CONTROL SYSTEMS, ROBOTICS AND AUTOMATION - Vol. XII - Lie Bracket - Kurt Schlacher

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1 LIE BRACKET Kurt Schlacher Deartmet for Automatc Cotrol ad Cotrol Systems Techology,,Johaes Keler Uversty Lz, Austra Keywords: olear cotrol, cofgurato sace, state sace Cotets. Itroducto 2. Bascs of Mafolds ad Budles 2.. Mafolds 2... Fbered Mafolds ad Budles 2.2. Flow, Taget Vectors ad Taget Budle 3. Le Dervatves ad the Le Bracket 4. Dstrbutos ad the Theorem of Frobeus 5. A Short Examle 6. Cocludg Remarks Glossary Bograhcal Sketch Summary The Le bracket s a ma whch assgs a thrd taget vector feld to two gve taget vector felds, all defed o a abstract mafold. Therefore, ths chater starts wth a troducto to the cocet of -dmesoal abstract mafolds ad secal ars of mafolds, called budles. These costructos are ecessary to derve the taget budle of a mafold, the cotaer of all taget vector felds. The Le dervatve of a smooth fucto s a lear ma that resects the Lebz rule; ts commutator s the Le bracket. But the Le bracket s also a dfferetal oerator that acts o vector felds, that tells us whether flows o mafolds commute, that hels to fd hdde costrats systems of lear artal dfferetal equatos. The last roerty s cotaed the theorem of Frobeus. After the dscusso of these tocs ad ther sgfcace for cotrol, a short examle cocerg the cotrollablty of a olear system shows how the develoed mathematcal machery hels to solve roblems olear cotrol.. Itroducto Whe egeers vestgate systems whch are descrbed by a set of ordary dfferetal equatos, they have to face the roblem to fd a sutable sace whch cotas the state ad the ut of the system. Ufortuately, the choce of a Eucldea sace s ot ossble geeral, ad oe has to ass to a more abstract cocet. E.g. oe caot descrbe all rotatos the 3-dmesoal sace globally by 3 coordates, although a local descrto s always ossble. Abstract mafolds overcome ths roblem, sce they behave locally lke a ece of a Eucldea sace, but they take to a cout the global olear behavor. Ths aroach mles that oe has to exted the calculus from the Eucldea sace to mafolds. Choosg secal coordates for a roblem

2 allows us to erform all the vestgatos the usual way, but oe has to face the roblem that a caocal choce of coordates does ot exst. Roughly seakg, all the calculatos must be meagful for ay coordate system. Ths dea leads to a coordate free reresetato of dyamc systems. It wll tur out that the systems uder cosderato ca be rereseted by the hel of secal mathematcal obects, called taget vector felds o abstract mafolds. Ths mles that dfferetal geometry becomes the ma tool for the descrto ad the desg of olear systems. The Le bracket s a ma whch assgs two taget vector felds to a thrd oe. The mortace of ths ma follows from the fact that ths ma aears roblems whch do ot seem to be lked to each other. E.g. the Le bracket measures the chage of a vector feld movg alog a curve, t s the commutator of two dfferetal oerators, t descrbes the hdde costrats systems of lear artal dfferetal equatos, t measures the lack of commutablty of secal mas called flows, etc. For cotrol urose, the Le bracket comes to lay, f we look for olear statead/or ut trasformatos. To get a feelg for ths roblem, we take a look at the smle -dmesoal lear tme varat system x = ax, =,...,, wth z = a T x = T z. Let us cosder the state trasform =, (). By takg the tme dervatve of () we get, of course = T x =. (2) The mortat observato s that we ca choose the trasformato (2) for the dervatves of the coordates freely ad derve (), the trasformato for the coordates, a trval maer. Let us cosder the olear system x = f ( x), =,..., wth smooth fuctos f together wth the olear state trasform z =ϕ ( x). (3) Takg aga the tme dervatve of (3) we get

3 = z = T ( x ) x, (4) where the fuctos T meet T ( x ) = ϕ ( ) x x. (5) Here, t s ot ossble to choose the fuctos T (4) freely, sce the fuctos T must satsfy (5). Therefore, gve (4) we have to check, f s ossble to fd (3). Addtoally, we have to solve a system of artal dfferetal equatos to derve ϕ fromt. Now, the Le-bracket allows us to costruct admssble choces for T ad tells us, whether these artal dfferetal equatos have a o trval soluto. Ths chater s orgazed as follows. Secto 2 gves a troducto to the cocet of abstract mafolds ad secal ars of mafolds, called budles, ad develos the bascs for dog calculus o mafolds. Secto 3 resets the Le bracket ad shows that ths bracket belogs to a famly of dfferetal oerators that oerate o geometrc obects defed o a mafold. Although the Le bracket s a mortat ma the mortace for cotrol s strctly related to the theorem of Frobeus whch s reseted Secto 4. A short alcato to the roblem of cotrollablty of olear systems s gve Secto 5. Fally, ths chater fshes wth some remarks cocerg dfferetal geometry ad the lterature. 2. Bascs of Mafolds ad Budles Curves ad surfaces the Eucldea sace were studed sce the earlest days of geometry. However, the dscoveres of Gauss rofoudly altered the course of dfferetal geometry ad oted the way to the cocet of a abstract mafold. Therefore, we gve the basc deftos ad reset some essetal results cocerg - dmesoal abstract mafolds. Oe ca costruct more comlex mafold from smler oes. Fbered mafolds are a mort case; t wll tur out that the geeralzatos of the taget lae of a surface, the so called taget budle s a secal fbered mafold. Therefore a short troducto to fbered mafolds ad budles s also gve Secto 2.. The subsecto after the ext resets the cocet of taget budles, taget vectors for abstract mafolds ad troduces mortat mas o mafolds, called flows. 2.. Mafolds A mafold s, roughly seakg, the geeralzato of a -dmesoal smooth surface the sace m wth m. Although oe ca show that every smooth -dmesoal 2+ mafold ca be embedded, whch was rove by Hassler Whtey, we wll gve a more abstract defto of a smooth mafold that avods the referece to ay

4 embeddg the sace k for some k. Defto A smooth -dmesoal mafold s a set M, together wth a coutable collecto of subsets U M, the coordate charts, ad oe-to-oe mas φ : U V oto coected oe subsetsv ν, the local coordate mas, such that the followg roertes are satsfed:. The coordate charts cover M, orm = U s met. 2..Let U deote the tersecto U U = U {}.The comoste ma φ φ : φ ( U) φ ( U) s a smooth (ftely dfferetable) fucto. 3. For ay ar of dfferet ots U, q U, qthere exst oe subsets W, W are met. such that φ ( ) W V, φ ( q ) W V ad φ ( W ) φ ( W ) {} = Some facts are worth metog at ths stage. The coordate charts allow us to defe a toology for M by declarg the sets φ ( W ) to be oe for ay oe W φ ( U). I terms of ths toology, the thrd requremet the defto above says that M has the Hausdorff searato roerty. The degree of dfferetablty of the trasto fuctos φ φ determes the degree of smoothess of the mafold M. For the sake of smlcty, we wll cosder oly smooth mafolds here. Besdes the basc coordate charts φ : U V, oe ca add addtoal coordate charts φ : U V subect to the requremet that the trasto fuctos φ φ, φ φ are smooth for ay o the overla φ ( U U ). I ths case the chart φ : U V s sad to be comatble wth the basc charts. Furthermore, the maxmal collecto of all comatble charts s called a atlas of M. The smlest case of a -dmesoal mafold s the sace. Let x = ( x,..., x ) be a coordate system of ad let ϕ be the coordate fuctos of whch ma a ot q to by x =ϕ ( q). Gve a -dmesoal mafold M wth coordate charts U ad coordate mas φ, the the comoste mag ϕ φ mas M to by x =ϕ φ ( ). We call the fuctos x =ϕ φ ( ) coordate fuctos ofm. Although we deote dfferet obects, here the coordates ad the coordate fuctos, by the same symbol, there should be o cofusos. Furthermore, ths allows us to dsese the exlct referece to a local coordate chart. We wll say, x = ( x,..., x ) s a local coordate system of M, whch s a abbrevato that there s a local coordate

5 ma φ : U V U wth a oe subset V ad a coordate chart U such that each has the local reresetato x = φ ( ). Of course, we kow the reresetato of ay other comatble chart by the codtos of defto. Let us cosder the -dmesoal ut sheres the -dmesoal sheres + = 2 ( x ) = S, the t s straghtforward to see that + are smooth mafolds embedded, sce the sets { x + x 0}, { x + U = S > U = S x < 0} are coordate charts + + wth the roectos o the laes x = 0 as coordate mas φ, φ + +. Whereas the 2- dmesoal coe ( x ) ( x ) ( x ) = 0 3 embedded s o mafold, because the org does ot have a eghborhood U, 2 whch ca be maed oe-to-oe oto a oe subset of. A less trval examle of a + smooth -dmesoal mafold s the set of all les through the org 0, whch s called the real roectve sace P. Havg the cocet of mafolds at our dsosal, we may cosder fuctos f : M o the -dmesoal smooth mafold M. Let U be a coordate chart wth coordate ma φ, the f ( x) = f( φ ( x)) s a ma f : φ ( U ). We call f dfferetable (smooth) at U, f ad oly f (ff) f s dfferetable (smooth) at φ ( ). It s easy to see that ths roerty does ot deed o the choce of U. The mortat set of smooth fuctos o M s deoted by C ( M ). Obvously, we have φ C ( M ) for a smooth mafold. The reader s asked to reflect twce o ths costructo. Roughly seakg, we ca do calculus lke, f we cofe our calculatos to the sutable charts ad take care that the calculatos are meagful also other charts. Now, we are ready to exted ths dea to mas betwee mafolds. Let M, N be two smooth m- ad -dmesoal mafolds. A ma f : M N s sad to be smooth, ff ts local reresetato s smooth for every coordate chart. I other words, let U,V be charts of M ad N wth coordate mas φ, φ such that, f ( U ) V s met, the f s smooth f the comoste ma

6 f =ϕ f φ : φ ( U ) ϕ ( V ) s smooth. The smooth ma f ca be used to trasfer a smooth fucto g C ( N ) from N to M by f ( g) = g f : M. (6) M The fucto f ( g) C ( ) s also called the ullback of g by f. Oly f f : M N s a local dffeomorhsm, f has a smooth verse f locally, the we ca trasfer fuctos from M ton. m Let x = ( x,..., x ) ad y = ( y,..., y ) be coordate systems for the two smooth m- ad -dmesoal mafolds M, N, the we may rewrte the ma f : M N as y = f ( x) usg these coordates. Followg the deas above, we are able to defe the rak of f as the rak of the Jacoba [ ]. Aga, the reader may f x covce hmself/herself that ths defto of the rak s deedet of the choce of the charts. Addtoally, the followg theorem tells us that the mas f betwee mafolds wth costat rak, also called regular mas, admt a very smle form. Theorem 2 Let M, N be two smooth m-ad -dmesoal mafolds. Let k be the rak of the regular ma f : M N at M, the there exsts coordates m x = ( x,..., x ) ear ad coordates y = ( y,..., y ) ear f ( ) such that f takes the caocal form k y = x,..., x,0,...,0 k x, y. Ths theorem s a easy cosequece of the mlct fucto theorem, ad wll ot be rove here. Oe-to-oe mas betwee mafolds ca be used to arameterze submafolds, lke we arameterze curves ad surfaces 3. Let N be a smooth mafold, the a submafold of N s a subsetm N together wth a mafold M ad a smooth oe-tooe ma f : M M wth maxmal rak. I artcular the dmesos of M, M cocde ad do ot exceed the dmeso of N. Ufortuately, ths defto of a submafold admts some rregulartes that we wsh to avod. Therefore, we defe ow the more restrctve regular submafold. Defto 3 Let N be a smooth -dmesoal mafold ad M be a submafold ofn. We callm a regular m-dmesoal submafold, ff for each M there exsts a chart

7 U, Uwth the coordate maφ such that local coordates codtos x = ( x,..., x ) the m + m + x = φ ( q) = 0, =,..., m are met for all q U. The coordates of defto 3 are also called flat coordates. Furthermore, the arameterzato of the submafold s relaced by the atural cluso Fbered Mafolds ad Budles Let us cosder the two smooth mafolds M, N ad a smooth ma f : M N. Frequetly, oe cosders the grah of f stead of f tself. The grah of f s the ew fucto gr f : M M N defed by gr f ( ) = (, f( )), M. The roduct M N s called the total sace. Ths set cotas the doma M ad the co-doma N of the ma f, the doma s also called the base sace. Ths dea ca be exteded to a ew structure. Defto 4 A fbered mafold s a trle ( EB,, π ) wth the mafolds E, dm( E) = m+, B, dm( B) = m ad a ma π : E B that s oto wth rak m. The mafold E s called the total sace, the ma π the roecto ad the mafold B the base sace. The subset F = π ( ) of E s called the fber over B. I may cases the dea of a fbered mafold wthout ay addtoal restrcto s slghtly too geeral. E.g. dfferet fbers may have totally dfferet toologcal structures. Ths roblem may be resolved by sstg that the fbered mafold look rather lke a roduct of mafolds ad the resultg obect s called a budle. Defto 5 A fbered mafold ( EB,, π ) s a budle, ff there exsts a mafold F, called the tycal fber, ad a ma Ψ : π ( U ) U F defed o a eghborhood U of B such that r Ψ = π(r ( q, ) = q, F ) ad F, F are dffeomorhc for all. O a budle we ca troduce adated coordates ( x, u) at least locally, where x, =,..., mare coordates of the base B ad u, =,..., are coordates for the tycal fber. We get eve a smler cture, f we look at x as the deedet ad u as the deedet coordates. Obvously, we have π ( x, u) = x. A fbered mafold ( EBπ,, ) whch s dffeomorhc to the budle ( B F, B,r ) s called trval. The well kow Möbus bad ca be rereseted by a fbered mafold whch s, of course, otrval. If the tycal fber of a budle s a lear sace, the the budle s called vector budle. A mortat examle of a vector budle s, e.g. the taget budle

8 T ( M) of the mafold M whch wll be troduced the followg subsecto. Ths subsecto started wth the grah of a ma betwee mafolds. Now, we are ready to gve the adequate defto for fbered mafolds. Defto 6 Let ( E,B, π ) be a fbered mafold. A ma σ : B E s called a secto of π, f t satsfes π σ = db o ts doma wth d B as the detty ma o B. The set of all sectos of π wll be deoted by Γ( π ). It s worth metog that we do ot requre that a secto s globally defed. Furthermore, there are mafolds that do ot admt global smooth sectos, whch are 2 3 zero owhere. Let us look at the ut shere S, the oe ca mage that t s 2 mossble to assg a o zero taget vector to ay ot of S a smooth way. Ths fact s exressed by the Hary Ball theorem that states, smly soke, you caot comb a hary ball a smooth way, or more recsely that ay smooth taget vector 2 feld o S must vash somewhere. Lke we cosdered mas betwee mafolds, we ca troduce mas for budles that reserve the budle structure. Defto 7 Let ( EB,, π ),( EB,, π ) be two budles. A budle ma s a ar f = ( f B, f E ) of mas fb : B B, fe : E Esuch that π fe = fb πs met o the doma of f. Let ( x, u) ad ( x, u ) be adated coordates of the budles ( EB,, π),( EB,, π ), the a budle ma f has the local reresetato x = f ( x), ( x, u) = f ( x, u) B E, where fb, fe deote the reresetato of ( f B, f E) the adated coordates. From ths reresetato t s easy to see that a budle ma f = ( f B, f E ) allows us to trasfer a secto σ Γ( π) to a secto σ Γ( π) by f ( ) f f σ = σ = E σ B, (7) ff f B s a dffeomorhsm. We call f ( σ ) also the ushforward of σ by f.

9 - - - TO ACCESS ALL THE 2 PAGES OF THIS CHAPTER, Clck here Bblograhy Mchael Svak, 979, Dfferetal Geometry, Vol. to 5, Publsh or Persh, Ic., Housto, Texas, [A stadard referece cosstg of fve books whch gves a comrehesve overvew o dfferetal geometry] Theodore Frakel, 998, The Geometry of Physcs, A Itroducto, Cambrdge Uversty Press, Cambrdge, UK, [A troducto to dfferetal geometry ad mathematcal hyscs wth may alcatos] Wllam M. Boothby, 986, A Itroducto to Dfferetable Mafolds ad Remaa Geometry, Academc Press, Ic., Orlado, USA, [A mathematcal troducto to the bascs of dfferetal geometry Hek Nmeer ad Ara, 990, Nolear Dyamcal Cotrol Systems, Srger, New York, [A stadard textbook o geometrc cotrol ad ts alcatos cludg mechacal olear cotrol systems Alberto Isdor, 995, Nolear Cotrol Systems, Srger, Lodo, UK, [The frst stadard textbook o geometrc cotrol ad ts alcatos] Shakar Sastry, Nolear Systems Aalyss, Stablty ad Cotrol, Srger, 999, New York, [A stadard textbook whch gves a overvew o olear systems, Lyauov theory, dfferetal geometry ad ther alcatos] Bograhcal Sketch Prof. Kurt Schlacher Kurt Schlacher was bor Graz, Austra, o the 6th of August the year he started to study electrcal egeerg at the Techcal Uversty of Graz, ad fshed 979 wth the dloma degree cum laude. I the year 980 he dd hs oblgatory atoal servce. I the year 98 he oed the deartmet of automatc cotrol at the Techcal Uversty of Graz, where he receved hs Ph.D. cum laude the year 984 ad hs habltato for automatc cotrol the year 990. I 992 he moved to Lz at the Johaes Keler Uversty, Austra, where he got the osto of a full rofessor for Automatc Cotrol that he holds resetly. Aart from several academc ostos, he serves as a Assocate Edtor of the IEEE Trasactos o Cotrol Systems Techology. He s also member of the scetfc commttees of the followg ourals: IFAC Iteratoal Joural of Automato Austra, Automatserugstechk (Oldebourg-Verlag, Germay), as well as member of the IFAC Techcal Commttees o Cotrol Desg ad Mechatrocs. Sce 2002 he s member of the IFAC coucl ad of EUCA coucl. Furthermore, he s head of the Chrsta Doler Laboratory for Automatc Cotrol of Mechatroc Systems Steel Idustres. Hs ma terests are modelg ad cotrol of olear systems wth resect to dustral alcatos alyg dfferetal geometrc ad comuter algebra based methods. He s author of more tha 80 ublcatos ublshed atoal ad teratoal roceedgs ad ourals, as well as co-author of the book Dgtale Regelkrese (Oldebourg-Verlag) together wth Prof. Hofer ad Prof. Gausch.