6.3 APPC Schemes: Polynomial Approach

Transcription

1 Page 6C. Chater 6. Cleentary Material Chater 6 Cleentary Material 6.3 APPC Shee: Plynial Arah Prf f here Ste. We etablih the exrein (6.48) We rewrite the ntrl law (6.39) and the nralized etiatin errr a ˆ ˆ LQ u = P ( y y) ε = z θ φ= ˆ y ˆ u. R Z (6C.) () () are ni Hurwitz lynial f degree n+ q n reetively n Rˆ () ˆ = + θ α = θ α (). a n Z b n (6C.) he rf i ilified if withut l f generality we deign () = () () (6C.3) where () i an arbitrary ni Hurwitz lynial f degree q if q and q () = fr q<. We huld int ut that the ae analyi an al be arried ut with q being Hurwitz but therwie arbitrary at the exene f e additinal algebra. We leave thi re general ae a an exerie fr the reader. Let u define u f u yf y (6C.4) q and write (6C.) a Py ˆ + LQ ˆ u = y f f ˆ ˆ Rqyf Z quf = ε (6C.5) where ˆ. y P y L (6C.6)

2 Page 6C. Chater 6. Cleentary Material Ue the exrein Rˆ θ Zˆ θ n+ q () q() = + αn+ q () () q() = αn+ q () ˆ n+ q ˆ n+ q P () = + αn+ q () LQ () () = + l αn+ q () (6C.7) (6C.8) in (6C.5) t btain ( n+ q ) yf = θ αn+ q () yf + θ αn+ q () uf + ε (6C.9) ( n+ q ) uf = ( θ ) αn+ q ( ) yf ( θ + l ) αn+ q ( ) uf ε + y. Define the tate ( n q ) ( n q ) + + f y f f f f. f u x y y u u hen (6C.9) an be exreed in the fr x = Atx () + b() tε + by (6C.) where θ θ In+ q Ο( n+ q ) x( n+ q ) At () = θ θ l ( n+ q ) x( n+ q ) I Ο n+ q (6C.) where ( n+ q ) x( n+ q ) Ο i an ( n+ q ) ( n+ q ) atrix with all eleent equal t zer. Sine ( n+ q ) u () = uf = uf + λ αn+ q uf y = y = y + λ α () y (6C.) ( n+ q ) f f n+ q f where λ i the effiient vetr f n q () + it fllw that u = [ ] x + [ λ ] x n+ q n+ q n+ q y = [ ] x + [ λ ] x. n+ q n+ q n+ q (6C.3)

3 Page 6C.3 Chater 6. Cleentary Material Ste. Etablih the e.. f the hgeneu art f (6C.) We an hw that fr eah frzen tie t det( I A( t)) = Rˆ LQ ˆ + Pˆ Z ˆ = A (6C.4) q q q where the lat equality i btained by uing the Dihantine equatin (6.38). Sine q A are Hurwitz A(t) ha eigenvalue with negative real art at eah tie t; i.e. A(t) i table at eah frzen tie t. We nw need t hw that A () t L A() t L fr the rertie θ L and θ L whih are guaranteed by the adative law (6.37). In a iilar way a in the knwn araeter ae the athing equatin (6.38) ay be exreed a β ˆ = (6C.5) ˆ * S l l al where S ˆl i the Sylveter atrix f R ˆ. Q Z ˆ ; ˆ ˆ [ ˆ] ˆ [... ˆ β l = lq lq = l]. * al i a defined in (6.3); and Uing the autin that the lynial RQ ˆ and Z ˆ are trngly rie at eah tie t we nlude that their Sylveter atrix S ˆl i unifrly nningular; i.e. ˆ det( S l ) > ν fr e ν >. herefre θ L ilie that Sl Sl L. Fr (6C.5) we have ˆ * l l q β ˆ l = S a (6C.6) whih ilie that β ˆ l i u.b. On the ther hand beaue θ L and θ L it fllw fr the definitin f the Sylveter atrix that S ˆ l L. Nting that ˆ ˆ ˆ ˆ * = Sl S lsl al l β we have ˆ β L whih i ilied by ˆ ˆ l S L l S l and ˆ S () t L l. Beaue the vetr θ θ l are linear binatin f θ ˆ β and all eleent in A() t are u.b. we have l A () t ( ˆ β () t + θ ()) t l whih ilie that A () t L. Uing here A.8.6 it fllw that the hgeneu art f (6C.) i e.. Ste 3. Ue the rertie f the L σ nr and B G lea t etablih bundedne Fr larity f reentatin let u dente the L δ nr a. Fr Lea A.5. and (6C.) we have x ε + x() t ε + (6C.7) fr e δ >. We define the fititiu ignal + y + u. (6C.8) f

4 Page 6C.4 Chater 6. Cleentary Material Uing (6C.) (6C.) (6C.7) and Lea A.5. we an etablih that φ f and i.e. f ε ε f + x + + (6C.9) t ( t ) f + f δ e τ ε ( τ) ( τ) ( τ) dτ. Uing the rerty ε L guaranteed by the adative law uing the bundedne f f and alying the B G lea we an etablih that f L. he bundedne f the ret f the ignal fllw fr f L and the rertie f the L δ nr that i ued t hw that f bund t f the ignal fr abve. Ste 4. Etablih that the traking errr nverge t zer Let u tart with the end equatin in (6C.) i.e. ε ˆ ˆ = R y Z u. Filtering eah ide f the abve equatin with LQ where q Lt l ( ) n + αn ( ) and l = [ l ] i the effiient vetr f Lt ˆ( ) we btain ( ) ˆ ˆ ε = R Z. q q LQ LQ y u (6C.) Uing =q and alying Lea A.. (Swaing Lea ) we btain the fllwing: where Q Q Q ˆ ˆ ˆ Q R ˆ y = R y+ r Z u = Z u+ r q q (6C.) αn () αn () r W() Wb () y θ a r W() Wb () u θ b (6C.) and W W b are a defined in Swaing Lea with W =. q Beaue u y L and θa θ b L it fllw that r r L. Uing (6C.) in (6C.) we btain Q Q Q LQ = L y u + r r ( ) ˆ ˆ ε R Z q (6C.3)

5 Page 6C.5 Chater 6. Cleentary Material whih due t Q y = Q( e+ y) = Qe bee Q Q LQ L e r r L u ( ) ˆ ˆ ε = R + Z. q (6C.4) Alying Lea A..4 (Swaing Lea 4) (i) t the end ter in the right-hand ide f (6C.4) we btain ˆ Q Q L Z u Z Lˆ() t u = + r3 (6C.5) where Q Z = αn () θb r3 = αn () F( l θb) αn () u () and Fl ( θ ) l ( + θ ). Uing the identity L ˆ () tq () = Lt ˆ (). Q () and the b b ntrl law given by the firt equatin in (6C.) we btain ˆ Q ˆ Q ˆ Lt () u = Lt (). u = P e. (6C.6) Fr (6C.6) (6C.5) (6C.4) we btain Q LQ L ˆ e Z Pˆ e L r r r ( ε ) = R + + ( ) 3. q (6C.7) Uing Lea A..4(ii) with f = / = we an btain the exrein Q L ˆ ˆ ˆ e R = L(). t R() t Q e+ r4 ˆ ˆ ˆ Z P e = Z(). t P() t e+ r5 r α () G( e l θ ) r α () G ( e θ ) 4 n a 5 n b (6C.8) where GG are a defined in Lea A..4 and Xt () α n ()() θ tdente the waed lynial Xt () θ tαn(). Due t e L and l θ a θ b L it fllw fr the exrein f GG that GG L. Uing (6C.8) in (6C.7) we btain ( ) LQ ( ε ) = LtR () () tq + Z () tpt () e + Lr ( r ) r r r ˆ ˆ ˆ ˆ q (6C.9) Uing the athing equatin (6.38) and the fat that effiient f A * () we an write * * A () = A () due t the ntant Lˆ t Rˆ t Q + Zˆ t Pˆ t = A = A (6C.3) * * (). () (). () () ().

6 Page 6C.6 Chater 6. Cleentary Material Subtituting (6C.3) in (6C.9) we btain r A e = υ υ LQ ( ε ) L( r r ) + r r r (6C.3) * q e = *. A υ (6C.3) Uing the fat that L () = α () l l = [ l ] we have n Q υ= αn () l ε r r + + r3 r4 r5. q herefre (6C.3) ay be written a Q e = α () l ε + w * n A q w = [ α * n ( ) l( r r) + αn ( ) w αn ( ) w] A Q w = F( l θb) αn ( ) u w = G+ G. (6C.33) Sine l u L ; F G G r r L ; and * α () () A n αn A * are tritly rer and table it fllw fr Crllary A.5.5 that w w w L and w a t. Let u nw aly Lea A.. (Swaing Lea ) t the firt equatin in (6C.33) t btain α e l W ()[ W () ] l w (6C.34) * n Q = ε + b ε + A q Q where W () W () nit f tritly rer table tranfer funtin. Sine all the b tranfer funtin in (6C.34) are rer with inut whih are bunded and in L it fllw that e L L. We an write the lant equatin Ry = Zu a r y + ( R ) y = Z u ( R ) ( R ) y = y + Zu y = y + Zu. Sine i ni table with the ae degree a R all the tranfer funtin in the abve equatin are rer and the bundedne f y u ilie that y i bunded. In turn thi ilie that e i bunded whih tgether with e L L ilie that e a t.

7 Page 6C.7 Chater 6. Cleentary Material 6.4 APPC Shee : State-Sae Arah Prf f here Ste. Devel the led-l tate errr equatin We exre the traking errr equatin ZQ e = u RQ in the tate-ae fr I n q e + θ = e+ θu e = C e (6C.35) where n q C = [ ] R + and θ θ are the effiient vetr f n q e R + R Q Z Q reetively. n+ q Let e e eˆ be the tate-bervatin errr. Uing (6.67) and (6C.35) we btain ˆ () ˆ ˆ e = A t e+ K C e e = Ae + θe θu (6C.36) where A a I n+ q i a table atrix; A () ˆ ˆˆ t A BK; and θ θ θ θ θ θ. Sine the lynial Z Q RQhave n n untable zer it fllw uing here A..5 that there exit lynial X() Y() f degree n+ q with X() ni whih atify the equatin R Q X+ Z QY = A (6C.37) *. A () i a Hurwitz lynial f degree ( n+ q) and ntain all the n table zer f Z Q RQif any. Dividing eah ide f (6C.37) with A () and uing it t erate n u we btain RQ X QYZ u. * + u * = u (6C.38) A A Uing Qu = Qu and Z u = Ry in (6C.38) we btain RXQ QYR u = u. * + y * (6C.39) A A he lant utut atifie

8 Page 6C.8 Chater 6. Cleentary Material y = C e + C eˆ + y. (6C.4) Ste. Etablih e.. fr the hgeneu art f (6C.36) Fr eah frzen tie t we have det( ) det( ˆ ˆˆ I A = I A + BK) = A( ) (where the lat equality i btained uing (6.63)) whih ilie that A () t i table at eah frzen tie t. If Z ˆ () () tq and Rˆ () t Q () are trngly rie i.e. ( Aˆ Bˆ ) i trngly ntrllable at eah tie t the ntrller gain K ˆ ay be alulated at eah tie t uing Akerann frula [3] i.e. K ˆ [ ] G A ( A) ˆ = where ˆ ˆ ˆ ˆ n+ q G [ B AB A Bˆ] i the ntrllability atrix f the air ( Aˆ Bˆ ). Beaue ( Aˆ Bˆ ) i aued t be trngly ntrllable and A ˆ B ˆ L due t θ L we have K ˆ L. Nw ˆ ˆ d K [ ] ( ) ( ˆ = G G G A A + G A A). dt Beaue θ L and θ L it fllw that Kˆ ( t) L whih in turn ilie that A ( t) L. Fr A being intwie table and A ( ) t L we have that A () t i a u.a.. atrix by alying here A.8.6. Beaue A i a table atrix the hgeneu art f (6C.36) i e.. Ste 3. Ue the rertie f the L δ nr and the B G lea t etablih bundedne Alying Lea A.5.9 A.5. t (6C.36) (6C.39) (6C.4) we btain eˆ C e y C e + eˆ + C e + u eˆ + y C e + (6C.4) where. dente the L δ nr fr e δ >. We relate the ter C e with the etiatin errr by uing (6C.36) t exre a C e C e = C ( I A) ( e u ). θ θ (6C.4) Sine ( CA ) i in the berver annial fr i.e. where A () = det( I A ) we have * α n + q () A () C ( I A ) =

9 Page 6C.9 Chater 6. Cleentary Material n n i C e = ( ie iu ) A θ θ i= where n = n+ q θi= [ θi θi θin] i=. We aly Lea A.. (Swaing Lea ) t eah ter under the uatin in (6C.43) t btain and n i n i () Q() ie i e Wi () ( Wbi () e) i A() θ A() θ Q() θ = + n i n i Q() iu i u Wi () ( Wbi () u ) i A() θ A() θ Q() θ = + where W W i = n+ q are tranfer atrie defined in Lea A.. with i bi n i ( Q ) ( ) W () =. herefre C e an be exreed a where n i n i () Q() n C e = e u r i= θ θ + A () i i () Q() () Q() Q() α () α () = + n+ q n+ q e r u A () θ θ () Q() () Q() (6C.43) Q() r W () W () e W () u. n ( ) i bi ( ) i bi i A () θ θ i= Fr the definitin f θ we have θ αn+ q () = θ αn+ q () θ αn+ q () = Rˆ ( tq ) () R() Q() + ( ) n+ q n+ q = Rˆ ( t ) R() Q() = θ aαn () Q() (6C.44) where θ a θa θ i the araeter errr. Siilarly a α () = θ α () Q () (6C.45) θ n+ q b n where θ b θb θ b. Uing (6C.44) and (6C.45) in (6C.43) we btain Q() Q () C e e r = aαn () bα () n u A () θ θ + Q () () () () Q() Q() Q() aαn y bαn u θ θ r A () Q() () () Q() = () () + (6C.46)

10 Page 6C. Chater 6. Cleentary Material where the end equality i btained uing Q () e = Q () y Q () u = Q () u. Uing the identitie α () u = φ α () y = φ n n () () and Lea A.. we btain the fllwing equalitie: Q () Q () () () Q Q θ aφ θ aαn y W = q Wbq φ + a () () () θ Q () Q () () () Q Q θbφ θbαn u W = q Wbq φ + b () () () θ where W W are a defined in Swaing Lea A.. with q bq abve equalitie in (6C.46) we btain () Q() C e = θ φ + r A () Q ( ) () Q ( ) W=. Uing the (6C.47) where Q() r r ( Wq () Wbq () φ Wq () Wbq () φ θ ). a + b A () θ + he nralized etiatin errr atifie the equatin ε = θ φ whih an be ued in (6C.47) t btain () Q() C e = ε + r. A () (6C.48) Uing the definitin f the fititiu nralizing ignal f + u + y and Lea A.5. we an hw that φ/ f / f L fr e δ >. Uing thee rertie f f alying Lea A.5. in (6C.48) we btain C e ε + θ f f Fr (6C.4) and in the definitin f we have f. (6C.49) f = + u + y C e + whih tgether with (6C.49) ilie

11 Page 6C. Chater 6. Cleentary Material r + + f ε f θ f g + f f where g ε + θ and g L due t the rertie f the adative law. If we nw aly the B G lea t the abve inequality we an hw that f L. Fr f L and the rertie f the L δ nr we an etablih bundedne fr the ret f the ignal. Ste 4. Cnvergene f the traking errr t zer Sine all ignal are bunded we an d etablih that ( ε ) L dt whih tgether with ε L ilie that ε() t() t and therefre θ () t a t. Fr the exrein f r r we an nlude uing Crllary A.5.5 that r L and r () t a t. Uing ε r L and r a t in (6C.48) and alying Crllary A.5.5 we have that C e L and C e () t a t. Cnider the firt equatin in (6C.36) i.e. ˆ e = A () t eˆ + K C e. Sine A () t i u.a.. and C e L C e a t we an nlude that eˆ a t. Sine e y y = C e + C eˆ (ee (6C.39)) it fllw that e () t a t and the rf i lete Rbut APPC: Plynial Arah Prf f here Ste. Exre u y in ter f the etiatin errr Fllwing exatly the ae te a in the rf f here 6.3. fr the ideal ae we an hw that the inut u and utut y atify the ae equatin a in (6C.) exreed a x = A() t x+ b () t ε + b y y = C x+ d ε + d y u = C x+ d ε + d y 3 4 (6C.5) where x At ( ) b( t) b and Ci i= ; dk k= 34 are a defined in (6C.) and ˆ y = P y. he deling errr ter due t Δ () d d nt aear exliitly in ( ) (6C.5). Due t the undeled dynai εε A () t n lnger belng t L a in the η η d ideal ae but belng t S( + f ). Beaue Δ + we have u d εε A () t S Δ + + f.

12 Page 6C. Chater 6. Cleentary Material Ste. Etablih the u.a.. rerty f A() t A in the ideal ae the APPC law guarantee that det( I A( t)) = A ( ) fr eah tie t where A () i Hurwitz. If we aly here A.8.6(ii) t the hgeneu art f (6C.5) we an etablih that A() t i u.a.. whih i equivalent t e.. rvided f dτ +Δ + < μ (6C.5) t + d t t any and e μ > where i a finite ntant. Beaue > nditin (6C.5) ay nt be atified fr all Δ f unle d the uer bund fr the inut diturbane i zer r uffiiently all. A in the MRAC ae we an deal with the diturbane in tw different way. One way i t dify = + n t = + nd + β where β i hen t be large enugh that that fr μ ( f +Δ ) < d d μ β < ay nditin (6C.5) i alway atified and A() t i e.. he ther way i t kee the ae and etablih that when tate-tranitin atrix Φ ( t τ) f () hi rerty f A() t (when hythei that grw large ver an interval f tie I = [ t t] ay the k A t atifie ( ) ( t τ Φ t τ k e ) t τ and t τ I. grw large) i ued in Ste 3 t ntradit the uld grw unbunded and nlude bundedne. We reed a fllw. Let u tart by auing that grw unbunded. Beaue all the eleent f the tate x are the utut f tritly rer tranfer funtin with the ae le a the rt f () and inut u y and the rt f () are lated in Re[] < δ/ it x fllw fr Lea A.5. that L. Beaue y ε L it fllw fr (6C.5) y u that L. Beaue u y are bunded fr abve by it fllw fr the equatin fr that annt grw fater than an exnential i.e. k ( t t ) () t e ( t) t t fr e k >. Beaue () t i aued t grw unbunded we an find a t > α > fr any arbitrary ntant α > t t uh that k ( t ) > αe α. We have d αe < ( t ) e ( t ) k α k( t t) whih ilie that t > α+ k α t t ln ( ) ln [ ( )]. Beaue α > t t and t ( t α t ) it fllw that

13 Page 6C.3 Chater 6. Cleentary Material ( t ) > α t ( t α t ). Let t = uτ t arg( ( ) )} τ = α. hen ( t) = α and () t α t [ t t) where t t α i.e. t t α. Let u nw nider the behavir f the hgeneu art f (6C.5) i.e. Y = A() t Y (6C.5) ver the interval I [ t t ) fr whih () t α and t t α where α > i an arbitrary ntant. Beaue det( I A( t)) = A ( ) i.e. A() t i a intwie table atrix the Lyaunv equatin A () tpt () + PtAt () () = I (6C.53) ha the lutin P(t) = P () t > fr eah t I. We nider the Lyaunv funtin Vt () = Y () tptyt () (); then V = Y Y + Y PY Y Y + P () t Y Y. Uing (6C.53) and the bundedne f P A we an etablih that P () t A () t. Beaue λy Y V λy Y fr e < λ < λ it fllw that i.e. V ( λ λ A ( t)) V; t ( λ λ A ( )) d τ Vt () e V( τ) t τ. Fr the interval I = [ t t) we have () t α and therefre herefre t d A ( τ) dτ Δ + f+ ( t τ) +. τ α Vt () e V( τ) t τ [ t t) (6C.54) λ ( t τ ) and t τ rvided f d α +Δ + < λ (6C.55) λ where λ =. Fr (6C.54) we have λy () t Y() t Y () t PY() t e λ Y ( τ) Y( τ) λ ( t τ )

14 Page 6C.4 Chater 6. Cleentary Material whih ilie that λ λ () ( t τ Yt e ) Y( τ) t τ [ t t) λ whih in turn ilie that the tranitin atrix Φ ( t τ) f (6C.5) atifie Φ ( t τ) β e t τ [ t t ) (6C.56) α ( t τ ) λ λ λ where β = α =. Cnditin (6C.55) an be atified by hing α large enugh and by requiring Δ f t be aller than e ntant i.e. λ ( f +Δ ) < 4 ay. In the next te we ue (6C.56) and ntinue ur arguent ver the interval I in rder t etablih bundedne by ntraditin. Ste 3. Bundedne uing the B G lea and ntraditin We aly Lea A.5. t (6C.5) fr t [ t t). We have x e x( t ) + ( ε ) + / δ ( t t ) tt tt where () dente the L δ nr () defined ver the interval [ t t) fr any tt < δ< δ < α. Beaue x tt δ L it fllw that x e ( t ) + ( ε ) +. / δ ( t t ) tt tt Beaue y u x + ( ε ) + it fllw that t t t t t t t t y u e ( t ) + ( ε ) +. / δ ( t t ) tt tt tt Nw () t = + n () t and d δ( t t ) d() = d( ) + t t +. δ t t δ n t e n t y u Beaue () tt δ () tt fr δ δ it fllw that δ( t t) = + d + + t t + t t () t n () t e ( t ) y u t t. Subtituting fr the bund fr y u we btain t t t t r δ( t t ) ε t t () t e ( t ) + ( ) + t t

15 Page 6C.5 Chater 6. Cleentary Material t ( t t ) ( t ) δ δ () t + e ( t ) + e τ ε ( τ) dτ. (6C.57) t Alying B G lea 3 (Lea A.6.3) t t t d ( ) t ( ) d δ t t ε τ δ t ε τ δ + + () t ( ( t )) e e e e d t t. t Fr t [ t t) we have t d ε dτ f Δ + + ( t ) +. α By hing α large enugh that d α < and by requiring δ 4 δ ( Δ + f) < 4 we have t δ ( t t) δ ( t ) δ ( t t) δ ( t ) ( + ( t )) e + e d ( + ( t )) e +. t Sine t t α ( t ) = α and ( t ) > α we have α< t + α e δα + ( ) ( ). herefre we an he α large enugh that ( t ) < α whih ntradit the hythei that ( t ) > α. herefre L whih ilie that x u y L. he nditin fr rbut tability i therefre fr e finite ntant >. λ δ ( f +Δ ) < in δ 4 Ste 4. Etablih bund fr the traking errr A bund fr the traking errr e i btained by exreing e in ter f ignal that are guaranteed by the adative law t be f the rder f the deling errr in the... he traking errr equatin ha exatly the ae fr a in the ideal ae and i given by n () Q() () αn () = ε + e v A () A ()

16 Page 6C.6 Chater 6. Cleentary Material where v i the utut f rer table tranfer funtin whe inut are eleent f θ η ultilied by bunded ignal. Beaue θ ε S( + f ) and η ( Δ + d) due t L it fllw fr Crllary A.5.8 that e S( Δ + d + f) and the rf i lete. 6.9 Exale Uing the Adative Cntrl lbx he PPC and APPC algrith reented in Chater 6 an be ileented uing a et f MALAB funtin and the Siulink blk Adative Cntrller rvided in the Adative Cntrl lbx. Fr ileentatin f the APPC algrith thee tl need t be ued tgether with the PI funtin r blk intrdued in Chater 3. In thi etin we dentrate the ue f the Adative Cntrl lbx in variu PPC and APPC rble via a nuber f iulatin exale PPC: Knwn Paraeter he Adative Cntrl lbx rvide an eay way f deigning and iulating a wide range f PPC hee. Fr the lynial arah the MALAB funtin ly an be ued t lve (6.3) fr a given lant f the fr (6.8) and deign lynial A * () and (). Denting the effiient vetr f Z() R() A * () () Q () a ZRALabdaQ reetively [KuKyREYPE] = ly(zralabdaq) return the effiient vetr f K u and lving re i ueful and a if it fail. Cnider the SPM K y. REYPE i returned a if the equatin z θ φ = (6C.58) fr the lant Z () b + + b+ b y = u u n n R () = + an + + a + a where θ = [ b b a a ] n z= y () n n φ = u u y y () () () () and () i a ni Hurwitz deign lynial f rder n. hi SPM i al ued in araeterizatin fr adative PPC in etin 6.9..

17 Page 6C.7 Chater 6. Cleentary Material he MALAB funtin uly r the Siulink blk Adative Cntrller with arriate tin and araeter an be ued t generate the ignal φ and z f the SPM (6C.58) and the ntrl ignal u a dentrated in Exale 6.9. and 6.9. belw. Exale 6.9. Cnider the lant y= u. We want t he u that the ledl le are equal t the rt f ( ) A () = + and y trak the referene ignal y =. he referene inut y atifie Q() y = fr Q () =. he ntrl ignal an be generated a K () () u Ky u = u ( y y ) () + () () = +.5. K () u and K () an be alulated uing the fllwing de: y Z = ; R = [ -]; A = [ ]; Labda = [.5]; Q = [ ]; [Ku Ky REYPE] = ly(zralabdaq) he reult i Ku () =.5 K () =.5.5. y Exale 6.9. In Exale 6.9. we have alulated the ntrl lynial a Ku () =.5and Ky () =.5.5. Nw we an generate the linear del and ntrl ignal uing the fllwing de (added t the de f Exale 6.9.): Labda = Labda; dt =.5; % ie inreent fr iulatin (e). t = [:dt:]; % Pre tie (e). lt = length(t); y = ne(lt); % Referene ignal. % State initializatin fr lant: [ntate x] = ufilt('init'zr); x(:) = x; % State initializatin fr linear araetri del: [ntate xl] = uly('init'[ ]LabdaLabdaQ); xl(:) = xl; % Signal initializatin: y() = ; % Pre fr k = :lt u(k) = uly('ntrl'xl(:k)[y(k) y(k)][ku Ky]...LabdaLabdaQ); [z(k) hi(:k)] = uly('utut'xl(:k)[u(k) y(k)][ ]...LabdaLabdaQ); dxl = uly('tate'xl(:k)[u(k) y(k) y(k)][ ]...LabdaLabdaQ); xl(:k+)=xl(:k) + dt*dxl; dx = ufilt('tate'x(:k)u(k)zr); x(:k+)=x(:k) + dt*dx;

18 Page 6C.8 Chater 6. Cleentary Material y(k+) = ufilt('utut'x(:k+)zr); end % Outut: y = y(:lt); he reult are hwn in Figure 6C. and 6C.. Anther way f iulating the abve hee i t ue the Siulink blk Adative Cntrller. We an ue the Siulink hee hwn in Figure 6C.3 and he the arriate tin and enter the arriate araeter in the enu f the Adative Cntrller blk. Figure 6C. Outut and ntrl ignal f Exale 6.9..

19 Page 6C.9 Chater 6. Cleentary Material Figure 6C. SPM ignal f Exale Clk t tie - Plant y y(t) [ -] theta Adative Cntrller Adative Cntrller. u u(t) z z(t) hi hi(t) Cntant y_ y(t) Figure 6C.3 Siulink ileentatin f the PPC Shee in Exale Fr the PPC hee baed n the tate-feedbak arah where the tate vetr e f the tate-ae rereentatin (6.5) fr the lant (6.5) i.e. et () = Aet () + Bu() t e = C e i available fr eaureent the MALAB funtin v an be ued t ute the tate-feedbak gain K f the PPC law

20 Page 6C. Chater 6. Cleentary Material u= K e fr a given deign lynial A * (). Denting the effiient vetr f * Z() R() A () by Z R A reetively [KREYPE] = v(z R A) return the feedbak gain K. REYPE i returned a if the equatin lving re i ueful and a if it fail. If the tate vetr f (6.5) i nt available fr eaureent then the value f the Luenberger berver gain K in (6.55) whih i required in etiatin f the tate vetr e an be uted uing the MALAB funtin tateet. Denting the deired harateriti lynial f A KC by A * () and the effiient vetr f A * () by A [KREYPE] = tateet(c R A) return the berver gain K. REYPE i returned a if the equatin lving re i ueful and a if it fail. he MALAB funtin urf r the Siulink blk Adative Cntrller with arriate tin and araeter an be ued t generate the ignal φ and z f the SPM (6C.58) and the ntrl ignal u a dentrated in Exale Exale Cnider the lant y= u. We like t he u that the ledl le are equal t the rt f A () = ( + ) and y trak the referene ignal y =. hi rble an be nverted t a regulatin rble by nidering the traking errr equatin ( + ) e = y y = u ( ) u = u + where we want e t nverge t. Auing that the tate f the annial rereentatin e= e+ u ε = [ ] e are available fr eaureent we an deign a tate-variable ntrller + u= u u = Ke. he value f K an be alulated a fllw: R = [ - ]; Z = [ ]; A = [ ]; [K REYPE] = v(z R A)

21 Page 6C. Chater 6. Cleentary Material he reult i K = [.5]. If the tate vetr e i nt available fr eaureent we ue a tate etiatr. Let u L ha the harateriti deign a tate etiatr with gain vetr L that [ ] lynial * A = ( + ). he fllwing de an be ued fr thi ure: R = [ - ]; C = [ ]; A = [ 4 4]; [L REYPE] = tateet(c R A) he alulated berver gain i L = [ 5 4]. Exale In Exale we have alulated the feedbak gain a K = [.5]. Nw we an generate the linear del and ntrl ignal uing the fllwing de: Q = [ ]; Z = ; R = [ -]; Q = [ ]; A = [ ]; ba = [Z R()]; [nu den] = urf('augent'[ ]QQba); [K REYPE] = v(nudena) Labda = Q; A = [ 4 4]; dt =.5; % ie inreent fr iulatin (e). t = [:dt:]; % Pre tie (e). lt = length(t); y = ne(lt); % Referene ignal. % State initializatin fr lant: [ntate x] = ufilt('init'zr); x(:) = x; % State initializatin fr tate ae and linear araetri del: [ntate xl] = urf('init'[ ]AQQLabda); xl(:) = xl; % Signal initializatin: y() = ; % Pre fr k = :lt [u(k) ubar(k)] = urf('ntrl'xl(:k)kqqlabda); [z(k) hi(:k)] = urf('utut'xl(:k)[u(k) y(k)][ ] AQQLabda); dxl = urf('tate'xl(:k)[u(k) ubar(k) y(k) y(k)][ ] AQQLabda ba); xl(:k+)=xl(:k) + dt*dxl; dx = ufilt('tate'x(:k)u(k)zr);

22 Page 6C. Chater 6. Cleentary Material x(:k+)=x(:k) + dt*dx; y(k+) = ufilt('utut'x(:k+)zr); end % Outut: y = y(:lt); he reult are ltted in Figure 6C.4 and 6C.5. Anther way f iulating the abve hee i t ue the Siulink blk Adative Cntrller. One an ue a hee iilar t the ne hwn in Figure 6C.3 and he the arriate tin and enter the arriate araeter in the enu f the Adative Cntrller Siulink blk. Figure 6C.4 Outut and ntrl ignal f Exale

23 Page 6C.3 Chater 6. Cleentary Material Figure 6C.5 Linear araetri del ignal f Exale Fr the linear quadrati (LQ) ntrl the MALAB funtin lq an be ued t ute the tate-feedbak gain K fr the lant (6.5) with the tate-ae rereentatin (6.7) and a given enalty effiient λ. Denting λ and the effiient vetr f Z() R() by labda and Z R reetively [KREYPE] = lq(z R labda) lve the Riati equatin (6.73) and return the feedbak gain K. REYPE i returned a if the equatin lving re i ueful and a if it fail. he MALAB funtin urf r the Siulink blk Adative Cntrller with arriate tin and araeter an be ued t generate the ignal φ and z f the SPM (6C.58) and the ntrl ignal u a in the ae f PPC uing a tate-feedbak arah. Exale Cnider the lant y= u f Exale We want t he u that the led-l lant i table and y trak the referene ignal y =. A befre thi rble an be nverted t a regulatin rble by nidering the traking errr equatin ( + ) e = y y = u ( ) u = u + where we want e t nverge t. Auing that the tate f the annial rereentatin (r the tate uefully etiated; ee Exale 6.9.3) e= e+ u [ ] ε = e

24 Page 6C.4 Chater 6. Cleentary Material are available fr eaureent we an deign the LQ ntrller + u= u u = Ke whih iniize the t funtin J = ( e +. u ) dt. he rrending feedbak gain K an be alulated a fllw: R = [ - ]; Z = [ ]; [K REYPE] = lq(z R.) he reult i K = [.736.5]. If we aly thi gain in Exale intead f [.5] we btain the reult hwn in Figure 6C.6 and 6C.7. Figure 6C.6 Outut and ntrl ignal f Exale

25 Page 6C.5 Chater 6. Cleentary Material Figure 6C.7 Linear araetri del ignal f Exale APPC An APPC hee in general i ed f a PPC law that an be btained uing ne f the three arahe entined in etin 6.9. and a PI algrith t etiate the unknwn lant araeter whih an be hen fr a wide la f algrith reented in Chater 3. he Adative Cntrl lbx funtin ly v lq an be ued t generate the ntrl ignal a dentrated in etin 6.9. with unknwn lant araeter. he funtin uly and urf r the Siulink blk Adative Cntrller with arriate tin and araeter inrrated with araeter identifiatin funtin (r the Paraeter Etiatr blk) an be ued t generate the ignal φ and z f the linear del (6C.58) and the ntrl ignal u. Exale Cnider the lant f Exale Aue that the lant araeter are unknwn. We want t aly the ae ntrl algrith. We an ue the linear del * z = θ φ where z= y ( + ) * θ = [ ] φ = u u y y ( + ) ( + ) ( + ) ( + ) t etiate the unknwn lant araeter. he Siulink blk Paraeter Etiatr and Adative Cntrller an be ued fr iulatin. Seleting the PI algrith t be LS with frgetting fatr β = and P = I and initializing θ

26 Page 6C.6 Chater 6. Cleentary Material a[ ] the reult ltted in Figure 6C.8 and 6C.9 dentrate that bth the etiatin and traking bjetive are ueful. Figure 6C.8 Outut and ntrl ignal f Exale Figure 6C.9 SPM araeter etiate θ = [ bˆ ˆ a] f Exale

27 Page 6C.7 Chater 6. Cleentary Material Bibligrahy []. KAILAH Linear Syte Prentie-Hall Englewd Cliff NJ 98. [] P. A. IOANNOU AND J. SUN Rbut Adative Cntrl Prentie-Hall Englewd Cliff NJ 996; al available at htt://www-rf.u.edu/ ~iannu/rbut_adative_cntrl.ht. [3] B. D. O. ANDERSON Exnential tability f linear equatin ariing in adative identifiatin IEEE ran. Autat. Cntrl (977)