, rst we solve te PDE's L ad L ad n g g (x) = ; = ; ; ; n () (x) = () Ten, we nd te uncton (x), te lnearzng eedbac and coordnates transormaton are gve

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1 Freedom n Coordnates Transormaton or Exact Lnearzaton and ts Applcaton to Transent Beavor Improvement Kenj Fujmoto and Tosaru Suge Dvson o Appled Systems Scence, Kyoto Unversty, Uj, Kyoto, Japan suge@robotuassyoto-uacjp Abstract Ts paper s concerned wt te use o reedom n te exact lnearzaton tecnque Frst, te reedom o te coordnates transormaton and nonlnear state eedbac n te exact lnearzaton s clared as te preparaton or ts use Second, we consder te transent beavor mprovement n te presence o mpulsve dsturbances Fnally, we gve a numercal example and an experment to sow ow to use te reedom and to sow te eectveness o ts metod Introducton Te exact lnearzaton tecnque[, ], wc s to transorm te orgnal nonlnear system nto a lnear controllable one wt state eedbac and coordnates transormaton, s one o te most eectve metods n te eld o nonlnear control In ts metod, te necessary and sucent condton or te tecnque to be applcable s well nown However, we ave to solve a seres o PDE's (partal derental equatons) n order to obtan suc eedbacs and coordnates transormatons Snce tere exst certan degrees o reedom n te soluton o te PDE's, we ave te same degrees o reedom n coosng te eedbac and coordnates transormaton So ar te reedom n te exact lnearzaton tecnque s neglected However, t may be possble to mprove te eedbac system n some sense by usng te reedom n lnearzaton[, ] Te purpose o ts paper s to sow tat ts s actually te case, tat s, we can tae advantage o ts reedom In secton, te reedom o te coordnates transormaton and nonlnear state eedbac n te exact lnearzaton s clared as te preparaton or ts use In secton, observng some propertes wc are nvarant under te reedom, we conder te regulaton o te output aganst an mpulsve dsturbance Furtermore we gve a numercal example and an experment to sow ow to use te reedom and to sow te eectveness o ts metod Exact Lnearzaton and Freedom n Coordnates Transormaton Ts capter dscusses te reedom o te coordnates transormaton or exact lnearzaton as te preparaton or te system desgn usng ts reedom wc s mentoned n capter wt some examples Exact Lnearzaton or te Sngle-nput Systems In ts secton, we consder sngle-nput control systems descrbed by _x = (x) + g(x)u () were x = [x ; x ; ; x n ] T, and g are smoot vector elds and x = x s an equlbrum pont Te Exact Lnearzaton Problem s, gven a system (), to nd ( possble) a eedbac u = (x) + (x)v and a coordnates transormaton = 8(x) suc tat te correspondng closed loop usng te eedbac n te new coordnates s lnear and controllable as te ollowng equaton _ = A + bv () Concernng to te problem, te ollowng result s well nown Teorem [, ] : Exact Lnearzaton Problem or te system () s solvable near x and only te ollowng condtons are satsed () te matrx g(x ) ad g(x ) ad n g(x ) as ran n () te dstrbuton span nvolutve near x n g; ad g; ; ad n I te system () satses te condtons o Teorem g o s p

2 , rst we solve te PDE's L ad L ad n g g (x) = ; = ; ; ; n () (x) = () Ten, we nd te uncton (x), te lnearzng eedbac and coordnates transormaton are gven by te ollowng equatons u = (x)+(x)v := Ln (x) L g L n (x) + L g L n (x) v () = 8(x) := (x); L (x); ; L n (x) T () Toug, wt ts procedure, we can get one par o exactly lnearzng coordnates transormaton and eedbac, tere exst nnte pars o coordnates transormatons and eedbacs n act snce te PDE's () and () ave not only one soluton but nnte ones However t s easy to see tat, we get one soluton o te PDE's () and (), ten all te solutons o te PDE's are gven by te ollowng lemma Lemma : I a uncton (x) solve te PDE's () and () by substtutng (x) or (x) near x, ten te uncton (x) (locally) satses te PDE's and only (x) = ((x)) () d = (8) d old or some smoot uncton n a negborood o x Proo s stragtorward by Frobenus Teorem Toug ts lemma s elementary act, t allows us to regard te uncton as a ree parameter and we can use t or te controller desgn Actually ts result can be extended to te global verson Teorem : I a uncton (x) gves te globally exactly lnearzng eedbac and coordnates transormaton, e te coordnates transormaton s a global deomorpsm on R n and te eedbac s regular or all x, by substtutng or n () and (), ten te uncton (x) gves a globally exactly lnearzng eedbac and coordnates transormaton by () and () and only (x) = ((x)) (9) olds or some smoot uncton, were : R!R s a global deomorpsm Proo : see Appendx Exact Lnearzaton or Mult-nput Systems In ts and ollowng secton, we consder multnput control systems descrbed by _x = (x) + mx g (x)u =: (x) + G(x)u () were x = [x ; x ; ; x n ] T, and g are smoot vector elds and x = x s an equlbrum pont And suppose G(x ) as ran m and te lnear approxmaton o ts system around x s controllable For te mult-nput case, te condton or exact lnearzaton s gven as ollows Lemma [] : Te system () s exactly lnearzable and only te system () as some vector relatve degree r ; r ; ; r m g at x st r + r + + r m = n Toug ts lemma seems to be n a slgtly derent manner rom Teorem, te essence o tem are te same Now let p > p > > p q be ndependent numbers among te vector relatve degree r ; ; r m g and let m be te repeated degree o p Ten te ollowng equatons old m = n = qx mx m () r = qx m p () I te system () satsy te condtons o lemma, tere exsts a coordnates transormaton = 8(x) and te system can be wrtten by te ollowng normal orm n te new coordnates _ = () + mx g ()u =: () + G()u () () := T () (); ; T (m) () ; ; ; T q() (); ; q(mq)t () () (j) () := g () := (j) ; ; (j) ; (j) () p T T (); ; g T (m ) () g () ; ; ; g q()t (); ; g q(mq )T () () g (j) () := ; ; ; g (j) () T = ; ; m p

3 were te elements o are dened as ollows := (j) := () T ; ; (m) T ; ; ; T T q() ; ; {z q(mq) () } T (j) ; (j) ; ; p (j) Here te ndex (j) corresponds to te j-t one o te subsystems wc ave relatve degree p Now we dene two more matrces ~() := () ; ; {z (m) } ; ; q(); ; {z q(mq) } ~G() := g () g m () g(m ) g(m m ) g q() g m q() gq(m q ) gq(m m q ) () (8) were ~ () R m, ~ G() R mm and ~ G() s nonsngular near = 8(x ) It s easy to see tat te lnearzng eedbac s gven by u = ~G(8(x)) ~ (8(x)) + v (9) Te Freedom or te Mult-nput Case Te prevous secton sows te reedom or multnput case s tat o te coordnates transormaton = 8(x) wc transorms te orgnal system nto a normal orm wt vector relatve degree r ; r ; ; r m g To obtan te new coordnates, rst we ave to nd ndependent output unctons (x); ; m (x) wc ave vector relatve degree r ; r ; ; r m g Ater ndng tem, let (j) (x) be te j unctons wc ave relatve degree p among tem Ten te coordnates transormaton 8 : x! s gven by (j) := L (j) (x); = ; ; ; p () Smlar to te sngle nput case, we only ave to sow te reedom o te unctons (j) (x) to ndcate tat o coordnates transormaton Smple examnaton gves te ollowng lemma Lemma : I a set o unctons () (x),, q(mq )(x) gves te exactly lnearzng coordnates transormaton by substtutng (j) (x) or (j) (x) n (), ten a set o unctons (j) (x) gves a (locally) exactly lnearzng coordnates transormaton by () and only, or eac = ; ; ; n m, (j) (x) = (j) ( () ; ; (m); L w s(t) g) (); ; (m ) = () ; ; (m) old or some smoot unctons (j) n a negborood o x, were L w s(t) g denotes all te unctons wose ndex s; t; w satsy te restrcton 8 < : s = ; ; ; t = ; ; ; m s w = ; ; ; p s p + () Proo s omtted Smlar to te sngle-nput case, we can obtan te ollowng global result Teorem : I a set o unctons () (x),, q(mq)(x) gves te globally exactly lnearzng coordnates transormaton by substtutng (j) (x) or (j) (x) n (), ten a set o unctons (j) (x) gves a globally exactly lnearzng coordnates transormaton by () and only, or eac = ; ; ; n m, (j) (x) = (j) ( () ; ; (m); L w s(t) g) () old or some smoot unctons (j), were te mappng col( () ; ; (m ))( ; ; ; L w s(t) g) : R m!r m s a global deomorpsm Proo as te same essence as tat o Teorem and s omtted Applcaton to Transent Beavor Improvement Ts capter treats te controller desgn or mpulsve dsturbance attenuaton as an example usng te reedom n coordnates transormaton Ten ts metod wll be appled to te magnetc levtaton system and ts eectveness wll be sown by experments p

4 Transent Beavor o te Orgnal System In ts secton, we consder te ollowng snglenput system _x = x x n n (x) + g n (x) u =: (x) + g(x)u () Ts system s already n normal orm and ave relatve degree n Now we desgn a stablzng controller or ts system usng exact lnearzaton tecnque wt lnear eedbac and, under a partcular stuaton, try to sape te transent response o te state x, were te uncton s wrtten by (x) = (x ) wt ree parameter by Lemma Here let () = and we assume d dx () = () wtout loss o generalty Ten smple examnaton gves x = [; ; ; x n ] T, = [; ; ; x n ] T () and we mae one more (not general) assumpton or some ( < < n) d dx ten te ollowng olds () = = d dx () = (8) x = [; ; ; x n+ ; ; x n, = [; ; ; x n+ ; ; x n ] T (9) ] T Now consder te beavor o te state x or te ntal condton x() = [; ; ; ; ; ] T () were denotes possbly nonzero ntal state Ts ntal state can be regarded to be set up by some mpulsve dsturbance I te ree parameter satses te assumptons () and (8), ten te ntal state () o te lnearzed system s equal to te orgnal one x() rrespectve o te coce o te parameter Tereore wen te trajectory o In desgnng a stablzng controller usng te exact lnearzaton tecnque togeter wt a lnear eedbac, (x) = (x) and (x) = (x) ( = ) yeld te same controller under ts ntal condton s (t), te trajectory o x s gven by x (t) = ( (t)) () and ts trajectory can be mproved by coosng te ree parameter subject to te condton () and (8) Followng sectons treats te magnetc levtaton system and sow te controller desgn usng above act And muc more concrete way o desgnng wt ts reedom wll be gven Descrpton o Magnetc Levtaton System In ts secton, we consder a magnetc levtaton system wc s to levtate a steel ball at te speced poston by electromagnet sown n Fgure Te parameters are dened as ollows; x p [m] be te gap between te magnet and te ball, [A] be te current, e [V] be te nput voltage, R = : [] be te resstance o te electromagnet, M = : [g] be te mass o te ball and G = 9:8 [m/sec ] be te acceleraton o gravty Wt some more constants Q = : [Hm], X = : [m], X = : [m] and L = : [H], let te state be x = [x p + X ; _x p ; ] T and let te nput be u = e ten we obtan te state space representaton _x = (x) + g(x)u () (x) := g(x) := x Qx G M(X + x ) x Qx R(X + x ) g (X + x )Q + L (X + x )g X + x Q + L (X + x ) () () Ts system satses te condton o Teorem, e exactly lnearzable, and one soluton satsyng te PDE's () and () s gven by (x) = x Easy calculaton sows tat te new coordnates x = [x p + X ; _x p ; x p ] gve te normal orm () And n te ollowng secton, x denotes ts new coordnates Improvement o Transent Beavor Wt te observaton n secton, we now consder te system wll ave an mpulsve dsturbance so tat te states _x p and may jump nstantaneously Ten ntal condton x() may be set up to x() = [; ; ] T, and we coose te ree parameter satsyng te condtons () and (8) or = Consderng tat te trajectory x (t) s gven by (), we sould coose () wc maps nto smaller doman suc as te uncp

5 ton arcsn Furtermore we coose te uncton () as tan or example, ten te trajectory x (t) ave to stay n te bounded doman x (; ) Hence we mae two controllers or te cases (x ) = x () (x ) = sn x () wc bot satsy te condtons () and (8) wt a lnear state eedbac v = [; ; ] () and responses x (t) or te ntal condton x() = [; ; ] T by smulaton s sown n Fgure In te gure, wen x s near te orgn, two trajectores loo te same However, wen x s away rom te orgn, te derence between two trajectores appears Experment Ten we perorm te expermental evaluaton As n te prevous secton, we mae two controllers or te cases (x ) = x (8) (x ) = sn x (9) wc are bot satsy te condtons () and (8) wt a lnear dynamc output eedbac : 9 (s+:)(s+8:+:)(s+8::) s(s+8:)(s+:9+:)(s+:9:) () We eep te steel ball levtated at te equlbrum pont by eac controller, ten we add te mpulsve dsturbance o [m/sec] to te state x, e te velocty o te ball Te responses x (t) or ts ntal condton x() = [; :; ] T are sown n Fgure Sold lne denotes te response or te case (x ) = x and Dased lne denotes te response or te case (x ) = (=) sn x Te result sows tat (x ) = (=) sn x mae te nuence by te dsturbance smaller tan tat n te case (x ) = x as n te prevous secton Te results obtaned n secton and sow tat t s useul to use te reedom n coordnates transormaton or mprovement o transent responses Conclusons In ts paper, we ave studed te use o reedom n te exact lnearzaton tecnque Frst, we ave clared te reedom o te eedbac and coordnates transormaton n te exact lnearzaton procedure as te preparaton or te system desgn usng ts reedom Second, we ave sown tat te reedom o coordnates transormaton can be used to sape te transent beavor n te presence o mpulsve dsturbances Fnally, we ave gven a numercal example and an experment to sow ow to use te reedom and to sow te eectveness o ts metod Snce every lnear controllable system s exactly lnearzable, we can apply ts metod to lnear systems n desgnng nonlnear controllers Reerences [] R Su : On te Lnear Equvalents o Nonlnear Systems, Systems & Control Letters, {, 8/ (98) [] A Isdor : Nonlnear Control Systems, Sprnger-Verlag, Berln, trd edton (99) [] H Njmejer and A J van der Scat : Nonlnear Dynamcal Control Systems, Sprnger- Verlag, New Yor (99) [] T Suge and K Fujmoto : On te Use o Freedom n te Cange o Coordnates n Exact Lnearzaton, Trans SICE, {, 8/8 (99) (n Japanese) [] K Fujmoto and T Suge : Exact Lnearzaton n te Presence o Input Step Dsturbances, submtted to Trans SICE (99) (n Japanese) Appendx Proo o Teorem : Te necessty s obvous and we only sow te condton s sucent By te assumpton, te coordnates transormaton = 8(x) := (x); L (x); ; L n (x) () s a global deomorpsm on R n In ts coordnates, te system () can be descrbed as te normal orm _ = n a n () + b n () T u =: a() + b()u () were a n () := L n (8 ()) and b n () := L g L n (8 ()) Ten a n () and b n () are contnuous and b n () = or all, because te eedbac () by substtutng (x) or (x) s regular p

6 Next we consder anoter set o coordnates produced by te output uncton (9), e = (x) := ((x)); L ()(x); ; L n ()(x) T() and consder a coordnates transormaton 9 :! gven by Ten we consder te eedbac () Te equatons L n a () = n() + ( ) a n () (9) L b La n () = ( ) b n () () old Te unctons L n a and L b La n are contnuous and L b La n = or all, ence te eedbac () s regular or all Ts proves te teorem = 9() = ( ); L a (); ; La n T () () = ( ) ( ) + ( ) () R e n ( n ) + ( ) n were denotes d =d, te sets o coordnates 's are dened by := [ ; ; ] T and te unctons 's are sutable smoot unctons snce s sucently smoot, e as contnuous dervatves or any order Ten we ave to sow s te coordnates transormaton 9 :! s a global deomorpsm and te correspondng eedbac u = Ln a () + v L b L n a () () s regular or all, because = (x) = 9 8(x) and 8 : x! s a global deomorpsm Let 9 :! denote te upper mappng truncated rom te mappng 9 :! Ten we sow te mappng 9 n 9 s a global deomorpsm on R n by nducton For = Te mappng 9 s equvalent to and s a global deomorpsm on R For = m Suppose te mappng 9 m s a global deomorpsm on R m Te mappng 9 m+ s gven by m m+ m+ = = 9 m ( m ) m ( m ) + ( ) m+ () ten te nverse mappng 9 m+ : m+! m+ s m # 9 m ( m ) m (9 m ( m )) + m+ ( ( )) (8) Hence te mappng 9 m+ s a global deomorpsm on R m+, snce te nverse mappng s contnuous (and smoot) or all m+ Te above argument proves te mappng 9 :! s a global deomorpsm on R n Te unctons 's are dened by and + := L a ( + ) + () + x (t) x M Fgure : Magnetc levtaton system tme (sec) L φ tan (x ) φ re (x ) Fgure : Responses o x wen x() = [; ; ] T x p (t) - (smulaton) x- - tme (sec) φ sn (x ) φ re (x ) Fgure : Responses o x or a dsturbance x = : [m/sec] (experment) p