Input-Output Linearisation of Nonlinear Systems with Ill-Defined Relative Degree: The Ball & Beam Revisited. D.J.Leith, W.E.

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1 Input-Output Lnearsatn f Nnlnear Systems wth Ill-Defned Relatve Degree: The Ball & Beam Revsted D.J.Leth, W.E.Lethead Dept. Electrncs & Electrcal Engneerng, Unversty f Strathclyde, 5 Gerge St., Glasgw G QE, U.K. Emal. dug@cu.strath.ac.uk, Tel , Fa Abstract In ths nte t s establshed that fr SISO systems lack f well-defned relatve degree s nt an bstacle t eact nversn. Suffcent cndtns fr the estence f eact lnearsng nputs are establshed. Eact trackng slutns f a number f eample systems, ncludng the ball and beam system studed by Hauser and thers, are analysed n detal. It s shwn that lnearsng nputs may be () hghly nn-unque, () dscntnuus and nclude mpulses etc., () fragle. A framewrk s establshed fr apprmate lnearsatn whereby eact lnearsng nputs are frmulated as the lmts f sequences f realsable, arbtrarly accurate lnearsng nputs. Ths framewrk s cnstructve n nature and f key mprtance when the eact lnearsng nputs are, fr eample, unrealsable. Ths s qute dfferent frm apprmate lnearsatn appraches prevusly cnsdered n the lterature where the degree f accuracy s generally dffcult t prescrbe.. Intrductn The desgn f lnearsng cntrllers has receved sme attentn n the lterature. In partcular, nput-utput lnearsatn (see, fr eample, Isdr 995) s a wdely advcated apprach fr accmmdatng plant nnlneartes. Unfrtunately, current nput-utput lnearsatn methds are strctly cnfned t systems whch are bth mnmum-phase and have well-defned relatve degree. Many systems d nt satsfy these cndtns and ths has mtvated many attempts t etend lnearsatn methds t a wder class f systems. Whlst the lterature manly cncentrates n relang the mnmum-phase requrement, the present paper addresses the lnearsatn f SISO systems wth lldefned relatve degree. In SISO systems, lack f welldefned relatve degree s asscated wth varatns acrss the peratng space n the number f dfferentatns requre fr the utput t be drectly cupled t the nput. (In the MIMO case, relatve degree may als be lldefned due t the cuplng matr havng less than full rank; when the rank s cnstant acrss the peratng space ths may be addressed usng dynamc etensn methds, Isdr 995). Wth regard t SISO systems, Hauser et al. (99) (and, mre recently, Tmln & Sastry 997) cnsder applyng nput-utput lnearsatn methds t a ball and beam system whch des nt have well-defned relatve degree. The prpsed appraches accmmdate systems wth ll-defned relatve degree at the cst f achevng nly apprmate lnearsatn. Furthermre, the requrement fr well-defned relatve degree s s fundamental t the estng nput-utput lnearsatn thery that t s far frm clear that eact lnearsatn s generally even pssble fr systems wth ll-defned relatve degree. The purpse f the present paper s, therefre, t nvestgate whether the lack f a well-defned relatve degree s ndeed a fundamental bstacle t achevng eact lnearsatn f the ball and beam system and ther systems wth ll-defned relatve degree. The dscussn s nt, hwever, cnfned slely t establshng when an eact lnearsng nput ests n specfc cases. General estence cndtns are sught and ne man bjectve f the paper s t establsh sme f the new ssues whch must be reslved n rder t acheve a slutn t the ll-defned relatve degree lnearsatn prblem. It s nted that the cst ncurred by adptng an apprmate, rather than eact, lnearsatn strategy s frequently tlerable prvded the accuracy f the apprmatn s effectvely a desgn parameter and can be chsen t be arbtrarly gd. Hwever, ths s nt generally the case wth the fregng appraches fr whch the degree f accuracy acheved s dffcult t prescrbe. An addtnal purpse f ths nte s therefre t make a frst step twards determnng a systematc apprach fr achevng arbtrarly accurate lnearsatn f systems wth ll-defned relatve degree.. A frst eample The requrement fr well-defned relatve degree s s fundamental t the estng nput-utput lnearsatn thery that t s far frm clear that eact lnearsatn s even pssble fr systems wth ll-defned relatve degree. Cnsder, therefre, the nnlnear system = - + u, y = + g(u) () wth g(u) =.5u u u < u < 5u + u + ().. 5. u < u (Ths system s nt n the cntrl affne frm nrmally cnsdered n the nput-utput lnearsatn lterature but becmes s when augmented wth a sutable nput flter). The relatve degree f the system s when u < and therwse, hence t s nt glbally well-defned. Nevertheless, wng t the relatve smplcty f ths system, t s pssble t analytcally derve an eactly lnearsng nput. Namely, prvded the ntal cndtn f the system s cmpatble wth ntalsng u t be g - (v-), an eact lnearsng nput (such that y=v) s

2 v + v - = & v + - u = (3) g ( v ) v - & - v v + v - = & v + - whch crrespnds t u, - u and u - respectvely. The swtch n u frm v + t g - (v-) trggered by u reachng frm abve r frm belw s cntnuus, but the swtch n u frm g - (v-) t v + trggered by u reachng frm belw r frm abve s dscntnuus n general. In addtn, whle u>, = v and s v-=k =; whle u<-, = v and v-=k =-. Of curse, the ntal cndtns fr the system may be ncmpatble wth ntalsng u t g - (v-) and/r cmputatnal neacttude n the slutn t () s sure t cause k t depart frm and k t depart frm. Hence, the lnearsng system, (3), lacks rbustness. Dsturbances and msmatches n the ntal cndtns may be accmmdated by augmentng the deal nput (3) t v + v - = & v + - g ( v ) v - & - v u = (4) v + v - = & v + - u err therwse where the addtnal term, u err, s selected t ensure that the requred trajectry, v, s attractve; fr eample, v + - λ(v- y) -, v + - λ(y - v) v + - λ(v- y) uerr = (5) + v- λ + + v sgn( y- v) therwse where λ and λ are pstve cnstants wth λ >. The asscated clsed-lp errr dynamcs y -v+ λ(y-v) =! ' λ "( sgn( v + - λ (v- y) -, v +- λ (v-y) y-v v y-v) = therwse cnfrm that the reference trajectry s attractve as requred. Hence, t s clearly demnstrated by the abve eample that the lack f a well-defned relatve degree des nt, n tself, prevent the eact nput-utput lnearsatn f a nnlnear system. Remark. Dscntnuty f eact lnearsng nput. It shuld be nted that the eact lnearsng nput n ths eample s generally dscntnuus fr trajectres n an peratng regn where the relatve degree s ll-defned. Ths bservatn prvdes cnsderable nsght nt the restrctn f cnventnal nput-utput lnearsatn thery t systems wth well-defned relatve degree. It appears t be asscated wth requrng the lnearsng # %%$ &% % (6) nputs t be cntnuus and can be relaed by etendng cnsderatn t nclude mre general types f nput. Remark. Dscntnuty mples generalsed functns. By augmentng the abve system wth ntegratrs at the nput, t fllws mmedately that nvertng nputs fr systems wth ll-defned relatve degree may nvlve mpulses and hgher dervatves f steps. Of curse, such nputs are nt physcally realsable. Hwever, smlarly t the stuatn wth lnear systems, ths ssue can be addressed prvded the nvertng nput can be frmulated as the lmt f an apprprate sequence f realsable nputs such that arbtrarly accurate lnearsatn may be acheved usng a realsable nput. Nte that ths stuatn s qute dfferent frm the apprmate lnearsatn appraches prevusly cnsdered n the lterature where the degree f accuracy acheved s generally dffcult t prescrbe. 3. Eactly lnearsable systems wth ll-defned relatve degree Cnsder the SISO nnlnear system Σ: = f ( ) + g( ) u, y = h ( ) (7) wth slutn defned n tme nterval [,T] (fr clarty, the analyss whch fllws assumes a fnte tme nterval but t may be readly etended t the nfnte nterval subject t the addtnal cndtn that the nternal zer dynamcs are stable). The relatve degree at an peratng pnt at whch equals s r( ). Let Φ R n dente the peratng regn f nterest and let r nf r( ) and = Φ r = sup r( ). Clearly, < r r and t s assumed that r s Φ fnte (whch smply crrespnds t a requrement that the utput f the nnlnear system s ndeed cupled t the nput at every peratng pnt n Φ, Isdr 995). The relatve degree s well-defned n Φ when r() s the same at every peratng pnt; that s, r = r. Cnventnal nput-utput lnearsatn thery s strctly cnfned t peratng regns wthn whch the relatve degree s well defned and nvlves dfferentatng the utput, y, the mnmum number f tmes necessary fr the nput t appear drectly n the utput equatn (namely, dfferentatng the utput r = r tmes). In the present cntet, hwever, the requrement fr the relatve degree t be well-defned s relaed; that s, cnsderatn s etended t accmmdate peratng regns wthn whch r need nt be equal r. Snce the relatve degree may vary but nevertheless has an upper bund, r, cnsder the r th dervatve f the utput, y. It fllws by dfferentatng (7) that y ( r) s f the frm (eplct epressns are gven by Lamnabh-Lagarrgue & Cruch 988)

3 @ (r) r ( j) ) ( ) j j= y = Lf h + c ( ) u + F(, u,, u ) * +, where c j ( ), F(,,, ) are nnlnear functns; fr eample, (8) y = L h+ L L h+ L L L h u+ L L h u+ L L h u (9) (r+) r+ r r r r f g f f g f g f g f Whlst the epressn fr y ( r) s, n general, rather cmple, t s nevertheless nted that the hghest dervatve f u nvlved s u ( ) and, furthermre, the nnlnear functn F s ndependent f u ( r r). It fllws mmedately frm (8) that the slutns (f any) t (r- r) ( r) r ( j) c ( ) u = v ( Lf h + c j( ) u j= () ( ) + F(, u, -, u ) lnearse the system (7) n the sense that y ( r) ( r) = v () The c j, j=... r r are nether unfrmly zer nr unfrmly nn-zer. Hence, () s a type f sngularly perturbed dfferental equatn fr whch the determnatn f slutns s generally nt straghtfrward. In partcular, t shuld be nted frm the prevus dscussn that t may generally be necessary t cnsder nn-classcal types f weak slutn nvlvng dscntnutes, mpulses and s n. (Whle such slutns may be well defned, they are weak n the sense that they are nt classcally dfferentable. Nte that theretcal results by, fr eample, Hrschrn & Davs (987) are restrcted t stuatns where the lnearsng nput s classcally dfferentable). Snce these slutns are nt physcally realsable, the requrement s t frmulate them as the lmt f an apprprate sequence f realsable nputs such that arbtrarly accurate nversn can be acheved usng a realsable nput. Ths ssue s addressed by the fllwng result. Therem Suppse that fr a target trajectry, v, a slutn (n the pecewse sense), u, ests satsfyng. = f ( ) + g( ) u / (r) r ( j) (r- r) y = Lf h + c ( ) u + c ( ) u () j= j ( ) (r- r) ( r) r ( j) = f + c j( j= + F(, u,, u ) c ( ) u v ( L h ) u (3) ( ) + F(, u,, u ) wth 34 c ( ) c ( ) ε c ( ) = (4) ε sgn c ( ) c ( ) < ε where c r r ( ) s defned by (8) and ε s a pstve fnte cnstant frm sme sequence {ε :ε as }. Suppse, n addtn, that there are a fnte number f ntervals [t j,t j+ ], j=,,... n whch c ( ) c ( ) s nn-zer and that t j+ τ ( r r ) t j t u j ( τ ) dτ dτ (5) r j s 7 bunded unfrmly n. Let y dente the slutn t = f( ) + g( ) u, y = h( ) wth ntal cndtns y (k) (t )=v (k) (t ), k=,, r. Under the fregng cndtns, the trackng errr y -v as ε. That s, an arbtrarly accurate lnearsng nput can be fund. Prf The relatve degree f the system () s welldefned and equal t r. Cnsequently, an nput-utput lnearsng nput fr ths system s defned by (3). A sequence f systems () and asscated sequence f nputs {u } are defned as ε. It fllws frm (8), () and (3) that (r) (r) ( ) y v = c ( ) c ( ) u 8 9 (6) r>? k ( k) ( k) t t k= k! t ( r r) +: >? τ c ) ) u ( ) d d t t ( c ( τ τ τr On ntegratng (6), y ( t) v( t) = y ( t ) v ( t ) ( ) r ( k) ( ) ( ) ( k) t t y t v t k= k! t tj j + <=< τ t j ( ) k j+ ε u ( τ ) dτ dτ The result nw fllws mmedately. r (7) Remark 3. Fllwng the cnventnal nput-utput lnearsatn apprach, the requrement that the ntal cndtns, y (k) (t )-v (k) (t ), k=,, r, are zer can be relaed prvded v s selected t ensure asympttc trackng f the reference trajectry. Remark 3. The cndtn, (5), s a generalsed bundedness requrement. It shuld s emphassed that there s n requrement fr the u themselves t be unfrmly bunded. Fr eample, eact lnearsng nputs may nvlve mpulses, n accrdance wth Remark.. Remark 3.3 Fllwng cmmn practce n the swtched system lterature, cnsderatn s, fr smplcty, cnfned t a pece-wse slutn t (3). Ths restrcts attentn t ntal cndtns fr the state and trajectres, v, such that there are a fnte number f ntervals [t j,t j+ ], j=,,... n whch c ( ) c ( ) s nn-zer. Hwever, ths may be relaed by emplyng a mre general type f slutn defntn; fr eample, Flppv (964). Remark 3.4 When the relatve degree s well-defned, then ths therem reduces t classcal nput-utput lnearsatn.

4 B A A E A A CDDDD W H IJJJJ T U K L N OPPPP Q R b ( ` a Many f the cndtns n the fregng therem are, unfrtunately, dffcult t test eplctly at present. Despte the cnsderable bdy f lterature relatng t dscntnuus (especally swtched) systems, nncnservatve estence cndtns are lackng. Wth regard t the present cntet, unfrm satsfactn f the bundedness cndtn, (5), s at present als dffcult t test n general. Nevertheless, the fregng results are cnstructve n nature and may therefre be nvestgated numercally by, fr eample, smulatn. Wth regard t the eample n sectn, t s straghtfrward t cnfrm by smulatn that the sequence f nputs defned by ()- (3) cnverges numercally t the eact nvertng nput, (3) and ths eample s nt pursued further. In the remander f ths paper, nsght nt the nature f these cndtns s sught va sme llumnatng eamples. 4. Ball and beam re-vsted: Eact trackng The ball and beam an nterestng nnlnear system whch s wdely used as a benchmark eample. The analyss f ths system s, cnsequently, f cnsderable nterest n ts wn rght and has been the subject f numerus papers. In partcular, the ball and beam s a SISO system wth ll-defned relatve degree and prevusly studed by Hauser et al. (99) and, mre recently, Tmln & Sastry 997. Cnsder the ball and beam system llustrated n fgure. The beam s made t rtate by applyng a trque t the beam at ts centre f rtatn and the ball s free t rll alng the beam wth dynamcs descrbed (Hauser et al. 99) by GGGG= F B 4 G sn 3 MMMM+ SSSS=, y (8) u T T where 3 4 = r rv θ θ V, r s the pstn f the ball, θ the angle f the beam, u s related t the trque by a nn-sngular transfrmatn and the values f the parameters B and G are, respectvely,.743 and 9.8. It s assumed that the ball remans n cntact wth the beam at all angles; that s, n practcal terms there ests gudes ensurng cntact (Hauser et al 99 requre a restrctn n angle and acceleratn t ensure cntact). It shuld be nted that t s nt the physcal behavur f an actual ball and beam, whch anyway the equatn (8) nly apprmates, that s f nterest here but the prpertes f the mathematcal mdel, (8), tself. On dfferentatng (8) y = WW y = B G sn ( 3) X Y (9) 4 3 y = B + B u BG cs Evdently, n a neghburhd f each peratng pnt at whch and 4, the ceffcent f u n the y ( 3) equatn s nn-zer. Hence, at these peratng pnts the ball and beam system has a well-defned relatve degree f three and t fllws mmedately frm cnventnal npututput lnearsatn thery (see, fr eample, Isdr 995) that chsng ( 3) Z [ v B4 BG4 cs3 u = () B4 lnearses the system n the sense that y ( 3 ) v ( 3 = ) Hwever, at peratng pnts at whch the ball pstn,, and/r beam angular velcty, 4, are zer, the relatve degree s nt well-defned snce t s nt cnstant wthn any pen neghburhd f such an peratng pnt. Cnsequently, cnventnal nput-utput lnearsatn methds are nt applcable at these peratng pnts. Mrever, snce the ball and beam system s nt nvlutve, eact feedback lnearsatn methds cannt be appled (Hauser et al. 99). In rder t address the afrementned dffculty, Hauser et al. (99) prpse drppng the term nvlvng u n the epressn fr y ( 3) t btan the apprmate utput equatn \ ( 3) y = B4 BG 4 cs3 () On dfferentatng (), ] ( 4) 4 y = B + BG( B) sn + B BG cs u 4 4 ^ _ () Assumng that the ceffcent f u n () s nn-zer n the relevant peratng envelpe, t can be seen that ths apprmate system has well-defned relatve degree f fur. Hence, t fllws frm cnventnal nput-utput lnearsatn thery that ( 4) 4 4 v B + BG( B) 4 sn 3 u = B4 BG cs 3 lnearses the apprmate system n the sense that 4) ( 4) (3) y = v (4) Hauser et al. (99) apply the nput defned by (3) t the rgnal ball and beam system and present smulatn results demnstratng that apprmate trackng f a requred trajectry s acheved. In a smlar ven, Tmln & Sastry (997), prpse a swtched cntrller whch utlses an apprmately lnearsng chce f nput n a neghburhd abut the peratng pnts at whch and/r 4 are zer, and elsewhere swtches t the eact lnearsng nput defned by (). Whlst Hauser et al. (99) and Tmln & Sastry (997) bth cnsder specfc chces f nput whch apprmately nput-utput lnearse the ball and beam system, nether address the fundamental ssue f whether eact nput-utput lnearsatn s, n fact, pssble nr d

5 j c they cnsder systematc methds fr prescrbng the accuracy f an apprmate lnearsatn. 4. Estence f eact trackng nputs Intally, fllwng Hauser et al. (99), cnsder the task f achevng eact utput trackng f the trajectry, y ref = 3cs(πt/5). Snce the requred trajectry passes thrugh zer, eact trackng f ths trajectry necessarly requres cnsderatn f an peratng regn wthn whch the relatve degree s ll-defned. Owng t the relatve smplcty f the ball and beam eample, t s pssble t derve eplct nfrmatn cncernng the frm f nput whch ensures eact trackng f the trajectry. It s evdent frm (8) that the beam angle, θ (.e. state 3 ), s related t the nput, u, smply by a duble ntegratr. Cnsder, therefre emplyng θ as a vrtual nput t the system. Ths change f nput fcuses attentn n the prmary dynamcs f the ball and beam. Assume, fr the mment, that the utput, y, equals y ref. It fllws frm the ball and beam equatns, (8), that alng ths utput trajectry the vrtual nput, θ, must be a slutn, f ne ests, f wth π Byθ y BG sn θ 5 = def gh + (5) =klm np qsr y π 9 y 5 (6) The ssue f the estence f a slutn t the eact trackng prblem s thus refrmulated as the estence f a slutn, θ, t (5) and (6) fr all values f utput, y, crrespndng t the reference trajectry, y ref. The equatns (5) and (6) specfy a secnd-rder nnlnear dfferental equatn and s can be nvestgated usng standard phase-plane technques. A detaled phase plane analyss s prvded n the Append. Evdently, the analyss establshes that eact trackng nputs d est fr the ball and beam system. Mrever, the analyss hghlghts a number f mre general ssues relatng t the cndtns n the therem prved n sectn 3. Remark 4. Nn-unqueness f lnearsng nputs. Nt nly d eact trackng nputs est fr the ball and beam system, there ests nfntely many such nputs sme f whch are cntnuus and thers whch are dscntnuus. The nput derved by Hauser et al. (99) apprmates that based n the cntnuus slutn marked A n fgure b. Remark 4. Nn-unqueness mples pssble nn-smth cnvergence. Althugh the analyss n sectn 3 establshes cndtns under whch the utput sequence {y } cnverges smthly (t v), the crrespndng nput sequence {u } need nt cnverge n the same manner. Rather, snce there may est many eact trackng nputs, the u may swtch frm apprmatng ne partcular nput t anther (and ndeed ths s bserved n numercal smulatns). Remark 4.3 Charactersatn f eact lnearsng nputs. Nn-smth cnvergence makes t dffcult at present t establsh a general charactersatn f the lmtng set f (eact trackng/lnearsng) nputs and ths remans an pen prblem. Charactersatn f the lmtng set s, nevertheless, pssble fr certan classes f system. Fr eample, let Σ dente any set f systems fr whch the lnearsng nputs may cntan step dscntnutes (e.g. Eample abve and/r the Hauser ball and beam) and cnsder the class f systems frmed by augmentng the memebre f Σ wth ntegratrs at the nput. It fllws mmedatedly that the lnearsng nputs fr ths class are just the dstrbutns (ncludng steps, mpulses etc.). Nte that the generalsed bundedness requrement n therem f sectn 3 des nt preclude such unbunded nputs. Such nputs are nt physcally realsable but ths ssue can be addressed prvded the nvertng nput s frmulated, as n the present analyss, as the lmt f an apprprate sequence f realsable nputs such that arbtrarly accurate lnearsatn may be acheved usng a realsable nput. 5. Ball & beam cntnued: fraglty f trackng Phase plane analyss establshes that the lack f welldefned relatve degree s nt an bstacle t eact trackng f the ball and beam. Hwever, these results apply t an dealsed stuatn where the ntal cndtns f the system ensure that the respnse t an apprprate nput eactly tracks the trajectry. Even when cnsderng a cmputer smulatn where mdellng errrs and measurement nse can be elmnated, the system may be numercally perturbed away frm the requred trajectry and, therefre, rbust trackng whch can accmmdate neact ntal cndtns and perturbatns s requred. It fllws frm t (8) that when there ests a slutn, θ, t tt Byθ = v + BG sn θ (7) then the ball and beam system s lnearsed n the sense that uuvuu y = v (8) Owng t the square term n the left-hand sde f (7), t s ww clear that there can nly est slutns when the sgn f v + BG sn θ s the same as the sgn f y. It fllws frm nspectn f the phase prtrats n fgure that v+ BG snθ and yθ are bth zer at the etreme pnts f the requred trajectry. At these tmes, the system s at a bundary pnt and the eact trackng slutn s etremely senstve t any perturbatn n v zz. Indeed, at such bundary pnts an nfntesmally small change n v{{ can result n the sgn f the rght hand sde f (7) beng ppste t that f y n whch case a slutn ceases t est. (Slutn here refers t an nput whch acheves eact trackng f the reference trajectry. In practcal terms, nn-estence f such a slutn mples that nly

6 ƒ ŠŠŠŠ Ž naccurate trackng s acheved). It shuld be nted that ths stuatn s eacerbated by the estence f nputs whch, whlst ensurng trackng f the requred trajectry, cease t est befre the utput magntude reaches 3; fr eample, the slutn marked B n fgure b. Hence, althugh there ests an nput whch ensures eact trackng f the reference trajectry under deal cndtns, the trackng cannt be epected t be nn-fragle. 5. Hauser ball & beam Adptng the apprach n sectn 3, n smulatns (wth v (4) =d 4 /dt 4 (3cs(πt/5))) the trackng errr, even between y (4) and v (4), s fund t rapdly ncrease after a shrt ntal perd f accurate trackng. It s necessary t augment the system, wth an uter feedback lp by selectng v (4) accrdng t ( 4) ( 4) ( 3) ( 3) v = yref + λ 3( yref y ) + λ ( } y ref y) (9) + λ(~~ yref y) + λ ( yref y) The uter feedback lp s requred t be strng wth the ceffcents selected s that the ples f the nmnal errr dynamcs are placed at s=-. Tme hstres f the utput trackng errr, y-y ref, are shwn n fgure 5 fr decreasng values f perturbatn, ε. On clser nspectn f fgure 5, t can be seen that the magntude f the trackng errr generally decreases as ε decreases, as mght be antcpated, ecept fr the rather pecular dsturbances whch can be seen at tmes near 4,, 4 and 9 secnds and whch determne the peak trackng errr bserved (n partcular, between y (4) and v (4 ). The rapd ncrease n trackng errr prevusly bserved n the absence f the restrng actn f the uter feedback lp s asscated wth these dsturbances. It s straghtfrward t verfy that the dsturbances bserved crrespnd t perds when v + BG sn θ s f ppste sgn t y and thus durng whch there ests n eact trackng slutn fr the Hauser ball and beam system. These perds ccur when a slutn bundary s reached and, n accrdance wth the prevus dscussn, the fragle nature f the eact trackng slutn cmbned wth the apprmate nature f the numercal slutn subsequently leads t the bundary beng vlated. Remark 5. Generalsed bundedness cndtn. Ths s an eample where the generalsed bundedness cndtn n the therem f sectn three s vlated and llustrates the nn-trval nature f ths cndtn. 5.. Blnear ball & beam The nn-cnvergence f the sequence f apprmate lnearsng nputs n the case f the Hauser ball and beam s asscated wth the sngular nature f the system whereby there des nt est n the phase plane an pen set f trajectres whch ncludes 3cs(πt/5) and fr whch each member f the set can be eactly tracked by ths system. Ths s evdently a sngular stuatn. T clarfy the analyss, and cnfrm that the dffculty les wth the square term n the dynamc equatns f the Hauser ball and, beam, cnsder a mdfed ball and beam system 3 4 = ˆ B G sn Œ, u = y (3) Ths system s dentcal t the Hauser ball and beam ecept fr the term 4 replacng 4. The ssues asscated wth the sngular nature f the square term n the left hand sde f (7) are thereby avded. Nevertheless, n dfferentatng (3) t can be seen that y = y = B G sn ( 3) (3) 4 3 y = B 4 + Bu BG 4 cs 3 Hence, smlarly t the rgnal ball and beam system, the relatve degree f the mdfed system s stll ll-defned n peratng regns where may be zer. Fr the csne trajectry as cnsdered prevusly, phase plane analyss agan establshes that (nfntely many) eact trackng nputs est. The trackng s nw nn-fragle snce the sngularty asscated wth the square term n the rgnal ball and beam s n lnger present. Adptng the apprach n sectn 3, the utput trackng errr between y (4) and v (4 s pltted versus ε n fgure 4. It can be seen that the trackng errr (as measured by the ntegral f the magntude f the pnt-wse errr) decreases mntncally as ε decreases. As usual, t remve the nfluence f ntal cndtns and ensure that y asympttcally tracks v (n addtn t y (4) trackng v (4) ), t s necessary t augment the system wth an uter feedback lp. Hwever, ths feedback can be very weak snce t s n lnger requred remedate breakng f the slutn bundary as n Cnclusn Prevus wrk n SISO systems wth ll-defned relatve degree has been cnfned t apprmate lnearsatn methds, wth lttle prvsn t prescrbe the apprmatn errr nr cnsderatn f whether eact lnearsng nputs may n fact est. The cntrbutn f ths nte s fur-fld, namely. It s establshed that fr SISO systems lack f welldefned relatve degree s nt an bstacle t eact nversn.. In the case f general SISO cntrl-affne nnlnear systems, suffcent cndtns fr the estence f an arbtrarly accurate lnearsng system are establshed. Whle the cndtns nvlved are presently dffcult t test analytcally, the dervatn s cnstructve n nature and the cndtns may therefre be evaluated numercally.

7 3. Eact trackng slutns f a number f eample systems, ncludng the ball and beam system studed by Hauser and thers, are analysed n detal. In vew f the wdespread use f, n partcular, the ball and beam as a benchmark eample, ths analyss s f cnsderable nterest n ts wn rght. Valuable nsght s ganed nt the fregng lnearsablty cndtns and the nature f lnearsng nputs fr ll-defned relatve degree systems generally ncludng that nputs may be () hghly nn-unque, () dscntnuus and nclude mpulses etc., () fragle. 4. A framewrk s establshed fr apprmate lnearsatn whereby eact lnearsng nputs are frmulated as the lmts f sequences f realsable, arbtrarly accurate lnearsng nputs. Ths framewrk s cnstructve n nature and f key mprtance when the eact lnearsng nputs are, fr eample, unrealsable. Nte that t s qute dfferent frm apprmate lnearsatn appraches prevusly cnsdered n the lterature where the degree f accuracy s generally dffcult t prescrbe. Acknwledgement D.J.Leth gratefully acknwledges the supprt prvded by the Ryal Scety fr the wrk presented. References FILIPPOV, A.F., 964, Dfferental Equatns wth Dscntnuus Rght-Hand Sde. Amer. Math. Sc. Translatns, 4, HAUSER,J., SASTRY,S., KOKOTOVIC,P., 99, Nnlnear Cntrl va Apprmate Input-Output Lnearzatn: The Ball and Beam Eample. IEEE Transactns n Autmatc Cntrl, 37, HIRSCHORN,R., DAVIS,J.,987, Output Trackng fr Nnlnear Systems wth Sngular Pnts. SIAM J. Cntrl & Optmsatn, 5, ISIDORI, A., 995, Nnlnear Cntrl Systems. (Sprnger, Lndn). KHALIL, H.K., 99, Nnlnear Systems. (Macmllan, New Yrk). LAMNABHI-LAGARRIGUE,F.,CROUCH,P., 988, A Frmula fr the Tme Dervatves f the Output f Nnlnear Systems, Systems & Cntrl Letters,, pp-7. LEITH, D.J., LEITHEAD, W.E., 999, Input-Output Lnearsatn by Velcty-Based Gan-Schedulng. Internatnal Jurnal f Cntrl, 7, TOMLIN,C.J., SASTRY,S., 997, Swtchng Thrugh Sngulartes. Prceedngs f Cnference n Decsn and Cntrl, San Deg. Append Befre cnstructng the phase prtrat f the ball and beam slutn n detal, the fllwng features can be deduced mmedately.. The sngular pnts, at whch many trajectres ntersect n the phase prtrat fr (5) and (6) are ndcated n š fgure a. š They are θ + πn,, π + πn,, š š θ + πn, 3, π θ + πn, 3, (3) š š π + θ + πn, 3, π θ + πn, 3 : n Ζ wth θ 3 π œž Ÿ = œ ž Ÿ sn BG 5. Owng t the square term n the left hand sdes f (5) and (6), a slutn ests nly when the sgns f the rght hand sdes are apprprate. It s knwn a prr that 3cs(πt/5) s a slutn t (6) wth 3 y 3 and thus the rght hand sde f (6) nn-negatve as requred. Hwever, t s evdent that a slutn t (5) can nly est n the regn n the phase plane wthn π whch the sgn f y + BG sn θ s dentcal t 5 the sgn f y. Ths cnstrant arses frm the square term n the left hand sde f (5). The regn n the phase plane wthn whch a slutn t (5) and (6) cannt est s ndcated n grey n fgure a. The bundary s the curve π ª «y + BG θ = y 5. sn, {, 3, 3} (33) 3. Any slutn wth y=3cs(πt/5) must traverse cntnuusly frm sngular pnts wth y=-3 t sngular pnts wth y=3 and vce versa va sngular pnts wth y=. 4. Owng t the square terms n bth (5) and (6), t s necessary t cnsder slutns crrespndng t bth the apprprate pstve and the negatve square rts. Ths leads t fur pssble cmbnatns; that s, y and θ bth pstve, y and θ bth negatve, y pstve and ±θ negatve, ²y negatve and ³θ pstve. The phase prtrat f the slutns t (5) and (6) fr y and θ bth pstve has the fllwng features () There s a sngle trajectry cnnectng the sngular pnt (,) t the sngular pnt (θ,3). () All the trajectres t the left f the trajectry descrbed n () cnnect the sngular pnt (,) t a pnt n (33) wth <y<3. () T the rght f the trajectry descrbed n (), there are nfntely many trajectres cnnectng the sngular pnt (,) t the sngular pnt (π-θ,3). (v) All the trajectres belw thse descrbed n () cnnect the sngular pnt (,) t a pnt n (33) wth <y<3. The phase prtrat fr -π θ s btaned frm the prtrat fr θ π by the transfrmatn (θ,y) (-θ,-y) and θ π wth

8 reversng the arrws. Sme trajectres are depcted n fgure b. T btan the slutns wth µ µ y and θ bth negatve the arrws n fgure b are smply reversed. The phase prtrat f the slutns t (5) and (6) fr θ π wth y negatve and θ pstve has the fllwng features () There s a sngle trajectry cnnectng the statnary pnt (π-θ,3) t the sngular pnt (π,). () All the trajectres t the rght f the trajectry descrbed n () cnnect the sngular pnt (π,) t a pnt n (33) wth <y<3. () T the left f the trajectry descrbed n (), there are nfntely many trajectres cnnectng the sngular pnt (θ,3) t the sngular pnt (π,). (v) All the trajectres belw thse descrbed n () cnnect a pnt n (33) wth <y<3 t the sngular pnt (π,). The phase prtrat fr -π θ s btaned frm the prtrat fr θ π by the transfrmatn (θ,y) (-θ,-y) and reversng the arrws. Sme trajectres are depcted n fgure c. T btan the slutns wth y pstve and ¹θ negatve the arrws n fgure c are smply reversed. Whle the phase prtrat n fgures b and c s nly shwn fr the regn -π θ π, wng t the snusdal term n (5), t s perdc wth perd π (see fgure a). It can be seen mmedately, by nspectn f the phase prtrat, that there est an nfnte number f nputs, θ, fr whch the utput, y, eactly tracks any quarter perd f the trajectry 3cs(πt/5). The phase prtrat nly depcts slutn fragments and these must, f curse, be peced tgether n rder t btan a slutn ver a full perd f the reference trajectry wng t the sngular pnts n the phase prtrat. T ensure that the actual nput, u, s cntnuus, θºº must be cntnuus at all pnts ncludng thse fr whch y= and y=±3. Snce,»» d θ»¼»» θ θ = + dy y d dy y (34) cntnuty at y=±3 requres cntnuty f dθ/dy. In addtn, cntnuty at y= requres cntnuty f d θ/dy. The nput derved by Hauser et al. (99) and pltted n fgure 3 can be seen t apprmate that based n the cntnuus slutn marked A n fgure b. r θ Fgure Schematc f ball and beam system y=bgsn(θ)(5/π) y=3 y (m) y (m) 3 - y=-3 sngular pnt θ (rads) (a) A B C θ (rads) (b)

9 y (rads) y-y ref (m) ε= - ε= -3 ε= θ (rads) (c) Fgure Phase prtrats f ball and beam alng utput trajectry 3cs(πt/5): (a) vervew wth regns wthn whch n slutn ests ndcated n grey, (b) detaled trajectres wth½ y,¾ θ pstve, (c) y negatve, Àθ pstve. Fr ncreased clarty, th ebegnnng/end f the trajectres are marked by n (b) and (c). Á (m/s 3 ) tme (s) Fgure 5 Trackng errr n Hauser ball and beam eample fr decreasng perturbatn, ε. T ( 4) ( 4) y v ds ε (unts) Fgure 4 Trackng errr vs. ε fr mdfed (blnear) ball & beam.