14. MRAC for MIMO Systems with Unstructured Uncertainties We consider affine-in-control MIMO systems in the form, x Ax B u f x t

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1 Lectue MAC o MIMO Systes wth Ustuctued Ucetates We cosde ae--cotol MIMO systes the o, ABu t (14.1) whee s the syste state vecto, u s the cotol put, B s kow costat at, A ad (a dagoal at wth postve ukow eleets) ae ukow costat atces. Also (14.1), : epesets the atched state-depedet (possbly olea) ucetaty, ad t s a bouded te-vayg ukow dstubace, whose uppe boud t a (14.) s kow. I the pevous sectos, MAC tackg desgs wee caed out, assug that the atched olea uceta ucto could be eactly paaetezed the o egesso vecto N, wth costat ukow coecets N ad a peselected kow, (wth locally Lpchtz cotuous copoets). I ths secto, we assue that the state-depedet uceta ucto ca be appoated usg a BF NN, wth kow bass uctos, (such as Gaussas wth ed cetes). Speccally, usg the Uvesal Appoato popety o BF-s, we assue that the ukow appg ca be appoated by a BF NN wth N N ed euos ad deal ukow costat weghts at, (14.3) o a kow copact set. Wthout loss o geealty, we dee, : (14.4) to epeset a sphee o a te ad kow adus. We also assue that sde the sphee, the deal (ukow) appoato ca be acheved wth soe kow appoato toleace a 0, (14.5) a, Outsde o, we assue that the olea state-depedet ucetaty uppe bouded by a kow possbly ubouded ucto. a, a ca be (14.6) he cotol obectve s to desg a state eedback Model eeece Adaptve Cotol (MAC) syste, whch guaatees boudedess o all sgals the coespodg closed-loop syste, whle ocg the syste state t tack the state e t o the desed epoetally stable eeece odel, 65

2 e Ae e Be t whch s dve by a bouded eeece sgal (14.7) t. Note that such a cotolle ust opeate the pesece o the syste stuctued ad ustuctued ucetates, whee the latte ae epeseted by: a) he ucto appoato eo (14.5), ad b) he bouded dstubace t (14.). t s uoly bouded, the We edately ote that whle the dstubace te appoato eo becoes bouded oly the syste state t s located sde the sphee. hs obsevato suggests a cotol law the o: u K 1 K u u 1 u u (14.8) u u whee u K (14.9) s the adaptve stablzg te, wth adaptve ga K,, u K K (14.10) s the adaptve tackg copoet wth adaptve paaetes K s the state odulato ucto, ad u epesets the state lte. ad N, hs cotolle wll be desged to ucto as ollows. he adaptve stablzg te u wll povde closed-loop stablty. he state lte u wll oce the syste state t ete te te ad ea sde o the appoato set, whee the adaptve tackg copoet u, coupled wth u, wll oce the syste to ollow a desed eeece odel. At the sae te, the odulato ucto t wll be chose to allow the cotolle (14.8) soothly tasto betwee the adaptve tackg ad the state ltg odes o opeato. I ode o the soluto to ests, the odel atchg codtos ust hold: ABK Ae (14.11) BK Be I, (14.11), K, K ae the deal eedback ad eedowad ga atces, espectvely. Note that oly estece o the deal gas s assued, whle the kowledge s ot equed to peo the desg. eak 14.1 I (14.10), deotes the ucto appoato. It s easy to see that the ucto appoato eo, 66

3 (14.1) depeds lealy o the paaete estato eo. (14.13) Usg odel atchg codtos (14.11), the syste dyacs (14.1) ca be ewtte as: Ae Be BuK K t (14.14) Substtutg (14.8) to (14.14), yelds: Ae Be Bu 1u u K K (14.15) Ae Be Bu K 1 u K u K Usg (14.9) ad (14.10), uthe gves: Ae Be B u K 1 B K K K K (14.16) K K o, equvaletly: Ae Be (14.17) B K 1K u K whee K K K K K K (14.18) ae the paaete estato eos. Let, e e (14.19) deote the syste tackg eo. Subtactg (14.7) o (14.17), gves the tackg eo dyacs, e Ae e t (14.0) B K 1K u K Let P P 0 be the uque soluto o the algebac Lyapuov equato. PAe Ae PQ, QQ 0 (14.1) Cosde the Lyapuov ucto caddate, V e, K, K, e Pet K K K K (14.) whee 0, 0, 0 ae the ates o adaptato. he te devatve o V, alog the taectoes o the eo dyacs (14.0), s gve by: 67

4 1 1 1 V e Pe e Pe t K K K K e Qe ee P (14.3) epbk 1K u K t K K K K egoupg the tes, yelds: V e QeeP epb1 u K 1 epbk t K K (14.4) 1 1 epbk t K K 1 1 epb t Usg the tace detty, oe ca wte: epbk t K epb b b a a epbk tk epb (14.5) b b a a epb t epb b a b a Substtutg (14.5) to (14.4), esults : V eqeep epb1 u K 1 t K K e PB (14.6) 1 t K K 1 e PB 1 t 1 epb I ode to keep the adaptve gas K,, K uoly bouded, adaptve laws ae chose the o: 68

5 K Po K, e PB Po K K, 1 e PB Po, 1 epb whee Po, N N atces 1 N ad Y y1 yn to the N Po, Y, ad t s deed colu-wse. Po, Y Po 1, y1 Po N, yn (14.7) u v deotes the Poecto Opeato. he opeato aps two N at (14.8) he opeato copoets ae: y y, 0 y 0 Po, y (14.9) y, ot whee : s a cove ucto. Gve a the au allowable agtude o the vecto, ad 0, ths ucto s gve below. a a (14.30) Usg the adaptve laws (14.7), t s easy to see that the devatve o the Lyapuov ucto (14.6) satses the ollowg equalty: V eqeep epb 1 u K (14.31) Beoe poceedg ay uthe, we dee the state odulato ucto, a 0, 1, 1 (14.3) whee 0 1 s a costat. hs ucto s a to the oe the cotuous deadzoe odcato. It s show the gue below. 1 0 Fgue State Modulato Fucto. 69

6 By deto, 0, 1, ad dees the wdth o a aulus sde whee 0 1. (14.33) Suppose that. he 1 ad (14.31) becoes: V e Qe e P e PB u K a a Q e e P e PB u e PB K (14.34) I ode to ake ths epesso o-postve, the state ltg cotol u s chose as, u k sg B Pe (14.35) whee k 0 epesets the state lte ga, ad the sg ucto s udestood copoet-wse. Usg (14.35), gves: (14.36) epbu epb u k epb 1 1 Substtutg (14.36) to (14.34), esults : V Q e e P k e PB e PB K a a Q e e P a a 1 1 epb k sg epb K a (14.37) We ow choose the state lte ga to be, k K a a a (14.38) whee K K, ad a s the au coad value. he, a P V Q e e a Pa Q e e a 0 (14.39) Q outsde o the copact set: a P E0 e : e a e0 (14.40) Q E, te te. So o all t, heeoe, et etes 0 e0 e t t e t t e t (14.41) ad hece, t e0 e t e0 e a (14.4) whee the uppe boud e a ca be eay coputed based o a ad the popetes o the eeece odel (14.7). Futheoe, assug that the appoato set s lage eough, 70

7 ples, o all t atewads. t, that s the syste state e0 e a (14.43) t etes Isde, the appoato eo te-devatve o (14.31) ca be uppe-bouded as: V e QeeP 1 epb (14.44) te te, ad eas thee becoes sall, ad the Lyapuov ucto Q e e P B a a a a (14.45) a P Q e e a B a a Q Cosequetly, V 0 outsde o the copact set: a P E1 e : e a B a a e1 (14.46) Q heeoe, et etes E 1 te te 1, ad eas sde ths set atewads. Sla to (14.41) ad (14.4), ad ode to esue that equalty (14.43) ad assue: t e1 e a (14.47) t eas sde, t s sucet to stegthe the e1 e a (14.48) At the sae te, due to the Poecto Opeato, all the adaptve paaetes ae UUB. Cosequetly, all taectoes o the closed-loop syste (14.1)-(14.8)-(14.7) ae UUB. Moeove, oe ca show that the tackg eo e e etes a eghbohood o the og, te te. he adus o the eghbohood (.e., the tackg eo ultate boud) s deteed by the au level set o the Lyapuov ucto V, whch esdes sde the set E, whee V 0: E e : e e 1 a a a K : K K 1, 1 K : K K 1, 1 N : 1, 1 (14.49) hs copletes the desg ad aalyss o the MAC cotolle o MIMO dyacs wth both stuctued ad ustuctued ucetates. 71