Lectue 4 8. MRAC Desg fo Affe--Cotol MIMO Systes I ths secto, we cosde MRAC desg fo a class of ult-ut-ult-outut (MIMO) olea systes, whose lat dyacs ae lealy aaetezed, the ucetates satsfy the so-called atchg codtos, ad f the full state s easuable, (.e., avalable o-le as the syste outut). Secfcally, we cosde th ode MIMO systes the fo, ABu f (8.1) whee R s the syste state, u R s the cotol ut, B R s kow, whle A R ad R ae ukow atces. I addto, t s assued that s dagoal, ts eleets ae ostve, ad the a A, B s cotollable. he ucetaty s toduced to odel cotol falues. he ukow, ossbly olea, fucto : f R R eesets the so-called syste atched ucetaty. It s assued that the fucto ca be wtte as a lea cobato of N kow bass fuctos, wth ukow costat coeffcets. f (8.) N I (8.), R N s the ukow costat at, ad R deotes the kow egesso vecto. I ode to guaatee estece ad uqueess of the syste tajectoes, t s assued that s locally Lschtz. he cotol objectve of the MIMO tackg oble s to choose the cotol ut u such that all sgals the closed-loo syste ae bouded, ad the state follows ef R the state of a efeece odel, secfed by the LI syste, ef Aef ef Bef t (8.3) whee R s Huwtz, R, ad t R s a bouded efeece ut Aef Bef (eteal coad). Note that the efeece odel dyacs ad the eteal ut t ust be chose so that ef t eesets the desed tajectoy fo t to follow. I suay, gve a bouded coad t, the cotol ut u eeds to be chose such that the tackg eo globally asytotcally teds to zeo. l t t 0 (8.4) t If the atces A ad wee kow, oe could have aled the cotol law, u (8.5) ad obta the closed-loo syste: AB B (8.6) ef 3
Coag (8.6) wth the desed dyacs (8.3), t follows that the deal (ukow) at gas ust be chose to satsfy the so-called atchg codtos: AB Aef (8.7) B Bef Assug that the atchg codtos take lace, t s easy to see that the closed-loo syste s the sae as the efeece odel, ad cosequetly, asytotc (eoetal) tackg s acheved fo ay bouded efeece ut sgal t. Reak 8.1 Gve the atces A, B,, Aef, Bef, o, ay est to satsfy the atchg codtos (8.7) dcatg that the cotol law (8.5) ay ot have eough stuctual fleblty to eet the cotol objectve. Ofte actce, the stuctue of A s kow, ad the efeece odel atces Aef, B ef ae chose so that (8.7) has a soluto fo,. Assug that, (8.7) est, cosde the followg cotol law: u (8.8) whee,, N R R R ae the estates of the deal ukow atces,,, esectvely. he estated atces wll be geeated o-le. Substtutg (8.8) to (8.1), the closed-loo syste dyacs ca be wtte. AB B (8.9) Subtactg (8.3) fo (8.9), closed-loo dyacs of the desoal tackg eo et t t ca be obtaed. vecto ef e AB B A B (8.10) ef ef ef Usg atchg codtos (8.7) futhe yelds: e A ef B Aef ef B B (8.11) A ef eb Let,, ad eeset the aaete estato eos. I tes of the latte, the tackg eo dyacs becoes: e Aef eb (8.1) Vecto ad at os Befoe oceedg ay futhe, ecall that gve a at Fobeus o s defed by A a R j, the at 33
wth j A t A A a (8.13) F t deotg the tace oeato. O the othe had, gve ay vecto -o, the duced at o s defed by A, j A su (8.14) 0 Collecto of Facts about vecto ad at os, (ove t). Fo vecto 1-o 1, the duced at o s equal to the au 1 absolute colu su, that s: A a 1 aj. 1 Fo vecto -o 1 j 1 sgula value of A, that s: A A Fo vecto -o absolute ow su, that s: 1, the duced at o s equal to the au a A. a, the duced at o s equal to the au a a. 1 j1 he duced at o satsfes: A A, ad fo ay two coatbly desoed atces, A ad B, oe also has: A B A B. he Fobeus o s ot a duced o of ay vecto o, but t s coatble wth the -o the sese that: A A. F Fo ay two coatbly desoed atces A ad B, the Fobeus e oduct s defed as: AB, tace F A B. Accodg to the Schwatz equalty oe has: tace A B A, B A B j F F F Fo ay two co desoal vectos a ad b, the tace detty takes lace: a b ba t Let 0, 0, 0. Gog back to aalyzg the tackg eo dyacs (8.1), cosde the Lyauov fucto caddate: 1 1 1 V e,,, e Pet (8.15) whee P P 0 satsfes the algebac Lyauov equato, PA A P Q (8.16) ef ef 34
fo soe QQ 0. he the te devatve of V, evaluated alog the tajectoes of (8.1), ca be calculated. 1 1 1 V e Pe e Pe t ef ef A eb Pe ep A eb 1 1 1 t e A PPA ee PB ef ef 1 1 1 t Usg (8.16), futhe yelds: 1 V e Qe e PB t 1 epb t 1 epb t Usg the tace detty, oe gets epb t epb b b a a epb t epb b b a a epb t epb a b b a Substtutg (8.19) to (8.18), esults : 1 V e Qe t e PB 1 1 t e PB t e PB If the adatve laws ae chose as, e PB t e PB epb the the te-devatve of V becoes egatve se-defte. (8.17) (8.18) (8.19) (8.0) (8.1) 35
V e Qe0 (8.) heefoe, the closed-loo eo dyacs ae stable, that s the tackg eo et ad the aaete estato eos t, t, t ae ufoly bouded fuctos of te. Cosequetly, the aaete estates t, t, t ae also bouded. Sce t s bouded the ef t ad ef t ae bouded. Hece, the syste state t s bouded ad the cotol ut ut (8.8) s bouded. he latte les that t s bouded ad thus et s bouded. Futheoe, the d te devatve of V t V e Qee Qe (8.3) s bouded, ad so V t s a ufoly cotuous fucto of te. he latte couled wth the facts that V t s lowe bouded ad V t 0 les (Babalat s Lea) that. hus, et lv t 0 t l 0 ad the MIMO tackg oble s solved. t Reak 8. (Hoewok: Pove ths stateet) If soe of the dagoal eleets of the ukow dagoal at ae egatve ad the sgs of all of the ae kow, the the adatve laws e PBsg t e PBsg (8.4) e PBsg solve the MIMO tackg oble, whee sg dagsg 1,, sg. 36