1 (11-b) L L y [ 1 ] θ ω I quto (11-), vrbl β s rltd to th scrw g, α, whch vrs wth th scrw s oprtg codto. It modls th rlto btw th lod of th prkg cbl d th motor s gl s lr. C w s th sprg costt of th prkg cbl. C fs s th sprg costt of th forc ssg sprg. N GR s th gr box s gr rto d p s th scrw ptch. 1 p C C β, 1,2,4,5 N C C w fs 2 GR 2πα w + 2 fs (12) 14249

4 . CONROL SYSEM DESIGN.1 Prformc Rqurmt h followg prformc spcfctos r rqurd for th EPB systm grl (Jum, t l., 24): -h systm must brk th cr wthout powr supply. -h systm must grt th dmdd brkg forc wth spcfd tm. -h mchcl dsg hs to b durbl d robust. -It s dvsbl tht th EPB systm b slt. o stsfy th frst spcfcto, th EPB systm uss th slf-lockg prcpl of th scrw-ut structur. hrfor, t s sstl to lmt th drcto of th cotrollr output wh pplyg th EPB. hs lmtto prvts th trrupt of th scrw slf-lockg mchsm by rvrs rotto of th motor. h rqurd prkg cbl tso s dtrmd by th wght of th cr d th clto of th rod. Bcus lctroc compots r ddd to th EPB systm, th robustss d sfty ssus bcom mor mportt th covtol prkg brk systms. Prformc of thr cotrollrs (o-off, lr P d olr P) ws comprd by L, t l., (27). h olr P cotrollr shows good robustss prformc comprd wth othr cotrol mthods. ssumpto 1. Wth th oprto rg whr th cotrol output s xpotl fucto of th gl rror, th mgtud of th motor s gulr vlocty s lrgr th or qul to 1[rd/s]. I othr words, for r > δ. [ rd s] θ 1 / (15) Rmrk 1. For our systm, r s bout 8[rd], δ s roud 2[rd] d EPB should b lockd wth 1[sc]. It s clr tht ssumpto 1 s physclly rsobl d th codto (15) s stsfd oc th powr s ppld. ssumpto 1 s dd to stsfy th stblty of ths systm. Dtls o stblty wll b xpld scto 4. r δ u δ r u u p _ l f (, α, δ ).2 Nolr P Cotrollr Dsg h olr P cotrollr (1) uss th olr fucto f(, l, α, δ). It ppls hgh g for smll rror d smll g for lrg rror. o vod xcssv hgh g d uwtd vbrtg rspos th ghbourhood of th qulbrum pot, t uss lr g for smll rror (Jg d Go, 21) s show Fg. 4. h output drcto of th cotrollr should b mtd th sm drcto s th motor s movmt to prvt th motor from rottg rvrs. I rl systm, du to physcl lmttos, th cotrol output s boudd. h cotrol output s dfd by (, l,, ) u f α δ (1) whr s th motor gl rror d l, α d δ r usr dsg prmtrs. h olr fucto s gv by Fg. 4. h fuctos of th Lr P d th Nolr P cotrollr. 4. SBILIY NLYSIS 4.1 Stblty of th Nolr P Cotrollr EPB Systm h output of th cotrollr s olr fucto dpdt o th stt vrbl x 1 θ. Iput voltg bcoms (16) d fucto f s dscrbd (14). h org of s tk s th qulbrum pot wthout loss of grlty (hll, 2). (, _,, ) V f α δ (16) p l Hr, (, p l, αδ, ) f α 1 lδ, δ, δ < r. _ α 1 l l α, 1, θ θ, >, δ >, r > d (14) h stt-spc modl of th EPB systm wth olr cotrollr c b dscrbd s combto of lr d olr lmts dscrbd by ( c, ), x + B x x x (17) 1425

5 d B x c, c, 1 β J J R for x / Ω l α 1 b δ L L L l α 1 R δ L L for x Ω, δ, δ < r l α 1 α 1 ( δ ) L t (18-) (18-b) (19) From th root locus lyss, w c sly show tht thr xst p,m d p,mx, such tht σ ( c, ) for α 1 lδ p,m p,mx (, ) x Ω, 1,2,4,5 whr σ( c, ) s th spctrum of th systm mtrx c,. commo Lypuov fucto s show (21) l α 2 x Px+ δ p1x1, δ L 1 l α + 2 V ( x) x Px+ δ p1 L 2 l α 1 + sg ( x1) δ p1x1, δ < r L (2) (21) Rmrk 2. Gv th st of c,, s log s w r bl to fd P, whch s th soluto of th commo Lypuov fucto, p 1, p 2 d p r postv. If P P > for som Q Q > stsfyg c, P+P c, -Q, x Ω, 1,2,4,5. Bcus sg(x 1 )x 1 >. It s clr tht V(x) s loclly postv dft d cotuously dffrtbl. h th drvtv of V(x) s gv by V( x) xqx l α + 2 δ p1x1 x2, δ L xqx + 2xB( x) Px l α sg ( x1) δ p1x1 x2, δ < r L for x Ω, 1, 2, 4,5 (22) Wh δ, t s clr tht V( x) s gtv dft sc w hv x 1 x 2 from bl 1. Wh δ< r, th scod trm of V( x) qul to (2). 2x B x Px α α ( δ ) l 1 1 x1 p1x1 p2x2 px L, 2x B(x) Px, s (2) (2) s lwys gtv f th l p 1 x 1 2 (δ α-1 - α-1 )/L trm s lmtd s show (24) sc x 1 x 2 d x 1 x th oprto rg wthout loss grlty. l x B( x) Px 2 p1x1 ( δ ) L 2 + ( δ ) l 1 1 x1 p2x2 px L (24) W s tht (25) holds, sc sg(x 1 )x 2 1 from ssumpto 1. Lt p P( m, ) m ssumpto 2. h (, 1), (, 2) d (, ) compots of th P mtrx must b grtr th zro. p >, p >, p > ( x ) ( δ ) l p1x1 L < 2sg l α δ p1x1 x2 L (25) 14251

6 From (24) d (25), w s tht 2x B x Px 2sg x p x x + l α δ L l < 2x B( x) Px 2 p1x1 ( δ ) L (26) Bcus V( x) x Q x, x Ω, th stts covrg to Ω xpotlly. 4.2 Slf-lockg Stt Stblty I scto 4.1, w showd tht, f x strts / Ω, x pprochs Ω s t. Sc V( x) x Q x, x Ω (27) I th slf-lockg stt, th oprtg rgo s dfd s (28) d x1 x2, x Ω. { x x x, F Fs, x } Ω < (28) 1 2 W kow tht f x1 x2, x Ω th Lypuov fucto (21) bcoms V( x) 2 x1 x p1 + x x p p R 2 x1 + x ( x1p1 + xp) L L p Rp Rp p th th drvtv of p p x1 x x1x L L L L whr p l δ α-1. Df th st M, such tht 2 2 { p } p (29) M x p x + R p x + x x R p + p > () h closur of th st M s o th x 1 -x pl R. h t s clr tht d, V x < x M, V x x Ω M N whr { p 1 or 1 1 } N x x + R x p x + p x (1) h motor currt s mtd ftr th scrw s slf-lockd s Fg. 2. So th drvtv of th motor currt bcoms zro. I th slf-lockg stt, th lrgst vrt st bcoms Ω M N (2) h lrgst vrt st s th trscto of th closur of th st M () d th st N (1) o th x 1 -x pl. hrfor, th lrgst vrt st bcoms st of th two trscto pots. Usg LSll s thorm, w c show tht th qulbrum pot s loclly uform d ultmtly boudd. Fg. 5 shows th vlu of th Lypuov fucto. It s postv wth th oprto rg d ts drvtv wth rspct to tm s gtv. Lypuov fucto vlu.5 x Fg. 5. Lypuov Fucto Vlu 5. EXPERIMENL RESUL Fg. 6 shows th xprmtl rsult of cbl forc d olr cotrol sgl t ch oprto mod. It s cotrolld by PWM mthod, so mxmum vlu of cotrol sgl 1 [%]. () shows th cbl forc d (b) shows th cotrol sgl wh th EPB systm ppls brk forc. (c) d (d) show wh th brk forc s rlsd. h trgt forc of pply mod s ssumd to b 98[N] d th o of rls mod s ssumd to b [N]. I th ppld mod, lthough th cotrol sgl xsts, oc th stt trs th slf-lockg rgo, Ω, th cbl forc s mtd du to th scrw s frcto. I th rls mod, bcus th scrw frcto s smll, th rspos tm s fstr th th ppld mod. h cbl forc s dcrsd ftr th cotrol sgl bcoms zro du to lrg rt

7 Forc (N) Duty (%) Forc (N) () Prkg cbl forc (pply mod) (b) Cotrol sgl (pply mod) (c) Prkg cbl forc (rls mod) CONCLUSIONS hs ppr troducd EPB systm d dscrbd ts chrctrstcs. h EPB systm ws modlld s sttdpdt rbtrry swtchd systm. h cotrol rqurmts wr xpld d th olr P cotrollr ws proposd. h stblty of th closd-loop systm ws vstgtd by pplyg Lypuov d LSll s thory. h stts covrg to slf-lockg stt xpotlly d th qulbrum pot s loclly uform d ultmtly boudd. hrough xprmtl rsults, th prformc of th olr P cotrollr ws show. CNOWLEDGMEN hs ppr ws supportd prt by th Compots d Mtrls chology Dvlopmt Progrm of MCIE, Rpublc of or udr Grt hs work ws supportd by th Br or 21 Projct 27. REFERENCES Jum, P., J. Jord, Dr.. Jsus, Dr. C. Isml d F. Srg (24). Cocptos of ctutors for lctrc prkg brk (EPB). FISI tchcl ppr, Documt No. F24F182. Jg, F. d Z. Go (21). pplcto of Nolr PID Cotrol to Clss of ruck BS Problms. Procdgs of IEEE Cof. o Dcsos d Cotrol, 1, pp hll, Hss (22). NONLINER SYSEM (rd dto), pp. 112, Prtc Hll. L, Y.O., C. W. L, H.B. Chug, C. C. Chug, Y. So, P. Yoo d I. Hwg (27). Nolr Proportol Cotrollr for Elctrc Prkg Brk (EPB) Systms. 14th s Pcfc utomotv Egrg Cofrc, Documt No Nrdr,.S. d J. Blkrsh, (1994). Commo Lypuov Fucto for Stbl LI Systms wth Commutg -Mtrcs. IEEE rsctos o utomtc cotrol, 9(12), pp Olsso, H.,. J. Åström, C. C. d Wt, M. Gäfvrt d P. Lschsky (1998). Frcto modls d frcto compsto. Europ Jourl of Cotrol, 4, pp Duty (%) (d) Cotrol sgl (rls mod) Fg. 6. Exprmtl rsult: rgt forcs for () d (c) r 98[N] d [N], rspctvly. 1425