11. Adaptive Control in the Presence of Bounded Disturbances Consider MIMO systems in the form,

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1 Lecure 6. Adapive Corol i he Presece of Bouded Disurbaces Cosider MIMO sysems i he form, x Aref xbu x Bref ycmd (.) y Cref x operaig i he presece of a bouded ime-depede disurbace R. All he assumpios ad oaios from he previous secio apply here, ha is we assume ha Aref, B, Bref, C ref are kow, wih A ref beig Hurwiz. he sysem mached m m uceraiies are represeed by a diagoal posiive defiie marix R ad a N m cosa marix R. I addiio, we assume ha, (.) ad ha he disurbace upper boud is kow ad cosa. he corol goal is bouded rackig of he referece model dyamics, x ref Aref xref Bref ycmd (.3) yref Cref xref m drive by a bouded ime-depede commad R. Based o (.), he corol ipu is seleced as: u x (.4) where N m R is he marix of adapive parameers. Subsiuig (.4) io (.), gives, x Aref xb x Bref ycmd (.5) where, (.6) is he marix of esimaio errors. Le, e x x ref (.7) be he rackig error. Subracig he referece model dyamics (.3) from ha of he sysem (.), yields he rackig error dyamics: e Aref eb x (.8) he Lyapuov fucio cadidae is seleced as i he previous secio, ha is:, V e e Pe race (.9) ycmd where deoes cosa raes of adapaio, ad P P is he uique symmeric posiive defiie soluio of he algebraic Lyapuov equaio, Aref P PAref Q (.) wih QQ. ime-differeiaig V, alog he raecories of (.8), gives: 47

2 V e Qee PB x e P race (.) Applyig he race ideiy, yields: a b race ba (.) V e Qerace e PB e P (.3) Suppose ha we use he same adapive laws as before, ha is: x e PB (.4) he, V e Qee P mi Qe e P (.5) ad, cosequely V ouside of he se, P E e, : e e (.6) mi Q Hece, raecories e,, of he error dyamics (.8) coupled wih he adapive law (.4), eer he se E i fiie ime ad say here for all fuure imes. However, he se E is o compac i he e, space. I fac, i is ubouded sice is o resriced. Iside he se, V ca become posiive ad, as a cosequece, he parameer errors ca grow ubouded, eve hough he rackig error orm remais less he e a all imes. his is he parameer drif pheomeo. I is caused by he disurbace erm. I shows ha he adapive laws (.4) are o robus o bouded disurbaces, o maer how small he laer are. Dead-Zoe Modificaio I order o eforce robusess, we cosider adapive laws wih he Dead-zoe modificaio:, if x e PB e e (.7) Nm, if e e Proposed by B.B. Peerso ad K.S. Naredra i Bouded error adapive corol, IEEE rasacios o Auomaic Corol, v. 7, 6-68, 98, he dead-zoe modificaio sops he adapaio process whe he orm of he rackig error becomes smaller ha he prescribed value e. his assures ulimae uiform boudedess (UUB) of, (i addiio o UUB of e ). We are goig o formally prove his claim. Suppose ha e e he he adapive law is defied by (.4), ad i resuls i he upper boud (.5). Cosequely, e eers E i fiie ime : e e, for all. From ha ime forward, he adapive parameer dyamics are froze, ha is N m. herefore, raecories remai i he compac se, 48

3 for all e, E (.8). his proves UUB of all sigals i he correspodig closed-loop sysem. Remark. P I (.6), he rackig error boud e depeds o he raio. Is imum mi Q is achieved for Q I. hus, wih he dead-zoe modificaio, he wors achievable rackig error value is: e P (.9) Moreover, eve whe he disurbace vaishes, wih he dead-zoe modificaio beig acive, asympoic sabiliy of he rackig error cao be recovered. he dead-zoe modificaio is o Lipschiz, ad as such may cause chaerig (high frequecy oscillaios) ad oher udesirable effecs whe he rackig error is a or ear he dead-zoe boudary. A smooh versio of he dead-zoe modificaio was proposed by J.-J. E. Sloie ad J.A. Coesee i Adapive slidig coroller syhesis for oliear sysems, Ieraioal Joural of Corol, 986. Moivaed by his idea, we choose a cosa, ad cosider a Lipchiz-coiuous modulaio fucio i he form, e e e, mi, e (.) his fucio is show i he figure below. e e e e Figure.. Lipschiz-Coiuous Modulaio Fucio. he adapive laws wih coiuous dead-zoe modificaio are defied as, x e e PB (.) Wih hese laws of adapaio, oe ca use Lyapuov-based argumes o prove bouded rackig ad UUB of all sigals. Sigma-Modificaio he Dead-zoe modificaio assumed prior kowledge of a upper boud for he sysem disurbace. he modificaio scheme, proposed by P. Ioaou ad P. Kokoovic i Adapive sysems wih reduced models, New York: Spriger-Verlag, 49

4 983, does o require ay prior iformaio abou bouds for he sysem disurbace. he adapive law wih modificaio is defied as: x e PB (.) Basically, his modificaio adds dampig o he ideal adapive law (.4). o prove UUB of all sigals, cosider agai he Lyapuov fucio cadidae (.9). Is ime derivaive alog he raecories of he rackig error dyamics (.8) becomes: race V e Qe e PB e P By defiiio: where race e Qe e P e Qe race race e P (.3) N m race iii F mi (.4) i m F i is he Frobeius orm of, ad mi is he miimum i diagoal eleme of. Moreover, usig he Schwarz iequaliy, gives: race F F F F F (.5) Subsiuig (.4) ad (.5) io (.3), resuls i: V Q e e P P Q mi F mi F F F (.6) F F mi Q e e mi F F mi mi Hece, V ouside of he compac se, N m P F F e, R R : e F (.7) mi Q mi his proves UUB of all sigals i he closed-loop dyamics. I paricular, (.7) proves UUB rackig of he exeral commad ycmd by he sysem oupu y, while he sysem operaes i he presece of parameric uceraiies, ad oparameric bouded ime-varyig disurbaces. E-Modificaio he drawback of he modificaio is ha whe he rackig error becomes small he adapive parameers have a edecy o go back o he origi, ha is hey ulear he gai values ha caused he rackig error o become small. I order o overcome his udesirable effec, K.S. Naredra ad A.M. Aaswamy developed he e modificaio, [K.S. Naredra, A.M. Aaswamy, A ew adapive law for robus adapive corol wihou persisecy of exciaio, IEEE rasacios o Auomaic Corol, 3:34-45, Feb. 987]. 5

5 he e modificaio replaces he dampig gai i (.) wih a erm proporioal o e PB. he raioal for usig such a erm is ha i eds o wih he oupu error. Wih his modificaio, he adapive laws iheri error-depede dampig effecs. he adapive laws wih e modificaio are chose as: x e PB e PB (.8) As see from (.8), he e modificaio adds variable (rackig error depede) dampig o he adapive laws dyamics. Usig hese laws, we ow compue he imederivaive of he Lyapuov fucio cadidae (.9), alog he raecories of he rackig error dyamics (.8). From (.3), i follows: race V e Qe e PB e P race e Qe e PB e P e Qee P e PB e PB race race Usig bouds (.4) ad (.5), gives: V Q e e P mi F mi e PB F F F e PB P Q (.9) (.3) F F mi Q e e e PB mi F F mi mi Cosequely, V ouside of he compac se from (.7). his argume proves UUB of all closed-loop sysems raecories, ad i paricular, i proves UUB rackig of he commaded sigal by he sysem oupu. For large rackig errors, he dead-zoe ad he e modificaio slow dow (dampe) he adapaio process. his udesirable effec coradics he corol goal of reducig he rackig error as fas as possible. I wha follows, we will iroduce he Proecio Operaor. I allows fas adapaio i he presece of parameric ad oparameric uceraiies, ad a he same ime he operaor eforces uiform boudedess of he adapive parameers.. Proecio Operaor We begi wih basic defiiios of covex ses ad fucios. Defiiio.: A subse R is covex if x, y R x y z, (.) 5

6 Relaio (.) saes ha if wo pois belog o he covex subse he all he pois o he coecig lie also belog o. Defiiio.: A fucio f : R R is covex if f x y f x f y, (.) Iequaliy (.) is illusraed i Figure.. I implies ha he graph of a covex fucio mus be locaed below he sraigh lie, which coecs ay wo correspodig fucio values. f(x) f(y) f(z) x Figure.: A Covex Fucio z y Saeme.: Le f x R R R f is covex. : be covex. he for ay cosa he subse Proof: Le,. he f ad f x is covex he for ay, f f f herefore, f ad, cosequely, which complees he proof. f. Sice Saeme.: Le f x R R Choose a cosa ad cosider he subse R f ad assume ha f : be a coiuously differeiable covex fucio., (i.e., R. Le is o o he boudary of ). Also, le ). he he ad assume ha f, (i.e., ( is o he boudary of followig iequaliy akes place: f where f f f R (.3) is he gradie vecor of f evaluaed a. 5

7 Relaio (.3) is illusraed i Figure.. I shows ha he gradie vecor, evaluaed a he boudary of a covex se, always pois away from he se. f Figure.: Gradie ad Covex Se Proof: Sice f x is covex he f f f or equivalely: f f f f he for ay ozero : f f f f akig he limi as yields relaio (.3) ad complees he proof. Suppose ha, he rue parameer vecor, belogs o a covex se R f (.4) Iroduce aoher covex se: R f (.5) I is obvious ha. We may ow defie he Proecio Operaor, which we shall use i he adapive laws. y, if f Pro, y y, if f ad y f f f y y f, if o. f or equivalely: f f y y f, if f ad y f Pro, y f (.6) y, if o 53

8 By defiiio,, y Pro does o aler he vecor y if belogs o he covex se defied i (.4). I he se f, he Proecio Operaor subracs a vecor ormal o he boudary f from y so ha we ge a smooh rasformaio from he origial vecor field y for o a age o he boudary vecor field for. he Proecio Operaor cocep is illusraed o Figure.3. f Pro, y y f f Figure.3: Proecio Operaor Usig Saeme. ad iequaliy (.3), we ge he followig impora propery of he Proecio Operaor:, if f y Pro, y, if f ad y f (.7) f f y, if o. f f or, equivalely Pro, y y (.8) Based o (.6), we ca ow defie he Proecio Operaor whe boh Y ad are marices of he same dimesios: N Y y y R ad R (.9) N N N, Y Pro, y Pro, Pro N y N (.) hus for marices, he Proecio Operaor is defied colum-wise. I he previous secios, we desiged model referece adapive corol sysems such ha ime-derivaives of he seleced Lyapuov fucios became egaive semi-defiie ouside of a compac se. hese ime-derivaives were give he form: race V e Qe e PB e P (.) 54

9 he mai ask was o choose such ha he race erm i (.) was o-posiive, ad he adapive parameers became uiformly bouded fucios of ime. We show how o force he race erm i (.) o be semi-egaive by usig he Proecio Operaor (.) ad is propery (.8). m r e PB Pro, YY (.) Pro, Y Y Relaio (.) gives he followig adapive law: Pro, e PB (.3) I essece, he Proecio Operaor esures ha he colums of he adapive parameer marix do o exceed heir pre-specified bouds. A he same ime, he operaor coribues o he egaive semi-defiieess of he Lyapuov fucio (.). V e Qee P Q e e P P Q mi mi Q e e mi Cosequely, V ouside of he compac se, Nm P e, R R : e F mi Q where, m ad is he imum allowable boud for he h colum (.4) (.5) (.6). his proves UUB of all sigals i he correspodig closed-loop sysem ad, i paricular, i proves UUB rackig of ay exeral bouded commad by he sysem regulaed oupu. f f f m Nex we show how o defie covex fucio,, m ad m covex ses. Boh he fucio ad he se defiiios are based o he desired upper bouds ha are imposed colum-wise. For he h colum of he adapive parameer marix N m R, choose proecio olerace ad defie f as: f f (.7) 55

10 Usig (.7), he ses are defied as: N R f (.8) From (.8) i follows ha for each,, m : N : R (.9) N x R : he gradie of he covex fucio (.7) ca be compued as: f (.) Usig (.3), he adapive law for becomes: f f f, if e PB e PB f e PB f f (.) e PB, if o he adapaio process i (.) esures uiform boudedess of each colum of he adapive ime-depede parameer marix forward i ime, ha is:,, m (.) 56