Longitudinal Spacing Control of Vehicles in a Platoon 1

Transcription

1 9 ICASE: The Insttute of Control, Automaton and Systems Engneers, KOREA Vol., No., June, 000 ongtudnal Spang Control of Vehles n a Platoon Tae Soo No and Kl To Chong Abstrat: The yapunov stablty theorem s used to derve a ontrol law that an be used to ontrol the spang between vehles n a platoon. A thrd order system s adopted to model the vehle and power-tran dynams. In addton, the onept of "Expeted Spang Error" s ntrodued and used to form a yapunov funton. Then a ontrol law that always dereases the yapunov funton s seleted. A platoon of four vehles and varous senaros are used to demonstrate the performane of the proposed ontrol law. Smulaton results show that the slnky effets are essentally removed and the robustness to the unertanty of vehle dynams s substantally mproved over prevous results. Keywords: yapunov funton, vehle ontrol, longtudnal ontrol, platoon I. Introduton It has been shown that the onept of operatng vehles n a platoon an sgnfantly mprove the effeny of exstng road systems. In order to meet suh expetatons, t s neessary to develop advaned vehle ontrol systems (AVCS) whh form a part of ntellgent vehle/hghway systems (IVHS)[]-[4]. As shown n Fg., one of the many funtons of AVCS s longtudnal ontrol, n whh the preassgned spang H between the predeessor, M and the follower vehle, M n the platoon s to be tghtly mantaned. Most of the earler works assume that the follower vehle M has aess to the veloty v and aeleraton a of the platoon lead vehle M and an therefore, measure the relatve spang, relatve veloty and relatve aeleraton between tself and the vehle n front of t[5]-[8]. There are many strateges for desgnng longtudnal ontrol laws. To te a few, PID type ontrollers have been proposed by Shladover [5][6], Shekholeslam [7][8], and Fujoka et al [9]. Hedrk et al. [0][] have extensvely used the sldng mode ontrol methodology. In the lterature ted above, the general struture of the ontrol nput a to vehle M may be wrtten as a = f( x, v, a, x, v, a, v, a ) () where( x, v, a ) denote the absolute poston, veloty, and aeleraton of vehle M. It has been reported that m or less spang an be mantaned n a platoon[5]-[9] by usng suh ontrollers. Shekholeslam and Desoer []. showed that tght spang ontrol s stll possble wth no ommunaton of nformaton about the lead vehle. In ths paper, we propose another form of longtudnal ontrol law, vz., a = f( x, v, a, x, v, a, a ) () whh ompletely elmnates the need for a ommunaton lnk between the platoon s leadng vehle and the follower ㄷ vehles, and uses the loal lnk between the predeessor and follower vehles, as depted n Fg.. We wll show that the proedure used n ths paper for desgnng ontrol laws somewhat mtgates the ad-ho nature of seletng the varables for feedbak or the struture of the sldng surfae. Also, t effetvely aommodates wde varatons n vehle Manusrpt reeved: Marh. 0, 000., Aepted: June. 0, 000. Tae Soo No, Kl To Chong: Mehatrons Researh Center, Chonbuk Natonal Unversty dynams and s sutable for the spang ontrol of mxed vehles. X -, v - Follower a - Predeessor M M - X Referene ne H Communaton nk Measurement nk X- M X v, a Fg.. Platoon wth global lnk. Communaton nk Measurement nk X -, v - X, v a To M - a M M - M To a v, a a - M a a - a Fg.. Platoon wth loal lnk. ead vehle M x ead vehle M II. Vehle dynams model Varous models for vehle dynams have been used n the study of longtudnal ontrol of platoons. For a platoon travelng wth a onstant speed and dreton, we have adopted the followng thrd-order model [5][6], x& = v (3) v& = a (4) ( a& = a a ). Fg. 3. Blok dagram of vehle dynams. where, the tme onstant of the vehle propulson system, s a sngle parameter whh represents the haratersts of dfferent vehles. Shown n Fg. 3 s the blok dagram of vehle dynams. Jerk and aeleraton lmts are easly (5)

2 Transaton on Control, Automaton, and System Engneerng Vol., No., June, onsdered wth ths model. Referrng to Fg., the error n spang dstane between vehles M and M an be wrtten as δ = x x H (6). Then & δ and δ & represent the relatve veloty and aeleraton, respetvely. III. ongtudnal ontroller desgn The objetve of longtudnal ontrol s to mantan the spang error δ below a predetermned level or, f possble, at zero. For example, the PID type ontroller may be wrtten n ts general form as a = k δ + k & δ + k δ & + k ( v v ) + k ( a a ) (7) x v a v a where k, et, are ontroller gans. Wth ths ontrol law, the x ontrol ommand s determned usng the nformaton measured and/or estmated at the urrent tme. In other words, the ontroller s always workng, as long as there s a spang error at the urrent tme. The only tme the ontroller stops workng s when the followng ondton s satsfed: 0 = k δ + k & δ + k δ& + k ( v v ) + k ( a a ). (8) x v a v a However, f some predtve nature s norporated nto the ontroller, one an make t behave more ntellgently. Even f urrent δ s large, f t s expeted to be dereasng, no further ontrol aton would be neessary. By ontrast, some aton s requred when δ s expeted to nrease even though ts urrent magntude s below the predetermned level. Wth the above reasonng n mnd, we an ntrodue the onept of "expeted spang error" as follows: d = ( x x ) + ( v v ) t + ( a a ) t For the sake of brevty, we assumed H s zero. As one may easly understand, d s the spang error at tme t f both the predeessor and follower vehles keep ther respetve urrent aeleratons onstant for t. Therefore, d reflets not only the urrent spang error but also the relatve veloty and aeleraton. One may let t = t t (0) where t s the urrent tme and t f s the unspefed future tme. In ths paper, we have used the yapunov stablty theorem [3] and the expeted spang error to obtan the ontrol laws. Frst, let's defne a postve defnte salar funton, that s, the yapunov funton, as shown below. V = f d. (9) () Dfferentatng the above yapunov funton gves V & = d d& () and usng Eq. (9) d& = ( x& x& ) + ( v& v& ) t ( v v ) + ( a& a& ) t ( a a ) t ( = a& a& ) t (3) sne we have assumed engne and powertran dynams as n Eq. (5), Eq. (3) beomes d & = ( a a ) ( a a) t (4). To use yapunov stablty theorem, we selet a so that d & = N d (5) where N s a postve onstant. Then, Eq. () may be rewrtten as V & = N d (6) whh s a negatve defnte funton. Ths mples that the yapunov funton V s a strtly dereasng funton, meanng that yapunov stablty s assured. Fnally, the ontroller whh satsfes the above argument s N a = ( a a ) + a + d (7) t where t and N are left undetermned as desgn parameters. Whle the proposed ontrol law stll needs to measure the spang error and relatve speed/aeleraton, the man dfferene omparng wth prevous ontrol laws s that the follower vehles do not reeve any nformaton dretly from the platoon lead vehle exept the frst one. Instead, the undretonal ommunaton lnk s requred, for eah follower vehle to reeve the ontrol ommand from the preedng vehle. Another feature of the proposed ontrol law s that t systematally aommodates the wde varaton n dynam haratersts of the vehles partpatng n the platoon. Ths feature wll be benefal f a platoon s omposed of vehles of varous knds. Therefore, the follower vehle has full aess to the predeessor vehle for nformaton suh as ontrol ommands and dynam haratersts. Further, due to the asaded struture of the ontrol laws, the follower vehle knows what the urrent stuaton s, and what all the vehles n front of t are about to do. IV. Smulaton and performane analyss As shown n Fg. 4, a four-vehle platoon s hosen and used n the smulaton beause t must retan the essental haratersts of a platoon. Indvdual vehles are assumed to be dental and the desgn parameters t and N are seleted after some numeral experments. Just for omparson purposes, a PID type ontroller gven by Eq. (7) and a smulaton senaro were adopted from referene [6]. The relevant numbers used n the smulaton are summarzed n Table. The spang dstane H s set at zero for ease of presentng the smulaton results. Then, the ontrol objetve s that every vehle lnes up wth the lead vehle. Ths should make no dfferene n nterpretng the smulaton results.

3 94 ICASE: The Insttute of Control, Automaton and Systems Engneers, KOREA Vol., No., June, 000 Table. Vehle and ontroller haratersts. Vehle Charatersts: = 0. se PID Controller: k x = 3.6, kv = 0.9, ka = 0.0, kv =.4, ka = 0. 0 yapunov Controller: t =.0, N = 0 Fg. 4. Platoon smulaton envronment.. Response to abrupt speed hanges by the lead vehle In ths example, the lead vehle aelerates at m /se to speed up by m /se after seonds and there are no ntal spang errors. As one may see from Fg. 5, the PID-type ontroller performs just as human drvers do. In the early stage, the spang error between the lead and frst vehle s the largest. That s beause of the delay n propulson or engne dynams. Although the frst vehle notes the aeleraton o f Fg. 5. Response to sudden speed hange. the lead vehle earler than the other vehles behnd t, the dfferene n aeleraton and speed are stll great, produng a large spang error. But the spang error between the next two vehles gets smaller beause the dfferene n aeleraton and speed have not yet bult up enough to yeld noteable errors. However as tme es by, the frst vehle athes up wth the lead vehle faster than any of the other vehle and spang errors dsappear after about 6 seonds. However, the yapunov ontroller proposed n ths paper shows exeptonal performane n the sense that all vehles move together and aheve almost dental aeleraton and speed wth eah other. In ths ase, the follower vehle knows what ts predeessors wll do, and takes the same ontrol aton. Therefore, the spang errors reman vrtually unhanged.. Response to ntal spang errors In ths senaro, all the vehles are movng wth the same speed as that of the lead vehle, and there are ntal spang errors, as shown n Fg. 6. To lne up wth the lead vehle, t would be best f the frst vehle deelerated, the seond kept the urrent speed (no ontrol aton), and the thrd vehle aelerated. Unfortunately, the struture of the PID ontroller does not allow that sort of ntellgent maneuver. Beause the frst vehle s ahead of the platoon leader, t wll deelerate and beause the seond vehle s behnd the frst, t wll a- (ead Vehle) (ead Vehle) 3 Travel Dreton 3 Travel Dreton (a) PID ontroller (a) PID ontroller (b) yapunov ontroller (b) yapunov ontroller

4 Transaton on Control, Automaton, and System Engneerng Vol., No., June, Fg. 6. Response to ntal spang errors. elerate. Ths wll make the thrd vehle aelerate at a hgher speed than the seond vehle. The smulaton results learly support ths expetaton. However, the yapunov ontroller ompletely elmnates the need for seond vehle speed hanges and the frst and the thrd vehles adjust ther speed n a symmetr pattern usng muh lower aeleraton/deeleraton levels than wth the PID ontroller. By avodng any unneessary aeleraton/deeleraton, the yapunov ontroller has the effet of savng energy. It also should be noted that osllatons n response are vrtually removed. 3. Senstvty to vehle dynams varatons Most longtudnal ontrollers are desgned usng some nomnal vehle dynams -ether lnear or nonlnear. MMahon and Hedrk[4] reported that frst order lag model approxmatons to vehle dynams nludng engne and powertran nvolve a sgnfant amount of smplfaton and many assumptons. The tme onstant of frst-order models may be a funton of many fators suh as vehle speed and mass. Therefore, the ontroller must be desgned so that t an aommodate, or at least be robust to, varatons n vehle dynams. Shladover[6] has suggested the use of a gan shedulng tehnque. Alternatvely, veloty-dependent haratersts n the frst-order model may be taken nto aount durng the desgn proess[5]. The yapunov ontroller proposed n ths paper nherently adapts tself to varatons n the dynams of the predeessor and follower vehles. As one an see from Eq. (7), the tme onstants for both vehles expltly appear n the aeleraton ommand. The adaptve feature of the urrent ontroller may be shown by a smulaton example. Usng the same senaro used n the sudden speed hange smulaton and fxng all relevant gans Fg. 7. Effets of hanges n powertran tme onstant. and parameters, only the tme onstant s vared. In Fg. 7, the spang error between the lead vehle and the frst follower vehle s presented. The performane of the PID ontroller degrades as the vehle dynams get slower, but the yapunov ontroller shows no performane degradaton. For a platoon omposed of dfferent types of vehles, the longtudnal ontroller of ndvdual vehles may need to be redesgned, but the yapunov approah provdes a straghtforward and unfed proedure for obtanng a ontroller. 4. Effets of desgn parameters: N and t In the present ontroller, N and t are left undetermned and regarded as desgn parameters whh are to be seleted. Although ther theoretal effets on the performane of the proposed ontroller are not treated n ths paper, some rough dea of them may be obtaned from Eq. (7). Rewrtng Eq. (7) n ts expanded form usng Eqs. (6) and (9), we get δ & δ a = ( a a ) + a + N ( + + δ& ). (8) t t It may be argued that N plays the role of overall ontroller gan and t ats lke a weghtng fator between the urrent spang error δ and ts rate of hange &. δ In other words, a small t mples a more tght ontrol of spang errors. Therefore, t would be natural to put more weght on the urrent error. Also a large t means looser ontrol of spang errors, and a onsequent longer tme needed to redue the errors. Fg. 8 shows the smulaton results as N and t vary. As before, the same senaro was used and only the spang error between the lead and follower vehles are dsplayed. As one an easly see, the overshoot s redued as N nreases. Although any hoe of N and t assures overall stablty, too small N may lead to the apparent nstablty of spang errors between the lead vehle and the frst vehle. T h a t (a) PID ontroller (a) t = se. (b) yapunov ontroller

5 96 ICASE: The Insttute of Control, Automaton and Systems Engneers, KOREA Vol., No., June, 000 (b) N = 0 Fg. 8. Effets of desgn parameters, N and t. s beause the jerk haratersts or the equvalent aeleraton profle of the platoon leader vehle s not onsdered n the desgn of ontroller for the frst vehle, and too small N means almost no ontrol authorty. It an seen that t affets both the overshoot and the speed of response. 5. Mxed Controllers In ase of ommunaton lnk falure, the spang of the whole platoon wll be jeopardzed and therefore, the system s relablty s very mportant. Whle the PID ontroller requres a global lnk n the sense that every follower vehle should reeve nformaton from the lead ar, the yapunov ontroller s wholly dependent on the loal lnks. As suggested n Fg. 9, usng both ontrol approahes wll mean that they wll serve as a bakups to eah other beause t s easy to swth from on ontroller to another regardless of whh ontroller s used as the prmary one. One should note that the mxng of two dfferent knds of ontrol log s possble beause although the yapunov ontroller n the follower vehle needs to reeve the aeleraton ommand a from the preedng ar, t s not affeted by the knd of proedure that s nvolved n obtanng t. To llustrate ths, two ases were smulated usng the sudden speed hange senaro. Frst, the PID ontroller s used as the prmary ontroller whle, the seond ar uses the yapunov ontroller. In the seond ase, the yapunov ontroller s used as the prmary ontroller whle the seond ar uses the PID ontroller. As an be noted from Fg. 0, the overall performane seems not to be degraded and the yapunov ontroller outperforms the PID n terms of spang errors. X -, v - a - M M - a a - a - Communaton nk Measurement nk X, v To M a To M a a M v, a ead vehle M Fg. 9. Platoon wth global and loal ommunaton lnk. (a) Prmary: PID, Bakup: yapunov (b) Prmary: yapunov, Bakup: PID Fg. 0. Response of mxed ontrollers. V. Conlusons and remarks In ths paper, the onept of Expeted Spang Error was ntrodued and the yapunov stablty theorem was appled to the desgn of ontrol log that s used to ontrol ntervehular spang n a platoon. The proposed ontroller uses a loal ommunaton lnk between the predeessor and follower vehles. Due to the asaded struture of the ontroller, a vehle n the platoon has full aess to nformaton on the dynams and ontrol of all the vehles ahead of t. Numeral smulaton was used to ompare the performane of the yapunov ontroller wth that of a PIDtype ontroller. The smulaton results showed that slnky effets ould be ompletely removed and that the yapunov ontroller s robust to varatons n vehle dynams. Also, the effet of desgn parameters on the performane of the yapunov ontroller was nvestgated. For further researh, the effets of jerk and aeleraton lmts need to be studed beause they are not onsdered n the ontrol desgn and are known to ndue system nstablty. Also, the sope of the present work does not nlude the effets of sensor nose and atuator delays. Fnally, the possblty of applyng the results of ths researh to nonlnear vehle dynams needs to be addressed and the method of seletng ontrol parameters warrants further n-depth nvestgaton. Referenes [] Speal Issue on Intellgent Vehle Hghway Systems, IEEE Trans. on Vehular Tehnology, vol. 40, no., 99. [] P. Varaya, Smart ars on smart roads: problems of ontrol, IEEE Transatons on Automat Control, vol. 38, no., pp , Feb., 993. [3] J. K. Hedrk, M. Tomzuka, and P. Varaya, Control ssues n automated hghway systems, IEEE Control Systems, pp. 3. De., 994. [4] S. E. Shlodover, Revew of the state of development of advaned vehle ontrol systems, Internatonal Journal of Vehle System Dynams, vol. 4, pp , 995. [5] S. E. Shladover, ongtudnal ontrol of automated gudeway transt vehles wthn platoons, Journal of Dynams Systems, Measurement, and Control, vol. 00, pp , 978. [6] S. E. Shladover, ongtudnal ontrol of automotve vehles n lose-formaton platoons, Journal of Dynams Systems, Measurement, and Control, vol. 3, pp. 3 4, 99. [7] S. Shekholeslam and C. A. Desoer, ongtudnal ontrol of a platoon of vehles I: near model, PATH Researh Report UCB-ITS-PRR-89-3, Aug., 9, 989. [8] S. Shekholeslam, and C. A. Desoer, ongtudnal ontrol of a platon of vehles III: Nonlnear model, PATH Researh Report UCB-ITS-PRR-90-, Aprl, 990. [9] T. et al, Fujoka, ongtudnal vehle followng ontrol

6 Transaton on Control, Automaton, and System Engneerng Vol., No., June, for autonomous drvng, AVEC'96 Internatonal Symposum on Advaned Vehle Control at the Aahen Unversty of Tehnology, pp th-8th of June, 996. [0] J. K. Hedrk, et al, ongtudnal vehle ontroller desgn for IVHS systems, Proeedngs of Ameran Control Conferene, pp , 99. [] D. H. MMahon, et al, ongtudnal vehle ontroller desgn for IVHS: Theory and experment, Proeedngs of Ameran Control Conferene, pp , 99. [] S. Shekholesam, and C. A. Desoer, ongtudnal ontrol of a platoon of vehles wth no ommunaton of lead vehle nformaton, Proeedngs of Ameran Control Conferene, pp , 99. [3] J-J. E. Slotne, and W., Appled Nonlnear Control, Prente-Hall Internatonal Edton, 99. [4] D. H. MMahon, and J. K. Hedrk, ongtudnal model development for automated roadway vehles, PATH Researh Report UCB-ITS-PRR-89-5, Ot., 989. [5] A. S. Hauksdottr and R. E. Fenton, On the desgn of a vehle longtudnal ontroller," IEEE Transatons on Vehular Tehnology, vol. VT-34, no. 4, pp. 8 87, Nov., Tae Soo No Dr. Tae Soo No, urrently assstant professor of aerospae engneerng at Chonbuk Natonal Unversty, Korea, reeved hs B.S. form Seoul Natonal Unversty, and M.S. and Ph.D. form Auburn Unversty, Alabama, respetvely and 99. Then he worked for Korea Aerospae Researh Insttute as a spaeraft system engneer untl 995. Hs past researh nterest nludes dynams and ontrol of flght vehles, multbody dynams and smulaton, mssle gudane law, orbtal mehans and ontrol, et. Hs reent researh area s the modelng, smulaton ontroller desgn of the autonomous vehles for the ntellgent transportaton systems. Kl To Chong Kl To Chong reeved hs BS degree n Mehanal Engneerng from Oren State Unversty n 984, hs MS degree n Mehanal Engneerng from Georga Insttute of Tehnology n 986 and hs Ph.D n Mehanal Engneerng from Texas A&M Unversty n 993. Durng , he was a full tme leturer n the mehanal engneerng at Youngnam Unversty. He s an assstant professor n the shool of eletrons and nformaton at Chonbuk Natonal Unversty. Hs researh nterests nlude system dentfaton, ntellgent transportaton system and neural networks.