Power-Aware Scheduling under Timing Constraints for Mission-Critical Embedded Systems

Transcription

1 owr-wr Suln unr Tmn onstrnts or Msson-rtl Em Systms Jnn Lu, H. ou, Nr rz, F Kur Dpt. o Eltrl & omputr Ennrn Unvrsty o lorn Irvn, US jnnl, ou, nr, kur@.u.u Dpt. o Eltrl & omputr Ennrn Unvrsty o lorn t Irvn strt owr-wr systms r tos tt must mk t st us o vll powr. Ty susum trtonl low-powr systms n tt ty must l to not only mnmz powr wn t ut s low, ut lso lvr r prormn wn rqur. Ts ppr prsnts nw suln tnqu or supportn t sn n vluton to lss o powr-wr systms n msson rtl ppltons. It wll omput sul tt stss strnnt mn/mx tmn n mx powr onstrnts t ll tms. Furtrmor, t wll lso mk t st ort to stsy ts mn powr onstrnt n n ttmpt to ully utlz r solr powr or to ontrol powr jttr. Exprmntl rsults sow tt our utomt tnqus yl sns tt mprov prormn n ru nry ost smultnously ompr to nrt sns us n prvous mssons. Ts tool orms t ss o systm-lvl rmwork tt wll nl snrs to rssvly xplor mny mor powr-prormn tros wt onn. 1 Introuton owr mnmnt s omn on o t ntrl ssus n m systms. Ty r prtulrly rtl to systms tt must rry tr own powr sour n nnot rly on powr outlt on t wll. Wtout powr, t systm s uslss. In t onsumr sp, t onsqun my mn not n l to mk n mrny ll or otr mnor nonvnns; ut n msson rtl systms, su lur n ost mllons n vn umn l. Ts ppr nvstts ky ssus n powr mnmnt n msson-ornt systms. Our motvtn xmpl oms rom t NS Mrs tnr rovr vlop t JL [7]. It turs svrl ntrstn proprts tt wr not qutly rss y prvous work. Frst, su systm must sn to powrwr, rtr tn low-powr. Son, t s rtl tt t powr mnmnt sons must m t t systm lvl, rtr tn only t t omponnt lvl. 1.1 owr-wr vs. low-powr Trtonlly, mny omponnts n systms v n sn to low-powr. Howvr, w lv tr s rtl rn twn powr-wr n low-powr systms. owr-wr systms must mk t st us o tr vll powr, n ty susum low-powr s spl s. In t Mrs tnr s, ts snrs onstrut lowpowr sn. It norport som o t st low-powr sn ll pproprt ornztonl pprovls or t pulton o ts ppr v n otn. I pt, t utor(s) wll prpr t nl mnusrpt n tm or nluson n t onrn rons n wll prsnt t ppr t t onrn. tnqus t ll lvls o strton. T tnr rovr tsl s two powr sour: solr pnl n ttry. To strtly ontrol powr rw, t snrs srlz ll tsks, nlun rvn, strn, ostl tton, n tn motors. Ts low-powr sn llows t rovr to oprt or unrs o ys urn ylt, n t slps t nt. Howvr, ull srlzton lso mns t rovr movs s slowly s 1m pr mnut, n t n only tk totl o tr pturs pr y. powr-wr sn n rtly mprov t utlty o t rovr. W osrv tt t ttry s non-rrl, n tus solr powr woul wst not us wl t s vll. In t xstn sn, t rovr ollows t sm srl sul rrlss o t solr powr lvl, n smply rts t xss nry to tn t wls. rovr wt mor prlllsm n ts sul n prorm ttr (mor tsks, mor qukly) wl svn vn mor ttry nry tn t xstn low-powr sn t n tk vnt o t r powr, s vlt y our xprmnts n t rsults ston. 1.2 Systm-lvl powr-wr sn W lv tt powr-wr sns must on t t systmlvl, not just t t omponnt lvl. ml s lw ppls to powr s wll, not just prormn. Tt s, t powr svn o vn omponnt must sl y ts prnt ontruton n n ntr systm. I omponnt only rws 2% o t powr n systm, % ruton n ts powr mounts to mrly 1% svn n t systm. Tror t s rtl to nty wr powr s n onsum n t ontxt o systm, not just t omponnts n solton. In t s o t Mrs rovr, t turns out tt som o t st onsumrs r not vn n t tl omputr, ut ty lso nlu t wl motors, t strn motors, lsr-u ostl tton, n t trs. sussul powr-wr sn must onsr ts non-omputton omns n oornt tr powr us s wol systm. 1.3 ppro: sn tools Our ppro s to support powr-wr sn wt systm-lvl sn tool. On o t lssons lrn rom t rovr ws tt wtout tool, t snr s no opton ut to m mny powrmnmnt sons n t mplmntton. s rsult, ty wr or to sn onsrvtvly n oul not onsr mor tn on or two sn ltrntvs. T purpos o our tool s to nl t xplorton o mny mor ponts n t sn sp, so tt tonl knowl out t msson n norport to rn t sn wtout rqurn rmt rsn. T work prsnt n ts ppr rprsnts on o t or tools n ts lrr sn rmwork. T snr nputs -lvl

2 vorl spton o t sn n trms o ommuntn prosss n onstrnts. Ts prosss v n ssn to run on sp xuton rsours, tr ntrtvly or smutomtlly y t sn tool. T suln tool n ts ppr onstruts onstrnt rp n prorms powr-wr suln. T output s tn to notr tool tt prorms optmztons n syntss o powr mnrs t t rtturl lvl. Ts ppr s ornz s ollows. Ston 2 rvws rlt work, n Ston 3 srs t pplton xmpl n mor tl. W prsnt t prolm ormulton n Ston 4 n rps suln lortms n Ston. Tn, w suss xprmntl rsults n Ston 6 ollow y our onlun rmrks n utur work. 2 Rlt Work ror works v rss mnmzton o powr us t t systm lvl. Tr ommon ol s to mnmz powr us wl mntnn ststory lvl o prormn or mtn rltm onstrnts. Howvr, ts low-powr tnqus otn nnot rtly pt n powr-wr systms. 2.1 Susystm sutown Suttn own l susystms su s ntwork ntrs, r sks, n splys n sv snnt mount o powr n systm. T sutown son n s on l tms o nvul susystms, ltou su ppros r lss tn ststory. ropos mprovmnts tr ttmpt to mk t tmout ptv to t tul us pttrn, or us prols to lp prt t propr tm to sutown n powrup susystms. [, 3, 6] Wl t s mportnt to mn t powr o susystms, unortuntly ts tnqus v svrl lmttons. Frst, ty o not nl tmn onstrnts, nlun lns n mn/mx sprton. Son, ty r not powr-wr n t sns tt ty o not stnus twn r powr (su s solr sours) vs. xpnsv powr (non-rrl ttry). Ts powr mnrs o not ontrol tr worklo; nst, ty mk t st ort to mnmz powr y trtn t worklo s vn. 2.2 Rl-tm suln Mny rl-tm suln tnqus v n propos to t, ut only rntly v rsrrs strt to rss powr ssus wt t ojtv o mnmzn powr us. For xmpl, rt monoton suln s n xtn to suln vrlvolt prossors. T s to sv powr y slown own t prossor just nou to mt t lns. [4] Su tnqus v svrl lmttons. Frst, ty r U sulrs tt mnmz U powr, rtr tn powr mnrs tt ontrol susystms n tsk xutons. Son, n prt, t s xtrmly ult to tun t volt or rquny sl o su prossor. s rsult, t rsk o mssn lns my, vn t ontxt swtn ovr s tkn nto ount. lso, wl ts sulrs mt tmn onstrnts, ty o not nl onstrnts on powr us. 2.3 owr wrnss W lv powr-wr suln must v svrl ky turs. Frst, ty must nl ot tmn n powr strnntly s r onstrnts. Ts s unlk prvous work tt trts tm s srl y-prouts ut nnot lwys mk stron urnts. Son, omn-sp knowl out t powr sour, ttry mol, n otr oprtn ontons must xprssl n trms o support typs o onstrnts on t tmn n powr. T typs o onstrnts tt r suntly xprssv or our pplton r mn n mx tmn onstrnts on tsks, s wll s mn Oprton Durton(s) Tmn onstrnts Htn strn motors t lst s, t most s or strn Htn wl motors t lst s, t most s or rvn Hzr tton 1 t lst 1s or strn Strn t lst s or rvn Drvn 1 t lst 1s or nxt zr tton Tl 1: Tmn onstrnts n Mrs rovr s oprtons n mx powr onstrnts on t systm. Mn/mx tmn onstrnts susum lns n prn pnns n n xprss pnns ross susystms [2]. Mx powr woul trk t ut mpos y t urrnt powr sours. Mn powr onstrnts, strtly spkn, my ountr-ntutv n tt t ors t powr mnr to mntn rtn lvl o tvty. T prmry motvton s tt powr rom solr pnls or otr r sours tt nnot stor soul ully utlz rly, or ls ty wll wst. notr motvton s to ontrol t jttr n t systm-lvl powr urv n n ttmpt to optmz ttry us. Howvr, mn powr onstrnts r not mprtvly nor, n w ssum tt ty my volt osonlly or mt y suln kroun tsks. 3 Motvtn Exmpl To monstrt t tvnss n ppllty o t powrwr suln tnqus, w oos NS/JL Mrs rovr s our motvtn xmpl. Its msson s to prorm snt xprmnts n mn on Mrs sur. T rovr s ploy n oprt or t lst 7 sols (ys on Mrs). I t kps prormn wll t t n o t snt pro, n xtn msson my ontnu. T rovr s powr supply oms rom non-rrl ttry rry n solr pnl. lrly, t urton o msson s lmt y t mount o rmnn ttry nry. Tus, rul mnmnt o powr us my yl potntl nry svns, s wll s prormn spup. T rovr trvls mon rnt trt lotons or xprmnts n mn n prorm. Sn t tmprtur on Mrs sur s s low s ;8, rvn n low tmprtur rqurs mor powr onsumpton us t motors must t rom tm to tm. Ts t nts tt mnl n trml susystms r t mjor powr onsumrs. Tror, our mol trts t mnl n trml susystm unr typl msson snro wn t rovr s movn to t nxt loton. W v -lvl srpton o t rovr s oprtons. To strt snl stp o movmnt, t must tt ny ostls on t movn rton n oos s nl or t nxt stp. Tn t our strn motors r strt to turn to t rt rton. Fnlly, sx wl motors r rvn to prorm snl stp o movmnt. Tror, zr tton - strn - rvn must oprt n squn. T otr st o tmn onstrnt oms rom t rqurmnt to t t motors or strn n rvn. ll our strn motors n sx wl motors must t wtn rtn pro pror to mnl oprtons. T tmn onstrnts r summrz n Tl 1. T powr onsumpton o oprton vrs wt nvronmntl tmprtur. W suppos tt t tmprtur s losly rlt to t sunlt nsty tt n msur y powr output rom solr pnl. In orr to xmn ow t powr-wr suln tnqus nl rnt onstrnts, w nvstt tr ss: st s, solr powr output s 14.9W t noon tm; typl s, wn solr powr output s 12W; n worst s, solr powr output s 9W wn t sun s to o own. T mxmum supply powr s lmt y t trsol o ttry powr output. W ssum t mxmum ttry powr rw s 1W. Tror, n ll ss, t rovr n sly oprt only

3 Rsour Durton (s) owr (W) Solr powr ttry 1 mx 1 mx 1 mx Ht on motor Ht two motors Drv Str Hzr tton Tl 2: owr onsumpton o Mrs rovr s oprtons ts nstntnous powr onsumpton s lss tn vll solr powr plus 1W mxmum ttry powr output, w onsttuts t mx powr onstrnt. Tl 2 llustrts t powr sours n onsumrs. T purpos o sulr s to ssn tsks to tm slots su tt ll tmn n powr onstrnts r sts. Wtout n utomt tool, t xstn soluton y JL to nrt. It srlzs ll oprtons to mnmz powr rw rom t nonrrl ttry. T xstn sn s vry low-powr, ut s lso vry slow n n possly nur tonl nry ost n som ss. y ntroun powr-wr suln, not only oul w mprov prormn, ut lso sv non-rl nry y ttr utlzton o solr nry. Ts s n ontrst to t onvntonl tr-o twn nry n prormn, wr mprovmnt on on s t t ost o t otr. powr-wr ppro n wn ot t t sm tm. Ston 6 provs tl nlyss to s stuy on t Mrs rovr xmpl. 4 rolm Formulton Our prolm ormulton s s on n xtnson to onstrnt rp us n notr tm-rvn suln prolm. Ston 4.1 rvws t s ormulton n tn srs our xtnsons to nln powr onstrnts. Ston 4.2 prsnts wy o vwn t tm/powr suln prolm s two-mnsonl onstrnt prolm y rwn nlos rom t Gntt rt. 4.1 Input rprsntton T nput to t powr-wr suln tool s onstrnt rp G(V E), wr t vrts V rprsnt tsks, n t s E V V rprsnt tmn rltonsps. In ton, t nput nlus unton r(v) rom t oprtons to t xuton rsours; powr onsumpton unton p(v) or t stmt powr us. T snr lso nputs t mnmum n mxmum powr onstrnts on t sul Tmn onstrnt rp E vrtx s non-ntv wt (v) orrsponn to t urton o xuton. W ssum ll vrts rprsnt tsks wt oun xuton ly tt r non-prmptl. T sulr wll nrt sul y ssnn strt tm s(v) to vrtx v. T s twn prs o vrts spy tmn onstrnts onjuntvly. E (u v) s wt w(u v) tt onstrns t strt tms o u n v: t rqurs tt t strt tm o v must sul s lst w(u v) tm unts tr u s strt tm. Mor ormlly, s(v) ; s(u) w(u v). Dpnn on t wt, t s r torz s orwr s n kwr s. n (u v) wt nonntv wt s ll orwr, n t xprsss mnmum tm-sprton onstrnt. Otrws, n Ht wl 1 & 2 / Ht wl 3 & 4 / Ht wl & 6 / Stp 1: Hzr tton / 1 Stp 1: Str / Stp 1: Drv / Stp 2: Hzr tton / 1 Stp 2: Str / Stp 2: Drv / Ht str 1 & 2 / Ht str 3 & 4 / Fur 1: onstrnt rp or Mrs rovr s oprtons (u v) wt ntv wt s ll kwr, n t s mxmum tm-sprton onstrnt rom v to u [1] Rsour mppn To nl prlll xuton rsours tt onsum powr, t unton r : V! Zmps vrtx to rsour ID. Exmpls o xuton rsours nlu not only omputn rsours su s m mroprossor ut lso otr onsumrs o powr, nlun mnl susystms n trs. W urtr ssum tt two nos u n v r mpp to t sm rsour (ormlly, r(u)=r(v)), tn u n v must srlz n t nl sul. Fur 1 sows t tmn onstrnt rp or t Mrs rovr xmpl. It sows tsks tt spn tl, mnl, n trml omns. T U pross s onstnt kroun tsk wtout prnrltonsp to ny otr oprtons, so t s not nlu n t rp. To llow mor prlllsm, tr s no rstrton on ovrlppn oprtons. Unlss sp n t tmn onstrnt, no srlzton s nssry. notr xmpl s vll n ppnx owr onsumpton n powr onstrnts W lso ssum t vllty o t powr onsumpton unton, p : V! R>, w rturns t stmt powr onsumpton y t tsks rprsnt y ll t vrts. In prt, t powr onsumpton wll rn or n t orm o (mn, mx, typl), rtr tn n xt numr. Sn our ormulton n xtn trvlly to nl ntrvls, w wll ssum smpl numr p(v) to smply t susson r. T powr us o t systm s onstrn y two nput prmtrs, mx powr n mn powr. T mx powr s r onstrnt: t ny vn momnt, t totl powr onsumpton y ll runnn tsks must not x t mx powr. T mn powr s sot onstrnt: t sulr soul mk t st ort to mt t mn powr ol. Ts wll ontrol t mount o jttr n powr rw, s wll s nsurn ull utlzton o rnwl powr su s solr. In t Mrs rovr xmpl, t mount o vll solr powr n trnslt to t mn powr onstrnt to nsur ull utlzton o r powr. T mx powr onstrnt n st to solr powr plus mxmum powr rw rom ttry, ltou t snr n st lowr mx powr onstrnt to xplor mor onsrvtv sn pont. -

4 4.2 Output rprsntton T sulr omputs t tsks tt mp to tm slots s sul. sul S s t st o tupl (v s(v)), wr v 2 V n rp G, n non-ntv ntr s(v) s t strt tm ssnmnt to vrtx v n tm unts. ltou strt tm s(v) oul srv s n ttrut o vrtx v, w o ntn to sprt t rsults rom t nput st. Howvr, t onnton twn nput prolm st n rsults s somow vu. It s not sy to justy t proprts o t rsults y prormn, powr onsumpton, utlzton n t. rom t nton o sul. W ntrou t powr-wr Gntt rt s nw rprsntton or our powr-wr sul. It s propos or t lst two purposs: t s us y t sulr s rprsntton to t rsults, n t lso srvs s t unrlyn mol or vsulzton tool. Gntt rts r ommon wy o prsntn suls vsully. T xuton o tsk s rprsnt y orzontl n wos lnt orrspons to ts urton. Tmn onstrnts, n onpts su s suln slk n t tm mnson, tou normlly not sown, n lso ntutvly vsulz y sltvly ttn nnotton on t ns. W us t vrtl sz o t ns to not powr onsumpton. Not tt w lry v orzontl sz o t ns or t wt o no n tm mnson, t vrtl sz now rprsnts t wt n t powr mnson. tr sln t vrtl sz o t n wt t tsk s powr onsumpton, t r o t n nts ts nry xpntur, w s omn wt n nry omn. Ts nvstton rtly onnts t sul wt powr rtrsts: y ollpsn ll ns to t lowst orzontl xs, t xpt powr sur o t sul, ow t powr urv vrs wtn or wtout t mx-mn rn, n t mk-up o t powr ontrutors t tm n lrly vsulz. lortm T s to us rp lortms n suln prolms s s ollows. s t lortm prorms trvrsl to t rp, t onstrnts r nspt to quly t vlty o t urrnt prtl orrn. Wn omplt orrn s vr, t lortm susss wt sl soluton. Otrws, rnt orrs r ttmpt untl soluton s oun or ll omntons v l. Ts sms to strtorwr ppro to most prolms. s w us t powr-wr Gntt rt s t rprsntton o rsults, som nw turs rs s t ltrntv ormultons to t prolm st. Gntt rts sust tt t powr-wr suln prolm s n nloy to two-mnsonl n-pkn prolm. It my ttmptn to solv ts prolm n n-pkn mnnr. Howvr, t soluton sp o su N-r prolms rows ruptly s t nput prolm st nrss. W v not yt sovr n tv lortm tt n solv t prolm y snl run. On unmntl tur o t ntv suln prolm n y t rpl rprsntton s tt, t onstrnts n t prolm st soul not prortz vnly s two-mnsonl r ounrs n t n-pkn vw. T tmn onstrnt s t most rtl on tt must tkn r o rst. I no vl tm sul xsts, tr s no pont to onsr powr onsumpton. Tus, t orzontl mnson n vrtl mnson n t rpl vw r not qul n snn. Tror, solvn t two-mnsonl n-pkn prolm t t sm tm oms lss promsn sm. Ts susts n nrmntl ppro. Frst, s on prn rltonsps n onstrnt rp, w try to n sul tt s vl on tm mnson. owr onstrnts n powr wts r not us n ts stp. W xtn t xstn lortm to sovr ll sul rom onstrnt rp. T lortm s sr n Ston.1. MxowrSul(GrpG, vrtx nor, Mxowr) sul := rlllsul(g, nor, nor); (sul = FIL) rturn FIL; ts := xuton tm o sul; or (t := ; t ts ; t := t +1) S := st o tv vnts t t; powr := powr onsumpton o ll vnts n S; ExtnSul := FLSE; wl (powr > Mxowr or ExtnSul) : rpt v := most slk vnt n S; (slk(v)= )ExtnSul := TRUE; ly v; powr := powr ; p(v); S := S ;v; untl (powr Mxowr or S = /); (S = /) rturn FIL; lok strt tm o ll vrts n S; sul := MxowrSul(G nor Mxowr); (sul 6= FIL) rturn sul; ExtnSul := TRUE; Uno s sn stp ; /* nnr loop, wl(:::) */ /* outr loop, or(t :::) */ rturn sul; Fur 2: Suln lortm or mx powr onstrnt Son, s on vl sul on tm omn, powr wts o vrts n mx powr onstrnt s ppl to just t xstn sul. Ston.2 xplns t mx powr onstrnt suln lortm w s los to n-pkn ppro s on powr-wr Gntt rt rprsntton. To vo xustv sr n t soluton sp, som ursts r us to v nts so tt mor rsonl solutons r xmn rst. Fnlly, vn sul tt mts ot tmn n mx powr onstrnt, w mk urtr justmnts to mt t mn powr onstrnt n Ston.3. Smlr to stp two, som n-pkn ursts r us. Sn t mn powr onstrnt s sotonstrnt, t lortm os not urnt tt powr onsumpton wll lwys x mn powr lvl, ut rrns t powr sur to rs wtn t mn-mx powr rn s mu s possl..1 lortm or prlll suln n tm mnson Tm-onstrn suln s strtorwr xtnson to prvous srlzton lortm [1]. Rtr tn srlzn ll vrts, t nw lortm only srlzs tsks tt sr t sm xuton rsour. notr moton s tt vrtul s r twn vrts ross rsour ounrs to mntn topolol trvrsl to t rp. Smlr to t prvous stuy, t nw lortm n prov to lwys n vl sul on xsts. T lortm n n xmpl r llustrt n ppnx..2 lortm or mx powr onstrnt suln T ppro to mtn mx powr onstrnt s smlr to run t t o ns n n-pkn mto. T lortm s sown n Fur 2. It s tr prmtrs: rp G, vrtx nor, n slr onstrnt Mxowr. T prlll sulr s lwys ll rst to nsur vl sul n tm omn. T lortm sns t rturn sul to n t rst tm t wn t mx powr onstrnt s volt, w s rrr to s powr spk. To lmnt t spk t ts pont, svrl smultnous vrts r ly so tt t t o t powr urv s low

5 Mxowr. To ly vrtx v, w n rom nor to v, wt sr postv wt s t strt tm. T mto tsl s ll rursvly tr t spk s lmnt y lyn vrts. vl sul s oun t lortm nnot n ny spk n t outr loop. T sul s rturn tr t outr loop s omplt. I no sl soluton n oun, lur not s rturn sustn tt tr mor vrts n to ly or t lry ly ons v n norrtly slt. T ky ssu n ts lortm s to proprly slt vrts to ly. In t worst s wr t wron ons r ly vry tm, t tm omplxty o xustv numrton s O(n! xp(n)). T o o vrts to ly or spk lmnton s mportnt, sn ly osn vrts oul not only us t lortm to l, ut lso xtn t sul tm unnssrly, ln to poor prormn. oo o o vrtx soul (1) nur mnml tonl xuton tm (2) nsstt mnmum rsuln to n vl soluton. W propos urst or oosn vrts s on slk. In our ontxt, t slk o vrtx s t mxmum mount o tm y w vrtx n ly n vl sul wtout voltn ny tmn onstrnts. T onpt rs rom t onvntonl mnn s t stn o n vnt to ts ln. Gvn orwr n kwr s, t slk o vrtx n torz s orwr slk n kwr slk. I (v u) s orwr, t orwr slk o v rrn y u s n s s(u) ; s(v) ; w(v u). Ts mns v s ly y ts mount o tm unt, t sul stll rmns vl sn t mn onstrnt stll ols. I v s multpl outon orwr s, ts orwr slk s t mnmum slk n rrn to ll trt vrts. T kwr slk rrs to t mxmum ly llow to t vrtx wtout voltn mx onstrnts. I vrtx v s kwr to u wt wt ;w(v u), t kwr slk n rrn to u s w(v u);s(v)+s(u), w sns ow r v n ly wtout nvltn ts mx tmn onstrnt. Smlrly, vrtx v s multpl outon kwr s, t mnmum kwr slk s slt. Not tt slk s only mnnul wt rr to outon s, tr orwr or kwr, o vrtx. Dlyn vrtx v os not volt ny tmn onstrnts sp y nomn s o v. I v os not v ny outon orwr s or kwr s, t orrsponn slk s st to t postv nnt vlu tt mpls t vrtx s not oun y rltn onstrnt t ll. Fnlly, t slk o vrtx s t mnmum vlu o ts orwr slk n kwr slk. t pont wn mx powr s x, t lortm lwys lys t vrts wt mor slk untl powr onsumpton s lowr wtn t s rn. T llustrton or su ry urst s tt w prsrv t lst slk vrts to vo lonr xuton tm n op to n soluton mor qukly. Tr r ss wr no vrtx wt non-zro slk s vll, or, tr vn ly nou vrts to lowr own t powr urv vl sul s not oun. Ts susts tt mx powr onstrnt nnot mt wtout xtnn t sul tt ls to lonr xuton tm. Ts ss rr to turnn t ooln vrl ExtnSul to TRUE n t lortm. T slk-s ursts most lkly l to t orrt solutons. I tr s no vrts wt non-zro slk r vll, t sul must xtn. Howvr, tr r som unsrl ts n ts s. Wn non-slk vrtx v s to ly t tm t, t sul oul n to t vrts strtn tr t, ut lso t ons or t. T vrts tt r stntons o orwr s rom v my ly tr t. In ton, t vrts wt nomn kwr s rom v oul lso ly. Su vor rsults n loosn sul y lyn vrts trou kwr s. To prvnt ts, w oul urtr xtn our slk-s sortn prour so tt t vrts MnowrSul(Grp G, vrtx nor Mxowr Mnowr) sul := MxowrSul(G nor Mxowr); (sul = FIL) rturn FIL; ts := xuton tm o sul; or (t1:=;t1 ts; t1:=t1+1) S1 := st o smultnous vnts t t1; p1:= powr onsumpton o ll vnts n S1; (p1 < Mnowr) or (t2:=t1-1;t2 ; t2 :=t2-1) S2 := st o smultnous vnts t t2; p2 := powr onsumpton o ll vnts n S2; (p2 > Mnowr) rpt : v := nxt nt n S2; (v s qul to ly to t1) ly v to t1; S2:=S2-v; sul := MnowrSul(G nor Mxowr Mnowr); (sul 6= FIL n xuton tm o sul ts) rturn sul; Uno s sn stp ; untl (S2= /); /* (p2:::) */ /* nnr loop, or(t2:::) */ /* (p1:::) */ /* outr loop, or(t1:::) */ rturn sul; Fur 3: Suln lortm or mn powr onstrnt wt kwr s wll onsr lst. ut ts stll nnot vo t prolm ompltly wn ll nts r zro-slk on kwr. W us mu smplr urst. tr nou vrts r slt to ly n mx powr onstrnt s sts, w lok t strt tm o rmnn vrts n t nt st. T strt tm o vrtx v s lok to t y n two s, postv (nor v) wt wt t, n ntv (v nor) wt wt ;t. Tror, ts vrts r rtrrly or to strt t rtn tm n no urtr lys n prorm to tm unlss t xtr s r unon. Howvr, su ly s mntory to sl soluton, t lortm wll rturn l n ltr rursons n t lortm wll r on vrtx rom t st to mk urtr ly n ll t rurson n. It s notl tt n som xtrm ss, t mx powr onstrnt sulr my not l to n vl sul vn tou tr xsts on. T rson s tt wn t rurson rturns l, t lortm os not numrt ll possl omntons n sltn vrts rom t nt st. Howvr, n prt, our ursts prorm vry wll n strn to vl soluton wtout unnssry sr on totl sul tm. Ts s us our slks ursts mk stron nts towr t orrt rton to sl soluton. lso, t urst to lok t vrts n t nt st or lln t lortm rursvly n lp to ru t possl orrn n ot prlll suln lortm n ltr rursons o mx powr onstrnt sulr..3 lortm or mn powr onstrnt suln Smlr to mx powr onstrnt suln, t mto to stsy mn powr onstrnt s nloous to n-pkn prolm wt mnmum t rqurmnt. T lortm s sown n Fur 3. Four prmtrs r pss to t lortm: rp G, vrtx nor, slr onstrnts Mxowr n Mnowr. vl sul oms rom lln mx powr sulr t t nnn o t lortm. T lnt o sul s not n u to stsyn mn powr onstrnt, w s sot-onstrnt. T lortm ks t sul to n t rst tm t1 wn powr onsumpton s low Mnowr, w w rr to s powr

6 p. Tn t sul s nspt rvrsly n t tm omn to n t2 wr no powr p s prsnt. I tr s ny pproprt vrtx t t2 to ly to ll t p t t1, t lortm lys t nt n lls tsl rursvly. Only ly s n us t mx powr sulr prrs to ssn t rlst strt tm to vrts. Otrws no vrtx t t2 s qul to ll t p t t1, t nnr loop ontnus to k t sul kwr untl t nss. Tn t lortm pros orwr to n t nxt p. owr Sul or st s Ht str Ht wl Drv Str Hzr tton tr t wol sul s nspt, t outr loop omplts. T nl sul s rturn s sl soluton. Howvr, t rursv ll to MnowrSul rturns wors sul, t lortm wll rturn t ttr on tt s xst or t rursv ll. T lortm rturns lur not only mx powr sulr nnot n sul, wn t prolm ls on r-onstrnts. T ntrstn prt n ts lortm s n t nt slton to ly vrts. s w xpt, lyn vrtx to mt mn powr onstrnt woul (1) rn no xtr xuton tm (2) rsult mnmum n to t xstn sul (3) nl to nry svn. Our ursts on oosn ly nts r stll s on slk, s sr n t prvous lortm. Howvr, u to t rnt ntur o t mx n mn powr onstrnts, w pply rnt pols n ts lortm. W prorm mor omplx nsptons to wtr vrtx v t tm t2 s qul to ly to ll powr p t t1. Frst, v s non-slk vrtx, no ly soul prorm sn w o not xpt lonr sul tm. Son, t slk o v must t lst t1 ;t2. Otrws movn v to t1 wll lkly rsult n lonr sul. Nxt, ny vrtx u n st S1, w s t st o smultnous vrts t t1, srs t sm rsour wt v, v soul not t nt, sn lyn v to t1 wll t lst or u to rsul, n oul l to n xtn sul. Fnlly, t ly soul nl to nry svn or mor ln powr sur. Tt s, lyn vrtx v t t2 to powr p t tm t1 s nl, tr soul not om lrr p t t2 trwrs u to t sn o v. In t n-pkn prsptv, vrtx v s to pl n tr t2 wr t powr onsumpton s p2; p(v), or t1 wr powr p p1 s prsnt. Vrtx v soul poston to t pl wr t t o t n s lowr. Tt s, only p2 ; p(v) > p1, s t ly oo mov. T runnn tm o t lortm my row n som xtrm ss. Ts s us t prolm on mn powr onstrnt s potntlly rr tn t otr two prolms; n w o not try to lmt t soluton sp y prtlly rwrn t xstn sul. T xtr runnn tm s orl to ts sot-onstrnt prolm. t lst w lry v sl soluton tt mts ll rtl onstrnts n trms o tmn n mx powr rstrtons. t ts pont w s t vnt to rk t prolm nto svrl stps. W try to solv rtl prolms rst, n nssry, or som rsonl ssumptons n orr to n t soluton qukly. s r s t r-onstrnts r rsolv n sl soluton s prov, w o not mn llown t lortm to tk mor tm on rltvly non-rtl justmnts. In t, w o pply som otr sts o rstrtons to tvly lmt t N-omplx soluton sp. Our ursts only quly vrts rstrt y sp proprts to rorr. In prt, ompr wt mx powr onstrnt sulr, t tonl rspons tm o ts lortm s nsnnt n most ss. owr-wr suln rsults o t Mrs rovr xmpl n oun n Ston 6. Fur 11, 12 n 13 n ppnx llustrt t stps n mx n mn powr suln to notr xmpl. Two sts o mx/mn powr onstrnts r xmn: 2/1 n 1/. Our sulr vs sl suls n ot ss owr Tm Fur 4: Sul or t st s Sul or typl s Tm Fur : Sul or t typl s 6 Exprmntl Rsults U Ht str Ht wl Drv Str Hzr tton Ts ston prsnts suln rsults or t Mrs rovr oprtons n s stuy or vlutn our powr-wr suln lortms n msson snro. Fur 4, n 6 sow t rsults or tr ss tr pplyn powr-wr suln lortms. Fur 4 vs rst two trtons o t loop n t st s. To utlz t vll r nry, w mnully unroll t loop n nsrt two tn prosss to mprov loop ny on ttr solr nry utlzton. Tror t son trton n rpt wtout too mu nry ost. In otr ss only on trton s sown sn loop unrolln s not nssry. In t st s, us powr ut s sunt, st sul s vn y llown oprtons to ovrlp. In t typl s, prlll oprtons r stll possl wl som tn prosss r srlz. In t worst s, tt powr ut ors ll oprtons to srlz, ln to slow sul. It s nssry to look t t xstn sul n prt. To vo xn mx powr supply, JL v srlz sul tt s x n ll stutons, rrlss o vll solr powr n owr Sul or worst s Tm Fur 6: Sul or t worst s U Ht str Ht wl Drv Str Hzr tton U

7 Solr powr (W) ttry nry (J) Solr nry (J) % o solr nry Tm (s) Movn stn % 7 2 stps - 14m % 7 2 stps - 14m % 7 2 stps - 14m Solr powr / 16 Rltv sp 14 Exuton sp sls wt powr sour n powr-wr suln Solr powr Tl 3: rormn o t rovr unr xstn sul 12 1 Sp (JL) Solr powr (W) ttry nry (J) Solr nry (J) % o solr nry Tm (s) Movn stn / % 2 stps - 14m % 6 2 stps - 14m % 7 2 stps - 14m Tm Sp (owrwr) Tl 4: rormn o t rovr unr powr-wr suls Solr powr / 16 owr rw rom ttry Ru nry ost n powr-wr suln Solr powr 12 powr onsumpton n rnt ontons. T xstn sul ppns to xtly t sm s our sul or t worst s. ut t unrlyn stnton s tt, our sul s ompltly onstrnt-rvn; t xstn soluton rwrs srlz ppro wtout wrnss o unsty powr onstrnts. T prormn n nry ost o our suls n xstn sul r ompr n Tl 3 n Tl 4. W us xuton tm n non-rrl nry ost s t mtrs to vlut our suls n ompr wt t xstn soluton. T xstn sm only suls or t worst s; wl n otr ss, solr nry s unr utlz n potntl opportunts to prormn mprovmnt r ovrlook. Howvr, ts ls to smnly onom ppro sn t nry ost s low. Our suls, on t otr n, spup t rovr s movmnt t % n t st s n 2% n t typl s, wl rwn mor non-rrl nry rom t ttry. To vlut ts tr-o, w pply our suls n t xstn sul to msson snro wn t vll solr powr vrs ovr tm, n ompr t prormn n nry ost n ts r ptur. W suppos t msson s to trvl to t nxt trt loton, w s 48 stps wy rom t urrnt loton. T msson strts roun noon wn mxmum solr powr s prsnt. Durn t pro wn t msson s n prorss, t powr output rom t solr pnl rops rom 14.9W to 12W tr 1 mnuts, tn lls to t worst s t 9W 1 mnuts ltr. I t xstn sul s ppl, t rovr wll spn 1 mnuts wlkn vnly n t st s, typl s, n worst s sn ts xuton sp s not wr o powr onstrnts. Ts rsults n lon xuton tm n onsrl nry ost n t worst s. Wn our suls r us, t rovr nss % o work n t st s, 42% o work n t typl s, lvn rst 8% to t worst s. Sn our suls spup xuton t t st s n typl s, t rovr n ns t msson rlr or vn to work n t ostly worst s. T rsults o ts s stuy r sown n Tl. T nlyss sows our sulswn ot on prormn n nry svns onsrly. Fur 7 lts t proprty o t powr-wr sulr n omtrl vw. T top rt llustrts ow our soluton justs t xuton sp ptvly wt vll powr ut, wl t xstn sm nors t powr onstrnt n lwys JL owr-wr Tmrm Solr powr Trvl Tm Enry Trvl Tm Enry (s) (W) stn (s) ost (J) stn (s) ost (J) Tot l Improv mnt 24.4% 32.7% Tm ttry powr - (JL) ttry powr - (owrwr) Fur 7: pttv spup n powr-wr suln oprts t t lowst sp. T worklo s rprsnt y t r (ntrl) o t sp urv ovr tm. Tror our urv rs t vn worklo rlr us o r xuton sp or oprtn n t worst s. T ottom rt sows t powr rw rom ttry ovr tm n ow t ltrs s powr onstrnt vrs. T nry xpntur s symolz y t ntrl o powr urv ovr tm. Wn t msson s omplt, ot sp urv n powr urv stop. ltou our powr urv s r n most tm urn t msson, y ompltn rlr w vo urtr nry ost rom ntrtn powr urv wt lonr xuton tm. Tror, vn t sm worklo, t powr-wr sulr wns ot on prormn n nry svns to non-rnwl sour. 7 onluson n Futur Work owr-wr sn oms mor mportnt ssu n msson rtl systms tt rqur st us o vll powr sour n lvrl prormn t t sm tm. W trt t suln lortms to systms wt vrous powr onstrnts n rnt lsss o powr onsumrs, wr powr-wr tnqus v potntls to ot prormn mprovmnt n nry svns. In ts ppr, w prsnt onstrnt-rvn mol tt norports powr n tmn onstrnts n systm-lvl ntrprtton. W propos tr or lortms tt rk t powr-wr suln prolms nto stps. V n nrmntl ppro, w stnus t ntur o su-prolm n pply ursts to solv t onstrnts y rnt mtos. T s stuy to rl pplton monstrts tt our powr-wr mto s pl o mprovn prormn wl svn non-rnwl nry. Svrl ntrstn ssus n ts mnson n urtr ttnton. To xpn t ppllty o our lortms, mor tv ursts n to sovr. s our xmpl sows, utomt loop suln tnqus r nssry or t powr-wr sn to lvr prormn n ost-tv mnnr. W woul lso lk to norport mor novl powr mnmnt tnqus nlun volt/rquny sln nto ts tool to support mor tv powr-wr sn. Tl : omprson o xstn sul to powr-wr suls unr msson snro

8 rlllsul(grpg, vrtx nor, vrtx ) L := Snl sour lonst pts (G nor); (postv yl oun) rturn FIL ; := st o topolol sussors o nt ; ( = / ) rturn sul wt s() := L; D := ; wl (D 6= /) v := SltSussor(D); : or u 2 ;v (r(v)=r(u)) (v u) to G, wt wt w(v u) := Mx((v) L(u) ; L(v)); ls ( (v u) os not xst) vrtul (v u) to G; w := t most rntly sul vrtx, wr (r(w)=r(v)) (w 6= nl) (w v) to G, wt wt w(w v) := Mx((w) L(v) ; L(w)); sul = rlllsul(g nor v); (sul 6= FIL) rturn sul wt s() := L; Uno s sn stp ; rturn FIL ; Fur 8: rlll suln lortm Vrtx Rsour Durton owr /2 / /3 /4 7 8 /3 /2 / /2-6 /3 Fur 9: onstrnt rp o t xmpl 2 Rrns [1]. ou n G. orrllo. Sotwr suln n t osyntss o rtv rl-tm systms. In ro. Dsn utomton onrn, ps 1 4, Jun [2]. ou n G. orrllo. Intrvl suln: Fn rn o suln or m systms. In ro. Dsn utomton onrn, ps , Jun 199. [3] E.-Y. un, L. nn, n G. D. Ml. Dynm powr mnmnt usn ptv lrnn tr. In ro. Intrntonl onrn on omputr- Dsn, ps , [4] T. Okum, T. Isr, n H. Ysuur. Rl-tm tsk suln or vrl volt prossor. In ro. Intrntonl Symposum on Systm Syntss, ps 24 29, [] T. Smun, L. nn, n G. D Ml. Evnt-rvn powr mnmnt o portl systms. In ro. Intrntonl Symposum on Systm Syntss, ps 18 23, t Fur 1: rlll sul rsult [6] M. Srvstv,. nrksn, n R. rorsn. rtv systm sutown n otr rtturl tnqus or nry nt prormml omputton. IEEE Trnstons on VLSI Systms, 4(1):42, Mr [7] NS/JL s Mrs tnr Hom ttp://mrs3.jpl.ns.ov/mf/nx.ml ENDIX: Dtl lortm llustrton t Fur 11: suln owr urv o t xmpl or powr-wr

9 Most slk vnt s ly to mt mx powr onstrnt. s rsult s lso ly t (1.) Mx powr sulr(2): ly owr p (1.) vl sul tr mx powr sulr owr p t (2.) Mn powr sulr (2/1) stp 1: ly to ll t p ovr Most slk vnt, Mnmum ly to ll t slot unr mn powr owr p owr p t ( 2.) Rsult tr mn powr sulr (2/1) stp 1: tr ly, nw p pprs ovr Dly or wll nrs xuton tm. Sn mn powr s sot onstrnt, Dly to on t ly tm. ovrlp wt Sn n r on sm rsour, w n not ly to ovrlp wt, s not slt t (3.) Mn powr sulr (2/1) stp 2: ly to ll t p ovr ; notn to o to ll t p ovr owr p owr p owr p (3.) vl rsult tr mn powr sulr (2/1). No mor movs r vll ltou tr r stll ps. Fur 12: Stps o powr-wr suln, mx powr = 2W, mn powr = 1W t t 2 Dly mor slk no t (1.) Mx powr sulr (1): ly n to mt mx powr 1. Lonr xuton tm s nssry to mt mx powr onstrnt. 1 1 owr p owr p t (1.) Rsult tr mx powr sulr (1) 1 ot n r tt ( slk), lol ly must tkn. os to ly t. ( 2. ) Mn powr sulr (1/), stp 1: ly to ll t powr p ovr t Dly to ovrlp wt (2. ) n ntrmt stp tr stp 1: t wol rp must rsul. Ts s snpsot or nw sul rturn rom prlll sulr s pss to mx powr sulr t (3. ) Mn powr. sulr (1/), stp 2: n t nw sul, om t slk on to 2 ly or mx powr t (3. ) Rsult o mn powr sulr (1/), swp squn wt sn w o not put loks. Ts s stll vl sul tr stp 1, mx powr sulr s ll. Sn w on t rwr ny ssumptons, t sulr s no knowl o xstn sul. Tror, n mx powr sulr, oms most slk on to ly Fur 13: Stps o powr-wr suln, mx powr = 1W, mn powr = W