Chapter 5. Complementary Material

Page 5C Chater 5 Coleentary Material Chater 5 Coleentary Material 53 MRC Shee: Known Plant Paraeter Proo o Lea 534 For n = we have α () = Λ () = Hene (583) ilie θ = θ = and a unique olution or θ 3 For n onider the oni greatet oon divior H () o Z () and R() hen we an write Z() = Z() H() R() = R() H() or oe orie olynoial air Z() R() Sine Z() i Hurwitz H () alo ha to be Hurwitz Hene we an divide both ide o (583) by H () and obtain ( ) θ α() R() + k θ α() + θ Λ () Z() =Λ() R() Z() Λ R() 3 It an be eaily een that the degree o the right-hand ide i n + n rh where r h i the degree o H () Sine Z() R() are orie heore A5 (ee the Aendix) ilie that there exit unique olynoial a () b () o degree n n r reetively uh that h ar () () + bz () () =Λ() R() Z () Λ () R () One the unique olution a () b () o the above equation i alulated the general olution or θ α() k ( θ α() + θ Λ ()) an be obtained a 3 θ α() = β() Z () + a() ( 3 ) k θ α() + θ Λ () = β() R () + b() (5C) where β () i any olynoial o degree n n + rh One a olynoial β () o degree n n + rh i ixed θ θ θ 3 an be alulated equating the oeiient o both ide o (5C) I R Z are orie (ie r h = ) and n= n then () = and hene the olution o (5C) or θ θ θ3 i unique

Page 5C Chater 5 Coleentary Material 54 Diret MRAC with Unnoralized Adative Law 54 Relative Degree n = Proo o heore 54(ii) [] Sine r L ro (i) we have that ωω L Furtherore ω an be exreed a I ( F) gg ( y ) ( I F) gy ω = y r Sine y = y + e = W r+ e we have () ω= Hr () + ω where ω = H() e and I ( F) gg ( W ) ( ) I ( F) gg ( ) ( I F) gw ( ) ( I F) g H () = H() = W () H () i a roer traner atrix whoe ole are table and e L L and goe to zero a t Hene ro Lea 3C we have ω L L and ω() t a t hereore by deinition o PE ω i PE i H() r i PE Sine r i uiiently rih o order n alying heore 343 thi urther ilie that ω i PE i n rank[ H( jω) H( j ω) H( jωn )] = n on C or any ω ω ωn R with ωi ωj or i j H() an be rewritten a α() R() kz() α() kz () k Z () kz () Λ() R() () kz () kz() D () Λ Λ() kz () R() l l () = = H H where D() = k Z () Λ () R () l= deg( Λ ( ) Z ( ) R ( )) = n + q and H n ( l+ ) R i a ontant oeiient atrix hereore we have where [ H( jω ) H( jω )] = HΨ( ω ω ) ϒ( ω ω ) n n n

Page 5C3 Chater 5 Coleentary Material l l l n l l l ( jω) ( jω) ( jωn ) ( ω ωn ) Ψ = ( ω ω ) ( jω ) ( jω ) ( jω ) diag ϒ n = D( jω) D( jωn ) ϒ ( ω ωn ) i obviouly noningular Ψ ( ω ωn ) i alo noningular ine it i a ubatrix o the Vanderonde atrix and ωi ωk i k hereore rank[ H( j ω ) H( jω )] = n i and only i rank( H) = n Next we how that n rank( H) = n Aue to obtain the ontradition that rank( H) < n hen there exit n a nonzero ontant vetor C R uh that C H = Uing the deooition C = [ C C 3 4] where ilie that C C R ; R thi n 3 4 C H = C R k Z + C k Z k Z + Λ k Z k Z + Λ ( ) R ( ) k Z ( ) = l l α() () () α() () () 3 () () () 4 Oberving that 4 i the oeiient o the that or equivalently ie l ter we have 4 = ( α α 3 ) C () R () + C () k Z () + Λ () k Z () k Z () = ( α ) C α() R () + C () + Λ () k Z () = 3 hi urther ilie kz () C α() = R () Λ () + C α() 3 Sine deg( 3Λ ( ) + Cα( )) n thi ontradit the oriene o Z() R() hereore rank( H) = n and hene rank[ H( j ω) H( jωn )] = n hu we have hown that ω i PE Now onider (598) and (5) ie e = Ae+ Bρθ ω e() = e θ = Γe ωgn ρ (5C) e = C e ( )

Page 5C4 Chater 5 Coleentary Material and the Lyaunov-like untion in (599) ie we have θ θ V( θ e) = + Fro (5) e P e Γ ρ eqqe V ν = e L e ν e C C e= ν e e or oe ontant ν > noting that L > We an rewrite (5C) a e = A() t e e = C e where e A Bρω () t C e = A( t) = C = θ Γω()gn t ( ρ ) C We an alo rewrite Vt () a V = e Pe where P P = Hene we have V = e ( PA+ A P) e ν ee= νe CC e whih ilie that () PA t + A () t P + νcc i negative eideinite hereore uing heore A83 to how that θ and e onverge to zero aytotially and hene exonentially at it i uiient to how that ( CA ) i a UCO air By Lea A84 ( CA ) i a UCO air i and only i ( CA + KC ) i a UCO air where ρ Γ Kt () = gn ( ρ ) Γω( t) whih atiie the integral ondition in Lea A84 ine ω L he yte orreonding to ( CA + KC ) an be written a e A Bρω () t C e = A( t) = C = θ Γω()gn t ( ρ ) C x = Ax+ Bρω x x = y = C x Oberving that ρ C ( I A) B i table and iniu-hae ine ωω L and ω i PE ω= ρ C ( I A ) B ω i PE a well beaue o Lea 3C (ee Chater 3:

Page 5C5 Chater 5 Coleentary Material Coleentary Material) hereore uing Lea A85 we have that ( CA + KC ) i UCO and the roo i olete 54 Relative Degree n = Proo o heore 543(ii) [] It ha already been etablihed that y φ L Fro (53) and the at that e L and hene y L we have that φ L Next we how that φ i PE Siilar to the ae o n = we write φ a where φ = H () o e and φ= Hr () + φ ( I F) gg ( ) W( ) ( I F) gg ( ) ( I F) gw ( ) ( I F) g H () = H () = W () + + Sine e L L and e L it ollow ro Lea 3C that φ L L and φ a t Hene we an aly exatly the ae te in the roo o heore 54 to how that φ i PE Now onider the error equation (5) (5) whih have the ae or a (598) (5) A in the roo o heore 54(i) we ue Lea 3C and the reult above tating that φφ L and φ i PE and etablih that the adative law (5) guarantee that θ onverge to zero exonentially at Uing (5) thi alo ilie that e onverge to zero exonentially at 543 Relative Degree Greater than When the relative degree i greater than the roedure i very iilar but the ontrol law beoe ore olex In thi ubetion we reent the relative degree-3 ae to illutrate the aroah and olexity When the relative degree i n = 3 the traner untion W () o the reerene odel annot be hoen to be SPR beaue aording to aution M (tated in etion 53) the relative degree o the reerene odel W () and the lant G() hould be the ae hereore the araeterization (53) and the ontrol law = θωwill not lead to a deired error equation where the SPR harateriti an be eloyed In order to be able to eloy thee harateriti we rewrite (54) in a or that involve an SPR traner untion a ollow: where ( ) e = W ()( + )( + ) ρ u θ ω u

Page 5C6 Chater 5 Coleentary Material u H () u H () H () ( )( ) = ω= ω = + + and W () are hoen o that W() = W() H() = W()( + )( + ) with relative degree i SPR Without lo o generality let u onider W () = ( + )( + )( + ) or oe > o that e ( u ) ρ θ = + ω he etiate o ê o e i given by eˆ = ( u ) + ρ θ ω where ρθ are the etiate o ρ θ reetively he deign roedure or n = ugget uing u = θω to obtain ˆ = [] and e + e = ρθ ω However the uggeted ontrol law ilie + u = ( + )( + ) u = ( + )( + ) θω involving θ whih i not available or eaureent noting that θ ould be obtained uing the adative law hoen Hene the hoie o u = θωi not eaible he diiulty o not being able to extend the reult or n = to n 3 wa iruvented only ater the late 97 Ater that everal ueul MRAC hee were rooed uing dierent aroahe [36] In the ollowing we reent a deign very iilar to that in [3] We irt rewrite the equation or e and ê : e = ρ ( e + θ ω) eˆ = ρe e = u θ ω + + where θ = θ θ We elet u o that e goe to zero a t In thi ae the etiation error ε = e eˆ atiie ε = e eˆ = ( ρ θ φ ρ e ) +

Page 5C7 Chater 5 Coleentary Material ε ρ γ θ Γ θ Conidering the Lyaunov-like untion V = + ρ + law we deign the adative θ = Γ εωgn( ρ ) ρ = γεe (5C3) where Γ=Γ and γ > Let u now exre e a e e u θω θω = + where u = u ω = ( + ) ω = ω + + Subtituting or θ we obtain e = e + u + ω Γωε gn( ρ ) θ ω o ounterat the ter ω Γ ωε gn( ) ρ we ue a o-alled nonlinear daing ter [45] in u a ollow: u ( ) θω α ω ω e = Γ (5C4) where α > i a deign ontant Uing the above ontrol law we obtain ( ( ) ) e = + α ω Γ ω e + ω Γωεgn( ρ ) whih will ore e and hene e to onverge to zero o hek the ileentability o the ontrol law we ubtitute (5C4) into u = ( + ) u and obtain u = θω+ θω ( + ) α( ωγ ω) e = θω+ θω 4 αωγωω ( Γω ) e (5C5) Γ + 4 3 α ( ω ω) α α εgn( ρ ) ( ) e ( ω Γω) e ( ω Γω) where ω = ω Hene the ontrol law an be ileented without uing ( + )( + ) dierentiator he tability roertie o the above MRAC hee are uarized in the ollowing theore heore 546 he MRAC hee (5C3) (5C5) guarantee that (i) all ignal in the loed-loo lant are bounded and e () t e () t a t ; (ii) the etiate ρ onverge to a ontant ρ aytotially indeendent o the rihne o r I r i uiiently rih o order n and Z R are orie then the araeter error θ = θ θ and traking error e onverge to zero

Page 5C8 Chater 5 Coleentary Material aytotially Further i k i known then θ e onverge to zero exonentially at Proo [] (i) Oberve that ε = ε + ρ θ ω ρ e e = ( + α( ω Γ ω) ) e + ω Γωεgn( ρ ) θ = Γεφ gn( ρ ) ρ = γεe (5C6) he tie derivative o the Lyaunov-like untion ε Γ ρ e V = + ρ θ θ + + γ γ where γ > i a deign ontant to be eleted atiie V = ε γ e γ α e ( ω Γ ω) + γ εe ω Γωgn( ρ ) ε γ r γ α r ( ω Γ ω) + γ ε e ω Γω ε γ γe α γe( ω Γω) Chooing γ = α we have V ε e γ Hene ε e ρ θ L and ε e L and thereore eˆ e L L ; y L hi urther ilie that ω = H () ω L (5C6) together with ε ω L ε L ilie that θρ L L (5C6) urther ilie that ε y L and hene ωω L Sine θφ e L we have e L whih together with y = e + W () r and roerne o W () ilie that y ω L and hene ω= ( + )( + ) ω L hereore all the loed-loo ignal are bounded Sine e e L and e e L it ollow that e() t e() t a t t t (ii) Sine ε e L we have ρ dτ γ ε e dτ γ( ε dτ) ( e dτ) L ie ρ = ρ L and thereore ρρ onverge to a ontant a t It an be etablihed that ω i PE rovided that r i uiiently rih o order n and Z R are orie uing iilar arguent and te a in the roo o heore 543 I k i known then ρ = and (5C6) i redued to ε = ε+ ρ θ ω θ = Γεωgn( ρ )

Page 5C9 Chater 5 Coleentary Material Sine ωω L and ω i PE uing the ae te a in the roo o heore 36 we an how that θ and e onverge to zero exonentially at I k i unknown we onider the equation or ε θ in (5C6) ie ε = ε+ ρ θ ω ρ e θ = Γεωgn( ρ ) a an LV yte with ε θ a the tate and ρ e a the inut Sine or the above yte (a hown above) θ and e onverge to zero exonentially at when ρ e = and ω i PE and ine (a hown in art (i)) ρ L e L L and e () t a t it ollow ro Lea A59 (in the Aendix) that ε () t θ () t a t 55 Diret MRAC with Noralized Adative Law Lea 55 Conider the lant equation (57) and the MRAC hee (56) (59) here exit a δ > uh that the ititiou noralizing ignal + + u y where denote the L δ nor 4 atiie the ollowing: (i) ω/ ω / L (ii) I θ L and W () i a roer traner untion with table ole then in addition to (i) u/ y/ ω/ W() ω/ L (iii) I r θ L then in addition to (i) and (ii) y / ω / L Proo [] (i) Sine ω = u ω = y and the relative degree o eah eleent o α( ) α( ) Λ( ) Λ( ) α( ) i at leat the reult ollow ro Lea A59 and the deinition o Λ ( ) (ii) Sine θ L alying Lea A59 to (5) we obtain y () t + θ ω + ω where denote any inite ontant hereore ine ω ω + ω + y + r u + y + + we have y / ω/ L Sine u = θω and θ L thi urther ilie that u/ L Finally ine W () ω ω or oe δ > uh that W ( δ ) i table W () ω/ L 4 he ae notation i ued in the roo

Page 5C Chater 5 Coleentary Material (iii) he reult ollow by dierentiating (5) and (5) uing (56) and the inequality ω ω + ω + y + r and alying Lea A5 Proo o heore 55 Ste Exre the lant inut and outut in ter o the adatation error θω hi te i already deribed in detail in the outline o the roo For θ L whih i guaranteed by the adative law we aly Lea 55 to etablih that the ignal bound all ignal in the loed-loo adative yte and oe o their derivative Ste Ue the waing lea and roertie o the L δ nor to uer bound θ ω with ter that are guaranteed by the adative law to have inite L gain Uing Lea A we have ( ) F ( ) θ ω= F( α ) θ ω+ θ ω + ( α ) θ ω where n n α α α α α F ( ) = ( + ) F( ) = F ( ) α > i an arbitrary ontant and n i the relative degree o W () Siilarly uing Lea A we have ( ( )) θ ω= W () θ W () ω+ W ()( W () ω ) θ b where Wb() W() are a deined in Lea A or W () = W () Cobining the two waing lea equalitie above we obtain ( ) θ ω= θ ω+ θ ω + θ ω+ ω θ F FW () ( W W Wb ) Exreing the noralized etiation error in (59) a ubtituting in the above equation we obtain ε = ρ θ φ ρξ and ( ) ρ θ ω= F θ ω+ θ ω FW + ε ( ) + ξ+ W Wbω θ ρ ρ Uing Lea A we have ξ = θ φ+ u () () ( = θ W ω+ W θ ω= W Wbω ) θ Hene we an rewrite the exreion or θ ω a θ ω= F θ ω θ ω FW ε ρw( ( Wbω ) θ) + + + ρ Alying Lea A again we obtain

Page 5C Chater 5 Coleentary Material ( ) n θ ω θ ω + θ ω + α ( ω+ ε ) (5C7) α where ω= ρw (( W ω ) θ ) Uing Lea A5 and Lea 55 we have b ω θ θ ω θ θ ω ε ε hereore or α > and ε θ L g L g = ε + θ (5C7) lead to (54) where beaue o Ste 3 Ue the B G lea to etablih ignal boundedne hi te i already deribed in detail in the outline o the roo Ste 4 Show that the traking error onverge to zero Fro (59) we have e = ε + ρξ Sine ξ= θ φ+ W [ ] [ ] [ ] [ [ θ ω = W θ ω θ W ω = W Wb ω ] θ ] ε ε θ L and ω n L we have e ξ L Furtherore uing (58) θθ L (a guaranteed by the adative law (59)) Lea 55 and Lea A we have e L hereore uing Lea A47 we etablih that e () t a t Ste 5 Etablih that the araeter error onverge to zero By deinition we have ( I F) gu ( I F) gy φ = W () y r Subtituting u = G () y and y = W () r+ e into thi we have where φ = W () H () r+ W () H () e ( I F) gg () W() ( I F) gg ( ) ( I F) gw ( ) ( I F) g = = W () H () H () It wa etablihed in the roo o heore 54 that i r i uiiently rih o order n then H () r i PE Hene alying Lea 3C ine W () i table and iniu hae and e L it ollow that φ i PE Next we etablih the onvergene reult Fro (59) we have ε = ρ θ φ ρξ θ =Γ εφgn( k / k )

Page 5C Chater 5 Coleentary Material Uing u y L we have φφ L Hene ine φ i PE and ξ L alying the arguent in the roo o heore 39 we etablih that θ () t e() t a t 56 Indiret MRAC with Noralized Adative Law Proo o heore 56 Ste Exre the lant inut and outut in ter o the adatation error θω A in the roo o heore 55 we have ( θ ω) ( θ ω) y = W () r+ u = G () W () r+ Again a in the roo o heore 55 we ue to denote the L δ nor and deine the ititiou ignal + u + y or oe δ > to atiy Lea 55 Following the ae te a in the roo o heore 55 we obtain + θ ω Ste Ue the waing lea and roertie o the L δ nor to uer bound θ ω with ter that are guaranteed by the adative law to have inite L gain Note irt that baed on (533) and (534) it an be etablihed uing iilar arguent a in the roo o heore 55 that θ k ˆ θ L ; θ θ L Uing (53) we have W () θ W () ω + θ W () ω + θ W () y = W () u Z () Q() u k 3 Λ() W () + ( QR ( ) ( ) Λ ( R ) ( ) ) y k Λ() Siilarly uing (533) we obtain θ W () ω + θ W () ω + θ W () y = W () u Z ˆ ˆ W () ( t) Q( t) u k 3 ˆ Λ() ( ˆ ˆ W () + Qt ( ) R( t ) Λ ( R ) ( ) ) y kˆ Λ() aking the dierene o the two equation above we get θ W () ω + θ W () ω + θ W () y = e + e + e 3 3 where θ θ θ and i i i

Page 5C3 Chater 5 Coleentary Material W () ˆ W () ˆ ( ) ˆ ( ) ˆ( ) ( ) e = Qt R t y Qt Z t u ˆ i i k () () Λ Λ Λ R() e = W() y = ( ) y k kˆ Λ() Q () e3 = W() ( Z() u R() y) = k Λ() hereore or ω ω ω and θ θ θ we have [ y ( ) ] W y θ W () ω= e Furtherore alying Lea A3 to Qt ˆ( ) i Z ˆ ( t ) u ) we obtain W ( ) Λ( ) e = ( Q ˆ ( t) i Rˆ ( t) y W ( ) kˆ Λ( ) e = e e where ˆ ˆ W() ˆ W() e = Q( t) R( t) y Z( t) u kˆ () () Λ Λ W() W() e = q D () α () α () u αn () y θ n n () () Λ Λ ˆ( ) Qt = qα () n Sine ˆ ˆ W() ˆ W() e = Q( t) R( t) y Z( t) u kˆ () () Λ Λ ˆ W() Λ() ˆ W() Λ() = Qt ( ) θ φ = Qt ( ) θ φ kˆ () ˆ () k Λ Λ W () () ˆ( ) Λ ( )[ ( ) = Qt ε W Wb φ ] kˆ Λ() θ where Lea A i ued to obtain the lat equality uing Lea 55 the deinition o Qt ˆ( ) and the at that θ L and y = W ()( r+ θ ω) u = G () W ()( r+ θ ω) we obtain e e + e + (5C8) ε θ where denote a generi oitive ontant Furtherore ro (57) we have y y = W ( u ) θ ω W θ ω =

Page 5C4 Chater 5 Coleentary Material hereore θ ω θ ω θ ω ω ( ) θ ω = ( ) = ( ) W y r = ie θ ω= θ ω Hene uing Lea A we obtain θ ω= F ( α ) θ ω+ θ ω + F ( α ) θ ω (5C9) where n n α α α α α F ( ) = ( + ) F( ) = F ( ) α > i an arbitrary ontant and n i the relative degree o W () and atiie n F( α) F( α α ) W ( ) α or any δ δ α > δ> Alying Lea A we obtain θ ω W = θ W ω + W W b ω θ = W e + W W b ω θ (5C) Subtituting in (5C9) we get ( F F W e W Wb ) θ ω= α θ ω+ θ ω + α + ω θ ( ) ( ) () () () Sine i bounded ro below ie alying Lea A5 and Lea 55 we obtain ( ) θ ω θ + + α + θ (5C) α n e α Uing (5C8) (5C) together with θ k ˆ θ L ; θ θ L we obtain n g + α α θ ω where g ( θ α ( θ ε θ )) α / + n + + L Ste 3 Ue the B G lea to etablih ignal boundedne hi te i the ae a Ste 3 o (the outline o ) the roo o heore 55 Ste 4 Show that the traking error onverge to zero Uing θ ω= θ ω in (57) we have e = y y = W ( u ) θ ω W θ ω W θ ω = = Alying Lea A and ubtituting (5C) we obtain

Page 5C5 Chater 5 Coleentary Material e = ( e+ W() Wb() ω θ ) W() W () b ω θ Uing boundedne o all the ignal and araeter etiate (5C8) and ε θ θ L thi ilie that e L Uing iilar arguent a in the roo o heore 55 we have e L hereore uing Lea A47 we etablih that e () t a t 57 Robut Diret MRAC Lea 578 Conider the lant equation (566) and the MRAC hee (568) (569) For any δ > and any δ ( δ ] the ititiou noralizing ignal + + u y where () denote the L t δ δ nor 5 atiie the ollowing: (i) ω/ ω / ω / / L (ii) I θ L and W () i a roer traner untion with all it ole in Re[ ] < δ / then in addition to (i) u / y / ω/ W() ω/ u / y / L (iii) I r θ L then in addition to (i) and (ii) ω / L (iv) For δ= δ = Proo (i) Uing the ae arguent a in the roo o Lea 55(i) we have ω/ ω / ω / L Uing = + u + y + u + y we t δ t δ t δ t δ urther have / L (ii) Rewriting the equation or e = y y right beore (569) a y = ρ W () θ ω+ ρ η+ ρ d+ W () r and uing the exreion or η d the at that W WΔ are tritly roer with all their ole in Re[ ] < δ/ and θθ L we etablih that y () t ω + u + + ; ie y/ L Uing (i) thi urther ilie that ω/ L Hene ine and θ ω / L we alo have u u u = θω L Uing Lea A5 we urther have W () ω/ Finally y / L ollow by taking the derivative o both ide o y = ρ W () θ ω+ ρ η+ ρ d+ W () r 5 he ae notation i ued in the roo below

Page 5C6 Chater 5 Coleentary Material Λ θ α( ) oberving that η + d W () Δ () u + d + d and that Λ( ) δ W () i roer (iii) he reult i etablihed uing the ae arguent a in the roo o Lea 55(iii) (iv) he reult ollow by deinition Proo o heore 575 Ste Exre the lant inut and outut in ter o the araeter error ter θ ω Fro (537) (538) we have y u GΛ θ α Λ = r+ θ ω+ η Λ θ α G( θ α+ θ3λ ) Λ Λ GD = r+ θ ω + θ α G θ α θ3 θ α θ α θ3 η Λ ( + Λ) Λ G ( + Λ) where η Δ u + ( Δ + ) d u Subtituting (539) we obtain y = W r+ θ ω + η y u = G W r+ θ ω + η u θ α + θ Λ Λ θ α u y Λ Λ 3 where η W η η W η Sine W G W are table alying Lea A5 thi ilie Λ θ α y W u d W + θ ω + Δ δ + + θ ω + Δ δ Λ δ θ α+ θ Λ u W u d W 3 + θ ω + Δ δ + + θ ω + Δ δ Λ δ ie W where denote a generi oitive ontant + θ ω + Δ δ Ste Ue the waing lea and roertie o the L δ nor to bound θ ω ro above with ter that are guaranteed by the robut adative law to have all gain in the Uing the ae notation and the ae te a in the roo o heore 55 we have Uing (569) we get ( ) θ ω= F θ ω+ θ ω + FW ( θ Wω+ W Wbω ) θ (5C)

Page 5C7 Chater 5 Coleentary Material ( ) ( ) ( ) (( ) ) ε = ρ θ φ+ W θω+ η+ d ρθφ+ W θω= ρ θw ω+ η+ d ρw W ω θ b and hene Subtituting in (5C) we obtain ( ) ε ρ θ Wω = W ( ) Wbω θ η d ρ + ρ ( ) [ ] FW θ ω= F θ ω+ θ ω + ε ( ) + ρw Wbω θ FW η+ d ρ Following the ae te a in the roo o heore 55 we obtain ( ) ( ( b ) ) + + + W W + + d α k θ ω θ ω θ ω α ε ω θ η k g + + αδ + d α where θ g= + + Δ + α k d α ( ε θ ) S S and η Δ ine ε θ ( η + d ) Ste 3 Ue the B G lea to etablih boundedne hi te i exlained in detail in the outline o the roo Ste 4 Etablih bound or the traking error e It ollow ro (569) that e = ε + ρξ Sine εε ρ S ( η + d ) and ξ L it ollow that e S ( η + d ) By deinition o Δ thi ilie that e S ( Δ + d ) ie t+ t ed τ ( Δ + d) + t > Ste 5 Etablih onvergene o the etiated araeter and traking error to reidual et Fro the deinition o φ we have α () α () φ = W() u y y r Λ() Λ() Uing y = W () r+ e and (566) we have

Page 5C8 Chater 5 Coleentary Material u = G () y η = G ()( W () r+ e ) η y y where η =Δ ()( u + d ) + d and hene where y u u φ= φ + φe α () α () φ = W() G () W() W() W() [ r] () () Λ Λ α () α () α () φe = W() G () [ e] W() n+ η y Λ() Λ() Λ() Lea 36 ilie that φ i PE with level α > O( Δ + d) i r i doinantly rih and Z R are orie Hene there exit > uh that t+ t φ ( τ) φ ( τ) dτ α I t Furtherore ine hene e η S ( Δ + d ) and u y L we have φ S ( Δ + d ) and e t+ l t+ l t+ l ( ) ( ) d ( ) ( ) d e( ) e ( ) d l φ τ φ τ τ φ τ φ τ τ φ τ φ τ τ l l t t t α d I l ( Δ + ) or any oitive integer l hereore or any l atiying < and or α d 8 ( Δ + ) < we have l α 8 l t+ l t α φτφ τdτ I 4 ( ) ( ) whih ilie that φ i PE Alying the arguent in the roo o heore 3 we etablih that when φ i PE (569) guarantee that θ onverge to a reidual et whoe ize i o the order o the odeling error; ie θ atiie θ () t ( Δ+ d ) + r ( t) where li ( ) θ t r t = θ and the onvergene i exonentially at Fro (57) we have e () t = W() θ ω+ ηy hereore ine ω L and η y ( Δ + d) we obtain

Page 5C9 Chater 5 Coleentary Material e () t ( Δ+ d ) + r () t e where r e W r θ Exonential onvergene o r () t to zero ilie that r () θ e t exonentially a t a well hu we have etablihed that e θ onverge exonetially to the reidual et S deined in the theore 5 Exale Uing the Adative Control oolbox he MRC and MRAC algorith reented in thi hater an be ileented uing a et o MALAB untion and the Siulink blok Adative Controller rovided in the Adative Control oolbox For ileentation o MRAC algorith thee tool need to be ued together with the PI untion or blok introdued in etion 36 In thi etion we deontrate ue o the Adative Control oolbox in variou MRC and MRAC roble via a nuber o iulation exale 5 MRC In etion 53 we have een that or a given lant o the or (573) and a reerene odel o the or (576) i the lant araeter are known then an MRC hee an be ontruted to ore the loed-loo yte (the integration o the lant (573) and the MRC) to behave a the reerene odel he ontrol araeter o thi MRC hee are obtained by olving (584) he MALAB untion roly an be ued to olve (584) or a given deign olynoial Λ () Denoting the oeiient vetor o Z() R() Z() R() Λ () a ZRZRL reetively [thetauthetaythetarreype]=roly(zrzrl) return the araeter vetor θ u θ y θ r REYPE i returned a i the equation olving roe i ueul and a i it ail Exale 5 Conider the lant y = u It i neeary to hooe u o that y ( + 3) trak the reerene ignal y o the reerene odel y = 3 r with rt () = + in(4 t + ) he ontrol ignal or thi tak an be generated a π 6 α( ) α() y ( ) u= θu u+ θ Λ y + θrr α( ) = [ ] Λ() y ( + ) hooing the degree o Λ () =Λ () a n = Fixing θ u θ y θ r an be deterined uing roly: Λ =Λ = + + () () Z = ; R = [ 6 9 ]; Z = ; R = [ 3 3 ]; Labda = [ ]; [thetau thetay thetar REYPE] = roly(zrzrlabda)

Page 5C Chater 5 Coleentary Material he reultant araeter vetor are the reultant ontrol law i θ = [3 9] u θ = [5 36 35] y θ = and r 3 9 5 36 u= u+ y+ 35 y+ r + + + + 5 Diret MRAC When the lant araeter are unknown we an no longer olve (584) to obtain the ontroller araeter Intead we need to ue an adative ontrol hee whih rodue the etiate o the ontrol araeter diretly baed on an aroriate araetri odel involving thee ontrol araeter (diret adative ontrol) or whih irt rodue the etiate o the lant araeter baed on another aroriate araetri odel and then alulate the ontrol araeter uing thee lant araeter etiate (indiret) A et o diret MRAC hee have been introdued in etion 54 and 55 and oe indiret MRAC hee in etion 56 he diret MRAC hee with noralized adative law o etion 55 an be ileented uing the MALAB untion urdrl urdrb or the Siulink blok Adative Controller o the Adative Control oolbox inororated with the araeter identiiation untion (or the Siulink blok Paraeter Etiator) deribed in Chater 3 urdrl i ued or hee baed on SPM or DPM while urdrb i or B-SPM and B-DPM Exale 5 57 deontrate variou ue o thee tool Exale 5 Conider the lant and the reerene odel o Exale 5 he orreonding linear SPM an be written a z = θ φ where u y r θ = θ θ θ z= W () u φ = W() ( H() u) W() ( H() y) W() y y α( ) [ ] W () = H() = = 3 ( + ) Λ() + + In Exale 5 we have alulated the araeter vetor a θ [3 9] u = 3 9 5 36 θ y = [5 36 35] θ r = and the ontrol law a u= u+ y+ 35y+ r + + + + Now we an generate the ontrol ignal a well a the ignal in the SPM uing the ollowing ode (added to the ode o Exale 5): Labda = onv(z Labda); theta = [thetau(:);thetay(:);thetar(:)]; dt = ; % ie inreent or iulation (e) t = [:dt:]; % Proe tie (e) trev = [-5:dt:-dt]; ltrev =length(trev);

Page 5C Chater 5 Coleentary Material lt = length(t); r = in(4t + i/6) + ; % Reerene ignal rrev = in(4trev + i/6) + ; % State initialization or the lant: [ntate x] = uilt('init'zr); x(:) = x; % State initialization or the reerene odel: [ntate x] = uilt('init'zr); x(:) = x; or k = :ltrev dx = uilt('tate'x(:)rrev(k)zr); x(:)=x(:) + dtdx; end % State initialization or linear araetri odel: [ntate xl] = urdrl('init'[3 ]LabdaZR); xl(:) = xl; % Signal initialization: y() = uilt('outut'x(:)zr); y() = ; % Proe: or k = :lt u(k) = urdrl('ontrol'xl(:k)[y(k) r(k)][3 ]LabdaZRtheta); [z(k) hi(:k)] = urdrl('outut'xl(:k)[u(k) y(k)][3 ]LabdaZR); dxl = urdrl('tate'xl(:k)[u(k) y(k)]labdazr); xl(:k+)=xl(:k) + dtdxl; dx = uilt('tate'x(:k)r(k)zr); x(:k+)=x(:k) + dtdx; y(k+) = uilt('outut'x(:k+)zr); dx = uilt('tate'x(:k)u(k)zr); x(:k+)=x(:k) + dtdx; y(k+) = uilt('outut'x(:k+)zr); end % Outut: y = y(:lt); y = y(:lt); he reult are hown in Figure 5C and 5C Another ean o iulation i uing the Siulink blok Adative Controller One an ue the hee hown in Figure 5C3 and hooe the aroriate otion and enter aroriate araeter in the enu o Adative Controller

Page 5C Chater 5 Coleentary Material Figure 5C Outut and ontrol ignal o Exale 5 Figure 5C Linear araetri odel ignal o Exale 5

Page 5C3 Chater 5 Coleentary Material Clok t tie 3+6 +9 Plant y y(t) theta theta Adative Controller Adative Controller u u(t) z z(t) Relay Produt hi hi(t) Contant r(t) 3+3 +3+ Model y_ y(t) Sine Wave Figure 5C3 Siulink hee to generate the ontrol ignal in Exale 5 Exale 53 Conider again the lant and the reerene odel o Exale 5 hi tie let u aue that we don t know any inoration about θ exet that θ6 = k / k Now we need to ue a araeter identiiation algorith to etiate θ For thi tak we an odiy the Siulink hee o Figure 5C3 a hown in Figure 5C4 In thi hee Paraeter Etiator i inororated into Adative Controller he araeter identiiation algorith i eleted to be LS with orgetting ator β = and initial ovariane P = I Knowledge about θ 6 i ued or araeter rojetion and θ i initialized a [ ] he reult are lotted in Figure 5C5 and 5C6 Although the araeter etiation i not ueul a een in Figure 5C6 Figure 5C5 how that ontrol and traking are ueul Above one ay reer to ue the MALAB untion urdrl inororated with one o the MALAB untion or PI eg ugrad (ee Chater 3) intead o uing Siulink a well

Page 5C4 Chater 5 Coleentary Material Clok t tie 3+6 +9 Plant y y(t) hi Paraeter Etiator Paraeter Etiator Adative Controller Adative Controller u u(t) z z(t) theta Relay Produt theta(t) Contant r(t) 3+3 +3+ Model y_ y(t) Sine Wave Figure 5C4 Siulink hee to generate the ontrol ignal in Exale 53 Figure 5C5 Outut and ontrol ignal o Exale 53

Page 5C5 Chater 5 Coleentary Material Figure 5C6 Linear odel araeter etiate o Exale 53 Exale 54 Conider the alar lant x = ax + bu where a b are unknown araeter but b i known to be nonnegative he objetive i to hooe an aroriate ontrol law u uh that all ignal in the loed-loo yte are bounded and x trak the tate x o the reerene odel given by x = x + r or a known reerene ignal rt () b he lant traner untion i G () = and the reerene odel traner untion a i W () = So we an ontrut the orreonding MRC a + u= θ x+ θ r α( ) ine H () = = Here θ Λ( ) and θ an be etiated baed on the SPM where z = u φ = x x + + z = θ φ b a Uing z = θ φand x= u we an eaily ee that θ a = and θ b = Now let u b iulate the lant etiation o the araeter and generation o the MRC ignal or

Page 5C6 Chater 5 Coleentary Material π a= 5 b= x = 5 and rt () = + in(4 t+ ) he Siulink blok Paraeter 6 Etiator and Adative Controller are ued or iulation he PI algorith i eleted to be LS with orgetting ator β = and P = I θ i initialized a [ ] and araeter rojetion or θ i ued auing that θ a in Exale 53 he reult lotted in Figure 5C7 and 5C8 how that both the etiation and the traking are ueul Figure 5C7 Outut and ontrol ignal o Exale 54

Page 5C7 Chater 5 Coleentary Material Figure 5C8 Linear odel araeter etiate o Exale 54 Exale 55 hi exale i the bilinear ounterart o Exale 5 he B-SPM or the yte in Exale 5 an be ontruted a where z ( θ φ z ) = ρ + θ = θu θy θ r ρ = θr z= y y y = W( ) r φ = W() ( H() u) W() ( H() y) W() y y z = W() u α( ) [ ] W () = H() = = 3 ( + ) Λ() + + he ontrol ignal a well a the other ignal in the B-SPM an be generated uing the ollowing line intead o the orreonding line in Exale 5: [ntate xl] = urdrb('init'[3 ]LabdaZR); u(k) = urdrb('ontrol'xl(:k)[y(k) r(k)][3 ]LabdaZRtheta); [z(k)hi(:k)z(k)]=urdrb('outut'xl(:k)[u(k) y(k) r(k)][3 ]LabdaZR); dxl = urdrb('tate'xl(:k)[u(k) y(k) r(k)]labdazr);

Page 5C8 Chater 5 Coleentary Material he reult are hown in Figure 5C9 and 5C One an ue the Siulink blok Adative Controller or iulation a well Figure 5C9 Outut and ontrol ignal o Exale 55 Figure 5C B-SPM ignal in Exale 55 Exale 56 hi exale i the bilinear ounterart o Exale 53 he B-SPM ontruted in Exale 55 i ued or araeter etiation Intead o the LS algorith a gradient algorith with Γ= I γ = and = i ued and θ () = / ρ () i eleted to be he reult are hown in Figure 5C and 5C 6

Page 5C9 Chater 5 Coleentary Material Figure 5C Outut and ontrol ignal in Exale 56 Figure 5C B-SPM araeter etiate o Exale 56 Exale 57 hi exale i the bilinear ounterart o Exale 54 he B-SPM or the yte onidered in Exale 54 an be ontruted a z = ρ ( θ φ+ z ) where

Page 5C3 Chater 5 Coleentary Material θ θ θ ρ = = θ z= x x x = r + φ = x x z = u + + Intead o the LS algorith o Exale 54 a gradient algorith with Γ= I γ = and = i ued and θ () = / ρ() i hoen to be he reult are hown in Figure 5C3 and 5C4 Figure 5C3 Outut and ontrol ignal o Exale 57

Page 5C3 Chater 5 Coleentary Material Figure 5C4 Linear odel araeter etiate o Exale 57 53 Indiret MRAC he indiret MRAC hee o etion 57 an be ileented uing the MALAB untion roly uridr or the Siulink blok Adative Controller o the Adative Control oolbox inororated with the PI untion (or the Siulink blok Paraeter Etiator) deribed in Chater 3 A entioned beore eah indiret MRAC hee i ooed o a PI algorith to generate the etiate o the lant araeter and a ontrol law to generate the ontrol ignal uing thee lant araeter etiate he tak o the untion roly i to alulate the ontrol araeter in thi ontrol law baed on (533) uridr generate the ontrol ignal a well a the other ignal involved in the araetri odel ued in PI he ue o the indiret MRAC tool o the Adative Control oolbox are deontrated in the ollowing exale Exale 58 Conider the lant and the reerene odel o Exale 5 Let u aue that we don t know any inoration about the lant exet that k / k We want to eror the ontrol tak o Exale 53 uing the indiret MRAC We an ue the Siulink hee hown in Figure 5C4 hanging only the entrie o the Adative Controller ( Λ () i eleted to be Λ () = ( + ) Λ () ) and the Paraeter Etiator blok he PI algorith i again eleted to be LS with orgetting ator β = and P = I θ i initialized a θ = [ ] and araeter rojetion i not ued he reult are lotted in Figure 5C5 and 5C6

Page 5C3 Chater 5 Coleentary Material Figure 5C5 Outut and ontrol ignal o Exale 58 Figure 5C6 SPM araeter etiate θ = b ˆ ˆ ˆ ˆ a a a o Exale 58 A above one ay reer to ue the MALAB untion uridr inororated with one o the MALAB untion or araeter identiiation eg ugrad (ee Chater 3) intead o uing Siulink

Page 5C33 Chater 5 Coleentary Material Exale 59 Conider the alar lant o Exale 54 with the ae reerene ignal We want to eror the ae ontrol tak uing indiret MRAC Chooing Λ () = + araeter etiation i baed on the linear odel z = θ φ where z = y θ = [ b a] φ= u y + + + he Siulink blok Paraeter Etiator and Adative Controller are ued or iulation he PI algorith i eleted to be LS with orgetting ator β = and P = I θ i initialized a θ = [ ] and araeter rojetion i not ued he reult lotted in Figure 5C7 and 5C8 how that both the etiation and the traking are ueul Figure 5C7 Outut and ontrol ignal o Exale 59

Page 5C34 Chater 5 Coleentary Material Figure 5C8 SPM araeter etiate θ = [ bˆ ˆ a] in Exale 59 54 Robut MRAC Exale 5 Conider the alar lant o Exale 54 In thi exale we will μ onider the exitene o a ultiliative lant diturbane Δ () = ; ie we will + μ aue that b b μ x = ( +Δ ( ) ) u= u a a+ μ π Let a= 5 b= x = 5 and rt () = + in(4 t+ ) a in Exale 54 and let 6 μ = I we ue the PI algorith and the MRC law o Exale 54 a they are we obtain the reult hown in Figure 5C9 and 5C A an be een in thee igure θ and x diverge and the adative ontrol deign ail Next we aly the robut PI algorith with dead zone odiiation ( g = 5 = ) he reult lotted in Figure 5C and 5C how that the araeter etiate do not diverge and the traking error reain in an allowable range

Page 5C35 Chater 5 Coleentary Material Figure 5C9 Outut and ontrol ignal o Exale 5 without robutne odiiation Figure 5C SPM araeter o Exale 5 without robutne odiiation

Page 5C36 Chater 5 Coleentary Material Figure 5C Outut and ontrol ignal o Exale 5 with robutne odiiation Figure 5C SPM araeter etiate o Exale 5 with robutne odiiation

Page 5C37 Chater 5 Coleentary Material Bibliograhy [] PA IOANNOU AND J SUN Robut Adative Control Prentie-Hall Englewood Cli NJ 996; alo available online at htt://www-ruedu/~ioannou/ Robut_Adative_Controlht [] A FEUER AND AS MORSE Adative ontrol o ingle-inut ingle-outut linear yte IEEE ran Autoat Control 3 (978) 557 569 [3] AS MORSE A oarative tudy o noralized and unnoralized tuning error in araeter-adative ontrol in Pro IEEE Conerene on Deiion and Control 99 [4] I KANELLAKOPOULOS PV KOKOOVIC AND AS MORSE Syteati deign o adative ontroller or eedbak linearizable yte IEEE ran Autoat Control 36 (99) 4 53 [5] M KRSIC I KANELLAKOPOULOS AND PV KOKOOVIC Nonlinear and Adative Control Deign Wiley New York 995 [6] AS MORSE High-order araeter tuner or the adative ontrol o linear and nonlinear yte in Pro US-Italy Joint Seinar on Syte Model and Feedbak Cari Italy 99