Differential flatness and control of nonlinear systems

Transcription

1 Dfferental flatness an ontrol of nonlnear sstes Antone Droun, Sebastão Soes Cunha, Aleanre Carlos Branao-Raos, Fel Antono Clauo Mora-Cano To te ths verson: Antone Droun, Sebastão Soes Cunha, Aleanre Carlos Branao-Raos, Fel Antono Clauo Mora-Cano. Dfferental flatness an ontrol of nonlnear sstes. CCC, 3th Chnese Control Conferene, Jul, Yanta, Chna. pp ,. <hal-999> HAL I: hal Subtte on 7 Jul 4 HAL s a ult-splnar open aess arhve for the epost an ssenaton of sentf researh ouents, whether the are publshe or not. The ouents a oe fro teahng an researh nsttutons n Frane or abroa, or fro publ or prvate researh enters. L arhve ouverte plursplnare HAL, est estnée au épôt et à la ffuson e ouents sentfques e nveau reherhe, publés ou non, éanant es établsseents ensegneent et e reherhe franças ou étrangers, es laboratores publs ou prvés.

2 Proeengs of the 3th Chnese Control Conferene Jul -4,, Yanta, Chna Dfferental Flatness an Control of Nonlnear Sstes Antone Droun, Sebastão Sões Cunha,, Aleanre C. Branão Raos, Fel Mora-Cano,3. LARA-ENAC, Toulouse 355, Frane E-al: UNIFEI, Itauba 375-, Bral E-al: 3. LAAS u CNRS, Toulouse 377, E-al: ora@laas.fr Abstrat: The purpose of ths ounaton s to nvestgate the onneton between output relatve egrees an fferental flatness propert of non lnear sstes. Conserng the relatve egrees of the outputs of a non lnear sste, neessar an suffent ontons for fferental flatness are splae. Then, t s shown that the applaton of the non lnear nverse approah to an output observable an output fferental flat sste s ental wth the fferental flat ontrol approah. The applaton of the non lnear ontrol approah to a four rotor arraft s onsere. A traetor trakng sste base on a two laers non lnear nverse ontrol struture s then propose. Ke Wors: Dfferental Flatness, Nonlnear Inverse Control, Traetor Trakng Introuton In the last eae a large nterest has rsen for new non lnear ontrol approahes suh as non lnear nverse ontrol [,,3], baksteppng ontrol [4] an fferental flat ontrol [5,6]. These ontrol law esgn approahes present soe strong slartes whh have reane unlear untl toa. In ths ounaton relatons between the non lnear ontrol approah an the fferental flat ontrol approahes are takle through the onseraton of the relatve egrees of the trake outputs. Atonal assuptons an efntons appear to be opportune n ths stu, suh as effet nepenen of the nputs of a general non lnear sste, output observablt an output ontrollablt of a fferental flat sste. The applaton onsere n ths stu s about traetor trakng b a four rotor arraft. The flght ehans of rotorraft are hghl non lnear an fferent ontrol approahes (ntegral LQR tehnques, ntegral slng oe ontrol) have been onsere wth lttle suess to aheve not onl autonoous hoverng an orentaton, but also traetor trakng More reentl nonlnear analtal ontrol esgn tehnques have been apple to rotorraft traetor trakng [7,8]. It appears that the flght nas of the rotorraft present a two level fferental flat struture whh s ae apparent when a new set of equvalent nputs s efne. Ths allows to ntroue here a non lnear nverse ontrol approah wth two te sales, one evote to atttue, heang an alttue ontrol an one evote to horontal traetor trakng. Dfferental Flat Output an Control Conser a general non-lnear na ontnuous sste gven b: X f ( X, U ) () X Y h () where X n, U, Y, f s a sooth vetor fel of X an U an h s a sooth vetor fel of X. It s suppose here that the onsere nputs are nepenent whh eans that eah nput has an nepenent effet over the state nas: rank f / u,, f / u (3) so that U an be etrate fro () an t s possble to wrte: U g( X, X ) (4) where g s a sooth funton.. Relatve egrees of outputs for nonlnear sstes Aorng to [] the sste ()-() s sa to have wth respet to eah nepenent output Y, a relatve egree r f the output nas an be wrtten as: wth an Y Y Y ( r ) ( r) b b X, U ) ( X, U ) ( (5) a ( X ) s,, r,, (6) ( s) s, b ( X, U ) / U,, (7) The output nas (5)-(6) an be rewrtten globall as: Y A( X ) (8) an Y B( X, U ) (9) 643

3 where an Here wth ) ( r ) ( r Y ( Y Y Y Y )' () ) ( r ) ( Y ( Y,, Y r )' () a( X ) A( X ) (-) a ( X ) a ( X ) (-) a ( X ),, a, r ( X ) s n (7-) The fferentable flat sste () s sa output ontrollable f: et([ / Z ]) (7-3) In that ase too, t s eas to erve a ontrol law of orer s + wth respet to output b onserng an output nas suh as: Z C( Z, V ) (8-) where C s suh that the nas of Z are stable an where V R s an nepenent nput. Then: U ( Z, C( Z, V )) (8-) The relatve egrees obe (see []) to the onton: r n (3) When the strt equalt hols, vetor Y an be aopte as a new state vetor for sste (), otherwse nternal nas ust be onsere. Then an ontrol law base on output feebak wll be unable to aster these nternal nas an f these nternal nas are unstable, the ontrol shee wll be napproprate. When nternal nas are stable or on t est, t wll be worth to onser an output feebak ontrol law. Fro (9), whle B(X, U) s nvertble wth respet to U, an output feebak ontrol law suh as: an be aopte. U ( X ) B ( X ) Y (4). Dfferental flat sstes Now suppose that Z R s a fferental flat output for sste (), then fro [3] the state an the nput vetors an be wrtten as: X (Z ) (5-) U ( Z, Z ) (5-) wth ( s ) ( s Z ( Z,,,,,, ) Z Z Z )' (6-) an s s Z ( Z, Z, Z )' (6-) where (.) s a funton of Z an ts ervatves up to orer s, an (.) s a funton of Z an ts ervatves up to orer s +, for = to where the s are ntegers. It appears of nterest to ntroue here three new efntons: The fferental flat sste s sa output observable f : rank ( / Z) n (7-) The fferental flat sste s sa full flat fferental f: u s 3 Neessar an Suffent Conton for Output Dfferental Flatness 3. Flatness an nternal nas It appears fro relatons (8) an (9) that a suffent onton for sste () to be fferentall flat output observable an output ontrollable wth respet to Y gven b () s that A s nvertble wth respet to X an that B s nvertble wth respet to U. A neessar an suffent onton for the nvertblt of A s: r n (9) whle (3) s a neessar onton for the nvertblt of B wth respet to U. In that ase t s possble to efne funtons an b: ( p) X A ( Y ) ( Y, Y,, Y ) () an ( p) U Bu ( A ( Y ))( Y ) ( Y, Y,, Y ) () Here: s r to () Then we have got here a pratal wa to hek f a gven output vetor Y s a fferental flat output: t shoul be suh as the orresponng atres A an B are respetvel global nvertble an nvertble wth respet to U, whle onton (9) shoul be satsfe. Then, a suffent onton for fferental flatness of Z s that Z s a state vetor for sste (),.e. there are no nternal nas n ths ase. 3. Relatve egree of a flat output Suppose now that sste () s a fferental flat output observable an output ontrollable sste where Z are the flat outputs wth relatve egrees r, = to. Then for a full flat fferental sste, fro (5-) an (6-) we an wrte: Z ( X ) (3) 644

4 where s a appng fro R n to R n. Then, fro (5-) an (6-) we an wrte: Z ( Z, U ) (4) where s a appng fro R n to R. Then, takng nto aount (3): e ( X, U ) Z E( X, U ) e ( X, U ) where E s a appng fro R n+ to R. (5) Then, oparng (3) an (4) wth (8) an (9), all the relatve egrees r are superor or equal to the orresponng s. Suppose now that for soe,, we have e / U then r s neessarl strtl superor to s an we shoul have: r s (6) ontrolle nternal nas. Whle an output non ontrollable flat sste wll be unable to ake ts outputs follow, through an output feebak ontrol, eouple lnear nas of orer s +, = to. 4 Dfferental Flatness of Rotorraft Dnas The onsere sste s shown n fgure where rotors one an three are lokwse whle rotors two an four are ounter lokwse. The an splfng assuptons aopte wth respet to flght nas n ths stu are a rg ross struture, no wn, neglggble aerona ontrbutons resultng fro translatonal spee, no groun effet as well as neglgble ar enst effets an ver sall rotor response tes. It s then possble to wrte splfe rotorraft flght equatons [7]. The rotor fores an oents for the rotorraft splae n fgure 8 are gven b: F f,, 3, 4 (3) M k F k f,, 3, 4 (3) Conserng (3) an (7-), ths s possble. Then for a full flat fferental sste we have neessarl: r s, (7), 3.3 Output feebak ontrol for traetor trakng F 4 F p Fro the above onseratons t appears that there s no fferene between a fferental flat ontrol law an a non lnear nverse ontrol law when apple to an output observable an output ontrollable fferental flat sste wth the sae ontrol obetves for the respetve outputs. Then, ong bak to relaton (4), an supposng that the nonlnear sste ()-() s an output observable an output ontrollable fferental flat sste, a new ontrol nput v = [v,, v ] an be ntroue n plae of Y suh as: r k ( r ) ( k ) ( k ) v Y Y Y = to (8) k akng the th output to follow lnear nas of orer r towars the target value Y. Ths akes the nas of the trakng error gven b: e Y Y = to (9) be suh as: e ( r ) ( r ) r e e e (3) () where the oeffents k an be hosen to ake the output nas asptotall stable an ensure the trakng of output Y towars the referene output Y. Observe that n the present ase relaton () hols an there are no nternal nas. To ope wth the saturaton of the atuators, the hoe of the oeffents k shoul be the result of a trae-off between the haratersts of the transent nas of the fferent outputs an the etree soltatons of the nputs. An output non observable fferental flat sste, when ontrolle through output feebak wll present non F 3 Fg. : Referene frae an fores of a four rotorraft where f an k are postve onstants an s the rotatonal spee of rotor. Sne the nerta atr of the rotorraft an be onsere agonal wth I = I, the roll, pth an aw oent equatons a be wrtten as: p ( l ( F4 F ) k q r) / I (33-) q l ( F F ) k p r) / I (33-) ( 3 4 r ( k ( F F F4 F3 )) / I (33-3) Where p, q an r are the roll, pth an aw bo angular rates. Here k ( I I ) an k ( I I ), where I, I r 4 an I are the nerta oents n bo-as, an l s the length of the four ars of the rotorraft., an are respetvel the bank, pth an heang angles, then the Euler equatons relatng the ervatves of the atttue angles to the bo angular rates, are gven b: p tg( )(sn q os r) (34-) os q sn r (34-) (sn q os r) / os (34-3) F q 645

5 In ths stu t s assue that there s no wn. The aeleraton aa a a ' of the entre of gravt, taken retl n the loal Earth referene frae, s suh as: a ( / )((os( )sn( )os( ) sn( )sn( )) F) (35-) ( / )((sn( )sn( )os( ) os( )sn( )) F) (35-) a a g ( / )(os( ) os( ) F) (35-3) where, an are the entre of gravt oornates, s the total ass of the rotorraft an: F F F F (36) 3 F4 In equatons (33-) an (33-), the effet of the rotor fores appears as fferenes so, we efne new atttue nputs u q an u p as: u q F F 3 (37-) u p F 4 F (37-) In the heang an poston nas, the effets of rotor fores an oents appear as sus, so we efne new guane nputs u an u as: u ( F F4 ) ( F F3 ) (37-3) u F F F F3 F4 (37-4) Equatons 33-, 33- an 33-3 are rewrtten: p l u k q r) / (38-) ( p I q / (38-) ( l uq k4 p r) I r k u / (38-3) Fnall, the oton equatons of the rotorraft an be wrtten n non-lnear state for as: f (, u) (39-) where ( p, q, r,,,,,,,,, )' (39-) an u ( u p, uq, u, u )' (39-3) It appears that ontrols u q an u r an be ae to var sgnfantl wth u an u reanng onstant. Atttue angles an an be seen as vrtual ontrols for the horontal poston of the rotorraft. Here the atttue nas are onsere to be the fast nas, the are at the heart of the ontrol sste. The heang an heght nas are ntereate whle the nas of the horontal poston oornates are the slower. Ths an lea to a two-level lose-loop ontrol struture. In ths two level ontrol struture, the fnal outputs are the oornates of the enter of gravt of the rotorraft,, an ts heang whle the ntereate outputs are gven b vetor Z (,,, )'. Then the Euler equatons proves the epressons : p sn (4-) q os sn os (4-) r sn os os (4-3) whle u an be epresse b nverson of the set of equatons (38-), (38-), (38-3) an (35-3), or ore spefall: u I p k qr) l (4-) u I p ( / u q ( 4 / I q k pr) l (4-) u ( I r) / k (4-3) (( g) ) /(os os) (4-4) Then, t an be onlue that the atttue an heang nas as well as the vertal nas of the rotorraft are fferentall flat when onserng the nput-output relaton between u an Z. Here, the relatve egrees of,, an are all equal to whle the enson of the atttue, heang an alttue nas are of the 8 th orer, then relatons () hols whle t an be easl shown that these fferental flat nas are output observable an output ontrollable. When onserng outputs an fro entres an, where an pla the role of paraeters, t appears fro equatons (35-) an (35-) that these slow nas are also output observable an output ontrollable fferental flat wth relatve egrees equal to two for a 4 th orer nas. Ths leas to propose the ontrol struture splae n fgure., Feebak output fast t ll Atttue, heang, alttue nas, Horontal guane nas,,, Feebak output slow ontrol loop, p,q,,,r,, Fg. : Propose ontrol struture 5 Rotorraft Traetor Trakng Here we are ntereste n ontrollng the four rotor arraft of fgure 3 so that ts entre of gravt follows a gven path wth a gven heang whle atttue angles an rean sall. Man potental applatons requre not onl the entre of gravt of the eve to follow a gven traetor but also the arraft to present a gven orentaton. 5. Fast nas ontrol, u p,u q,u,u, Atttue, heang an alttue ontrol loop,,, Guane ontrol loop 646

6 We aopt for the flat outputs seon orer nas an ther seon ervatve shoul be suh as: = ( ) (4-) = ( ) (4-) = ( ) (4-3) = ( ) (4-4) arsn( (sn D os D ) / u ) (46-) arsn( (os D sn D )/( u os ) (46-) where D ( ) (47-) D 5.3 Cases stues ( ) (47-) Here we onser two ases: one where the obetve s to hover at an ntal poston of oornates,, whle aqurng a new orentaton, an one where the rotorraft s trakng the heloïal traetor of equatons: ( t) os t (48-) t ( ) sn t (48-) t (48-3) ( t) t / (48-4) where s a onstant raus an s a onstant path angle. Heang ontrol at hover Fg 3: The onsere rotorraft The epressons of the ontrol nputs n relatons (4-) to (4-4) are fe b p, q, r gven b relatons (4-) to (4-3) an b p, q, r gven b: p os sn (43-) q os sn os (43-) sn ( sn) os os r sn os os (43-3) os sn os os sn where,, an are gven b (4-) to (4-4) where appear the urrent target values for an, an, an the fnal target values of an, an. 5. Desgn of horontal guane ontrol law Now, onserng equatons (35-) an (35-), to nsure that an aopt seon orer nas suh as: ( ) (44-) ( ) (44-) followng the non lnear nverse ontrol approah, an ust be hosen suh as: Then : (/ )((os sn os sn sn ) u ) ( ) (/ )((sn sn( )os os sn ) u ) ( ) (45-) (45-) In ths ase we get the guane ontrol laws: I u u g (49) k wth the followng referene values for the atttue angles: an (5) Here the heang aeleraton s gven b: r ( ) (5) Startng fro an horontal atttue ( ()=, ()=), atttue nputs u q an u p rean equal to ero. Then, fgures 4 an 5 spla soe orresponent sulaton results. Traetor trakng ase In ths ase we get the guane ontrol laws: u u g (5) Here the peranent referene values for the atttue angles are suh as: (53) an F,F 3 F,F sn (54) 4 g Fg. 4: Hover ontrol nputs te 647

7 Fg. 5: Heang response urng hover an the esre guane an orentaton aeleratons are gven b: os( t) sn( t) (55), In fgures 6 to 8 sulaton results are splae where at (t) -.4 (t) Fg. 6: Evoluton of rotorraft horontal trak Fg. 7: Evoluton of rotorraft alttue te (t) te ontrol approah has been stue. Conserng the relatve egrees of the outputs of a non lnear sste, neessar an suffent ontons for fferental flatness have been splae. It has been shown that the applaton of the non lnear nverse approah to an output observable an output fferental flat sste leas to an output feebak ontrol law ental to the one erve fro the fferental flat ontrol approah. Then the non lnear flght nas of a rotorraft have been anale an t has been shown that these nas are fferental flat wth output observablt an output ontrollablt propertes. The applaton of the non lnear nverse ontrol approah to ths four rotor arraft has been onsere an a traetor trakng ontrol struture base on two non lnear nverse ontrol laers has been propose. Referenes [] Sngh, S. N. an Sh, A. A., Nonlnear eouple ontrol snthess for aneuvrng arraft, Proeengs of the 978 IEEE onferene on Deson an Control, Psatawa, NJ, 978. [] Hassan K. Khall, Nonlnear Sstes, Prente Hall, 3r E.,. [3] M. Fless, J. Lévne, P. Martn, an P. Rouhon, 995, Flatness an efet of non-lnear sstes: theor an eaples, Int. J. Control,, Vol. 6, No. 6, pp [4] Ghosh, R. an Toln,C. J., Nonlnear Inverse Dna Control for Moel-base FlghtProeeng of AIAA,. [5] Farrel J., M.Shara an M. Polarpou, Baksteppng-base Flght Control wth Aaptve Funton Approaton, Journal of Guane, Control an Dnas, 8(6),pp89-, Nov.-De. 5. [6] Lu, W.C., Duan L. Mora-Cano, F. an Ahabou, K., Flght Mehans an Dfferental Flatness. Dnon 4, Proeengs of Dnas an Control Conferene, Ilha Soltera, Bral, pp , 4. [7] L.Lavgne, F. Caaurang F. an Bergeon B., Moellng of Longtunal Dsturbe Arraft Moel b Flatness Approah, AIAA Guane, Navgaton, an Control Conferene an Ehbt, Teas Austn, USA, 3. [8] Droun A., Mquel T. an Mora-Cano F., Nonlnear Control Strutures an Rotorraft Postonnng, AIAA Guane Navgaton an Control onferene, Honolulu, August 8. F,F F 4 F te Fg. 8: Rotorraft traetor trakng nputs ntal te the rotorraft s hoverng. 6 Conluson In ths ounaton the relaton between fferental flatness an the effetve applablt of non lnear nverse 648