Lectue 6. Poectio Opeato Deiitio A.: Subset Ω R is cove i [ y Ω R ] λ + λ [ y = z Ω], λ,. Relatio. states that i two poits belog to the cove subset Ω the all the poits o the coectig lie also belog to Ω. Deiitio A.: Fuctio : R R is cove i λ + λ y λ + λ y, λ. Iequality. is illustated o Figue., ad it states that gaph o a cove uctio ust be located below the staight lie, which coects ay two coespodig uctio values. y z z y Stateet A.: Let R R the subset Ω = R Poo: Let Ω ay λ heeoe { } Figue.: A Cove Fuctio : be cove uctio. he o ay costat > is cove.,. he λ + ad. Sice λ λ + λ λ + λ = ad, cosequetly, Ω which copletes the poo. is cove the o
Stateet A.: Let R R Choose a costat > ad coside the subset Ω = R Ω ad assue that < : be cotiuously dieetiable cove uctio., i.e., { } R. Let is ot o the bouday o Ω. Also, let Ω. he the Ω ad assue that =, i.e., is o the bouday o ollowig iequality takes place:.3 whee = R is the gadiet vecto o evaluated at. Relatio.3 is illustated o Figue A.. It shows that the gadiet vecto evaluated at the bouday o a cove set always poits away o the set. Ω Poo: Sice is cove the Figue A.: Gadiet ad Cove Set λ + λ λ + λ o equivaletly: + λ + λ he o ay ozeo < λ : λ + λ < = akig the liit as λ yields elatio.3 ad copletes the poo. Suppose that, the tue paaete vecto, belogs to a cove set Ω
{ } Ω = R.4 Itoduce aothe cove set: { } Ω = R.5 It is obvious that Ω Ω. We ay ow deie the poectio opeato, which we shall use i the adaptive laws. o equivaletly: Po, y y, i = y, i y ad y y, i ot. y y, i > ad y > Po, y =.6 y, i ot aely,, y i.4. I the set { } bouday { = λ} Po does ot alte the vecto y i belogs to the cove set Ω deied, the poectio opeato subtacts a vecto oal to the o y so that we get a sooth tasoatio o the oigial vecto ield y o λ = to a taget to the bouday vecto ield o λ =. he poectio opeato cocept is illustated o Figue.3. { = λ} Po, y y { } Figue.3: Poectio Opeato
Usig Stateet A. ad iequality.3, we get the ollowig ipotat popety o the poectio opeato: y Po, y, i =, i ad y y = λ, i ot..7 o, equivaletly Po, y y.8 Based o.6, we ca ow deie the poectio opeato whe both atices o the sae diesios: Y ad Θ ae Y y y R Θ = R = ad.9, Y = Po, y Po, Po Θ y. Relatio. iplies that o atices the poectio opeato is deied colu-wise. Fially, we show that the tace tes i the Lyapuov uctio deivative.6 becoe sei-egative due to the usage o the Poectio Opeato i oig the adaptive laws.7. Sice all the tace tes i.6 ae siila, we oly coside the ist oe. t K K e PB Γ + Λ = K K Po K, Y Y λ = K Po K, Y K Y. Usig. yields the sae adaptatio law o K as i.7, aely: K =Γ Po K, e PB. Basically, Poectio Opeato esues that the colus ati a K do ot eceed thei pe-speciied bouds K K o the adaptive paaete, ad at the sae tie the opeato cotibutes to the egative sei-deiiteess o the Lyapuov uctio.6. 3
et we show how to deie cove uctio = { Ω },, = i ad cove sets. Both the uctio ad the set deiitios ae based o the desied uppe a bouds K K that ae iposed colu-wise. Fo a th colu K o the adaptive paaete ati K R, choose poectio toleace ε > ad deie i as i.: Usig.3, the sets a K K a ε K = K =.3 Ω ae deied as: Ω = K R.4 { } Fo.4 it ollows that o each =,, : a K R K K Ω = a Ω = K R K K + ε.5 We eed to copute the gadiet o the cove uctio.3. = K = K a ε K ε a K.6 Usig., the adaptive law o K ca be witte colu-wise as: >, i e PB + e PB e PB =Γ < e PB, i ot K.7 he adaptatio pocess i.7 esues uio boudedess o each colu o the adaptive paaete ati K owad i tie, that is: 4
a { } a { ε } K K K t K +, t,.8 he est o the adaptive paaetes i.7 ae deied i a siila ashio. 3. Poectio Opeato based MRAC Desig o MIMO Systes with Ustuctued Ucetaities Usig the adaptive laws.7, it is easy to see that the deivative o the Lyapuov uctio. satisies the ollowig iequality: i λi εa V = e Qe+ e PBΛε λ Q e + PBΛ ε e = e Q e PBΛ a 3. At the sae tie, due to Poectio Opeato popeties, all the adaptive paaetes ae UUB. Cosequetly, usig heoe 5., we coclude that the taectoies o the closedloop syste.-.5-.7 ae UUB. Moeove, the tackig eo e= etes a eighbohood o the oigi, withi a iite tie. he adius o the eighbohood i.e., the ultiate boud is deteied by the iiu level set o the Lyapuov uctio V which cotais the set E whee V : e PBΛ ε E = e R : e λi Q a { a K R : K K ε, } a : K R K K ε, + { } a { : Θ R ε, } + Θ + 3. 4. Adaptive Augetatio o a Baselie Cotolle Coside MIMO plat dyaics: = A + B Λ u+ 4. p p p p p 5
whee p R is the -diesioal syste state vecto, u R is the -diesioal vecto o idepedet vitual cotol iputs, A p is the -by- kow costat ati, B p is the -by- kow costat ati, Λ is the -by- ukow costat diagoal ati with positive diagoal eleets, ad p is the -by- ukow possibly oliea state-depedet vecto. Assuig o ucetaities i the odel, that is I, p Λ= =, a baselie oial liea dyaic cotolle ca be desiged to povide coad tackig. he cotolle dyaics ca be witte i the o: = A + B B 4. c c c c p + c c whee c R is the c diesioal cotolle state vecto, R is the diesioal eeece desied coad sigal. he coespodig augeted platcotolle syste becoes: p Ap p Bp = u p B c c A + c Λ + + c Bc A B B 4.3 he baselie cotolle is deied as: u = K + K 4.4 bl whee K R ad K R atices, coespodigly, with = p + c. he eeece odel dyaics is chose as: ae the eedback ad eedowad oial gai = A + B + B u 4.5 e e bl o equivaletly e = A+ B K e + B + B K Ae Be 4.6 whee A e Ap = B c Ac 4.7 6
It is assued that the baselie cotolle is desiged such that Assuptio 4.: I Λ= I ad p i 4.4 yields asyptotic tackig, i.e., li e t = t A e is Huwitz. = the the baselie cotol eedback u bl. he cotol obective is to id u i 4. such that the state o the augeted syste 4.3 tacks the state e o the eeece odel 4.6 i the pesece o the syste ucetaities Λ ad p, while all sigals i the closed-loop syste eai bouded. o this ed, we itoduce the tackig eo vecto: e= e 4.8 otal cotol iput is deied as adaptive augetatio o the baselie cotolle: u = K + K + K + K = u + u p bl ad ubl uad 4.9 whee K R ad K R R is ae the iceetal adaptive gais ad the estiated ucetaity. Substitutig 4.9 ito 4.3, yields: = A+ B Λ K + K + B + B Λ K + K BΛ p p 4. A B p Assuptio 4.: Give a costat diagoal ati Λ, thee eist "ideal" ati gais K R ad K R such that the ucetaity atchig coditios take place: p Ae = A+ B Λ K Be = B + BΛ K 4. ote that the kowledge o the tue gais will ot be equied. Fo 4. - 4., it iediately ollows that: 7