Transcript
W¡uP¡æf[
¨W üxÍ :ÿ Cq zG [¢ u Nonlinear Adaptive and Robust Flight Control Using the Backstepping Algorithm
[ 2Ú
2000
tСp ¡٠¨WÍsW¡Y ×
W¡uP¡æf[
¨W üxÍ :ÿ Cq zG [¢ u Nonlinear Adaptive and Robust Flight Control Using the Backstepping Algorithm
[
2000
tСp ¡٠¨WÍsW¡Y ×
¨W üxÍ :ÿ Cq zG [¢ u
Nonlinear Adaptive and Robust Flight Control Using the Backstepping Algorithm
êp½ ¥î³ f[ú W¡uP ¡æf[÷¿ C;¥ [ 10Ú
1999
tСp ¡٠¨WÍsW¡Y × × W¡uP ¡æf[ú u¥ [ 12Ú
1999
æÙ$ ÙæÙ$ æÙ
#À Æ f[t üxÍ ¨W Cqõ zG ªú CK °. Æ:÷¿ üxÍ ¨W Cqõ zG£ : ¨W ô:" 0 ÏôAéú ô ô"Y ÿ ô"÷¿ uÛ L èßð  ë:÷¿ Cqõ zG ,Y µý, Æ f[t backstepping ªú Ì ; èßðú ôè KAÜèÅ Cqõ z G °. ªù ¨W ô:"ú :<& ßÌ¢ Cq zGª 6, KAú < æ üÇì: Aú Ì M°. Z¢, üxÍ :ÿCq ÁY üxÍ
Cq Áú :Ì W³G½ ÝÝì ½ U& EÍê 9 Æ q# M ¨W Cq zGªú Cè °. üxÍ :ÿCqªú Ì ½ ?Ý ¡ò © f ¨ú \© êÀ êEç¿ @ êõ ºÜèÅ C q zGªú CK °. ýL üxÍ Cqªú Ì ?Ý ¡ò ½ © f ¨ ¾õ yý NL °L A L, ³ú \©èÅ Cq z Gªú CK °. CKý Cqª :Ìý ¨W èßð KAú òbd Áú Ì < ÷6, F-16 üxÍ ¨W ?Ýú Ì èqª ú ½±¥÷ ¿ CKý Cqª ú * °.
sÅq : üxÍ ü±Cq, backstepping ª, êEç¿, :ÿCq, Cq ¡¥ : 98416-525 i
ò» #À ò» ò» v ò» tÁ
i ii v vi
1
1.1 1.2 1.3 1.4 2
u E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . u ô³ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . u 4Ì . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . f[ u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Backstepping
2.1
2.2
2.3
1
ªú Ì¢ üxÍ ü±Cq
¨W ?Ý . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 ÏôAé . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 W³G½ ?Ý . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 ÏôAé uS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cq zG KA ©u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Æ A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Cq zG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 ©u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ½X èqª . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
1 2 4 5 6
6 6 7 8 12 14 15 20 21
iii
3
êEç¿ú Ì¢ üxÍ :ÿ ü±Cq 3.1 3.2 3.3
3.4 4
W³G½ ?Ý ¡ò ¢ ³ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . êEç¿ uS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cq zG KA ©u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Æ A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Cq zG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 ©u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ½X èqª . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
üxÍ ü±Cq 4.1
4.2
Cq zG KA ©u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Æ A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Cq zG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 ©u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ½X èqª . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
@Á ÷L[¶ ¨W ?Ý 5
A F-16
A.1 A.2 B
C
[?/! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W³G½ ?Ý . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
òbd KA Á Cq ÏôAé yÛ C.1 2, 3
$t PÌý Cq yÛ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
27 28 30 31 31 38 39 46
46 47 47 51 52 59
61
64
64 64 69
71
71
iv
$t PÌý Cq yÛ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ÏôAé yÛ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.2 4
72
C.3
73
Abstract
75
ò» Aú Ì¢ üxÍ ü± Cq uS . . . . . . . . . . . . . . . . . . . . .
2.1
Two-timescale
2.2
Simulation result: Backstepping controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2
Simulation result: Backstepping controller (continued) . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2
Simulation result: Backstepping controller (continued) . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2
Simulation result: Backstepping controller (continued) . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2
Simulation result: Backstepping controller (continued) . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.1
3-layer
3.2
Simulation result: Adaptive backstepping controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2
Simulation result: Adaptive backstepping controller (continued) . . . . . . . . . . . . . . . . . . 42
3.2
Simulation result: Adaptive backstepping controller (continued) . . . . . . . . . . . . . . . . . . 43
3.2
Simulation result: Adaptive backstepping controller (continued) . . . . . . . . . . . . . . . . . . 44
3.2
Simulation result: Adaptive backstepping controller (continued) . . . . . . . . . . . . . . . . . . 45
4.1
Simulation result: Robust backstepping controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.1
Simulation result: Robust backstepping controller (continued) . . . . . . . . . . . . . . . . . . . . 55
4.1
Simulation result: Robust backstepping controller (continued) . . . . . . . . . . . . . . . . . . . . 56
4.1
Simulation result: Robust backstepping controller (continued) . . . . . . . . . . . . . . . . . . . . 57
4.1
Simulation result: Robust backstepping controller (continued) . . . . . . . . . . . . . . . . . . . . 58
êEç¿ uS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
13
29
v ò» 2.1
ÏôAé u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
3.1
W³G½ e ?Ý ¡ò (%) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
4.1
W³G½ e ?Ý ¡ò (%) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
A.1 A.2
[?/!¿ uý ¨ A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W³G½ ?Ý º½ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vi
67 68
1
1.1
tÁ u E
Æ:÷¿ ¨W Cqèßðú zG æ©t ¨W "A¢ ÏÌ\× °L A L, ÏÌ\× Â© xÍÜý ?Ý Â©t xÍCqªú :Ì Cq õ zG¢°. ªù L;Cqª, LîWAª, xÍ/:Cqª # N² ° j¢ xÍCqªú PÌ£ ½ ° $> °. # xÍÜý ?Ýù zG ÏÌ\× t ¨W Ïô"ú üp: AÝ& #Í4 :[, Iù ÏÌ Â© ÆA¢ ú u æ©t ÏÌ\× Â Cqõ Ä©t zG L, zGý Cq ú ¨W ÏÌ\× 0t ºÜèb ¢°. ¢ ³>ú Ä æ ÞxÍܪ (feedback linearization) ú Ì¢ üxÍ ¨W Cq [¢ u ØqP°. ÞxÍܪù \׺½ Óù ;³º½õ Ì üxÍ èßðú ôÆ¢ xÍèßð÷¿ ºÞ Cqõ zG ª÷¿, ºÞ ý xÍèßð ¨W ÏÌ\× U M :[ ¨W ÏÌ\× Â © ÆA¢ ú Ã$£ ½ ³Æ Cqõ u£ ½ °. # ªú Ì æ©t üxÍ èßð ô:"ú AÝ& NL qb 6, zero dynamics õ Cq£ ½ { :[ nonminimum phase system 4ÙKAú Ã$£ ½ {° ³> °. Backstepping ªù / üxÍ :ÿCq ÿÌü8t ß& u ØqL üxÍ Cqª °. ªù GW: èßð © Ù èßð \׺½õ \ Cq³÷¿ zA \Ù èßðú Cq YAú Ä ,÷¿, nonminimum phase system KAú Ã$£ ½ ÷6, üxÍ :ÿ Cq Áú :Ì üx 1
2
1.2
u ô³
Í èßð ÝÝìú L²£ ½ ° $> °. ¨W üxÍ £ < I¦ ]ý: º½ U ºÜ : [ ô:"ú AÝ& ?Ý Í q·°. 0t ¨W Cqõ zG EÍ W³G½ ?Ý ¡ò, Z ½Ù ½ ³ú L² , |Å °. èßð ¡ò# ºÜ, ½Ù pú L² ³ú èÅ Cq Á÷¿ :ÿCq ÁY Cq Áú àú ½ °. :ÿCqªù èßð 4Ù º½# Ç& èßð u :¦¢ Cq ú ìè÷¿ 8A Cq uSõ ºÜèÅ ª÷¿ ½ ¢ ô:  ª L £ ½ °. / :ÿCqª ú  èÅ u® üxÍ èßð÷¿ :Ì Â¢ u ØqL ÷6, êEç¿ universal approximation "ú Ì¢ :ÿCqªê uüL °. Cqªù ½ Y èßð ?Ý ¡ò ¢ ³ "A¢ ©æ K ©°L A L, K t Æ¢ ú uú ½ Cqõ zG ª÷¿ ½ ¢ ½ô:  ª L £ ½ °. 1.2
u ô³
ÞxÍܪù Á:÷¿ Aüq ÷6 Ûb ÿÌüL üxÍ è ßð Cqª÷¿ üxÍ ¨W Cq[Cê Gý PÌü}°. Meyer ® Hunt Àý â Cq[C, Lane Y Stengel ù ¨W Cq[C ;³ ÞxÍܪú :Ì " A¢ ;³º½õ Cq °. Hedrick Y Gopalswamy sliding method õ :Ì ?Ý ¡òõ L² °. # ÞxÍܪú ¨W Cq[C ? :Ì£ EÍ, zero dynamics õ Cq£ ½ { :[ nonminimum phase system KAú Ã$£ ½ { L, W³G½ 2ò Z 3ò \ yÛ¨ Å ° [C> f¢°. ÞxÍܪú üxÍ ¨W Cq[C :Ì Z °ô ªù ¨W ô:" ú Ì ÏôAéú timescale ¿ uÛ L, 4ÙØ® ½ÙØ C 1, 2
3
1
tÁ
3
qõ u ª °. Two-timescale ú Ì Cqõ zG YAù ¾ 3 ³G¿ uÛ£ ½ °. $9 ½ÙØt 4ÙØ \׺½ p; q; r ú Cq³ ÷¿ zA , ; ; sq :ú 8[ êÀ p; q; r ¢ u:ú GS¢°. 4 ÙØt \׺½ p; q; r ½ÙØt fý u:ú 8[ êÀ ìC Cq ³ Æe; Æa; Ær ú GS¢°. :, 4ÙØ Ïô" ½ÙØ ü© Í ò :[, 4ÙØ \׺½ p; q; r ¿ sq u:ú è { AÝ& 8[£ ½ qt, ½ÙØt p; q; r è ÿ ¢ ³ #Í# M°L A¢ °. 0t ªú PÌ æ©t 4ÙØ Ïô" ½ÙØ Ïô" ü © Í b ¢°. # Ç&« ÂÙÛ ¨W Cq [¢ ut "»¢ ½ ¡: ©uYA { 4ÙØ Cq ú ½ÙØ Cq ð ?Û& À ÷¿ zA , ¨W ¢ two-timescale ú T¢°L A °. ¨W Cq[Ct two-timescale [¢ ½¡: ©uù Schumacher ® Khargonekar © ü ØqP°. ù 4ÙØ èßðt AÝ¢ ºÞ ØqP °L A L ,ú ½ÙØ Â ½ÙØ \׺½® Cq³÷¿ uý èßðú u °. ýL òbd ¥½õ Ì èßð KAÜü æ¢ 4Ù Ø / Cq ú @A °. # ut zA¢ ª ¢ KA © uù Í Ä! L conservative :[, èßð KAú Ã$ æ©t GSý 4 ÙØ / Cq Í v3 ý°. 0t ©u@Y 0 Cqõ zG£ EÍ Cq³ jÜü$# [C f£ m°. ù Z¢ Cq z G ©uYAt ¨W Cq8 © fý *, |³ 8³ ¢ ³ú Xè # ° ù Aú °. ÞxÍܪú :Ì YAt Z °ô q²Óù CqÂ\ ü ¨W èß ðú AÝ& NL qb ¢° , °. # ¨W W³G½ ù º½ U ºÜ :[ ¨W èßð ô:"ú AÝ& ?Ý ,ù Í q²Ï Æ °. Two-timescale ú Ì¢ Cq EÍ, "A¢ W³G½ ÝÝì © ; 4, 5, 6
7
4
1.3
u 4Ì
èßð KAú 3 ý° , uý °. èßð ?Ý ¡ò# ½ ³ú L² üxÍ Cq Á÷¿t, :ÿCqÛbt Krsti¢ , CqÛbt Qu backstepping ª #¢ uõ ½± °. Funahashi, Hornik #ù °W êEç¿ üxÍ ¥½õ AÝê¿ vÇ£ ½° ,ú½¡:÷¿ < L, Farrell, Lewis #ù ¢ universal approximation "ú Ì , ³WY °W êEç¿ú üxÍ :ÿCq¿ (nonparametric nonlinear direct adaptive controller) Ì uõ ½± °. ¨W CqÛbt Singh # wing-rock motion ú Cq Ú, Kim Y Rysdyk ¨Wõ Cq Ú :ÿ êEç ¿ú PÌ °. 8
9, 10
11, 12
13, 14
15
16, 17
1.3
u 4Ì
Æ f[t backstepping ª © üxÍ ¨W Cqõ zG ªú C K °. ÞxÍܪú Ì¢ Æ: üxÍ ¨W Cq® µý, two-timescale Aú PÌ ML ³Æ òbd ¥½õ Ì ½¡:÷¿ KAú < °. Z¢, Cq8 # ê © f W³G½ * Ûú ? L² ÷6, KA ©u °. üxÍ ¨W EÍ minimum phase "ú ½¡:÷¿ < ,ù Í q²Ï Æ 6, Z¢, Ç L ¨W nonminimum phase "ú EÍ °. , ª ù zero dynamics KAú Ã$ q·° ³> °. ¢`, é (2.17), (2.18) Y v 2.1 ú U_Ã8, x ; x èßð ¢ = "ú >£ ½ °. é (2.17) ú Ã8 ¨W êõ y \׺½ x x ú Cq Cq ³÷¿ PÌþ ½ üú N ½ °. x èßð ¢ x ³±µ g ; g a v 2.1 #Í& , ¤ ¾ h ü© Í ¾6 sC# ÆA ½u \ ¾õ . 0 t x èßðú Cq Cq³÷¿t u ð x Ì :¦ ° ,ú N ½ °. é (2.18) Y v 2.1 ú Ã8 x èßð ¢ u Cq³ ±µ g ¾ê Í ¾ :[, u x èßðú Cq Cq³÷¿ :< ° ,ú Ý £ ½ °. , x èßð 1
1
1
1
1
1
1
1
1
1
1
2
2
1
2
1
1
1
1
1
2
2
2
2
1
2 Backstepping
xd1
+
+
ªú Ì¢ üxÍ ü±Cq
k1
Outer loop Controller
xd2
+
+
k2
13
Inner loop Controller
Flight Dynamics
u
x1 ; x2
x2 x1
2.1
Two-timescale
Aú Ì¢ üxÍ ü± Cq uS
¢ Cq³÷¿ x , x èßð ¢ Cq³÷¿ u ͽ¢ "ú ¨°. Z¢, v 2.1 ú Ã8, x èßð [´ý ¨ ¾ , x èßð [´ý ¨ ¾Ã° Í ¾° ,ú Ý £ ½ °. ,ù ]ý:÷¿ x èßð x èßð ü© ò : [ èßð timescale uÛþ ½ ° ,ú y¢°. Menon, Bugajski, Snell #ù ¢ two-timescale ú Ì ¨W Cqõ u uõ ½± °. ù ¨W èßðú ô ô"ú èßðY ÿ ô"ú èßð÷¿ Ûý L, èßð ¢ Cqõ 0¿ zG °. ª t ¨W ô:" 0 ô ô"ú èßð÷¿ x èßðú, ÿ ô" ú èßð÷¿ x èßðú zA¢°. Cq ü# Ø¿ uüÚ, ½ ÙØt \׺½ x õ è { ¿ ºÜèÈ ½ °L A L, x < º xd õ 8[ êÀ ÿ ô"ú èßð ¢ Cq³ xd õ zG¢°. 4ÙØ t x ½ÙØt fý <º xd õ 0 êÀ ô ô"ú èßð ¢ Cq³ u õ zG¢°. 2.1 ù two-timescale Aú Ì¢ Cq uSõ #Í6°. ªù x èßð ¢ Cq³÷¿ x õ PÌ :[ ÞxÍܪð ¨W ô:" Ì :¦¢ Cq zGª L £ ½ °. Æ:÷¿ Cqõ zG£ : u õ Cq³÷¿ zA£ ½ ,ù Cq u ú ¿ ºÜèÈ ½ :[ °. Two-timescale ú Ì¢ ¨W Cq zGYA 2
2
1
2
2
1
4, 5, 6
2
1
2
1
1
2
2
2
1
2
14
2.2
Cq zG KA ©u
t ÿ ô"ú èßð Cq³÷¿ \׺½ x õ zA£ ½ ,ù, x è ßð Í ò :[ ÿ ô"ú èßðt x õ ¿ ºÜèÈ ½ îYõ uú ½ °L A° :[ °. 0t Cqªú :Ì¢ èßð KAú ½¡:÷¿ Ã$ æ©t ô ô"ú èßð Cq ú Í À ÷¿ z A©b ¢°. # ¨W EÍ, ¢ ª ¢ KA ©u@Y uù À ¾ C q ú :Ì£ EÍ, Cq³ jÜ $# ½¡:÷¿ ?Ýü Mù mù s7½ èßðú [´ý [C f£ ½ °. ¢ [C>ú Ä æ Æ f[t two-timescale Aú Ì ML backstepping ªú Ì üxÍ ¨W Cqõ zG ªú CK¢°. Backstepping ªú Ì Cqõ zG YAtê, ½ÙØt x õ Cq æ x õ \ ³÷¿ PÌ 6, 4ÙØt x õ Cq æ© u õ Cq³÷¿ PÌ¢°. # two-timescale ú Ì¢ Cqªt 4ÙØ ô:" ½ÙØ ü©t Í ò :[ ½ÙØt ³ #Í# M°L A ,Y µý, Æ f[t 4ÙØ èßð ô:"Y è ÿ ¢ ³ú L² Cqõ zG L, x ; x © positive de nite ¢ òbd ¥½õ zA KAú <¢°. 0t ªù x õ Cq Cq³÷¿ x õ PÌ¢° Q8t ¨W ô:" :¦¢ Cq zGª 6, x ô"ú L² :[ ; èßð KAú Ã$ æ üÇì: Aú Ì M° $>ú L °. 2
2
2
1
2
2
1
2
1
2
2
2.2.1
Æ A
Cq zG KA ©u PÌý Æ A ú Aý 8 °üY °. A u: xd = [ d; d; d]T î¢ ° qE \½ cd > 0  °ü éú T¢° 2.1
.
1
,
.
d
x ; 1
x_ d1 ; xd1 cd
(2.20)
2 Backstepping
ªú Ì¢ üxÍ ü±Cq
15
t kk ù vector Óù matrix 2-norm ú y¢°. A ü± ³ ôSù ÆA ° .
2.2
V_
= 0; q_ = 0
(2.21)
A °ü 0 U&¢° j j < m; j j < m ? ; © f ; f g ; f ; f a ; f ; g ; g a ; g ; h ù î¢ 6 ; © yÛ ° A X ¾ °üY C¢ý° .
2.3
1
1
2
2
3
1
1
2
,
1
.
.
2.4
jj m < 2 2.2.2
(2.22)
Cq zG
°üù Cq zGYA Å¢ ÃSAý °. ÃSAý °ü 0 U&¢° j j m; j j m ; jj m õ T ? ; ; ®  g ( ; ; ; ) ù ° æ ÃSAý 2.1 ù g t¿ xÍëÆ ?ÛS&ú u <£ ½ °. Cqõ zG æ °üY ù ¡ò \׺½ z ; z 2 R õ ꢰ. t xd ½Ùt sq x ¢ <º 6, xd 8ó zGþ x 8[©b < º °. .
2.1
invertible
1
.
1
1
1
1
3
2
2
2
z1 = x1 xd1
(2.23)
z2 = x2 xd2
(2.24)
é (2.17), (2.18) 0 ¡ò ô¡ù °üY uý°. z_1 = x_ 1 x_ d1
= f ( ; ) + g ( ; ; ; )x + g a ( ; )x + h ( ; )u + f g ( ; ; ; ) x_ d 1
1
2
1
2
1
1
1
(2.25)
16
2.2
Cq zG KA ©u
z_2 = x_ 2 x_ d2
= f ( ; ; p; q; r) + f a ( ; )x + g ( ; )u x_ d 2
2
2
2
(2.26)
2
Xt tÁ¢ AY ÃSAýõ #¿ Cqõ zG¢ @Y °ü Aý¿ vÇý°. Aý é èßð ©
(2.25), (2.26)
2.1
÷¿ vÇü èßðt Cq³ u °üY Aü8 °
uniformly ultimately bounded
u = B2 1
t xd; A \½ ° 2
2
,
k2 z2
.
g1a ( ; )T z1
g1 ( ; ; ; )T z1 A2
(2.27)
2 R31 ; B2 2 R33 °üY Aü6, k1 ; k2 Cq ÷¿t j
.
xd2 = g1 ( ; ; ; )
1
k1 z1 f1 ( ; ) f1g ( ; ; ; ) + x_ d1
(2.28)
A2 = f2 ( ; ; p; q; r) + f2a ( ; )x2 @xd2 f1 ( ; ) + g1 ( ; ; ; )x2 + g1a ( ; )x2 + f1g ( ; ; ; ) @x1 @xd2 f3 ( ; )x2 g1 ( ; ; ; ) 1 k1 x_ d1 + xd1 @x3 @xd2 B2 = g2 ( ; ) h ( ; ) @x1 1
(2.29) (2.30)
Z¢ ¡ò ½¶¢G k ; k õ S<¥÷¿ ¾ 4¿ C¢£ ½ ° ,
1
.
2
< °üY ù òbd ¥½õ L² . .
V = 21 z1T z1 + 12 z2T z2
(2.31)
$9, A 2.1 Y 2.4 õ Ì 8 ÃSAý 2.1 © V < d Æ : g invertible ¢ j \ ½ d U&¢°. 0t V < d Æ :, °ü Ù#éú T \½ cg U&¢°. 1
1
g1
( ; ; ; )
1
cg
1
1
1
(2.32)
2 Backstepping
ªú Ì¢ üxÍ ü±Cq
17
Z¢, A 2.3 © °ü Ù#éú T j \½ c U&¢°. kg1a ( ; )k cg a
(2.33)
kh1 ( ; )uk ch
1
(2.34)
kf1 ( ; )k cf
1
(2.35)
1
f1
g
( ; ; ; ) cf g
(2.36)
d
x
(2.37)
1
_ cxd _1
1
æ ét kh ( ; )uk ¾õ u YAt u jÜ (saturation) ü :[ î¢ ° Pìú Ì °. é (2.32), (2.35), (2.36), (2.37) ú é (2.28) Ì 8,
xd
° ü Ù#éú T¢° ,ú ÃÆ ½ °. 1
2
d
x 2
cg
h
1
1
k1 kz1 k + cf1 + cf1g
+ cxd
i
(2.38)
_1
æ Ù#é (2.38) ù °ü KA 0 æ©t c < 1 ú T©b ¢°. c ù g a ; g norm U÷ ¿ ØqK Ú, é (2.17) t g a ê © f W³G½ ¨÷¿ Í ù m U© :[ ¾ Í °. Æ f[t PÌý W³G½ ?Ýú Ì ½X:÷¿ GS¢ @Y c < 0:13 }°. é (2.44) t V > kc c c c Æ : V_ < 0 t¿, z ; z î¢ 6 °ü ¦ D ¿ ½ :÷¿ ½¶¢°. 1
1
1
1
1
1
1
( 1 2 + 3 )2 4 1 (1 1)
(
D = z1 ; z1
1
2
c c +c ) 2 R kz k + kz k 2(k (1 c ) 3
1
2
2
2
1 2 1
3
2
1
)
(2.49)
Z¢, c ; c ; c k ; k ë t¿, k ; k õ S< ½¶©æõ ¿ 3 ½ °. æ Aý W³G½õ AÝ& NL EÍ üxÍ ¨W ; ; <ºú 8[£ ½ Cqõ zG£ ½ üú ^¢°. ýL Cqõ :Ì 8 <º8[ ¡ò D ¿ ½¶ 6, ¡ò ¾õ zG S<£ ½ ° Pìú Ãu°. U u® µý æ Aý xù <º:ú y¢°. t Ã" CKý Cqèßð :Ìý ¨W ü  <ºú Í 0 L ÷6, y¤ù Is ù ¡ò©æ 4t <º ú 8[ L üú N ½ °. 1
2
22
2.3
½X èqª
500 480
V (ft/s)
460 440 420 400 380
0
2
4
6
8
10 time (sec)
12
14
16
18
20
0
2
4
6
8
10 time (sec)
12
14
16
18
20
0
2
4
6
8
10 time (sec)
12
14
16
18
20
12 10 8 α (deg)
6 4 2 0 −2 −4
0.08 0.06 0.04 β (deg)
0.02 0 −0.02 −0.04 −0.06 −0.08
2.2
Simulation result: Backstepping controller
2 Backstepping
ªú Ì¢ üxÍ ü±Cq
23
1.5 1
p (rad/sec)
0.5 0 −0.5 −1 −1.5
0
2
4
6
8
10 time (sec)
12
14
16
18
20
0
2
4
6
8
10 time (sec)
12
14
16
18
20
0
2
4
6
8
10 time (sec)
12
14
16
18
20
0.3 0.2
q (rad/sec)
0.1 0 −0.1 −0.2 −0.3 −0.4
0.15
r (rad/sec)
0.1 0.05 0 −0.05 −0.1
2.2
Simulation result: Backstepping controller (continued)
24
2.3
½X èqª
60 50
φ (deg)
40 30 20 10 0 −10
0
2
4
6
8
10 time (sec)
12
14
16
18
20
0
2
4
6
8
10 time (sec)
12
14
16
18
20
0
2
4
6
8
10 time (sec)
12
14
16
18
20
30 20
θ (deg)
10 0 −10 −20 −30 −40
70 60 50 ψ (deg)
40 30 20 10 0 −10
2.2
Simulation result: Backstepping controller (continued)
2 Backstepping
ªú Ì¢ üxÍ ü±Cq
25
15
δe (deg)
10 5 0 −5 −10
0
2
4
6
8
10 time (sec)
12
14
16
18
20
0
2
4
6
8
10 time (sec)
12
14
16
18
20
0
2
4
6
8
10 time (sec)
12
14
16
18
20
15
δa (deg)
10 5 0 −5 −10
20 15
δr (deg)
10 5 0 −5 −10 −15
2.2
Simulation result: Backstepping controller (continued)
26
2.3
½X èqª
7000 6000 5000 px (ft)
4000 3000 2000 1000 0
0
2
4
6
8
10 time (sec)
12
14
16
18
20
0
2
4
6
8
10 time (sec)
12
14
16
18
20
2
4
6
8
10 time (sec)
12
14
16
18
20
5000 4000
py (ft)
3000 2000 1000 0 −1000
4
1.1
x 10
1.08
h (ft)
1.06 1.04 1.02 1
0
2.2
Simulation result: Backstepping controller (continued)
êEç¿ú Ì¢ üxÍ :ÿ ü±Cq
3
$t ¨W èßð Ì ½ # W³G½ ÝÝìú L² ML Cqõ zG °. ¨W ô:"ù üxÍ 6, ¨W Ïô ]ý: Ç\ Ä! 3 Ì :[ ¨W AÝ¢ ½¡: ?Ýú u ,ù Í q²Ï Æ °. Z¢, two-timescale ú Ì¢ üxÍ ¨W Cq EÍ W³G ½ ÝÝì © Cqèßð ÝKA©° , uü}°. 0t ¨W C qõ zG£ : ½ ÝÝì ³ú L² , |Å °. $t : ÿCqªú êEç¿ ÿÌ ¢ ³ú èÅ Cqõ zG¢°. 2
8
W³G½ ?Ý ¡ò ¢ ³
3.1
$t ê ÏôAé ¢ ?Ý ¡òõ L²¢°. é (2.18) t A ý f ( ; ; p; q; r); f a ( ; ); g ( ; ) ¢ ÝÝìú L² , 8Aý ¥½ ú f^ ( ; ; p; q; r); f^ a ( ; ); g^ ( ; ) L 8, 2 $t zG¢ Cq³ù °üY vÇ ý°. 2
2
2
2
2
u^ = B^2 1
2
h
k2 z2 g1a ( ; )T z1 g1 ( ; ; ; )T z1
A^2
i
A^2 = f^2 ( ; ; p; q; r) + f^2a ( ; )x2 @xd2 f1 ( ; ) + g1 ( ; ; ; )x2 + g1a ( ; )x2 + f1g ( ; ; ; ) @x1 @xd2 f3 ( ; )x2 g1 ( ; ; ; ) 1 k1 x_ d1 + xd1 @x3 @xd2 B^2 = g^2 ( ; ) h ( ; ) @x1 1 27
(3.1)
(3.2) (3.3)
28
3.2
êEç¿ uS
æ éú é (2.42)  8, èßð ?Ý ¡ò L²ü}ú : V_ °üY vÇ ý°. V_ = k1 kz1 k2 + z1T g1a ( ; )xd2 + z1T h1 ( ; )^u
+ zT g a ( ; )T z + g ( ; ; ; )T z + A + B u^ + B u 2
=
1
1
1
1
2
2
2
B2 u
k1 kz1 k2 k2 kz2 k2 + z1T g1a ( ; )xd2 + z1T h1 ( ; )^u + z2T B2 [^u u]
, k kz k 1
1
k2 kz2 k2 + z1T g1a ( ; )xd2 + z1T h1 ( ; )^u z2T
2
æ ét = B [u u^] ¿ Aü}÷6, ,ù èßð ?Ý ¡ò © f ¨ °. , f ( ; ; p; q; r); f a ( ; ); g ( ; ) ¢ ?Ý ¡ò ©t, KA 0 ©t °ü [Gõ T ù§W ½ N ® \: @ ê W 2 RN N ; V 2 RN N U&¢° 2
3.1
(Universal Approximation Theorem)
1
,
2 +1
3
3
2
1 +1
2
= W T ~(V T xnn) + (xnn);
.
k(xnn )k N
8 xnn in some input space
(3.7)
< ÷L[¶ 11, 12 ÷S. êEç¿ ¢ "ù universal approximation ÷¿ N²K °. ,ù êEç¿ º½ U Ä!¢ üxÍ ¥½õ :ù º½õ Ì AÝ& vÇ£ ½ .
30
3.3
Cq zG KA ©u
° ,ú y 6, êEç¿ Ûb ÿÌüL $ |Å¢ î °.
3.3
Cq zG KA ©u
:ÿCqªù èßð ;³ [Gõ Ì "A¢ º½õ ìè÷¿ 8A ¥÷¿, èßð \×# ½Ù ÞE ºÜ Ìê Æ¢ ú uú ½ êÀ Cq èßð uSõ ºÜèÅ Cqª °. ªù 8A º½ 0 ¾3 ? :ÿCqª (direct adaptive control) Y ? :ÿCqª (indirect adaptive control) ¿ uÛ £ ½ °. ? :ÿCqªù èß𠺽õ ìè÷¿ 8A¢ °ü, 8Aý ìC º½L A L Cq º½õ @A ª 6, ? :ÿCq ªù ? Cq º½õ ìè÷¿ 8A ª °. ªú Ì Cq õ zG£ : Cqèßðú èßð# Cq º½¿ º½Ü YA j¥ý°. 0t ªù ³¢ xÍèßðú |î÷¿ ;ü}°. Æ f[t üxÍ :ÿCqª (nonparametetric nonlinear adaptive control) ú Ì Cqõ zG¢°. ªù èßð ;³ [Gõ Ì "A¢ ¥½ P (function approximator) ¿ üxÍ ô:?Ýú vÇ ª °. Cqõ zG£ : Cq èßðú º½Ü M :[, ªù Ä!¢ üxÍ èßð Cq Ì :¦ °. X 3.1 <t W³G½ ÝÝì © ¨ 8 ý° ,ú ÃL, 3.2 < t êEç¿ ¥½õ AÝ& vÇ£ ½ ° ,ú ð. Æ f[t ¥ ½ P¿ êEç¿ú zA , êEç¿ ;³ ÝÝì © f ¨ ú \© êÀ êEç¿ º½õ ºÜèÇ°. ªù èßð ]ý: º ½ I¨ ¥½ P º½õ 8A¢° Q8t ? :ÿCqª¿ Ûêþ ½ ê , Æ: ? :ÿCqªY µý Cqèßðú º½Ü vÇ M° " °.
3
êEç¿ú Ì¢ üxÍ :ÿ ü±Cq
31
Æ A
3.3.1
:ÿCqõ zG YAt 2 $t PÌ¢ A 2.1 2.5 ® ¥Í °üY ù Aú Ì¢°. A êEç¿ ³÷¿ xnn = [xd; x ; x ]T ú PÌ 6 ³ù qE N  é ú TèÇ° A é ú T \: @ ê î¢ 6 \¢©æ WM ; VM ú NL ° 3.1
1
(3.7)
3.2
1
,
2
.
(3.7)
,
.
kW kF WM ; kV kF VM
(3.8)
t kkF ±µ Frobenius normy ú y¢°. 3.3.2
Cq zG
Aý 3.1 © êEç¿ W³G½ ÝÝì © f ¨ú AÝ& vÇ êÀ \: @ ê U&¢°. # Cqõ zG YAt ÝÝì © f ¨ [¢ AÃõ uú ½ { :[, Aý 3.1 ú T \:
@ ê W; V õ GS£ ½ {°. 0t, Cqèßð é (3.7) ú T \: ^ ; V^ ú Ì 6, KAú Ã$£ ½ êÀ zGý :ÿªY © W; V ¢ 8A W ^ ; V^ ú ºÜèÇ°. W Cq 8A W^ ; V^ PÌüt¿ \: @ ê® 8A ò ¢ ³ f¢°. °ü ÃSAý KA ©uYAt \: @ ê® 8A ò © f ³ú #Í6°. ÃSAý Aý ú T \: @ ê W; V ¢ 8A¡òõ W~ = ^ ; V~ = V V^ L A L Z = [W; V ] ¿ A W W p 3.1
3.1
diag
y
Frobenius norm:
k kF = A
[
T A]
tr A
.
32
3.3
Cq zG KA ©u
8Aý @ ê W^ ; V^ ¢ êEç¿ ;³¡ò °üY vèý° ^ T ~(V^ T xnn) W
h
= W~ T ~(V^ T xnn) 0 (V^ T xnn)V^ T xnn ^ T 0 (V^ T xnn)V~ T xnn + w W
.
i
(3.9)
t 0 (^z) = d~dz z z L w 2 R °üY Aý° 3
,
.
=^
~ T (V^ T xnn)V T xnn W T O(V~ T xnn) (xnn) w(t) = W 0
(3.10)
Z¢ kwk ù qE j \½ Ci (i = 1; 2; 3; 4) © °ü Ù#éú T¢° ,
.
kwk C1 + C2
Z~
F + C3
Z~
F kx1 k + C4
Z~
F kx2 k
(3.11)
< sq ³ xnn  ù§W (hidden layer) ;³¡ò °üY vèý°. .
~~ = ~ ~^ = ~(V T xnn ) ~(V^ T xnn )
(3.12)
é (3.12) ͺ ¨ú V^ T xnn ú u÷¿ ìÆ ½ (taylor series) õ ; 8 °üY °.
d~ T nn ) = ~ (V^ xnn ) +
~(
V Tx
dz z=V^ T xnn
V~ T xnn + O(V~ T xnn )
(3.13)
æ éú é (3.12)  8 ù§W ;³¡ò °üY vÇý°. ~~ = (V^ T xnn )V~ T xnn + O(V~ T xnn ) 0
(3.14)
: ;³W (output layer) ;³¡ò °üY ý°. ^ T ~(V^ T xnn) W
= W^ T ~(V^ T xnn) W T ~(V T xnn) (xnn) h i = W~ T ~(V^ T xnn) W T ~(V T xnn) ~(V^ T xnn) (xnn)
é (3.15) é (3.14), W = W~ + W^ ; V~ = V V^ ú Â 8 °üY °. ^ T ~(V^ T xnn ) W
= W~ T ~(V^ T xnn ) W~ T 0 (V^ T xnn)V~ T xnn
(3.15)
3
êEç¿ú Ì¢ üxÍ :ÿ ü±Cq
33
^ T (V^ T xnn)V~ T xnn W
W T O(V~ T xnn ) (xnn )
0
h
= W~ T ~(V^ T xnn ) 0 (V^ T xnn )V^ T xnn ^ T 0 (V^ T xnn)V~ T xnn + w W
i
(3.16)
0t é (3.9) <ü}°. Z¢, sigmoid ¥½® yÛ (); ddzz sC# î¢ ° PìY A 3.1 ú Ì 8, é (3.13) t Lò¨ O °ü Ù#éú T¢° ,ú ÃÆ ½ °. ( )
O
(V~ T xnn)
~(V T xnn) +
~(V^ T xnn )
+
0 (V^ T xnn)V~ T xnn
c + c + c
V~
kxnn k
c
V
c+ ~
F
+ c kx k
V~
F + c kx k
V~
F F 1
(3.17)
2
t c j \½õ a 6, kAxk kAkF kxk "ú Ì °. C, é (3.10) æ éú Â L A 3.1 ú Ì 8 °ü [Géú uú ½ °.
kw(t)k
W~
F cVM c + c kx1 k + c kx2 k
+ WM c + c
V~
F + c kx k
V~
F + c kx k
V~
F + N
C + C
Z~
+ C
Z~
kx k + C
Z~
kx k 1
1
2
3
F
F
2
1
4
F
(3.18)
2
æ ÃSAý @ ê 8A¡ò © f êEç¿ ;³¡ò é (3.9) ¿ vÇü6, ÆÙ w ¾ é (3.11) ¿ C¢ý° ,ú Ãu°. °ü ÃSAý Cq PÌü & [´ý ¨ v õ A L, KA < Å¢ é (3.22) õ <¢°. ÃSAý é Y °üY Aý w 2 R ; v 2 R ; 2 R õ L² 3.2
3
(3.10)
v=
z2
kz2 k +
3
.
(3.19)
34
3.3
= kv ZM +
Z^
F
Cq zG KA ©u
(kx k + kx k) 1
(3.20)
2
kv max fC3 ; C4 g
(3.21)
t ù j \½ L kv æ ú T \½ ° : °ü Ù#é ¢° ,
,
.
i
h
z2T (w + v) kz2 k C1 + C2
Z~
F
+
(3.22)
< $9 é (3.19) õ Ì zT (w + v) õ vÇ 8 °üY °. .
2
(kz k ) kz k +
z2T (w + v) = z2T w
2
2
(3.23)
2
æ é é (3.11) ú Â 8 °ü éú uú ½ °. h
z2T (w + v) kz2 k C1 + C2
Z~
i
+ C
Z~
F kx k + C
Z~
F kx k F 3
1
4
2
(kz k ) kz k + 2
2
2
(3.24)
¢`, Z~ = Z Z^ A® é (3.20), (3.21) ú Ì 8 °ü [Géú uú ½ °.
C3
Z~
kx1 k + C4
Z~
F kx2 k kv
Z Z^
F (kx1 k + kx2 k) F
(3.25)
é (3.25) õ é (3.24) Â Aý 8 °üY °. h
h
C1 +
i C2
Z
kz2 k C1 +
i C2
Z
z2T (w + v) kz2 k C1 + C2
Z~
= kz k 2
h
~
~
+ F
i
(kz k ) kz k + 2
2
2
F
+ kzkzk2k + 2
F
+
(3.26)
0t é (3.22) <ü}°. Xt tÁ¢ AY ÃSAýõ #¿ :ÿCqõ zG YA KA ©u ¢ 4Ìú °ü Aý¿ vÇ¢°.
3
êEç¿ú Ì¢ üxÍ :ÿ ü±Cq
Aý
3.2
é
35
÷¿ vÇü èßðú L² : Cq³ u ® :ÿªY ú°üY A 8 èßð©®êEç¿@ ê °
(2.25), (2.26)
(adaptive law)
,
ultimately bounded
u = B^2 1
h
.
uniformly
.
g1a ( ; )T z1 g1 ( ; ; ; )T z1 A^2
k2 z2
æ ét A^ ; B^ ; v é ê W^ ; V^ ù °ü :ÿªY © GS¢° 2
(2.29), (2.30), (3.19)
2
i
+ W^ T ~(V^ T xnn) + v
(3.27)
¿ Aü}° Z¢ êEç¿ @ .
,
.
^_ W V^_
h
=
w ~(V^ T xnn )z2T
=
^ z2
v xnn (V^ T xnn )T W
0
(V^ T xnn )V^ T xnn z2T 0
T
i
^ w W
(3.28)
v V^
(3.29)
t ; w ; v j \½¿ ? zGº½ ° Z¢ ¡ò ½¶¢G k ; k ; ú S<¥÷¿ ¾ 4¿ C¢£ ½ ° < °üY ù òbd ¥½õ A . .
,
1
2
.
.
V = 12 z1T z1 + 12 z2T z2 + 2 1
w
h i ~ ~ + 1 tr V~ T V~ 2 v
h i tr W T W
(3.30)
¥½õ ÏôAé (2.25), (2.26) Y :ÿªY [Gé (3.28), (3.29) ©t yÛ¢ ù é (2.42) õ Ì °üY u£ ½ °. V_ = k1 kz1 k2 + z1T g1a ( ; )xd2 + z1T h1 ( ; )u
+ zT g a ( ; )T z + g ( ; ; ; )T z + A + B u + B u h i h i + 1 tr W~ T W~_ + 1 tr V~ T V~_ 2
w
1
1
1
1
2
2
2
B2 u
v
(3.31)
æ ét u èßð ?Ý ¡ò { EÍ \: Cq³ú #Í46 °üY Aý°. u = B2 1 k2 z2
g1a ( ; )T z1 g1 ( ; ; ; )T z1 A2
+ W^ T ~(V^ T xnn) + v
(3.32)
36
3.3
Cq zG KA ©u
é (3.32) õ é (3.31) Â 8 °üY °. V_ = k1 kz1 k2 k2 kz2 k2 + z1T g1a ( ; )xd2 + z1T h1 ( ; )u z2T + z2T W^ T ~(V^ T xnn ) h i h i + zT v + 1 tr W~ T W~_ + 1 tr V~ T V~_ 2
w
(3.33)
v
æt = B [u u] ¿ Aü}°. æ é é (3.9), (3.28), (3.29) õ Â 8 °üY °. 2
V_ = k1 kz1 k2 k2 kz2 k2 + z1T g1a ( ; )xd2 + z1T h1 ( ; )u
+ zT
h
h
~ T ~(V^ T xnn) (V^ T xnn)V^ T xnn W
2
h
0
i
^ T (V^ T xnn )V~ T xnn + w + v W 0
h
+ tr W~ T ~(V^ T xnn)zT 0 (V^ T xnn)V^ T xnn zT + W^ T + tr V~ T xnn 0 (V^ T xnn)T W^ z + V^ 2
i
ii
2
(3.34)
2
Trace
" tryxT = xT y ú :Ì 8, æ éù °üY ©°. h
i
V_ = k1 kz1 k2 k2 kz2 k2 + z1T g1a ( ; )xd2 + z1T h1 ( ; )u + tr Z~T Z^ + z2T (w + v) h
i
h
Z¢, tr Z~T Z^ = tr Z~T Z 8 °üY ý°.
i
h i tr Z T Z
~ ~
Z~
ZM F
2
Z
~
F
Y é (2.43), (3.22) õ Ì
2 2 + c3 ) +
Z~
ZM V_ k21 (1 c1 ) kz1 k2 k2 kz2 k2 + (2ck1 c(1 c) F
+ kz k 2
h
C1 +
k1
i C2
Z
~
F
1
+
k2
k2
1
1
2
1
1 2
2
æ ét C = 5
2
cc c k c
2
3
1
( 1 2 + 3 )2 2 1 (1 1)
2
2
2 1
1
C1 2 k2 2 ZM
= 2 (1 c ) kz k 2 kz k 2 kz k
+ C kz k
Z~
F + (2ck c(1+ cc)) + 2Ck + + 2 2
~
2
Z
2
F
2
+ Ck + ZM + ¿ A 8 °üY °. 2 1
2 2
(3.35)
2
2
2 V_ k21 (1 c1 ) kz1 k2 k22 kz2 k2 + C2 kz2 k
Z~
F 2
Z~
F + C5
2
Z
~
F
h
~
Z
2
F
ZM
i2
(3.36)
3
êEç¿ú Ì¢ üxÍ :ÿ ü±Cq = k2 (1
c1 ) kz1 k2
1
k2
4 kz k 2
2
4
37
~
3T 2
2
1 6 kz2 k 7
2
Z
2 4
Z~
F
6 5 4
F
32
3
k2
C2 7 6 kz2 k 7
C2
2
2
5 4 5
Z
~
+C
5
F
(3.37)
0t k 4C > 0 ú T êÀ k ; õ zA 8, æ é P¥« ¨ j¥ý ±µ positive de nite t¿ °ü éú uú ½ °. 2 2
2
2
2 V_ k21 (1 c1 ) kz1 k2 k42 kz2 k2 4
Z~
F + C5
: 0 < < min k (1 ¿ °ü éú u3 ý°. 1
2
c1 ) ; k42 ; 4 min f w ; v g
(3.38)
õ T êÀ õ xØ 8 /[:÷
V_ 2V + C5
(3.39)
æ ét V > C Æ : V_ < 0 t¿, z ; z ; Z~F î¢ 6 °ü ¦ D ¿ ½:÷¿ ½ ¶¢°. 2
D = z1 ; z1
5
1
2 R ; Z~F 2 RN 3
2
N2 +2N2 +N3 kz
1+
1
1
2
Z
k + kz k + max f ; g ~ F C w v 2
2
2
5
Z¢, c ; c ; c ; C ; ZM ; k ; k ë t¿, k ; k õ S< ½¶©æõ ¿ 3 ½ °. æ Aý W³G½ ?Ý ÝÝì EÍ êEç¿ú Ì ; ; <º ú 8[£ ½ Cqõ zG YAú #Í6°. Cqõ :Ì 8 <º8[ ¡ ò êEç¿ @ ê ¡ò D ¿ ½¶ 6, ¡ò ¾õ zG S<£ ½ ° Pìú Ãu°. Æ:÷¿ :ÿCqªY :Ìý èßðù ½, unmodeled dynamics èß𠺽 ô ºÜ © ÝKA© ½ ° , uü}°. ¢ Ç\ù Ù 0t parameter drift, high-gain instability, fast adaptation, high-frequency instability ¿ uÛý 1
2
3
1
1
2
1
2
38
3.3
Cq zG KA ©u
°. Æ f[t ¢ ³>ú Ä æ :ÿªY (robust adaptive law) ¢ [ê -modi cation ªú Ì °. é (3.29), (3.28) t w W^ ; v V^ ¨ù -modi cation ª © 8 ý ¨÷¿t èßð ?Ý ¡ò © :ÿªY º½ W^ ; V^ S ,ú £ú 6, ³ú S< zGº½ °. -modi cation ª ½ switching-, -modi cation ª #ú :Ì£ ½ ÷6 :Ìý ª 0t ½¶Y KA < YA SB µ°. 20
20, 21
©u
3.3.3
òbd¥½õ Ì X<tzGýCq©¡ò\׺½ z = [zT ; zT ]T ® êEç¿ @ ê 8A¡ò Z~ ©æõ îê . Aý é ¿ vÇü èßðt \׺½® êEç¿
iT h ¡òº½ za = kzk ;
Z~
F °ü Ù#éú T¢° 1
2
(2.25), (2.26), (3.27), (3.28), (3.29)
3.3
.
kza (t)k
r
1 21 (t t0 ) kza (t0 )k + e 2
p
2 C h1 2 3 1
i 2 e 1 (t t0 )
5
2
(3.40)
t = max f1; w ; v g; = min f1; w ; v g ° Z¢ ¡òº½ za L1 üY ° 1
.
2
,
norm
ù°
.
(
p
C kza (t)k1 = max kza (t )k ; 22 3 1
0
<
.
1 2 1
)
5
(3.41)
2
kza k2 V 21 kza k2 t¿, é (3.39) © °ü Ù#é ¢°. 2
V_ 2V + C5 kza k2 + C5 1
æ é ÙÀ Aýüq Aý B.1 ú :Ì¢°. é (B.6) c = ; = C ; = 0 ú Â 8 °ü Ù#éú u°. 1
1
1 2 1
; c2
5
kza (t)k
r
1 21 (t t0 ) e kza (t0 )k + 2
p
2 C h1 2 3 1
2
5
i 2 e 1 (t t0 )
=
1 2 2
; c3
=
3
êEç¿ú Ì¢ üxÍ :ÿ ü±Cq
39
p2
æ ét N ½ " kza(t)k # kza(t )k t ½¶ C ÷¿ ³S , Z ³S¢°. 0t kza(t)k / ù # Y ½¶ / Y ÆX¢°. 0
2
3 1
5
2
é (3.40) ú Ã8, ¡ò \׺½ ½¶êõ m æ©t õ ¾3 zA©b ¢° ,ú N ½ °. 3.4
½X èqª
zGý Cq ú æ©t ½X èqª ú ½± °. # \׺ ½ <º:ù 2.3< 4ÌY ÆX¢°. zGº½ k = 3; k = 8; = 0:2; kv = 0:153; = 0:001; ZM = 0:6142; w = v = 30; N = 30 ÷¿ zA °. W³G½ ?Ý ¡ò ¿ f L ¾ v 3.1 Aýüq °. 3.2 v 3.1 Aýý W³G½ ?Ý ¡ò U& EÍ èqª @Y õ #Í6°. t >xù <º:ú y 6, ¡ >xù 2 $t îê¢ backstepping ªú Ì¢ Cq èqª @Y °. ìxù $t îê¢ :ÿCqõ PÌ ¢ EÍ èqª @Y °. t Ã" W³G½ ÝÝì ¢ ³ú L² ML Cqõ zG ú :, W³G½ ?Ý ¡ò¿ © Cqèßð 9 ý° ,ú N ½ °. 8 êEç¿Y :ÿCqõ @¦¢ üxÍ :ÿCq W³G½ ?Ý ¡ò © 9 { u:ú 8[ L üú N ½ °. 1
2
2
40
3.4
v
3.1
W³G½ e ?Ý ¡ò (%)
W³G½ ¡ò W³G½ ¡ò W³G½ ¡ò Cl
79.6
Cm
207.0
Cn
180.1
Clp
16.0
Cmq
77.6
Cnp
86.7
Clr
148.9
CmÆe
146.2
Cnr
94.9
ClÆa
141.8
CnÆa
48.3
ClÆr
69.8
CnÆr
228.5
½X èqª
êEç¿ú Ì¢ üxÍ :ÿ ü±Cq
41
520 500 480 V (ft/s)
460 440 420 400 380 360
0
2
4
6
8
10 time (sec)
12
14
16
18
20
0
2
4
6
8
10 time (sec)
12
14
16
18
20
0
2
4
6
8
10 time (sec)
12
14
16
18
20
12 10 8 α (deg)
6 4 2 0 −2 −4
1
0.5 β (deg)
3
0
−0.5
−1
3.2
Simulation result: Adaptive backstepping controller
42
3.4
½X èqª
1.5
p (rad/sec)
1 0.5 0 −0.5 −1
0
2
4
6
8
10 time (sec)
12
14
16
18
20
0
2
4
6
8
10 time (sec)
12
14
16
18
20
0
2
4
6
8
10 time (sec)
12
14
16
18
20
0.3 0.2
q (rad/sec)
0.1 0 −0.1 −0.2 −0.3 −0.4
0.15 0.1
r (rad/sec)
0.05 0 −0.05 −0.1 −0.15 −0.2
3.2
Simulation result: Adaptive backstepping controller (continued)
êEç¿ú Ì¢ üxÍ :ÿ ü±Cq
43
60 50 40 φ (deg)
30 20 10 0 −10
0
2
4
6
8
10 time (sec)
12
14
16
18
20
0
2
4
6
8
10 time (sec)
12
14
16
18
20
0
2
4
6
8
10 time (sec)
12
14
16
18
20
30 20
θ (deg)
10 0 −10 −20 −30 −40
70 60 50 ψ (deg)
3
40 30 20 10 0
3.2
Simulation result: Adaptive backstepping controller (continued)
44
3.4
½X èqª
15
δe (deg)
10 5 0 −5 −10
0
2
4
6
8
10 time (sec)
12
14
16
18
20
0
2
4
6
8
10 time (sec)
12
14
16
18
20
0
2
4
6
8
10 time (sec)
12
14
16
18
20
25 20
δa (deg)
15 10 5 0 −5 −10 −15
15 10
δr (deg)
5 0 −5
−10 −15 −20
3.2
Simulation result: Adaptive backstepping controller (continued)
êEç¿ú Ì¢ üxÍ :ÿ ü±Cq
45
7000 6000
4000
x
p (ft)
5000
3000 2000 1000 0
0
2
4
6
8
10 time (sec)
12
14
16
18
20
0
2
4
6
8
10 time (sec)
12
14
16
18
20
2
4
6
8
10 time (sec)
12
14
16
18
20
5000
3000
y
p (ft)
4000
2000 1000 0
4
1.1
x 10
1.08 1.06 h (ft)
3
1.04 1.02 1
0
3.2
Simulation result: Adaptive backstepping controller (continued)
üxÍ ü±Cq
4
$t ½# W³G½ ÝÝì ¢ ³ú èÅ æ êEç¿ ú Ì¢ :ÿCqªú PÌ °. $t üxÍ Cqªú Ì ½ ÝÝì &¢ ü± Cqèßðú zG¢°. 3
4.1
Cq zG KA ©u
Cqªù ÝÝì \½, è, Z \׺½ ¥½¿ vÇü "A¢ 4 ©°L A L, 4t Æ¢ ú uú ½ LAý Í× C qõ zG ª °. 0t Cqªú Ì Cqõ zG£ : ÝÝ ì ½ ¢ ³ ¾õ yý @A©b ¢°. ÝÝì ¢ ³ Cqõ zG£ : 8A¢ ©æõ «q# EÍ Cqèßð KAú Ã$£ ½ {°. :ÿCq èßð ;³ [Gõ Ì Cq º½õ { ºÜèÅê À zGü ,Y µý, Cq Cqèßð uSõ ºÜèÅ M°. ¨W Cq[Ct ÏÌ\× ¦  H1 ® ù xÍ Cqª÷ ¿ zGý Cq ú ¨W \׺½ 0 à PÌ ß (gain scheduling) ª Gý PÌü}°. ® µý Æ f[t ÷L[¶ 10 @Yõ Ì üxÍ ü±Cqõ zG °. Æ f[t zG¢ Cq¨ù W³G½ Ý Ýì © f ¨ú \© êÀ ºÜ " °. 46
4
üxÍ ü±Cq
47
Æ A
4.1.1
Cqõ zG YAt 2 $t PÌ¢ A 2.1 2.5 ® ¥Í W³G½ ÝÝì ½ © f ¨ ¾õ C¢ °ü Aú Ì¢°. A W³G½ ÝÝì ½ © f ¨ i ¾ yÛ ¢ ¥½ Æi (x) : R 7! R Z \½ Æi ¿ C¢ü6 ¥½ Z \½ Æi õ NL ° ,
4.1
3
3
,
.
ki k Æi ; i = 1; 2 4.1.2
(4.1)
Cq zG
°ü ÃSAý ÝÝì © f ¨ú \© êÀ ºÜ Cq¨ ú #Í46 KA 0 t¿
k k k k + ' 1
¢°. 0t éú Ì 8 é (4.2) Tý° ,ú Ö3 ÃÆ ½ °. é (4.3) ú < æ© °üY EÍõ L² . c, kk ' 8 L, kk > ' 8
k k 3 '3 = k k 3 '3 k k ' k k3 + 3 '3 k k3 + 3 '3
k k 3 '3 = 3 '3 k k3 ' ' 2 ' k k k k3 + 3 '3 k k2 k k3 + 3 '3
t¿, é (4.3) <ü}°.
(4.2) (4.3)
48
4.1
Cq zG KA ©u
W³G½ ÝÝì ½ © f ¨ú i (i = 1; 2) ¿ A 8, é (2.25), (2.26) © ¡òô¡ù °üY vÇý°. z_1 = f1 ( ; ) + g1 ( ; ; ; )x2 + g1a ( ; )x2 + h1 ( ; )u
+ f g ( ; ; ; ) x_ d + 1
1
(4.4)
1
z_2 = f2 ( ; ; p; q; r) + f2a ( ; )x2 + g2 ( ; )u x_ d2 + 2
(4.5)
Xt tÁ¢ AY ÃSAýõ #¿ Cqõ zG¢ @Yõ Aý 8 °ü Aý® °. Aý é ¿ vÇü èßðt Cq³ u °üY Aü8 è ßð © ° (4.4), (4.5)
4.1
,
exponentially attractive
u = B2 1
t xd; A \½ ° 2
2
k2 z2
.
g1a ( ; )T z1 g1 ( ; ; ; )T z1 A2 + v2
(4.6)
2 R31 ; B2 2 R33 °üY Aü6, k1 ; k2 Cq ÷¿t j
.
xd2 = (g1 ( ; ; ; ) + g1a ( ; ))
1
k1 z1 f1 ( ; ) f1g ( ; ; ; ) + x_ d1 + v1
A2 = f2 ( ; ) + f2a ( ; )x2 @xd2 f1 ( ; ) + g1 ( ; ; ; )x2 + g1a ( ; )x2 + f1g ( ; ; ; ) @x1 @xd2 f ( ; )x2 @x3 3 @v1 d d @v1 1 (g1( ; ; ; ) + g1a ( ; )) k1 I3 + @xd x_ 1 + x1 + @' '_ 1 @xd2 B2 = g2 ( ; ) h ( ; ) @x1 1
(4.7)
(4.8) (4.9)
& [´ý ¨ v ; v °üY Aý° 1
.
2
v1 =
1 k1 k2 Æ k1 k3 + 3 '3 1
(4.10)
4
üxÍ ü±Cq
49
2
v2 =
k2 k + ' Æ2
(4.11)
t = z Æ ; = z Æ ; ' = exp t 6 ; zGº½¿t j \½ ° ýL Æ ; Æ A ú T ¥½ Z \½ ° 1
1
1 1
2
,
2 2
4.1
2
.
.
< °üY ù òbd ¥½õ L² . .
V = 12 z1T z1 + 12 z2T z2
(4.12)
é (4.4), (4.7) ú é (4.12) Â 8 °üY °. @V f1 ( ; ) + g1 ( ; ; ; )xd2 + g1a ( ; )xd2 + h1 ( ; )u + f1g ( ; ; ; ) x_ d1 + 1 V_ = @z 1 @V + (g ( ; ; ; ) + g ( ; )) x xd + @ V z_
@z1
@V = @z [ 1
=
1a
1
2
@z2
2
k1 z1 + h1 ( ; )u + 1 + v1 ] +
2
@V (g ( ; ; ; ) + g1a ( ; )) x2 @z1 1
xd2
@V + @z z_ 2
2
k1 kz1 k2 + z1T (h1 ( ; )u + 1 + v1 ) + z1T g1 ( ; ; ; )z2 + z1T g1a ( ; )z2 + z2T z_2 (4.13)
æ é é (4.5), (4.8), (4.9) õ Â 8 °üY °. V_ = k1 kz1 k2 + z1T (h1 ( ; )u + 1 + v1 ) + z1T g1 ( ; ; ; )z2 + z1T g1a ( ; )z2
+ zT f ( ; ; p; q; r) + f a ( ; )x + g ( ; )u 2
2
2
@xd2 x_ @x1 1
=
2
@xd2 x_ @x3 3
@xd2 d x_ @xd1 1
2
@xd2 d x @ x_ d1 1
@xd2 '_ @'
k1 kz1 k2 + z1T (h1 ( ; )u + 1 + v1 ) + z1T g1 ( ; ; ; )z2 + z1T g1a ( ; )z2
+ zT [A + B u] 2
=
2
2
k1 kz1 k2 + z1T (h1 ( ; )u + 1 + v1 )
+ zT g ( ; ; ; )T z + g a ( ; )T z + A + B u + 2
1
1
1
1
2
2
2
(4.14)
50
4.1
Cq zG KA ©u
æ é é (4.6) ú Â 8 °üY °. V_ = k1 kz1 k2 k2 kz2 k2 + z1T (h1 ( ; )u + 1 + v1 ) + z2T (2 + v2 )
=
k1 kz1 k2 k2 kz2 k2 + z1T
0
+ v + zT ( + v ) 1
1
2
2
(4.15)
2
t 0 , h ( ; )u + °. æ ét h ( ; )u õ y¨÷¿ s 0 ú A Ú, h ( ; )u ]ý:÷¿ ¨W Cq8 © f * Ûú y :[ ¾ Í It y¨÷¿ L²©ê Cqèßð À ³ú yX M °. æ é é (4.10), (4.11) ú  L, A 4.1 ú :Ì 8 °üY °. 1
1
1
1
1
1
z1T 1 k1 k2 Æ k1 k3 + 3 exp 3t 1
V_ = k1 kz1 k2 k2 kz2 k2 + z1T 01 + z2T 2 k1 kz1 k2 k2 kz2 k2 + k1 k
=
k1 kz1 k2 k2
k1 k4
z2T 2 k2 k + exp t Æ2 k2 k2 k2 k + exp t
+ k k t t kz k + k k exp t + kkkk+ exp exp t k k + exp 2
2
1
1 3
k1 k + 3 exp 3
3
3
3
3
2
t
2
3
2
(4.16)
æ é ÃSAý 4.1 ú :Ì 8 °üY °.
V_ k1 kz1 k2 k2 kz2 k2 + 2 exp
t
(4.17)
: 0 < < min fk ; k g õ T êÀ õ xØ 8 /[:÷¿ °ü éú u3 ý°. 1
2
V_ 2V + 2 exp
t
(4.18)
0t èßð © exponentially attractive °. æ Aý W³G½ ?Ý ÝÝì EÍ üxÍ Cqªú :Ì ; ; <ºú 8[£ ½ Cqõ zG YAú #Í6°. Z¢, ÝÝì ½ © f ¨ ¾õ yý 8A£ ½ ú :, Cqõ :Ì 8 <º8[ ¡ò 0 ÷¿ ½¶¢° ,ú Ãu°. ¢`, æ Aýt °üY ù A 8 üq P Ìü}°.
4
üxÍ ü±Cq
51
A °ü 0 U&¢° j j m; j j m; jj m õ T ? ; ; ® Â g ( ; ; ; )+ g a ( ; ) ° .
4.2
1
invertible
1
.
ÃSAý 2.1 t N ½ " g ( ; ; ; ) invertible °. Z¢, g a ( ; ) ê p; q; r © f W³G½ *Û÷¿t, v Aýý , ¤ g ( ; ; ; ) ü© ¾ Í :[ g ( ; ; ; ) + g a ( ; ) ¨t ³ Í °. 0t æ ® ù Aú Ì£ ½ °. 1
1
1
1
4.1.3
1
©u
òbd¥½õ Ì , X<tzGýCq©¡ò\׺½ z = [zT ; zT ]T ©æõ îê . 1
Aý é ü Ù#éú T¢° 4.2
(4.4), (4.5), (4.6), (4.7), (4.8), (4.9)
2
¿ vÇü èßðt ¡ò \׺½ z °
.
kz(t)k e
(t t0 ) kz
p (t )k + 22 1 e 0
t0 h 2
e
(t t0 ) 2
e (t t0 )
i
(4.19)
< é (4.18) © °ü Ù#é ¢°. .
V_ 2V + 2 exp
t
æ é ÙÀ Aýüq Aý B.1 ú :Ì¢°. é (B.6) c = c = 2; = ú Â 8 °ü Ù#éú u°. 1
kz(t)k e
(t t0 ) kz
p (t )k + 22 1 e 0
t0 h 2
e
(t t0 ) 2
2
e (t t0 )
1 2
; c3
= ; =
i
é (4.19) õ Ã8, ¡ò \׺½ ½¶êõ m æ©t ; õ ¾3 zA©b ¢° ,ú N ½ °.
52
4.2
4.2
½X èqª
½X èqª
zGý Cq ú æ©t ½X èqª ú ½± °. # \׺½ <º:ù 2.3< 4ÌY ÆX¢°. zGº½ k = 3; k = 8; = 0:05; = 0:1 ¿ z A °. W³G½ ?Ý ¡ò ¿ f L, ¾ v 4.1 Aýüq °. 4.1 ù v 4.1 Aýý W³G½ ?Ý ¡ò U& EÍ èqª @Y õ #Í6°. t >xù <º:ú y 6 ¡ >xù 2$t îê¢ backstepping ªú Ì¢ Cq èqª @Y °. ìxù $t îê¢ Cqõ PÌ ¢ EÍ èqª @Y °. t Ã" Cqª ¢ Cqèßðù W³ G½ ?Ý ¡ò U& EÍê 9 { u:ú 8[ L üú N ½ °. 1
2
4
üxÍ ü±Cq
53
v
4.1
W³G½ e ?Ý ¡ò (%)
W³G½ ¡ò W³G½ ¡ò W³G½ ¡ò Cx
63.6
Cy
109.4
Cz
118.2
Cxq
30.0
Cyp
70.0
Czq
46.2
CxÆe
63.6
Cyr
55.7
CzÆe
114.2
Cl
53.1
CyÆa
110.4
Cn
60.6
Clp
51.2
CyÆr
108.3
Cnp
136.3
Clr
45.9
Cm
74.2
Cnr
46.1
ClÆa
118.2
Cmq
54.9
CnÆa
129.5
ClÆr
46.2
CmÆe
72.2
CnÆr
58.7
54
4.2
½X èqª
550 500
V (ft/s)
450 400 350 300 250
0
2
4
6
8
10 time (sec)
12
14
16
18
20
0
2
4
6
8
10 time (sec)
12
14
16
18
20
0
2
4
6
8
10 time (sec)
12
14
16
18
20
20
α (deg)
15 10 5 0 −5
0.6
β (deg)
0.4 0.2 0 −0.2 −0.4
4.1
Simulation result: Robust backstepping controller
üxÍ ü±Cq
55
0.8 0.6
p (rad/sec)
0.4 0.2 0 −0.2 −0.4 −0.6 −0.8
0
2
4
6
8
10 time (sec)
12
14
16
18
20
0
2
4
6
8
10 time (sec)
12
14
16
18
20
0
2
4
6
8
10 time (sec)
12
14
16
18
20
0.3 0.2
q (rad/sec)
0.1 0 −0.1 −0.2 −0.3 −0.4
0.25 0.2 0.15 r (rad/sec)
4
0.1 0.05 0 −0.05 −0.1
4.1
Simulation result: Robust backstepping controller (continued)
56
4.2
½X èqª
60 50
φ (deg)
40 30 20 10 0 −10
0
2
4
6
8
10 time (sec)
12
14
16
18
20
0
2
4
6
8
10 time (sec)
12
14
16
18
20
0
2
4
6
8
10 time (sec)
12
14
16
18
20
40 30 20 θ (deg)
10 0 −10 −20 −30 −40
80
ψ (deg)
60 40 20 0 −20
4.1
Simulation result: Robust backstepping controller (continued)
üxÍ ü±Cq
57
15
δe (deg)
10 5 0 −5 −10
0
2
4
6
8
10 time (sec)
12
14
16
18
20
0
2
4
6
8
10 time (sec)
12
14
16
18
20
0
2
4
6
8
10 time (sec)
12
14
16
18
20
15
δa (deg)
10 5 0 −5 −10
25 20 15 δr (deg)
4
10 5 0 −5
−10 −15
4.1
Simulation result: Robust backstepping controller (continued)
58
4.2
½X èqª
6000 5000
px (ft)
4000 3000 2000 1000 0
0
2
4
6
8
10 time (sec)
12
14
16
18
20
0
2
4
6
8
10 time (sec)
12
14
16
18
20
2
4
6
8
10 time (sec)
12
14
16
18
20
5000 4000
py (ft)
3000 2000 1000 0 −1000
4
1.15
x 10
h (ft)
1.1
1.05
1
0.95
0
4.1
Simulation result: Robust backstepping controller (continued)
@Á
5
Æ f[t üxÍ ¨W Cqõ zG ªú CK L, CKý zGª ¢ ¨W èßð KAú ½¡:÷¿ ©u °. "&, üxÍ :ÿCq ÁY üx Í Cq Áú :Ì ?Ý ¡ò ½ © &¢ "ú #Í4 ¨W C q zGªú Cè °. Z¢, Æ f[t CK¢ Cq zGªú F-16 üxÍ ¨W ?Ý :Ì ½X èqª ú ½±¥÷¿, ú * °. @Yõ A ý 8 °üY °. 1.
2.
¨W W³G½ ?Ý ¡ò ½ U& M°L A L, backstepping ª © üxÍ ¨W Cqõ zG ªú CK °. ªù U üxÍ ¨W Cq PÌüqµÏ ÞxÍܪY µý, W³G½ Lò y Û¨ú PÌ M÷6 nonminimum phase system ê :Ì£ ½ ° " °. Z¢, / PÌü timescale separation Aú Ì¢ ¨W Cq zGªY µ ý KAú < æ© üÇì: Aú PÌ M°. , Æ f[t CK ¢ üxÍ ¨W Cq zGªù ¨W ô:"ú :<& ßÌ¢ Cq zG ª 6, Cq zGª :Ìý ¨W èßð KAú ½¡:÷¿ <£ ½ ° $> °. üxÍ :ÿCq ÁY êEç¿ú Ì ¨W W³G½ ?Ý ¡ò ½ © &¢ üxÍ ¨W Cqõ zG ªú CK °. ªù êEç¿ universal approximation "ú Ì êEç¿ ½ ?Ý ¡ ò © f ¨ú AÝ& vÇ£ ½ °L A L, êEç¿ ;³ ½ 59
60
?Ý ¡ò © f ¨ú \© êÀ êEç¿ @ êõ :ÿC qªY © ºÜèÅ ª °. W³G½ ?Ý ¡ò ½ U& E Íê <º8[ ¡ò uniformly ultimately bounded ° ,ú < °. 3.
üxÍ Cq Áú Ì ¨W W³G½ ?Ý ¡ò ½ © & ¢ üxÍ ¨W Cqõ zG ªú CK °. ªù ½ ?Ý ¡ò © f ¨ ¾ "A¢ ©æ K j¥ý°L A L, K t &¢ "ú #Í4 Cqõ zG ª °. W³G½ ?Ý ¡ò ½ U& EÍê <º8[ ¡ò exponentially attractive ° ,ú < °.
°üY ù u ³ó ½±üqb ¢°L fý°. Æ f[t üxÍ ¨W ?Ý Cqõ zG£ : ¨W ÏôAé üxÍ ¨ú \© (cancellation) L KA ú æ¢ xÍ Cq¨ú 8 °. ÏôAé ¨ù ¨W ô¡: "ú #Í4 L ÷t¿ üxÍ ¨ KAú ©X Å÷¿ Ì M°. 0t üxÍ ¨ú \© Í× Cqõ zG ð, \׺½ ºÜ 0ô W³G½ ºÜ " ú ßÌ $# °ô Í× òbd ¥½õ Ì ¨W ]ý: "ú ?Û& ßÌ Cqõ zG ªú u©b £ , °. Z¢, Æ f[t :ÿCq®
Cqªú Ì W³G½ ¡ò © &¢ "ú #Í4 Cqõ zG °. Cqèßð & ³ú yX Å¿ ÝÝì ½ unmodeled dynamics °. ¨W EÍ ô ô¡: " © JÜþ ½ ÷t¿, ¢ ³ © &¢ "ú #Í4 Cqõ zG ,ú 8ó u© b £ , °.
÷L[¶ [1] Meyer, G., Su, R., and Hunt, L. R., Application of Nonlinear Transformation to Automatic Flight Control, Automatica, Vol. 20, No. 1, 1984, pp. 103107.
[2] Lane, S. H. and Stengel, R. F., Flight Control Design Using Non-linear Inverse Dynamics, Automatica,
Vol. 24, No. 4, 1988, pp. 471483.
[3] Hedrick, J. K. and Gopalswamy, S., Nonlinear Flight Control Design via Sliding Methods, Journal of Guidance, Control, and Dynamics,
Vol. 13, No. 5, 1990, pp. 850858.
[4] Menon, P., Badgett, M., and Walker, R., Nonlinear Flight Test Trajectory Controllers for Aircraft, Journal of Guidance, Control, and Dynamics, Vol. 10, No. 1, 1987, pp. 6772.
[5] Bugajski, D. J. and Enns, D. F., Nonlinear Control Law with Application to High Angle-ofAttack Flight, Journal of Guidance, Control, and Dynamics, Vol. 15, No. 3, 1992, pp. 761767.
[6] Snell, S. A., Enns, D. F., and Garrard Jr., W. L., Nonlinear Inversion Flight Control for a Supermaneuverable Aircraft,
Journal of Guidance, Control, and Dynamics,
Vol. 15, No. 4,
1992, pp. 976984.
[7] Schumacher, C. and Khargonekar, P. P., Stability Analysis of a Missile Control System with a Dynamic Inversion Controller,
Journal of Guidance, Control, and Dynamics,
1998, pp. 508515.
61
Vol. 21, No. 3,
62
[8] Schumacher, C. and Khargonekar, P. P., Missile Autopilot Designs Using Gain Scheduling and Dynamic Inversion,
H1 Control with
Journal of Guidance, Control, and Dynamics,
Vol. 21, No. 2, 1998, pp. 234243. [9] Krsti¢, M., Kanellakopoulos, I., and Kokotovi¢, P.,
Nonlinear and Adaptive Control Design,
John Wiley and Sons, New York, 1995. [10] Qu, Z.,
Robust Control of Nonlinear Uncertain Systems,
John Wiley and Sons, New York,
1998. [11] Funahashi, K., On the Approximate Realization of Continuous Mapping by Neural Networks, Neural Networks,
Vol. 2, No. 3, 1989, pp. 183192.
[12] Hornik, K., Stinchcombe, M., and White, H., Multilayer Feedforward Networks are Universal Approximators, Neural Networks, Vol. 2, No. 5, 1989, pp. 359366. [13] Farrell, J. A., Stability and Approximator Convergence in Nonparametric Nonlinear Adaptive Control, IEEE Transactions on Neural Networks, Vol. 9, No. 5, Sept. 1998, pp. 10081020. [14] Lewis, F. L., Yesildirek, A., and Liu, K., Multilayer Neural-Net Robot Controller with Guaranteed Tracking Performance,
IEEE Transactions on Neural Networks,
Vol. 7, No. 2, March
1996, pp. 388399. [15] Singh, S. N., Yim, W., and Wells, W. R., Direct Adaptive and Neural Control of Wing-Rock Motion of Slender Delta Wings, Journal of Guidance, Control, and Dynamics, Vol. 18, No. 1, 1995, pp. 2530. [16] Kim, B. S. and Calise, A. J., Nonlinear Flight Control Using Neural Networks, Guidance, Control, and Dynamics,
Vol. 20, No. 1, 1997, pp. 2633.
Journal of
÷L [¶
63
[17] Rysdyk, R. T. and Calise, A. J., Adaptive Model Inversion Flight Control for Tilt-Rotor Aircraft, Journal of Guidance, Control, and Dynamics, Vol. 22, No. 3, 1999, pp. 402407. [18] Stevens, B. L. and Lewis, F. L.,
Aircraft Control and Simulation,
John Wiley and Sons, New
York, 1992. [19] Morelli, E. A., Global Nonlinear Parametric Modeling with Application to F-16 Aerodynamics, Tech. rep., NASA Langley Research Center, 1998. [20] Ioannou, P. A. and Sun, J., Robust Adaptive Control, Prentice Hall, New Jersey, 1996. [21] Narendra, K. S. and Annaswamy, A. M.,
Stable Adaptive Systems,
1989. [22] Khalil, H. K., Nonlinear Systems, Prentice Hall, New Jersey, 1996.
Prentice Hall, New Jersey,
Appendix A
F-16
¨W ?Ý
[?/!
A.1
F-16
¨W ôLA _vG (body xed axis) ¢ [?/! °üY °. Ix = 9496 slug ft2 Iy = 55814 slug ft2 Iz = 63100 slug ft2 Ixz = 982 slug ft2
Ii [?/!¿ uý ¨÷¿t v A.1 Aýüq °.
A.2
W³G½ ?Ý
W¡: *Y ?/! °üY ü, y¤, ê, Cq³ ¢ ¥ ½¿ vèý°. F-16 ¨W W³G½ ?Ýù ÷L[¶ 19 Òõ Ì ÷6, Æ f [t Cq³ W³G½ ?Ý xÍ:÷¿ vÇý°L A °. CxT CyT CzT ClT
= Cx( ) + CxÆe Æe + 2cqV Cxq ( )
(A.1)
= Cy + CyÆa Æa + CyÆr Ær + 2bpV Cyp ( ) + 2brV Cyr ( ) = Cz ( ; ) + CzÆe Æe + 2cqV Czq ( )
= Cl ( ; ) + ClÆa ( ; )Æa + ClÆr ( ; )Ær + 2bpV Clp ( ) + 2brV Clr ( ) 64
(A.2) (A.3) (A.4)
A F-16
¨W ?Ý
65
= Cm( ) + CmÆe ( )Æe + 2cqV Cmq ( )
CmT
(A.5)
= Cn( ; ) + CnÆa ( ; )Æa + CnÆr ( ; )Ær + 2bpV Cnp ( ) + 2brV Cnr ( )
CnT
(A.6)
W³G½ ¨ù °ü é÷¿ GSü6, º½ ½X v A.2 Aýüq °. W³G½õ GS£ : ü, y¤, Cq8 ºæ ³æ¿ radian ú PÌ¢ °. Cx = [ a1 ; a2 ; a3 ; a4 ][ 1; ; 2 ; 3 ]T CxÆe
=a
Cxq
= [ b ; b ; b ; b ; b ][ 1; ; ; ; ]T
5
1
2
3
4
2
5
3
4
Cy = c1 CyÆa
=c
2
CyÆr
=c
3
Cyp
= [ d ; d ; d ; d ][ 1; ; ; ]T
Cyr
= [ e ; e ; e ; e ][ 1; ; ; ]T
1
1
2
3
2
2
4
3
3
2
4
3
Cz = [ f1 ; f2 ; f3 ; f4 ; f5 ][ 1; ; 2 ; 3 ; 4 ]T (1 2 ) CzÆe
=f
Czq
= [ g ; g ; g ; g ; g ][ 1; ; ; ; ]T
6
1
2
3
4
2
5
3
4
Cl = [ h1 ; h2 ; h3 ; h4 ; h5 ; h6 ; h7 ; h8 ][ ; ; 2 ; 2 ; 2 ; 3 ; 4 ; 2 2 ]T Clp = [ i1 ; i2 ; i3 ; i4 ][ 1; ; 2 ; 3 ]T Clr
= [ j ; j ; j ; j ; j ][ 1; ; ; ; ]T 1
2
3
4
2
5
3
4
ClÆa
= [ k ; k ; k ; k ; k ; k ; k ][ 1; ; ; ; ; ; ]T
ClÆr
= [ l ; l ; l ; l ; l ; l ; l ][ 1; ; ; ; ; ; ]T
1
1
2
2
3
3
4
4
5
5
6
6
7
7
2
2
2
3
2
2
66
A.2
W³G½ ?Ý
Cm = [ m1 ; m2 ; m3 ][ 1; ; 2 ]T CmÆe
= [ m ; m ; m ][ 1; ; ]T
Cmq
= [ n ; n ; n ; n ; n ; n ][ 1; ; ; ; ; ]T
4
1
5
2
6
2
3
4
5
2
6
3
4
5
Cn = [ o1 ; o2 ; o3 ; o4 ; o5 ; o6 ; o7 ][ ; ; 2 ; 2 ; 2 ; 2 2 ; 3 ]T Cnp = [ p1 ; p2 ; p3 ; p4 ; p5 ][ 1; ; 2 ; 3 ; 4 ]T Cnr
= [ q ; q ; q ][ 1; ; ]T 1
2
2
3
CnÆa
= [ r ; r ; r ; r ; r ; r ; r ; r ; r ; r ][ 1; ; ; ; ; ; ; ;
CnÆr
= [ s ; s ; s ; s ; s ; s ][ 1; ; ; ; ; ]T
1
1
2
2
3
3
4
4
5
5
6
6
7
8
9
2
10
2
2
3
2
3
3
; 3 ]T
A F-16
¨W ?Ý
67
v
A.1
[?/!¿ uý ¨ A
2 Ixz I1 = Iz (IIxzIzIyI)+ 2 xz +Iz ) I2 = IxzI(xIIxz IIyxz 2
I3 = Ix IzIz Ixz 2 I4 = Ix IIzxzIxz 2 I5 = IzIyIx I6 = IIxzy I7 = I1y
2 Ixz I8 = Ix (IIxxIzIyI)+ 2 xz I x I9 = Ix Iz Ixz 2
7:7012 10 2:7548 10 1:0548 10 1:6415 10 9:6040 10 1:7594 10 1:7917 10 7:3361 10 1:5873 10
1 2 4 6 1 2 5 1 5
68
A.2
v a
b
1:943367 10
2
4:833383 10
1
2:903457 10
1
6:075776 10
1
c d e f
g
h
i j
1:145916 100 1:006733 10
1
1:378278 10
1
8:071648 10
1
8:399763 100 3:054956 10
1
4:126806 10
1
1
6:250437 10
2
1:463144 10
1
2:635729 10
2
1:4798 10
2
1:192672 10
1
k
l
m
n
o
2:978850 10
1
4:516159 10
1
8:0630 10
2
5:159153 10
2:993363 10
1
r
2
1:189633 10
1
4:354000 10
1
5:776677 10
1
1:189974 10
1
6:067723 10
1
4:073901 10
2
2:192910 10
2
7:4523 10
2
1
4:211369 100
3:464156 10
0
3:746393 10
1
4:928702 10
1
5:0185 10
1
2:677652 10
2
3:298246 10
1
3:698756 10
1
1:167551 10
1
2:107885 10 4:404302 10 3:348717 10
2
1:373308 10
1:588105 10 8:115894 10 3:337476 10
2:141420 10
4:276655 10
2
5:199526 10
1
1:004297 10
1
1:237582 10
0
1 2 1
1:156580 10
1:642479 10
2
9:035381 10
1
7:422961 101 1
4:260586 100
6:923267 100
4:177702 10
9:162236 100
3:292788 102
6:848038 102
0
4:775187 100
1:672435 10
1:026225 101
2
1:357256 10
1:098104 10
1:101964 10
9:100087 10
1
2:835451 10 1:247721 10
0
7:391132 10
0 0
3:253159 10
2
3:152901 10
3
3:7756 10
15
3:213068 10
1
1:579864 10
2
3:598636 101
0
2:411750 102
1
0
2
0
s
6:016057 10 8:679799 10
6:988016 10 1:131098 101
0
0
q
8:644627 100
6:594004 10
0
p
1
3:554716 10
0
4:120991 102
2:136104 10
1
1:058583 10
2:172952 10
W³G½ ?Ý º½
4:132305 10
1
4:080244 102
A.2
W³G½ ?Ý
1 0
4:851209 10
1
5:817803 10
2
5:2543 10
1
2:247355 102
2:003125 10
1
8:476901 10
1
1:926178 10
1
7:641297 10
1
6:233977 10
2
4:013325 100
6:573646 10
3
2:302543 10
3:535831 10
1
1
2:512876 10
1
2:514167 10
2
2:038748 10
1
òbd KA Á
Appendix B
A
°ü 0 ® T = T (Æ) > 0 U& 8 yÛAé x_ = f (t; x) © x(t) ° B.1
uniformly ultimately bounded
.
kx(t0)k < Æ =) kx(t)k B; 8t t0 + T; 8Æ 2 (0; d)
A
B.2
°ü 0 ® = (Æ) U& 8yÛAé x_ = f (t; x) x=0ù °
equilibrium point
exponentially attractive
kx(t0 )k < Æ =) kx(t)k (Æ)e
Aý
B.1
(B.1)
.
8t t0 ; 8Æ 2 (0; d)
(t t0 ) ;
\׺½ x 2 Rn ¢ °ü èßðú L²
(B.2)
.
x_ = f (t; x)
? x 2 Rn  V (x) °ü [Géú T òbd ¥½L 8
,
c1 kxk2 V (x) c2 kxk2
(B.3)
@V f (t; x) c3 kxk2 + e t @x
(B.4)
èßð © °ü Ù#éú T¢° (i) cc = EÍ
.
3 2
kx(t)k
r
c2 e c1
r
c t t ) 0 c
kx(t0 )k + c (t 2 t0 ) e 1
c t t ) 0 c
r kx(t0 )k + c c c2 c e 1 3 2
3 2 2(
c t c
3 2 2
(B.5)
(ii) cc 6= EÍ 3 2
r kx(t)k cc2 e 1
3 2 2(
69
t0 h 2
e
(t t0 ) 2
e
c t t )i 0 c
3 2 2(
(B.6)
70
t c ; c ; c ; ; j \½ ° < òbd ¥½ è ¢ yÛ V_ é (B.3), (B.4) © °ü Ù#éú T¢ °. 1
2
.
3
.
V_ cc3 V + e
^¿Ï ¥½ W = pV ¿ A 8 W_ =
t
1
pV_ V
2
t¿, W_ ù °ü Ù#éú T¢°.
p W_ 2cc3 W + 2 e
t 2
1
æ é Comparison Lemma õ :Ì 8 W (t) °ü Ù#éú T¢°. 22
W (t) e
c t t ) 0 c
p Zt W (t0 ) + 2 e
c t t ) 0 c
p W (t0 ) + (t2 t0 ) e
3 2 2(
(i) cc = EÍ 3 2
W (t) e
3 2 2(
t0
c t ) c e 2 d
3 2 2(
c t c
3 2 2
(ii) cc 6= EÍ 3 2
h i c W (t0 ) + p c c2 c e t e (t t ) e c (t t ) 3 2 Z¢, kx(t)k Wpc(t) L W (t0) pc2 kx(t0)k t¿ kx(t)k °ü Ù#éú T¢°.
W (t) e
(i) = EÍ
c t t ) 0 c
3 2 2(
2 0
0
2
3 2 2
0
1
c3 c2
kx(t)k
r
c2 e c1
c t t ) 0 c
3 2 2(
r
kx(t0 )k + c (t 2 t0 ) e 1
c t c
3 2 2
(ii) cc 6= EÍ 3 2
kx(t)k
r
c2 e c1
c t t ) 0 c
3 2 2(
r
kx(t0 )k + c c c2 c e 1 3 2
t0 h 2
e
(t t0 ) 2
e
c t t )i 0 c
3 2 2(
Cq ÏôAé yÛ
Appendix C
C.1
2, 3
2, 3
$t PÌý Cq yÛ
$t PÌý xd ¨ù é (2.28) t °üY zGü}°. 2
xd2 = g1 ( ; ; ; )
1
f1 ( ; ) f1g ( ; ; ; ) + x_ d1
k1 z1
, R (x ; x ) s (x ; x ) 1
1
3
1
1
3
æ ét xd õ R (x ; x ) 2 R ; s (x ; x ) 2 R ¿ A °. :, é (2.29), (2.30) t PÌý @x@xd 2 R ; @x@xd 2 R ¨ù °üY GSý°.y 1
2
= =
3 3
3
3 3
2 1
@xd2 @x1 @xd2 @x3
1
@R1 (x1 ;x3 ) s 1 @ @R1 (x1 ;x3 ) s 1 @
1
3
2 3
(x ; x )
1
1
2
@R1 (x1 ;x3 ) s 1 @
3
3 1
3
(x ; x ) 1
3
@R1 (x1 ;x3 ) s 1 @
@s (x ; x ) (x ; x ) 0 + R (x ; x ) @x 1
3
3
1
1
1
3
æ é yÛ¨ù °üY °.
@R1 (x1 ; x3 ) @ @R1 (x1 ; x3 ) @ @R1 (x1 ; x3 ) @ @R1 (x1 ; x3 ) @ @s1 (x1 ; x3 ) @x1 @s1 (x1 ; x3 ) @x3
y x 2 Rm ; y 2 Rn Æ:
,
@y @x
(
i; j
1
1
@s (x ; x ) (x ; x ) + R (x ; x ) @x 1
3
3
3
@g1 ( ; ; ; ) R1 (x1 ; x3 ) @ = R1(x1; x3) @g1 ( ;@ ; ; ) R1(x1; x3) = R1(x1; x3) @g1 ( ;@ ; ; ) R1(x1; x3) = R1(x1; x3) @g1 ( ;@ ; ; ) R1(x1; x3) ; ; ) = k1 I33 @f1@x( ; ) @f1g ( ; @x1 1 @f1g ( ; ; ; ) = @x3 @y ) ¨ @xi ±µú y¢° j
=
R1 (x1 ; x3 )
n
m
71
.
1
1
3
1
1
1
3
72
C.2 4
C.2
4
$t PÌý Cq yÛ
$t PÌý Cq yÛ
4
$t PÌý xd ¨ù é (4.7) t °üY zGü}°. 2
xd2 = (g1 ( ; ; ; ) + g1a ( ; ))
1
k1 z1 f1 ( ; ) f1g ( ; ; ; ) + x_ d1 + v1
, R (x ; x ) s (x ; x ) 2
1
3
2
1
3
æ ét xd õ R (x ; x ) 2 R ; s (x ; x ) 2 R ¿ A °. :, é (4.8), (4.9) t PÌý @x@xd 2 R ; @x@xd 2 R ¨ù °üY GSý°. 2
2
2 1
@xd2 @x1 @xd2 @x3
= =
1
3 3
@R2 (x1 ;x3 ) s 2 @ @R2 (x1 ;x3 ) s 2 @
3
3
2
1
3
3
1
3 2
2 3
(x ; x ) 1
3
3
@R2 (x1 ;x3 ) s 2 @
(x ; x ) 1
3
@R2 (x1 ;x3 ) s 2 @
@s (x ; x ) (x ; x ) 0 + R (x ; x ) @x 1
3
3
1
2
1
3
2
1
@s (x ; x ) (x ; x ) + R (x ; x ) @x 1
3
2
1
3
3
æ é yÛ¨ù °üY °. @R2 (x1 ; x3 ) @ @R2 (x1 ; x3 ) @ @R2 (x1 ; x3 ) @ @R2 (x1 ; x3 ) @ @s(x1 ; x3 ) @x1 @s(x1 ; x3 ) @x3
Z¢, v
1
= = = = = =
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:, @x@v
1 1
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2 R33 ù °üY GSý°. @v1 @x1
I33 + 21 1T 3 k1k3 1 1T Æ1 Æ 1 + 3 k1 k + 3 '3 (k1 k3 + 3 '3)2
= k k 1
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1 k1 k2 @Æ1 k1 k3 + 3 '3 @x1
3
2
1
1
3
C
Cq ÏôAé yÛ
C.3
73
ÏôAé yÛ
ÏôAé yÛ¨ù °üY GSý°. 2
3
6sin tan 0 @g ( ; ; ; ) 6 = 666 cos 0 @ 4 0 0
cos tan 7 7 sin 777 5 0
1
2
@g1 ( ; ; ; ) @
=
6 6 6 6 6 4 2
cos cos2
0 0
3
0 0 0
7 7
sin cos2
cos 777 5 0
0 60 6 @g ( ; ; ; ) 6 = 660 0 @ 4 0 cos tan 1
2
@g1 ( ; ; ; ) @
@g1a ( ; ) @
0 = 0 0 6 6 6 6 6 4
= 4Sm
0 0 2 6 6 6 6 6 6 6 6 6 6 6 6 4 2
@g1a ( ; ) @
=
6
S 6 6 4m 664
3
0 0
sin cos2
3
0 0
cos cos2
sin tan
7 7 7 7 7 5
7 7 7 7 7 5 0
0
1 Cz ( )c q C A sin @Cxq ( ) c+ cos @Czq ( ) c cos @ cos @ cos cos
B @ 0
yp b cos @C@ ( )
B @
Cx ( )c q
sin
sin Cxq ( )c
cos
sin
0 0 sin Cyp ( )b 0
cos
@Cxq ( ) @ c
3
sin cos
0
1
sin Czq ( )c C
sin
sin
A @Czq ( ) c @
0
0
cos sin Cxq ( )c+ cos2 Czq ( )c
sin sin cos2
cos
cos Cxq ( )c
0
cos
7 7 7 7 7 7 7 7 @C ( ) y r @ b7 7 7 7 5
sin
cos Czq ( )c
0 sin Cyr ( ) 0
3 7 7 7 b7 7 5
74
C.3
0
2
1 Cz ( ; )qS C A sin @Cx ( ) cos @Cz ( ; ) qS + qS cos @ cos @
cos cos
@f1 ( ; ) @
B 6 6 @ 6 6 60 6 6 sin sin [T +C ( )qS ] x 6B 6@ 6 @Cx ( ) qS 6 cos sin @ 6 4
1 = mV
0
2
@f1 ( ; ) @
60 6 6 6B 6@ 6 6 6 4
1 = mV 2
@f1g ( ; ; ; ) @
=
g V
=
g V
sin sin cos2
=
g V
=
g V
[T + Cx( )qS ] +
cos sin cos2
C
z
3
( )qS
3
(cos sin sin cos cos ) 7 7 sin sin sin + cos cos sin cos sin cos cos 777 5 0
6 6 6 6 6 4
3
(sin sin + cos cos cos ) 7 7 cos cos sin sin cos sin sin cos cos cos 777 5 0
6 6 6 6 6 4
sin cos2
3
(sin sin cos sin cos ) 7 7 cos sin sin + cos cos cos + sin sin sin cos 777 5 0
6 6 6 6 6 4
1 cos
2
@f1g ( ; ; ; ) @
7 7 7 7 17 7 7 cos sin Cz ( ; )qS 7 C7 A7 @Cz ( ; ) qS 7 sin sin @ 7 5
0
2
@f1g ( ; ; ; ) @
3
sin cos
17 7 7 cos cos [T +Cx ( )qS ] sin Cy ( )qS sin cos Cz ( ; )qS 7 C7 A7 @Cy ( ) @Cz ( ; ) qS 7 + cos @ qS sin sin @ 7 5
1 cos
2
@f1g ( ; ; ; ) @
[T +Cx ( )qS ]
ÏôAé yÛ
3
(sin cos cos cos sin ) 7 7 cos sin cos cos sin sin + sin sin cos sin 777 5 0
6 6 6 6 6 4
1 cos
Abstract
Nonlinear adaptive ight control law and nonlinear robust ight control law are proposed. First, backstepping controller is used to stabilize all state variables simultaneously without two-timescale assumption that separates the fast dynamics, involving the angular rates of the aircraft, from the slow dynamics that includes angle of attack, sideslip angle, and bank angle. The proposed method makes good use of the characteristics of the ight dynamics, and the closed-loop stability can be proved without unrealistic restriction. Uncertainties of the aerodynamic coef cients are also considered. An adaptive controller based on neural networks and a robust controller are used to compensate for the effect of the aerodynamic modeling error. The neural networks' parameters are adjusted to offset the error term by stable adaptive laws. A robust control law is designed to compensate for the error term with an assumption that the size of the error term is known. The closed-loop stability of the error states is examined by the Lyapunov theory, and it is shown that the error states exponentially converge to a compact set. Finally, a nonlinear simulation of F-16 aircraft maneuver is performed to demonstrate the performance of the proposed control laws.
Keywords
: nonlinear ight control, backstepping, neural networks, adaptive control, robust control
Student Number
: 98416-525
75
