ecure 8 7. Sabiliy Analyi For an n dimenional vecor R n, he and he vecor norm are defined a: = T = i n i (7.) I i eay o how ha hee wo norm aify he following relaion: n (7.) If a vecor i ime-dependen, hen i runcaed funcional norm i defined a: ( τ ) = (7.3) τ Noe ha in (7.3), he vecor norm i ued inead of he vecor norm. For a mari A, vecor induced and mari norm will be denoed by A and A, correpondingly. Suppoe ha Λ = Λ K = Kˆ ˆ () (7.4) are he parameer bound for he Projecion Operaor ued in he adapive law (6.9). Fir, we ae and prove he following f abou inpu-o-ae abiliy of a linear imeinvarian (TI) yem. emma 7. There ei ricly poiive conan ( A, A) n ( ) =, he ae vecor () R k λ, uch ha aring a any iniial condiion of he TI yem wih a Hurwiz mari A and an eernal inpu m = A+ Bu (7.5) u R, aifie 73
B k A u λa () + () (7.6) Proof: Since A i Hurwiz hen here mu ei ricly poiive conan (, ) k λ uch A A ha e ka e λ. In hi cae, runcaed norm upper bound for he oluion of (7.5) can be found. () A ( τ ) A A A ( τ ) ( τ) = e + e Bu dτ λa λ ( τ) kae + k B e u( τ) dτ k + B e u d k + B e d u λa ( τ) λa ( τ) A ( τ ) τ A τ ( τ) τ B ka + u λa Thi complee he proof. A A (7.7) Ne, we find ufficien condiion o guaranee uniform ulimae boundedne (UUB) of all he ignal in he yem (5.), which i conrolled by a filered MRAC conroller, in he form of (6.5), (6.) and (6.9). Toward ha end, inroduce he o-called ideal erence model c = A + B r (7.8) and le e = (7.9) repreen he error beween he ual erence model (6.) and i ideal arge (7.8). Then he erence model error dynamic can be compued a: Uing (6.) and (7.9), give ( ) ˆ T e = A e + BΛˆ I K (7.) m m uc ( ) ˆ T ( ) e = A e + BΛˆ I K e + + e (7.) m m where ˆΛ, K ˆ, and e are UUB. We will ue Small ain Theorem ype argumen o find ufficien condiion for inpu-o-ae abiliy of (7.). Thi yem can be 74
repreened by he wo feedback-conneced yem H and H, a hown in he blockdiagram: H ( ) I m m Λˆ ( ) ( ) n n I A B e uc Kˆ T ( ) + e H Figure : Reference Model Error Dynamic According o he figure, hee wo yem are defined a and ( ) e = H u e = A e + BΛˆ I u (7.) c m m c ˆ T c c u = H u = K e + + e (7.3) Noe ha H i driven by he commanded conrol inpu ˆ T c while H i driven by he yem ae vecor u = K e + + e (7.4) ( ) = e + + e Bounded (7.5) e H denoe he gain of he yem H. One can eaily how ha he gain value i finie. In f, applying emma 7. o (7.) give 75
B Λ e () k e ( ) + ( ( ) I ) u A m m c λa B Λ k e ( ) + ( ) u + u A c c λ A uc k k e + B Λ + u A ( ) ( ) A c λa (7.6) where i he. The inequaliy in (7.6) give finie upper bound for he gain H of he yem H : gain of he proper and able filer H k ( ) A B Λ + (7.7) λa Furhermore, he relaion ˆ T uc = K ( e + + e) K e () () e() + + e (7.8) K e + K + e () ( ) imply ha he gain H of he yem H i alo finie: K (7.9) H Finally, uing Small ain Theorem allow o claim inpu-o-ae abiliy of he yem (7.), if < (7.) H H Becaue of (7.9), i i ufficien o chooe he erence model mari proper able filer ( ) uch ha he mall gain condiion A and he < (7.) H K i aified. 76
Remark 7. The upper bound in (7.) provide guideline for elecing he erence model mari. Baically, hee quaniie need o be choen o minimize he A and he filer gain H of he yem H in (7.). Baed on he above argumen and coninuing from he inequaliy (6.), one can ae and prove he following lemma. emma 7. e he proper able filer bound (, ), he Hurwiz mari A, and he Projecion Operaor K Λ be choen uch ha (7.) hold. Then all rajecorie of he cloed-loop yem (6.6), he erence model (6.), and he racking error dynamic are uniformly ulimaely bounded. Moreover, he racking error dynamic are globally aympoically able. Proof: Since he eernal command r i bounded hen i UUB. Alo, becaue of (6.), he racking error e i UUB. Furhermore, ince he aumed inequaliy (7.) implie (7.), he erence model error dynamic (7.) are inpu-o-ae able and, conequenly e i UUB. Hence, a i follow from (7.9), i UUB. Thu, he definiion of he racking error in (6.), implie ha i UUB. All hee f allow for he applicaion of he Barbala emma o (6.), proving UUB of all he ignal and aympoic convergence of he racking error e o he origin. The proof i complee. We now ummarize he filered MRAC deign and i abiliy properie. Theorem 7. The filered MRAC conrol archiecure coni of he filered feedback ignal (6.5), he erence model (6.), ˆ T u = K = u uc c = A + BΛˆ u u + B r c c and he Projecion Operaor baed adapive law (6.9). 77
T Proj (, ), ( ) T T ( ( c) ) Kˆ = Γ Kˆ e PB Kˆ = K ˆ Λ = Proj Λˆ, BPeu u Γ, Λ ˆ = I B If he conrol filer ( ), he erence mari (, ) A Λ m m K Λ are choen o aify he mall gain condiion in (7.) < H K, and he Projecion Operaor bound where H i he gain of he yem H in (7.) hen: ( ) e = H u e = A e + BΛˆ I u c m m c The yem ae aympoically rack he ae of he erence model (6.), when he laer i driven by any bounded erence command r( ). The yem ae vecor and he conrol inpu u are UUB. The correponding cloed-loop dynamic remain able wih all of i inernal ignal bounded, uniformly in ime. Eample 7. Conider open-loop unable ( A a ) b =Λ=. Alo uppoe ha = > calar TI yem dynamic and uppoe ha = a+ u A = a = a+ k < and ha a low-pa filer ( ) i choen in he form: u a = ( ) = u + a c 78
wih a >. Then he yem H dynamic are e = H u e = a e u e = u Conequenly, c c c + a + a + a H = ( + a )( + a ) The yem impule repone funcion h ( ) can be direcly compued via he invere aplace ranform: () h = ( + a )( + a ) = = a e a e a a + a a a a a a a ( ) + a In hi cae, he gain of he yem H i Thu, for a given a (7.) i aified: a a H = () h d = a e ae d a a a a, i i ufficien o chooe a uch ha he mall gain condiion a < a k Conequenly, he filer conan a mu be choen ufficienly large, o ha: a a > k or, equivalenly a > k + a = a+ k 3 79
Suppoe ha he plan conain a order uaor model, whoe ranfer funcion i: = a + a ( ) where a > i he invere of he uaor ime conan. In hi cae, he H dynamic become: a e = H u e = u ( + a )( + a )( + a ) c c The impule repone funcion of H i: a e a e a e h = a + + Conider he cae when a a a ( a a )( a a ) ( a a)( a a) ( a a)( a a) a < a < a Then he impule repone funcion become: a a a a e a e a e h = a + a a a a a a a a a a a a Now, he yem gain can be upper bounded: H = () h d a + + a a a a a a a a a a a a a = ( a a )( a a) and, hu H a ( a a )( a a) 8
Since our goal i o minimize he gain of H, we chooe a uch ha he upper bound above i minimized. The laer i equivalen o imizing he following epreion: I i eay o ee ha he opimal oluion i: ( a a )( a a) min a a a = + a In hi cae, he gain upper bound become: 8a H ( a a ) A he ame ime, he mall gain condiion require ha 8a < ( a ) a k When given he uaor dynamic, he above inequaliy conrain he elecion of he erence model dynamic which i achievable under he filered MRAC deign. 8