Control of Mobile Platforms Using a Virtual Vehicle Approach

Transcription

1 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 46, NO., NOVEMBER Conrol of Mobile Plaform Uing a Virual Vehicle Approach M. Egere, X. Hu, an A. Soky Abrac Two moel inepenen oluion o he problem of conrolling wheel-bae mobile plaform are propoe. Thee wo algorihm are bae on a o calle irual ehicle approach, where he moion of he reference poin on he eire rajecory i goerne by a ifferenial equaion conaining error feeback. Thi, combine wih he fac ha he proen able conrol algorihm are baically proporional regulaor wih arbirary poiie gain, make he oluion robu wih repec o error an iurbance, a emonrae by he experimenal reul. Inex Term Conrol of mobile plaform, nonlinear feeback conrol, rajecory racking. I. INTRODUCTION In hi noe, he problem of conrolling wheel-bae mobile plaform i uie. Many inurial applicaion nee problem like hi o be ole in orer o hae goo an robu pah racking algorihm for ifferen ype of auomae ak. Naurally, hi ha been a well uie opic [3], [4], [6] [8], [], [2] [4], an quie a few meho hae been propoe o ole he problem, for example he curaure eering meho [5], [7], or he flane approach [8]. Howeer, many of hee meho eiher ue open loop conrol, which i quie eniie o meauremen error an iurbance, or are highly moel epenen, making he conroller ery complex an har o implemen in pracice, ince exac moeling of he plaform i ypically no an eay ak. Our oluion o hi problem coni of wo lighly ifferen conrol raegie for racking a reference rajecory bae on poiion an orienaion error feeback. Thee raegie are largely moel inepenen becaue hey proie only he roaional an ranlaional elociy conrol. In oher wor, hey are higher leel conrol. Naurally, for plaform ha o no hae irec conrol oer he elociie, one nee o eign he acuaor conrol o ha hee elociy conrol are realize. The implemenaion coul be ju a aic mapping, a in he car cae, or a ynamic regulaor. The fir of he wo raegie i eigne in uch a way ha i only require conrol in he laeral irecion, i.e., roaional conrol, while keeping he longiuinal elociy a a conan alue. Thi raegy i eelope for mobile plaform ha o no uppor fine an accurae ranlaional elociy conrol, uch a fairly cheap RC car. A a price one ha o pay for uing only one conrolle inpu, he algorihm only work locally. Therefore he econ of he wo raegie require boh roaional an ranlaional conrol, an will be hown o be globally able. The wo propoe algorihm are furhermore boh bae on a o-calle irual ehicle approach, where he moion of he reference poin (he irual ehicle) on he planne rajecory i goerne by a ifferenial equaion conaining error feeback. I can be iewe a a combinaion of he conenional rajecory racking, where he reference rajecory i parameerize in ime, an a ynamic pah Manucrip receie Noember 2, 998; reie Augu 23, 999 an Augu 4, 2. Recommene by Aociae Eior G. Bain. Thi work wa ponore in par by he Sweih Founaion for Sraegic Reearch hrough i Cenre for Auonomou Syem a KTH, an in par by he Sweih Reearch council for Engineering Science. The auhor are wih he Opimizaion an Syem Theory, Royal Iniue of Technology, SE- 44 Sockholm, Sween ( magnue@mah.kh.e; hu@mah.kh.e; oky@mah.kh.e). Publiher Iem Ienifier S ()35-X. Fig.. The general iea behin he irual ehicle approach. following approach [4], where he crierion i o ay cloe o he geomeric pah, bu no necearily cloe o an a priori pecifie poin a a gien ime. The main iea behin our approach can be een in Fig., an he reaon for calling he reference poin, ogeher wih he aociae ifferenial equaion, a irual ehicle i ha he reference poin i moing on he pah ha we wan he plaform o follow. A he ame ime i ha i own ynamic for ecribing he moion, an one of he aanage wih our approach i ha i i quie robu wih repec o meauremen error an exernal iurbance. Thi i ue o he fac ha he moion of he irual ehicle i goerne by racking error feeback, an baically only proporional conrol are ue. If boh he racking error an iurbance are wihin cerain boun, he reference poin moe along he reference rajecory while he robo follow i wihin he prepecifie look-ahea iance. Oherwie, he reference poin low own an wai for he robo. In hi noe, we uy he performance of he wo propoe pah following algorihm analyically an how how hey are implemene on wo ifferen plaform repeciely: a RC car an a Nomaic 2 mobile plaform. In orer o compare he wo algorihm, he abiliy analyi will be performe on kinemaic moel of he wo plaform, which are in eence he ame. Thi noe i organize a follow. In Secion II, we preen our wo conrol algorihm, followe by he abiliy analyi in Secion III. In Secion IV, he fir conroller i implemene on a mall RC car, an he econ algorihm i implemene on he Noma 2 in orer o re he fac ha our propoe oluion really are moel inepenen. Thee experimen alo how ha our propoe oluion o no only work in heory, bu in pracice a well. II. CONTROL ALGORITHMS A. Problem Formulaion Our main ak i o fin a laeral conrol f () an longiuinal conrol () ha make a robo follow a mooh reference pah, parameerize by a irual ehicle (), moing on he pah. The pah i gien by x = p() y = q(); ( f ) () where we aume ha p 2 () +q 2 () 6= 8 2 [; f ], an he ubcrip an for eire poiion. Furhermore, our conrol objecie are lim up ()! (2) lim up! j j (3) where () = x 2 +y 2, an x = x x; y = y y. In (3) i he yaw angle (orienaion of he ehicle), = aan2(y; x) i he eire orienaion, an (x; y) i a reference poin on he robo, for /$. 2 IEEE

2 778 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 46, NO., NOVEMBER 2 example he cener of graiy. Furhermore, > i a mall number ha, among oher hing, epen on he maximum curaure of he reference pah, an i he look-ahea iance ha will be more carefully efine laer. B. Conrol Algorihm In hi ecion, we preen he fir of he wo conrol algorihm. We aume ha he longiuinal elociy = _x 2 +_y 2 i conan, an we hu only conrol he laeral elociy. Thi conrol algorihm i eigne for plaform which o no hae goo acuaor, uch a our fairly cheap raio-conrolle car. Therefore, our purpoe here i o keep he longiuinal an laeral conrol a imple a poible. In orer o realize (2), we inrouce he parameer, an require [] which, afer iffereniaing, gie = ( ) (4) (x(_x _x) +y(_y _y)) = ( )+ _ : (5) Now, ince _x = p ()_; _y = q ()_, wehae _ =(xp ()+yq ()) 2 (x _x +y _y ( )+ _ ): (6) If we le enoe he angle beween he elociy ecor an he x-axi (for mo pracical yem, i almo ienical o ), hen x _x +y _y ( )+ _ = ( co( ) ( )+ _ ): Now, le in (4) be gien by = co( ), wih >, an ==, an le r enoe he orienaion angle of he angen o he reference cure a. Then (6) can be rewrien a _ = (2 co( p 2 + q 2 ) ): (7) co( r ) I can be een from (4) ha ()! if co( ) i kep poiie an co( r) ay nonzero, an hu he fir conrol objecie (2) woul be realize if hi were o hol. Howeer, co( r ) will be zero only if (x; y) T i normal o he cure a (p();q()). In orer o aoi hi ingulariy, we nee o make he following aumpion abou he reference cure. Aumpion 2.: The boun on he curaure of he reference cure i ufficienly mall. A ypical example here i a raigh line. Howeer, hi aumpion alone may no be enough. In fac, he laeral conrol ha o be eigne uch ha he following hol. Aumpion 2.2: 2 co( ) > for all. Remark: I i eay o ee ha if he curaure of he reference cure i ufficienly mall, hen if iniially xp () +yq () >, i will ay poiie, proie ha Aumpion 2.2 i alo aifie (hink of a raigh line). The aifacion of Aumpion 2.2 epen boh on he iniial coniion an on he laeral conrol. In fac, i i eay o ee ha if Aumpion 2.2 i aifie iniially, an if, for ome >, he laeral conrol keep j () ()j =2 for all, hen Aumpion 2.2 i alway aifie. Thi alo implie ha he econ conrol objecie (3) woul be fulfille. For a car-like robo, we propoe he following proporional, laeral conrol: f = k( ); k > (8) an we ummarize he fir conrol algorihm a follow. Algorihm 2.: _ = p (2 co( ) )) p +q co( ) f = k( ) = con; where f an conrol he laeral an longiuinal elociie repeciely. We hu propoe a imple conrol algorihm for eering mobile plaform, uch a car-like robo, ha i gien by (7) an (8). Laer we will ue a kinemaic moel of he car o how ha Aumpion 2.2 i me by uing he laeral conrol (8), proie ha he reference cure aifie Aumpion 2.. Since Aumpion 2. an 2.2 can be aifie iniially only for a ube of he iniial coniion, hi algorihm i no ali globally. On he oher han, if one i able o ue finer longiuinal conrol, uch ingulariie can be aoie, a will be hown in he nex ubecion where he econ of our conrol algorihm i preene. C. Conrol Algorihm 2 For hi algorihm we aume ha he acuaor are accurae enough o ha fine longiuinal elociy conrol i aailable. In conra o he preiou algorihm, hi new algorihm will furhermore be efine globally. We houl fir poin ou ha in [6] an [2], racking conrol are eigne o ha he racking error en o zero globally. Howeer, he conrol here are complex an highly moel epenen, while our aim i o prouce conrol ha are quie inuiie an moel inepenen. From (), we, a alreay menione, hae _x = p ()_; _y = q ()_, or _x p () + _y q () =(p 2 ()+q 2 ()) _, which implie ha if he robo woul rack he pah perfecly, i.e., _x = _x an _y = _y,we woul hae _ = p () p 2 ()+q 2 () _x + q () (9) p 2 ()+q 2 _y: () () The reaon for expreing _ in hi way i ha p 2 ()+q 2 () i alway nonzero by our aumpion abou he moohne of he reference pah. If we now enoe = _x 2 +_y 2, an aume ha _, hen _ = = p 2 ()+q 2 (): On he oher han, hee expreion o no conain any poiion error feeback, which i imporan for robune. We hu a error feeback o _, an propoe he ynamic for he reference poin a follow: _ = ce p 2 ()+q 2 () () where an c are poiie number ha are o be eermine laer, an wih he appropriae choice of an c; will be he eire pee a which one wan he ehicle o rack he pah.

3 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 46, NO., NOVEMBER Since become unefine when =, in orer o aoi hi ingulariy, we efine ~ = (2 +3 )+ (2() +3() ) if > if where can be aken a, i.e., he eire racking boun. I i eay o ee ha ~ i alo well efine a =ince lim! ( )=. Conequenly, we can efine he ranlaional an roaional conrol a follow. Algorihm 2.2: If we plug in he laeral conrol law (8), we hae an() =k( l ): So, = an ((lk=)( )), where k houl be choen o reflec he eering range acceible by he robo. Now, we efine he error = ; = ; = r which gie u he following error ynamic: ce _ = p p ()+q () f = k ~ + ~ _ = (x co( )+y in( )); where boh an k are poiie, an ~ = ~. Remark: A = i hol ha (2) _ = co( ) _ = k + (in( ) + + an( ) co( )( )) _ = (in( ) + + an( ) co( )( )) + ()w (4) ~ j = = an _~ j = = _ an herefore he roaional conrol i coninuou eerywhere. Obiouly, he ranlaional conrol i alo efine an coninuou eerywhere. If 6=, hen he ranlaional conrol can be alo expree a = co( ) where =. I i obiou ha Algorihm 2.2 ju eer he robo owar he reference poin on he eire pah, wih a pee proporional o he iance racking error. III. STABILITY ANALYSIS Een hough our conrol algorihm are largely moel inepenen, he abiliy analyi ill ha o be one wih repec o mahemaical moel of he plaform. We apply he fir conrol algorihm o a RC car plaform, where fine conrol of he elociy i exremely ifficul o achiee. We hen apply our econ algorihm o a Noma plaform, where one can apply fine conrol o boh he longiuinal an he laeral elociie. A. Sabiliy Analyi of a Car-Like Plaform Conier a imple kinemaic moel for a car-like plaform, where we le (x; y) enoe he mile poin of he rear axle, an le enoe he eering angle. Then, i i well known (ee [3], for example) ha he moel can be wrien a _x = co( ) _y = in( ) _ = an() (3) l where l i he lengh beween he fron an he rear axle. In hi moel, =. In pracice, he range of he acceible pee i alway limie. One can replace he pee conrol = co( ) by, for example, = aan() co( ) in orer o ake care of hi auraion problem. where () =(q p p q =(p 2 + q 2 )) 3=2 i he curaure of he pah, an w =(= co( )) co( )( ). I i obiou ha he error ynamic (4) i no globally able, ince j j =(=2) or =(where i no well efine) are ingular poin. Howeer, if we conier w a an inpu or perurbaion o he yem efine in (4), hen we can how ha he ynamic i locally exponenially able if he inpu i e o zero. I i eay o compue he linearize marix A of (4) a (; ; ) a A = k + an raighforwar calculaion how ha A i a able marix for all > an k >. Therefore, (4) i a locally exponenially able yem rien by he inpu w. I i well known ha for uch a yem, if boh he iniial coniion ((); (); ()) an jwj are mall enough, hen ((); (); ()) will remain mall. Thi alo implie ha he ingular poin will no be reache. In orer o keep jwj mall, i uffice o aume ha he upper boun for () i mall enough. Remark: I i eay o ee ha wih hi moel, Aumpion 2.2 i aifie. We noe ha 2 co( ) = co( )(2 (= )), an () en o monoonically if j j i boune from =2. Therefore, if iniially, j ()j i mall enough an (2(()= )) >, hen Aumpion 2.2 i aifie. B. Sabiliy Analyi of an Omni-Direcional Plaform ) Vehicle Moel: The Noma 2, epice in Fig. 3, i a ynchrorie ri-wheeler, where he wo lae wheel are mechanically urne in he ame irecion a he aciely conrolle maer wheel. If we o no wan o moel ie lip, ue o he inene low pee applicaion ha he econ conrol algorihm wa eigne for 2, or he mechanical elay beween he wheel, we ge he moel _x = co( ) _y = in( ) _ = f : (5) 2 The algorihm wa eigne wih an inelligen erice agen applicaion in min [2], where low pee are ypically enough.

4 78 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 46, NO., NOVEMBER 2 I houl be poine ou ha for hi moel, he laeral conrol, f,i alreay he angular elociy of he ehicle an we aume he acuaor o no hae any auraion boun. 2) Sabiliy Analyi: Le u conier he error ynamic: _x = p ()_ (x co( )+yin( )) co( ) _y = q ()_ (x co( )+yin( )) in( ) _ ~ = k ~ (6) where _ i efine in (). Since hi conrol algorihm i globally efine, in conra o Algorihm 2., we o no nee o inclue in he error ynamic. Suppoe ha he eire pee of he robo,, i greaer han zero, an ha f =. (In pracice, hi mean ha he eire pah houl be long enough). From (6), i irecly follow ha: ~ = ~ ()e k : (7) Furhermore, ince = x 2 +y 2, hen if i hol ha 6= _ = (x _ x +y _ y) = (x co( )+y in( ))2 + xp +yq _ (8) p 2 + q2 = co 2 ( ) + ce co( r ): (9) Now, uppoe ha a = ; ( )=. Then uing L Hopial rule we hae lim! Similarly, we hae x x 2 +y 2 = lim! +( y x )2 = lim! y x 2 +y 2 = q (( )) : p 2 + q 2 p (( )) p 2 + q 2 : Remark: Thi alo implie ha een hough i no efine a ( )=; lim! exi an i equal o r. Since _( )=(c= p 2 + q 2 ), from (8) we hae _j ()! = c : On he oher han, a =, ince ( )=,wehae = (p(( )) p(( +))) 2 +(q(( )) q(( +))) 2 2 : So, by he chain rule we hae _j = = c an hu (8) i well efine. Since c >, hi implie ha = can happen only a an iniial coniion (or perurbe o here by iurbance, for example). Now, le u enoe a() = co 2 ( ()): We houl noice ha () = ~ () only if (). Now alo le 8(; ) enoe exp We can alway formally expre () =8(; )() + a( ) : 8(; )c co( () r()) : Due o he fac ha lim ()! exi an can be zero only a =, he aboe expreion i ali for all iniial coniion an <T, where T i eiher or a poible finie ecape ime. If () >, hen j8(; )j = exp = exp exp a( ) [ in 2 ( ~ )] ( ~2 ) = exp ( ) 2 () [e 2k e 2k ] 2k e ( ()=k()) ; 8T : (2) where T i uch ha (T )= an j8(; )j() + Thu j8(; )jc co( r ) e ( ()=k) () + () max ; e ( ()=k) () + ce ce ()=k : ()=k 8 : I i eay o ee ha afer he ranien ecay exponenially, he racking error,, can mee any a priori gien boun by uning ; an k.now, if () i mall enough hen r (ince lim! = r) hu ~!. Then, inpecing (9) gie u ha () houl be cloe o he eay-ae oluion of which i efine by _ = + ce = ce where he choice of c = e = gie ha =. Thu, one can inerpre a he eire pee, ince, by our eign, he acual pee i gien by = co( ). IV. IMPLEMENTATION A. Implemenaion of he Fir Conrol Algorihm We choe o implemen he fir conrol algorihm on a mall, raio conrolle car, where we imply connec he ranmier o a compuer. Howeer, our car yem i bae on a fairly cheap oy car wih coare A/D an D/A conerion a well a a ea zone in he ero yem. Therefore he eering i far from precie, making queion concerning robune ery imporan, o wha work in imulaion may no work a all here. Howeer, a een in Fig. 2, he experimen

5 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 46, NO., NOVEMBER 2 78 (a) (a) (b) (b) (c) Fig. 3. The Noma 2 [(a)] ogeher wih e run wih wo ifferen iniial poiion [(b) an (c)] are epice in orer o re he global conergence of he econ conrol algorihm. In hi example, a circular pah wa racke. (c) Fig. 2. In (a), he raio conrolle car ue for rying ou he fir conrol algorihm i epice. In (b), he racke rajecory (oe) an he fron poin (oli) on he acual car can be een, a well a he fron an he rear poin ploe ogeher. In (c), he orienaion of he car can be een. urne ou o be aifying, upporing he pracical feaibiliy of our fir conrol algorihm. B. Implemenaion of he Secon Algorihm on a Noma 2 Mobile Plaform In orer o re he fac ha our propoe conrol algorihm really are moel inepenen, we choe o implemen he econ algorihm on a Noma 2 mobile plaform. The ynamic of hi plaform iffer quie a lo from nonholonomic RC car, bu he approach ill work well, which can be een in Fig. 3, where we hae ploe ifferen e

6 782 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 46, NO., NOVEMBER 2 run on he Nerer, he Noma imulaor. Thee e run, where he robo rack a circle from ifferen iniial poiion an configuraion, clearly inicae ha our propoe econ conrol algorihm work globally in a able an robu way. I houl be emphaize ha if we were o ue he fir, local algorihm on hee e run, i woul fail ince he iniial poiion an orienaion woul make xp ()+yq () =, an hu _ woul no be efine anymore if he fir algorihm were o be ue. V. CONCLUSION In hi noe, wo inuiie, moel inepenen pah following conrol raegie are propoe, an he abiliy analyi i one wih repec o wo ifferen plaform. Wha i new here i ha by combining he conenional rajecory racking approach an he more recen geomeric pah following approach, we can eign a irual ehicle ha moe on he reference pah an i regulae in a cloe-loop fahion by exploiing he poiion error. In he fir algorihm, he elociy i kep conan, while he oher, global meho epen on he poibiliy of fine elociy conrol. Implemening hee iea on acual robo gie u ome experimenal aa ha how ha our conroller work in pracice a well a in heory, which i wha we were aiming for, ince our main eign raegy wa o keep he conrol algorihm moel inepenen an a imple a poible. REFERENCES [] J. Ackermann, Robu Conrol. Lonon, U.K.: Springer-Verlag, 993. [2] M. Aneron, A. Orebäck, M. Linröm, an H. I. Chrienen, Inelligen Senor Bae Roboic. Heielberg, Germany: Springer-Verlag, 999. Ch. ISR: An Inelligen Serice Robo, Lecure Noe in Arificial Inelligence. [3] R. W. Brocke, Aympoic abiliy an feeback abilizaion, in Differenial Geomeric Conrol Theory, R. W. Brocke, R. W. Millmann, an R. W. Suman, E. Boon, MA: Birkhauer, 983, pp [4] G. Campion, G. Bain, an B. D Anréa-Noel, Srucural properie an claificaion of kinemaic an ynamic moel of wheele mobile robo, IEEE Tran. Robo. Auoma., ol. 2, pp , Feb [5] C. Canua e Wi, A. D. NDoui-Likoho, an A. Micaelli, Feeback conrol for a rain-like ehicle, Proc. 994 IEEE In. Conf. Roboic Auomaion, 994. [6] C. Canua e Wi, B. Siciliano, an G. Bain, Theory of Robo Conrol. New York: Springer-Verlag, 996. [7] C. Canua e Wi, Tren in mobile robo an ehicle conrol, in Conrol Problem in Roboic, B. Siciliano an K. P. Valaani, E. Lonon, U.K.: Springer-Verlag, 998, pp Lecure Noe in Conrol an Informaion Science 23. [8] M. Flie, J. Léine, P. Marin, an P. Rouchon, Flane an efec of nonlinear yem: Inroucory heory an example, In. J. Conrol, ol. 6, no. 6, pp , 995. [9] E. Freun an R. Mayr, Nonlinear pah conrol in auomae ehicle guiance, IEEE Tran. Robo. Auoma., ol. 3, pp. 49 6, Feb [] J. Gulner an V. Ukin, Sabilizaion of nonholonomic mobile robo uing Lyapuno funcion for naigaion an liing moe conrol, in Proc. 33r Conf. Deciion Conrol, FL, Dec. 994, pp [] S. V. Gue an I. A. Makaro, Sabilizaion of program moion of ranpor robo wih rack laying chai, preene a he Proceeing of LSU, ol., 989. [2] Z. Jiang an H. Nijmeijer, Tracking conrol of mobile robo: A cae uy in backepping, Auomaica, ol. 33, no. 7, pp [3] R. Murray an S. Sary, Nonholonomic moion planning: Seering uing inuoi, IEEE Tran. Auoma. Conr., ol. 38, pp. 7 76, May 993. [4] N. Sarkar, X. Yun, an V. Kumar, Dynamic pah following: A new conrol algorihm for mobile robo, preene a he Proc. 32n Conference Deciion Conrol, San Anonio, TX, Dec [5] A. Soky, X. Hu, an M. Egere, Sliing moe conrol of a car-like mobile robo uing ingle-rack ynamic moel, preene a he Proc. IFAC 99:4h Worl Congre, Beijing, China, July 999. A Polynomial Approach o Nonlinear Syem Conrollabiliy Yufan Zheng, Jan C. Willem, an Cihen Zhang Abrac Thi noe ue a polynomial approach o preen a neceary an ufficien coniion for local conrollabiliy of ingle-inpu ingle-oupu (SISO) nonlinear yem. The coniion i preene in erm of common facor of a noncommuaie polynomial expreion. Thi reul expoe conrollabiliy properie of a nonlinear yem in he inpu oupu framework, an gie a compuable proceure for examining nonlinear yem conrollabiliy uing compuer algebra. Inex Term Common facor, conrollabiliy, ifferenial fiel, noncommuaie ring, nonlinear yem. I. INTRODUCTION Conrollabiliy i one of he cenral noion of moern conrol heory. The reul on conrollabiliy of linear yem hae been eminal in he eelopmen of he fiel, an he lieraure on conrollabiliy of nonlinear yem i a. See, for example, [6], [], [7], [8], [], [5], an [9]. Traiionally, conrollabiliy i efine for linear ae pace yem an refer o he poibiliy of ranferring a yem from any iniial o any erminal ae. For nonlinear ae-pace yem he noion of conrollabiliy or rong acceibiliy refer o he cae where he conrol can ac on he yem ae, bu may be inufficien o ranfer i o a pecifie erminal ae. Ofen, nonlinear yem conrollabiliy i efine in erm of yem ae equaion an ee by mean of Lie iribuion or heir ual form. The noion of conrollabiliy i recenly exene o yem in more general framework. For linear yem, conrollabiliy i iewe in [8] in erm of yem rajecorie which may no necearily be he yem ae. A yem i efine o be conrollable if one can wich from any feaible pa rajecory in he yem behaior o any feaible fuure rajecory, afer ome ime elay. I i obere ha he lack of behaioral conrollabiliy implie he exience of an auonomou yem oupu, which i a nonriial funcion of he yem ariable. I urn ou ha a linear ime-inarian inpu oupu yem i conrollable if an only if i oe no hae auonomou ariable in i behaior an if an only if he polynomial marice ha pecify he yem behaior are lef coprime. The noion of auonomou ariable i alo ue o ecribe conrollabiliy of nonlinear yem [], [5], [7], [9]. In [7], local conrollabiliy of nonlinear ae pace yem i ecribe in erm of he abence of local fir inegral which are auonomou ariable of he yem ae. In [] an [9], conrollabiliy of nonlinear ae-pace yem i ecribe by he abence of auonomou ariable in erm of ifferenial one-form. Moreoer, he nee for a conrollabiliy concep for nonlinear inpu/oupu yem i icue in [3], where a Manucrip receie April 2, 2; reie February, 2. Recommene by Aociae Eior H. Huijber. Thi work wa uppore by he Auralia Reearch Council an he Naural Science Founaion of China. Y. Zheng i wih he Deparmen of Elecrical an Elecronic Engineering, he Unieriy of Melbourne, Parkille, Vic. 3, Auralia, an alo wih he Iniue of Syem Science, Ea China Normal Unieriy, Shanghai 262, China ( y.zheng@ee.mu.oz.au). J. C. Willem i wih he Iniue of Mahemaic an Compuing Science, Unieriy of Groningen, 97 AV Groningen, The Neherlan ( J.C.Willem@mah.rug.nl). C. Zhang i wih he Deparmen of Elecrical an Elecronic Engineering, he Unieriy of Melbourne, Parkille, Vic. 3, Auralia ( c.zhang@ee.mu.oz.au). Publiher Iem Ienifier S () /$. 2 IEEE