Nonlinear Adaptive and Robust Flight Control Using the Backstepping Algorithm

W¡uP¡æf[
¨WŸ üxÍ :ÿ „ CqŸ
zG‚ [¢ u
Nonlinear Adaptive and Robust Flight Control
Using the Backstepping Algorithm
[ 2Ú
2000
tСp ¡Ù
¨WÍsW¡Y
× –
W¡uP¡æf[
¨WŸ üxÍ :ÿ „ CqŸ
zG‚ [¢ u
Nonlinear Adaptive and Robust Flight Control
Using the Backstepping Algorithm
[
2000
tСp ¡Ù
¨WÍsW¡Y
× –
¨WŸ üxÍ :ÿ „ CqŸ
zG‚ [¢ u
Nonlinear Adaptive and Robust Flight Control
Using the Backstepping Algorithm
“êp½ ¥î³
f[ú W¡uP ¡æf[÷¿ C;¥
[ 10Ú
1999
tСp ¡Ù
¨WÍsW¡Y
× –
ז W¡uP ¡æf[ú u¥
[ 12Ú
1999
æÙ$
ÙæÙ$
æÙ
#À
Æ f[‚t™ üxÍ ¨WŸ CqŸõ zG ™ “ªú CK •°. Ɗ:÷¿ üxÍ
¨WŸ CqŸõ zG£ :‚ ¨WŸ ô:"‚ 0† Ïô“Aéú ô ô"Y —
ÿ ô"÷¿ uÛ L èßð‚  Š ë:÷¿ CqŸõ zG ™ ,Y™ µý, Æ
f[‚t™ backstepping Ÿªú Ì Š ; èßðú ôè‚ KAÜèř CqŸõ z
G •°. Ÿªù ¨WŸ ô:"ú :<& ßÌ¢ CqŸ zGŸª6, KAú ’
< Ÿ æ Š üÇì: Aú Ì “ M™°. Z¢, üxÍ :ÿCqÁY üxÍ CqÁú :Ì Š W³G½ ÝÝì „ ½ˆ U& ™ EÍ‚ê  9 Æ
q#“ M™ ¨WŸ CqŸ zGŸªú Cè •°. üxÍ :ÿCqŸªú Ì Š ½ˆ
„ ?Ý ¡ò‚ © Œf ™ ¨ú \© êÀ êEç¿ @êõ ºÜèř C
qŸ zGŸªú CK •°. ”ýL üxÍ CqŸªú Ì Š ?Ý ¡ò „ ½ˆ
‚ © Œf ™ ¨ ¾Ÿõ yý NL °L A L, ” –³ú \©èř CqŸ z
GŸªú CK •°. CKý CqŸª :Ìý ¨WŸ èßð KAú òb’d’ Áú Ì Š ’< •÷6, F-16 üxÍ ¨WŸ ?Ýú Ì Š èqªŽú ½±¥÷
¿ CKý CqŸª  ú *’ •°.
sÅq : üxÍ ü±Cq, backstepping Ÿª, 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
Ÿªú Ì¢ üxÍ ü±Cq
¨WŸ ?Ý . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Ïô“Aé . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 W³G½ ?Ý . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 Ïô“Aé uS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
CqŸ zG „ KA ©u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 ŸÆ A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 CqŸ zG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3  ©u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
½X èqªŽ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
CqŸ zG „ KA ©u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 ŸÆ A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 CqŸ zG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3  ©u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
½X èqªŽ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
üxÍ ü±Cq
4.1
4.2
CqŸ zG „ KA ©u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 ŸÆ A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 CqŸ zG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.3  ©u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
½X èqªŽ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
@Á
÷L[¶
¨WŸ ?Ý
5
A F-16
A.1
A.2
B
C
[?/! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
W³G½ ?Ý . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
òb’d’ KA Á
CqŸ „ Ïô“Aé yÛ
C.1 2, 3
$‚t PÌý CqŸ 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 CqŸ yÛ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ïô“Aé yÛ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.2 4
72
C.3
73
Abstract
75
” ò»
Aú Ì¢ üxÍ ü± CqŸ 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
Ïô“Aé u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
3.1
W³G½ e’ ?Ý ¡ò (%) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
4.1
W³G½ e’ ?Ý ¡ò (%) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
A.1
A.2
[?/!¿ uý ¨ A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
W³G½ ?Ý º½ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vi
67
68
1
1.1
tÁ
u •E
Ɗ:÷¿ ¨WŸ Cqèßðú zG Ÿ æ©t™ ¨WŸ "A¢ ÏÌ\ׂ °L A L, ” ÏÌ\ׂ © xÍÜý ?݂ ©t xÍCqŸªú :Ì Š Cq
Ÿõ zG¢°. “ªù L;CqŸª, LîW“Aª, xÍ/:CqŸª # N²• °
j¢ xÍCqŸªú PÌ£ ½ °™ $> °. ” # xÍÜý ?Ýù zG ÏÌ\×
–‚t ¨WŸ Ïô"ú üp: AÝ& #Í4Ÿ :[‚, Iù Ï̖‹‚ ©
ÆA¢  ú uŸ æ©t™ Š ÏÌ\ׂ  Š CqŸõ ŠÄ©t zG L, zGý
Cqú ¨WŸ ÏÌ\ׂ 0†t ºÜèˆb ¢°.
¢ ³>ú •Ä Ÿ æ Š ˆÞxÍܟª (feedback linearization) ú Ì¢ üxÍ
¨WŸ Cq‚ [¢ u ØqP°. ˆÞxÍܟªù \׺½ Óù ;³º½õ Ì
Š üxÍ èßðú ôÆ¢ xÍèßð÷¿ ºÞ Š CqŸõ zG ™ Ÿª÷¿, ºÞ
ý xÍèßð ¨WŸ ÏÌ\ׂ U “ MŸ :[‚ ¨WŸ Š ÏÌ\ׂ Â
© ÆA¢  ú Ã$£ ½ ™ ³Æ CqŸõ u£ ½ °. ” # Ÿªú Ì Ÿ
æ©t™ üxÍ èßð ô:"ú AÝ& NL qb 6, zero dynamics õ Cq£ ½
{Ÿ :[‚ nonminimum phase system 4ÙKAú Ã$£ ½ {°™ ³> °.
Backstepping Ÿªù /– üxÍ :ÿCq‚ ÿÌü8t ߌ& u Øq“L ™ üxÍ CqŸª°. Ÿªù GW: èßð‚ © Ù èßð \׺½õ
\ Cq³÷¿ zA Š \Ù èßðú Cq ™ YAú ŠÄ ™ ,÷¿, nonminimum
phase system KAú Ã$£ ½ ÷6, üxÍ :ÿ „ CqÁú :Ì Š üx
1
2
1.2
u ô³
Í èßð ÝÝìú L²£ ½ °™ $> °.
¨WŸ™ üx́ £ < I¦† Š “ ]ý: º½‚ U Š ºÜ Ÿ :
[‚ ” ô:"ú AÝ& ?Ý Ÿ Í q·°. 0†t ¨WŸ CqŸõ zG ™
E͂™ W³G½ ?Ý ¡ò, Z™ ½Ù ½ˆ –³ú L² ™ , |Å °. èßð
¡ò# ºÜ, ½Ù pˆú L² Š ” –³ú ™èř CqÁ÷¿™ :ÿCq
ÁY CqÁú àú ½ °. :ÿCqŸªù èßð 4Ù º½# Ç& èßð
u‚ :¦¢ Cqú ìè÷¿ 8A Š CqŸ uSõ ºÜèř “ª÷¿
½ˆ‚ ¢ ô:  Ÿª†L £ ½ °. /–‚™ :ÿCqŸª ú ’Â
èř u® üxÍ èßð÷¿ :̂ ¢ u Øq“L ÷6, êEç¿
universal approximation "ú Ì¢ :ÿCqŸªê uüL °. CqŸªù ½ˆ
Y èßð ?Ý ¡ò‚ ¢ –³ "A¢ ©æ K‚ š©°L A L, ” –‹ K
‚t ’Æ¢  ú uú ½ ™ CqŸõ zG ™ “ª÷¿ ½ˆ‚ ¢ ½ô: Ÿª
†L £ ½ °.
1.2
u ô³
ˆÞxÍܟªù Á:÷¿ Aüq ÷6 Š Ûb‚ ÿÌüL ™ üxÍ è
ßð CqŸª÷¿ üxÍ ¨WŸ Cq[C‚ê Gý PÌü}°. Meyer ® Hunt ™ Àý–
â Cq[C‚, Lane Y Stengel ù ¨WŸ Cq[C‚ ;³ ˆÞxÍܟªú :Ì Š "
A¢ ;³º½õ Cq •°. Hedrick Y Gopalswamy ™ sliding method õ :Ì Š ?Ý
¡òõ L² •°. ” # ˆÞxÍܟªú ¨WŸ Cq[C‚ ”? :Ì£ E͂™,
zero dynamics õ Cq£ ½ {Ÿ :[‚ nonminimum phase system KAú Ã$£ ½ {
L, W³G½ 2ò Z™ 3ò \ yÛ¨ ›Å °™ [C> Œf¢°.
ˆÞxÍܟªú üxÍ ¨WŸ Cq[C‚ :Ì ™ Z °ô “ªù ¨WŸ ô:"
ú Ì Š Ïô“Aéú timescale ¿ uÛ L, 4Ùؒ® ½Ùؒ C
1, 2
3
1
tÁ
3
qŸõ u ™ “ª°. Two-timescale —ú Ì Š CqŸõ zG ™ YAù ¾
3 ³G¿ uÛ£ ½ °. $9 ½Ùؒ‚t™ 4Ùؒ \׺½ p; q; r ú Cq³
÷¿ zA Š, ; ; sq• ˆ:ú 8[ êÀ p; q; r ‚ ¢ Ÿuˆ:ú GS¢°. 4
Ùؒ‚t™ \׺½ p; q; r ½Ùؒ‚t fý Ÿuˆ:ú 8[ êÀ ìC Cq
³ Æe; Æa; Ær ú GS¢°. :, 4Ùؒ Ïô" ½Ùؒ‚ ü© Í òŸ :[‚,
4Ùؒ \׺½ p; q; r ¿ sq• Ÿuˆ:ú 蓍 { AÝ& 8[£ ½
qt, ½Ùؒ‚t™ p; q; r 蓍 Šÿ‚ ¢ –³ #Í#“ M™°L A¢
°. 0†t “ªú PÌ Ÿ æ©t™ 4Ùؒ Ïô" ½Ùؒ Ïô"‚ ü
© Í †b ¢°. ” # Ç&«“ ÂÙÛ ¨WŸ Cq‚ [¢ u‚t™ "»¢ ½
¡: ©uYA { 4Ùؒ Cqú ½Ùؒ Cqð ?Û& À ÷¿
zA Š, ¨WŸ ¢ two-timescale —ú T¢°L A •°.
¨WŸ Cq[C‚t two-timescale —‚ [¢ ½¡: ©uù Schumacher ® Khargonekar ‚ © ü ØqP°. ”ù 4Ùؒ èßð‚t AÝ¢ ‹ºÞ ØqP
°L A L ,ú ½Ùؒ‚  Š ½Ùؒ \׺½® Cq³÷¿ uý
èßðú u •°. ”ýL òb’d’ ¥½õ Ì Š èßð KAÜüŸ æ¢ 4Ù
ؒ /™ Cqú @A •°. ” # ” u‚t zA¢ “ª‚ ¢ KA ©
uù Í Ä! L conservative Ÿ :[‚, èßð KAú Ã$ Ÿ æ©t GSý 4
Ùؒ /™ Cq Í v“3 ý°. 0†t ©u@Y‚ 0† CqŸõ zG£ EÍ
‚™ Cq³ jÜü$# ‚ [C Œf£  m°. ”ù Z¢ CqŸ z
G „ ©uYA‚t ¨WŸ Cq8‚ © Œfý *, |³ „ 8³‚ ¢ –³ú Xè ™ # °™ ù Aú •°.
ˆÞxÍܟªú :Ì ™ YA‚t Z °ô q²Óù CqÂ\ ü™ ¨WŸ èß
ðú AÝ& NL qb ¢°™ ,°. ” # ¨WŸ W³G½ù Š “ º½‚ U Š ºÜ Ÿ :[‚ ¨WŸ èßð ô:"ú AÝ& ?Ý ™ ,ù Í q²Ï
Æ°. Two-timescale —ú Ì¢ CqŸ EÍ, "A¢ W³G½ ÝÝ쁂 © ;
4, 5, 6
7
4
1.3
u 4Ì
èßð KAú 3 ý°™ , uý † °. èßð ?Ý ¡ò# ½ˆ –
³ú L² ™ üxÍ CqÁ÷¿t, :ÿCqÛb‚t™ Krsti¢ , CqÛb‚t™
Qu
backstepping Ÿª‚ Ÿ#¢ uõ ½± •°.
Funahashi, Hornik #ù °W êEç¿ üxÍ ¥½õ AÝê¿ vÇ£
½°™ ,ú½¡:÷¿ ’< •L, Farrell, Lewis #ù ¢ universal approximation
"ú Ì Š, ³WY °W êEç¿ú üxÍ :ÿCqŸ¿ (nonparametric nonlinear direct adaptive controller) Ì ™ uõ ½± •°.
¨W CqÛb‚t™ Singh
# wing-rock motion ú Cq ™Ú, Kim Y Rysdyk ¨WŸõ Cq ™ Ú :ÿ êEç
¿ú PÌ •°.
8
9, 10
11, 12
13, 14
15
16, 17
1.3
u 4Ì
Æ f[‚t™ backstepping Ÿª‚ © üxÍ ¨WŸ CqŸõ zG ™ “ªú C
K •°. ˆÞxÍܟªú Ì¢ Ɗ: üxÍ ¨WŸ CqŸ®™ µý, two-timescale
Aú PÌ “ ML ³Æ òb’d’ ¥½õ Ì Š ½¡:÷¿ KAú ’< •°.
Z¢, Cq8# šê‚ © Œf ™ W³G½ * Ûú ? L² •÷6, KA
©u „ ’<YA‚t timescale separation Aú Ì “ MUŸ :[‚ KAú Ã$ Ÿ æ© Cq ¾Ÿõ ›Å \÷¿ C¢©b £ ›Å {qP°.
Z¢, ?Ý ¡ò# ½ˆ‚ ¢ –³ú ™èş æ Š üxÍ ¨WŸ èßð‚ Â
Š üxÍ :ÿCqŸ® CqŸõ zG L ”  ú üp •°. ÷L[¶ 14
@Yõ ÿÌ Š °W êEç¿ú :ÿCqŸ¿ u L êEç¿ @êõ :
ÿªY‚ 0† ºÜèÉ÷¿, ?Ý ¡ò® ½ˆ‚ © Œf ™ –³ú \© êÀ ü
xÍ :ÿ ü±CqŸõ zG •°. ”ýL ÷L[¶ 10 @Yõ ÿÌ Š ?Ý ¡ò®
½ˆ ¾Ÿõ yý NL °L A L üxÍ ü±CqŸõ zG •°.
1
tÁ
1.4
5
f[ u
Æ f[ uù °üY °. 2 $‚t™ ¨WŸ üxÍ Ïô“AéY W³?݂ ©
tÁ L, ?Ý ¡ò „ ½ˆ U& “ M™ E͂ backstepping Ÿª‚ © üxÍ ¨
WŸ CqŸõ zG ™ “ªú CK¢°. 3 $Y 4 $‚t™ W³G½ ?݂ ÝÝì
™ E͂ êEç¿ú Ì¢ üxÍ :ÿCqŸ® üxÍ CqŸõ zG L, òb’d’ ¥½õ Ì Š KAú ©u¢°. ”ýL 5 $‚t @Áú ™°.
2
Backstepping
Ÿªú Ì¢ üxÍ ü±Cq
Ÿªù GW: èßð‚  Š \׺½ ÆÙõ Cq³÷¿ zA Š CqŸõ zG ™ “ª°. $‚t™ F-16 ¨WŸ ?݂ [© tÁ¢ ó, W³?Ý
ú j¥¢ ¨WŸ Ïô"ú AÝ& NL °L A L, backstepping Ÿªú Ì Š
üxÍ ¨WŸ ?Ý CqŸõ zG ™ “ªú CK¢°.
Backstepping
2.1
2.1.1
¨WŸ ?Ý
Ïô“Aé
Æ f[‚t L²¢ ¨WŸ üxÍ 6 îê Ïô“Aéù °üY °. ¨WŸ™ 6 ÆA¢ ¾Ÿ |³Y W³, 8³ Ì¢°L A •°. “u ;‚ ¢ –
³ù L² “ MU°.
18
V_
= cos mcos [T + Fx ] + sinm [Fy ] + sin mcos [Fz ]
+ g [ cos cos sin + sin sin cos + sin cos cos cos ]
sin [T + Fx] + cos Fz
_ = cos tan p + q sin tan r
mV cos mV cos g
+ V cos
[sin sin + cos cos cos ]
cos sin [T + Fx] + cos Fy sin sin Fz
_ = sin p cos r
+ Vg
mV
mV
(2.2)
mV
[cos sin sin + cos cos sin sin sin cos cos ]
6
(2.1)
(2.3)
2 Backstepping
Ÿªú Ì¢ üxÍ ü±Cq
7
p_ = I2 pq + I1 qr + I3 L + I4 N
q_ = I5 pr I6 p2 r2
(2.4)
+I M
(2.5)
r_ = I2 qr + I8 pq + I4 L + I9 N
(2.6)
_ = p + tan (sin q + cos r)
(2.7)
7
_ = cos q sin r
_ = sin q + cos r
cos (2.8)
(2.9)
ŠŸt V; ; ™ šê, ‹ü, ˜y¤ú, p; q; r ù ôLA _v9 (body xed
axis) ‚ ¢ šê Ûú #Í6°. ”ýL ; ;
™ ôLA _vG® “uLA _
vG (earth xed axis) P ¡Æ êõ, Fx; Fy ; Fz ® L; M; N ù ôLA _v
9‚ ¢ * ÛY ?/! Ûú #Í6°. Ii ™ [?/!¿ uý ¨÷¿t, A® ½X ÙÀ A.1 ‚ Aýüq °.
2.1.2
W³G½ ?Ý
WŸ‹¡: *Y ?/!™ °üY ‹ü, ˜y¤, šê, Cq³‚ ¢ ¥
½¿ vèý°. Æ f[‚t™ ÷L[¶ 19 F-16 ¨WŸ W³G½ ?Ýú Ì •÷6,
Cq³ W³G½ ?݂ xÍ:÷¿ vÇý°L A •°.
Fx = CxT qS
=
Cx () + CxÆe Æe +
cq
2V Cxq () qS
Fy = CyT qS
=
Fz = CzT qS
=
bp
br
Cy + CyÆa Æa + CyÆr Ær + Cyp () + Cyr () qS
2V
2V
Cz (; ) + CzÆe Æe +
cq
2V Czq () qS
8
2.1
L = ClT qSb
=
M
bp
br
Cl (; ) + ClÆa (; )Æa + ClÆr (; )Ær + Clp () + Clr () qSb
2V
2V
= CmT qSc
=
N
Cm () + CmÆe ()Æe +
cq
2V Cmq () qSc
= CnT qSb
=
¨WŸ ?Ý
bp
br
Cn (; ) + CnÆa (; )Æa + CnÆr (; )Ær + Cnp () + Cnr () qSb
2V
2V
;³ ˆÞxÍܟª (input-output feedback linearization) Z™ two-timescale —ú Ì Š üxÍ ¨WŸ CqŸõ zG ™ E͂™, ¨WŸ Cq8 ôLA _vG
9‚ ¢ ?/! Ûú ŒfèÅêÀ zGü}°L A L Cq8‚ © Œf
™ * Ûú X袰. æ é‚ #Í& ,¤, Æ f[‚t™ Cq³Y šê‚ ©
Œf ™ W³G½ * Ûú ? L² Š CqŸõ zG L KAú ©u¢°. W
³G½ u: ½X?Ýù ÙÀ A.2 ‚ Aýüq °.
2.1.3
Ïô“Aé uS
Ÿªú Ì¢ CqŸ zG‚ :¦ êÀ, W³?Ýú Ïô“Aé‚ Â
Š ¨ ]ý:, ½¡: —‚ 0† °üY Aý •°.
Backstepping
2 3
2
6_ 7
1
_ = mV
_
6 7
6 7
6 7
6 7
4 5
2
+
6
6
6
6
6
4
6
6
6
6
6
4
3
[T + Cx()qS ] + Cz (; )qS
7
7
cos sin [T + Cx()qS ] + cos Cy qS sin sin Cz (; )qS 777
5
0
sin
cos
cos
cos
32 3
cos tan 1
sin tan 7 6p7
76 7
sin 0
cos 777 666q777
54 5
1
sin tan cos tan r
Ÿªú Ì¢ üxÍ ü±Cq
2 Backstepping
2
sin
cos
6
+ 4Sm 666cos Cyp ()b
4
0
2
sin
cos
6
6
+ V2mS 666
4
2 3
+
cos
cos
cos
cos
C
zq
0
C
zÆe
0
32
Ær
(sin sin + cos cos cos )
7
7
cos sin sin + cos cos sin sin sin cos cos 777
5
0
1
cos
3
(2.10)
3
2
_ 6 I pq + I qr 7 6I Cl(; )qSb + I Cn(; )qSb7
7
6
7 6
7 6
6
7
=
+
6
7
6
7
_ 6I pr I p r 7 6
I Cm ()qSc
7
4
5 4
5
I Cl (; )qSb + I Cn (; )qSb
r_
I qr + I pq
6p7
6 7
6 7
6q 7
6 7
4 5
2
1
5
6
3
2
2
2
4
6I3 Clp ()b + I4 Cnp ()b
V S 6
6
+
0
I Clp ()b + I Cnp ()b
6
6
4
2
4
0
6
6
qSb 6
6I7 CmÆ
e
6
4
()
2
60 cos 4 5=4
_
0 sin
cos
32 3
0
I3 Clr ()b + I4 Cnr ()b7 6p7
I7 Cmq ()c
76 7
7 6q 7
0
76 7
54 5
I Clr ()b + I Cnr ()b r
0
76 7
4
9
32
3
I3 ClÆa (; ) + I4 CnÆa (; ) I3 ClÆr (; ) + I4 CnÆr (; )7 6Æe 7
c
b
0
I ClÆa (; ) + I CnÆa (; )
4
2 3
_
6 7
9
9
0
2 3
4
7
8
2
+ 4
3
7 6 Æe 7
76 7
76 7
6 7
CyÆr 7
7 6Æa 7
54 5
3
6
6
6
6
6
4
2
C
xÆe
C
xq
cos sin CxÆe sin sin CzÆe cos CyÆa cos
0
0
0
2
+ Vg
32 3
0
()c +
()c
7 6p7
76 7
cos sin Cxq ()c sin sin Czq ()c cos Cyr ()b777 666q777
54 5
0
0
r
0
6
9
6p7
7
7
5 6q 7
6
7
cos 4 5
cos 9
76 7
76 7
7 6Æa 7
76 7
54 5
0
I ClÆr (; ) + I CnÆr (; )
4
9
Ær
(2.11)
3
sin 7 66
r
(2.12)
10
2.1
°üY \׺½õ x ; x
¢°.
1
2
¨WŸ ?Ý
2 R3 ; x3 2 R2 ÷¿ uÛ L, Cq³ u 2 R3 õ A
x1 = [; ; ]T
(2.13)
x2 = [p; q; r]T
(2.14)
x3 = [;
(2.15)
]T
u = [Æe ; Æa ; Ær ]T
(2.16)
Aý x ; x ; x ; u ‚ 0† é (2.10) (2.12) ‚ Aýý Ïô“Aéú °üY vÇ£ ½
°.
1
2
3
x_ 1 = f1 (; ) + g1 (; ; ; )x2 + g1a (; )x2 + h1 (; )u + f1g (; ; ; )
(2.17)
x_ 2 = f2 (; ; p; q; r) + f2a (; )x2 + g2 (; )u
(2.18)
x_ 3 = f3 (; )x2
(2.19)
æ é‚t f; g; h ™ é (2.10) (2.12) ¨ú ¿t‚ 0† A¢ ,÷¿t, f; g; h ¨
]ý: y® ¾Ÿ v 2.1 ‚ Aýüq °. ¨ ¾Ÿ™ V = 500 ft=sec; h = 10000 ft
EÍ, \׺½ °ü ©æ K‚t ºÜ£ : "X (singular value) /ÂY /™
÷¿ vÇ •°.
10Æ 30Æ
15Æ 15Æ
180Æ 180Æ
45Æ 45Æ
2 Backstepping
Ÿªú Ì¢ üxÍ ü±Cq
v
¨
f1
g1
x_ 1 g1a
h1
f1g
f2
x_ 2 f2a
g2
x_ 3
f3
2.1
11
Ïô“Aé u
]ý: y
8³ „ WŸ‹¡: *Û
šê¯â‚ ¢ šê –³
šê‚ U ™ WŸ‹¡: * Û
Cq8‚ U ™ WŸ‹¡: * Û
|³‚ ¢ –³
šê –³ „ WŸ‹¡: ?/! Û
šê‚ U ™ WŸ‹¡: ?/! Û
Cq8‚ U ™ WŸ‹¡: ?/! Û
¡Æ ‚ ¢ šê –³
max ()
5:910 10
1:729 10
7:785 10
9:333 10
6:880 10
2:731 10
1:128 10
5:649 10
1:414 10
1
0
2
2
2
1
1
0
1
min ()
1:294 10
1:710 10
0
0
1:516 10
8:101 10
4:576 10
2:241 10
1:000 10
0
0
2
1
2
1
3
12
2.2
CqŸ zG „ KA ©u
CqŸ zG „ KA ©u
2.2
Æf[‚t L²¢ CqŸ zG@:ù ¨WŸ \׺½ |‚t ‹ü, ˜y¤,
¾ ¿ sq• Ÿuˆ:ú 8[ êÀ ™ ,°.
$9, ;³ ˆÞxÍܟªú Ì Š CqŸõ zG ™ YAú L²© Ã. :,
CqÂ\ ü™ ;³º½™ x L, Cq³ù u °. é (2.17) ú Ã8 x èßð‚ Â
¢ u Cq³ ±µù h Ú, v 2.1 ‚t N ½ " h ¾Ÿ™ x èßð‚ ”?
–³ú yX™ °ô ¨‚ ü© ” ¾Ÿ °. c u õ Ì Š ”? x ú Cq¢°8
zGý Cq³ ¾Ÿ Í Á ,°. ” t¿, Ɗ:÷¿ ;³ ˆÞxÍܟªú
Ì Š ¨WŸ CqŸõ zG£ :‚™ Cq8 ºæ‚ © Œf ™ W³G½ *
Û Í °L A L, h = 0 ÷¿ zA¢°. ”ýL u ‚ ¢ Cq³ ±µú u
ú ½ ú :«“ x ¨ú è‚ Â© yÛ¢ °ü, GSý x ‚ ¢ Lò èßð‚
© CqŸõ zG¢°. 0†t “ªú :Ì Ÿ æ©t™ W³G½ 2ò Z™ 3ò \ yÛú Ì©b 6, ;³º½ x Y Cq³ u P [Gú L² Ÿ :[‚
nonminimum phase èßð‚ ©t™ 4Ù KAú Ã$£ ½ {°™ ³> °. üxÍ
¨WŸ E͂ minimum phase "ú ½¡:÷¿ ’< ™ ,ù Í q²Ï Æ6,
Z¢, Ç L ¨WŸ™ nonminimum phase "ú “™ EÍ °. Œ, Ÿª
ù zero dynamics KAú Ã$ Ÿ q·°™ ³> °.
¢`, é (2.17), (2.18) Y v 2.1 ú U_Ã8, x ; x èßð‚ ¢ = “ "ú Œ>£
½ °. é (2.17) ú Ã8 ¨WŸ šêõ 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& ,¤ ” ¾Ÿ h ‚ ü© Í ¾6 sC# ÆA ½u \ ¾Ÿõ °. 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 ¾Ÿê Í ¾Ÿ :[‚,
u x èßðú Cq ™ Cq³÷¿ :< °™ ,ú Ý£ ½ °. Œ, 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
+
+
Ÿªú 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± CqŸ uS
¢ Cq³÷¿™ x , x èßð‚ ¢ Cq³÷¿™ u ͽ¢ "ú “¨°.
Z¢, v 2.1 ú Ã8, x èßð‚ [´ý ¨ ¾Ÿ , x èßð‚ [´ý ¨ ¾ŸÃ° Í ¾°™ ,ú Ý£ ½ °. ,ù ]ý:÷¿ x èßð x èßð‚ ü© òŸ :
[‚ èßð timescale uÛþ ½ °™ ,ú y¢°.
Menon, Bugajski, Snell #ù ¢ two-timescale —ú Ì Š ¨WŸ CqŸõ u
 ™ uõ ½± •°. ù ¨WŸ èßðú ô ô"ú ™ èßðY —ÿ
ô"ú ™ èßð÷¿ Ûý L, èßð‚ ¢ CqŸõ 0¿ zG •°. “ª
‚t™ ¨WŸ ô:"‚ 0† ô ô"ú ™ èßð÷¿ x èßðú, —ÿ ô"
ú ™ èßð÷¿ x èßðú zA¢°. CqŸ™ ü# ؒ¿ uü™Ú, ½
Ùؒ‚t™ \׺½ x õ 蓍 { ¿ ºÜèÈ ½ °L A L, x <
º xd õ 8[ êÀ —ÿ ô"ú ™ èßð‚ ¢ Cq³ xd õ zG¢°. 4Ùؒ
‚t™ x ½Ùؒ‚t fý <º xd õ 0† êÀ ô ô"ú ™ èßð‚ ¢
Cq³ u õ zG¢°. ” 2.1 ù two-timescale Aú Ì¢ CqŸ uSõ #Í6°.
Ÿªù x èßð‚ ¢ Cq³÷¿ x õ PÌ Ÿ :[‚ ˆÞxÍܟªÃ° ¨W
Ÿ ô:"‚ Ì :¦¢ CqŸ zG“ª†L £ ½ °.
Ɗ:÷¿ CqŸõ zG£ : u õ Cq³÷¿ zA£ ½ ™ ,ù CqŸ u ú ¿ ºÜèÈ ½ Ÿ :[°. Two-timescale —ú Ì¢ ¨WŸ CqŸ zGYA
2
2
1
2
2
1
4, 5, 6
2
1
2
1
1
2
2
2
1
2
14
2.2
CqŸ zG „ KA ©u
‚t —ÿ ô"ú ™ èßð Cq³÷¿ \׺½ x õ zA£ ½ ™ ,ù, x è
ßð Í òŸ :[‚ —ÿ ô"ú ™ èßð‚t x õ ¿ ºÜèÈ ½ ™
îYõ uú ½ °L A°Ÿ :[°. 0†t CqŸªú :Ì¢ èßð KAú
½¡:÷¿ Ã$ Ÿ æ©t™ ô ô"ú ™ èßð Cqú Í À ÷¿ z
A©b ¢°. ” # ¨WŸ EÍ, ¢ “ª‚ ¢ KA ©u@Y uù À ¾Ÿ C
qú :Ì£ E͂™, Cq³ jÜ $# ½¡:÷¿ ?Ýü“ Mù mù s7½
–‹ èßðú • Š ‚ [´ý [C Œf£ ½ °.
¢ [C>ú •Ä Ÿ æ Š Æ f[‚t™ two-timescale Aú Ì “ ML
backstepping Ÿªú Ì Š üxÍ ¨WŸ CqŸõ zG ™ “ªú CK¢°. Backstepping Ÿªú Ì Š CqŸõ zG ™ YA‚tê, ½Ùؒ‚t™ x õ Cq Ÿ æ
Š x õ \ ³÷¿ PÌ 6, 4Ùؒ‚t x õ Cq Ÿ æ© u õ Cq³÷¿
PÌ¢°. ” # two-timescale ú Ì¢ CqŸª‚t 4Ùؒ ô:" ½Ùؒ‚
ü©t Í òŸ :[‚ ½Ùؒ‚t™ ” –³ #Í#“ M™°L A ™ ,Y™
µý, Æ f[‚t™ 4Ùؒ èßð ô:"Y 蓍 Šÿ‚ ¢ –³ú L² Š
CqŸõ zG L, x ; x ‚ © positive denite ¢ òb’d’ ¥½õ zA Š KAú
’<¢°. 0†t Ÿªù x õ Cq ™ Cq³÷¿ x õ PÌ¢°™ Q8‚t ¨W
Ÿ ô:"‚ :¦¢ CqŸ zGŸª6, x ô"ú L² Ÿ :[‚ ; èßð
KAú Ã$ Ÿ æ Š üÇì: Aú Ì “ M™°™ $>ú “L °.
2
2
2
1
2
2
1
2
1
2
2
2.2.1
ŸÆ A
CqŸ zG „ KA ©u‚ PÌý ŸÆ Aú Aý 8 °üY °.
A Ÿuˆ: xd = [d; d; d]T ™ î¢ ° Œ qE \½ cd > 0 ‚  Š °ü éú
T¢°
2.1
.
1
,
.
d
x ;
1
x_ d1 ; xd1 cd
(2.20)
2 Backstepping
Ÿªú Ì¢ üxÍ ü±Cq
15
ŠŸt kk ù vector Óù matrix 2-norm ú y¢°.
A ü± š³ „ ôSù ÆA °
.
2.2
V_
= 0; q_ = 0
(2.21)
A °ü <Cõ T ™ \½ m; m > 0 U&¢° jj < m; jj < m ?
; ‚ © f ; f g ; f ; f a ; f ; g ; g a ; g ; h ù î¢ 6 ; ‚ © yÛ °
A ˜X ¾Ÿ™ °üY C¢ý°
.
2.3
1
1
2
2
3
1
1
2
,
1
.
.
2.4
jj m < 2
2.2.2
(2.22)
CqŸ zG
°üù CqŸ zGYA‚ ›Å¢ ÃSAý°.
ÃSAý °ü <Cõ T ™ \½ m; m; m > 0 U&¢° jj m; jj m ; jj m õ T ™ ? ; ; ® ‚  Š g (; ; ; ) ù
°
æ ÃSAý 2.1 ù g Ž t¿ xÍëÆ ?ÛS&ú u Š ’<£ ½ °.
CqŸõ zG Ÿ æ Š °üY ù ¡ò \׺½ z ; z 2 R õ ꢰ. ŠŸt
xd ™ ½Ù‚t sq“™ x ‚ ¢ <º6, 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 uý°.
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
CqŸ 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
X‚t tÁ¢ AY ÃSAýõ Ÿ#¿ CqŸõ zG¢ @Y™ °ü Aý¿ vÇý°.
Aý é
èß𠩙
(2.25), (2.26)
2.1
÷¿ vÇü™ èßð‚t Cq³ u °üY Aü8 °
uniformly ultimately bounded
u = B2 1
ŠŸt 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<¥÷¿ ¾Ÿ 4¿ C¢£ ½ °
,
1
.
2
’< °üY ù òb’d’ ¥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&¢°. 0†t V < d Æ :, °ü Ù#éú T ™ \½ cg U&¢°.
1
1
g1
(; ; ; )
1
cg
1
1
1
(2.32)
2 Backstepping
Ÿªú 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 é‚t kh (; )uk ¾Ÿõ u ™ YA‚t™ u jÜ (saturation) üŸ :[‚ 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 ’<YA‚ Ìý°.
é (2.31) ‚t Aý V õ è‚ Â© yÛ 8 °üY °.
@V
@V
V_ = @z
z_1 +
z_
@z 2
1
(2.39)
2
é (2.25), (2.28) ú é (2.39) ‚  8 °üY ý°.
@V V_ = @z
f1 (; ) + g1 (; ; ; )xd2 + g1a (; )x2 + h1 (; )u + f1g (; ; ; ) x_ d1
1
+ @ V g (; ; ; ) x xd + @ V z_
@z1
1
@V
= @z
[
2
2
@z2
2
k1 z1 + g1a (; )x2 + h1 (; )u] +
1
@V
g1 (; ; ; ) x2
@z1
xd2
@V
+ @z
z_
=
k1 kz1 k2 + z1T g1a (; )x2 + z1T h1 (; )u + z1T g1 (; ; ; )z2 + z2T z_2
=
k1 kz1 k2 + z1T g1a (; )xd2 + z1T g1a (; )z2 + z1T h1 (; )u
+ zT g (; ; ; )z + zT z_
1
1
2
2
2
2
2
(2.40)
18
2.2
CqŸ zG „ KA ©u
é (2.26), (2.29), (2.30) ú é (2.40) ‚  Š Aý 8 °üY °.
V_ = k1 kz1 k2 + z1T g1a (; )xd2 + z1T g1a (; )z2 + z1T h1 (; )u + z1T g1 (; ; ; )z2
+ zT f (; ; p; q; r) + f a (; )x + g (; )u
2
2
2
@xd2
x_
@x1 1
=
@xd2
x_
@x3 3
2
2
@xd2 d
x
@ x_ d1 1
@xd2 d
x_
@xd1 1
(2.41)
k1 kz1 k2 + z1T g1a (; )xd2 + z1T g1a (; )z2 + z1T h1 (; )u + z1T g1 (; ; ; )z2
+ zT [A + B u]
2
2
=
2
k1 kz1 k2 + z1T g1a (; )xd2 + z1T h1 (; )u
+ zT g a (; )T z + g (; ; ; )T z + A + B u
1
2
1
1
1
2
(2.42)
2
é (2.28) ‚t f ù V ¥½t¿ é (2.41) ‚™ zT @x@Vd V_ ¨ j¥üqb “, Æ
f[‚t™ A 2.2 õ Ì Š ³¿Ü •°. C é (2.42) ‚ é (2.27) ú  L, é
(2.33), (2.34), (2.38) ú Ì 8 °ü [Géú uú ½ °.
1
2
2
V_ = k1 kz1 k2 k2 kz2 k2 + z1T g1a (; )xd2 + z1T h1 (; )u
k1 kz1 k2 k2 kz2 k2 + kg1a (; )k kz1 k xd2 + kh1 (; )uk kz1 k
h
k1 1 cg a cg
1
i
1
1
h
kz 1 k 2 + c g a c g
1
1
1
cf1 + cf1g
i
+ cxd + ch kz k
_1
1
k2 kz2 k2
1
é (2.45), (2.46), (2.47) ‚ Aý ci õ  8 °üY °.
V_ = k1 (1 c1 ) kz1 k2 + (c1 c2 + c3 ) kz1 k k2 kz2 k2
= k2 (1
k2 (1
1
c1 ) kz1 k2
k1
2 (1
c1 ) kz1 k
c1 c2 + c3
k1 (1 c1 )
2
+ (2ck c(1+ cc))
1 2
1
2
3
1
k2 kz2 k2
(c c + c )
(2.43)
2k (1 c )
õ é (2.48) Y A 8, /[:÷¿ V_ ‚ ¢ [Géú °üY u3 ý°.
V_ 2V + (2ck c(1+ cc))
(2.44)
1
c1 ) kz1 k2 k2 kz2 k2 +
1 2
2
3
1
1
1 2
1
2
3
1
2 Backstepping
Ÿªú Ì¢ üxÍ ü±Cq
19
æ é‚t ci (i = 1; 2; 3) ® ™ °üY Aü}°.
c1 = cg1a cg1 1
c2 = cf1 + cf1g
c3 = ch1
= min
(2.45)
+ cxd
(2.46)
_1
k1
2 (1
c1 ) ; k2
(2.47)
(2.48)
é (2.48) ‚t > 0 Ÿ æ©t™ c < 1 ú T©b ¢°. c ù g a ; g norm U÷
¿ ØqK ™Ú, é (2.17) ‚t g a ™ šê‚ © Œf ™ W³G½ ¨÷¿ Í ù m U©“Ÿ :[‚ ” ¾Ÿ Í °. Æ f[‚t PÌý W³G½ ?Ýú Ì Š ½X:÷¿ GS¢ @Y c < 0:13 }°.
é (2.44) ‚t V > kc c c c Æ : V_ < 0 t¿, z ; z ™ î¢ 6 °ü š¦ 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 ‚ ët¿, k ; k õ S< Š ½¶©æõ ¿ 3 ½
°.
æ Aý™ W³G½õ AÝ& NL ™ E͂ üxÍ ¨WŸ ; ; <ºú 8[£ ½
™ CqŸõ zG£ ½ üú ‹^¢°. ”ýL CqŸõ :Ì 8 <º8[ ¡ò
D ¿ ½¶ 6, ¡ò ¾Ÿõ zG S<£ ½ °™ Pìú Êu°. ŸU u®
™ µý æ Aý ’<YA‚t™ timescale separation Aú Ì “ MUŸ :[‚, KA
ú Ã$ Ÿ æ©t Cq ¾Ÿõ ›Å \÷¿ ¾3 zA©b £ ›Å {°. ¢
`, æ Aý ’<YA‚t™ °üY ù A 8 üq PÌü}°.
A B™
°
æ Aú*ù Ÿæ©té (2.30) úØL™é (2.17), (2.18) úU_Ã. é (2.11),
(2.18) ‚t g (; ) ™ šê p; q; r ‚ ¢ u Cq³ ±µ°. ¨WŸ Cq8ù 1
2.5
2
3
1
2
invertible
2
2
1
.
2
20
2.2
CqŸ zG „ KA ©u
ë:÷¿ 9 šêõ Cq êÀ zGü}Ÿ :[‚, sq• ¨WŸ W³Ò‚
© g (; ) ™ sC# invertible °. ,ù v 2.1 ‚t g (; ) /™ " 0 I
¦†™ Pì¿ê Ý£ ½ °. Z¢, é (2.10), (2.17) ú Ã8 h (; ) ™ Cq8‚ ©t
Œf ™ 9‚ ¢ * Û÷¿t, v 2.1 ‚t N ½ " g (; ) ‚ ü© ” ¾Ÿ Ÿ :[‚, é (2.30) ‚t B ‚ yX™ –³ Í °. 0†t B ™ ¨\ invertible °
L A£ ½ °. Æ f[‚t PÌ¢ W³Ò‚ 0† Š “ E͂  Š ½X:÷
¿ GS© Æ @Y B ™ sC# invertible °™ ,ú Ý£ ½ }°.
2
2
1
2
2
2
2
2.2.3
 ©u
òb’d’¥½õÌ ŠX<‚tzGýCqŸ‚©¡ò\׺½ z = [zT ; zT ]T
©æõ îê .
Aý é
÷¿ vÇü™ èßð‚t ¡ò \׺
½ z ™ °ü Ù#éú T¢°
1
2.2
2
(2.25), (2.26), (2.27), (2.28), (2.29), (2.30)
.
kz(t)k e
(t t0 ) kz
Z¢ ¡ò \׺½ z L1
,
norm
h
(t )k + p2ckc (1+ c c ) 1
1 2
0
3
1
1
ù °üY °
e (t t0 )
i
(2.50)
.
kz(t)k1 = max
(
kz(t0 )k ; p c1 c2 + c3
2k1 (1 c1 )
)
(2.51)
’< é (2.44) ‚ © °ü Ù#é ¢°.
.
2 + c3 )
V_ 2V + (2ck1 c(1
c)
2
1
1
æ é‚ ÙÀ‚ Aýüq ™ Aý B.1 ú :Ì¢°. é (B.6) ‚ c = c =
c c c ; = 0 ú  8 °ü Ù#éú u™°.
k
c
1
2
( 1 2 + 3 )2
2 1 (1
1)
kz(t)k e
(t t0 ) kz
h
(t )k + p2ckc (1+ c c ) 1
0
1 2
1
3
1
e (t t0 )
i
1
2
; c3
= ; =
2 Backstepping
Ÿªú Ì¢ üxÍ ü±Cq
21
æ é‚t N ½ " kz(t)k ™ #Ÿ kz(t )k ‚t ½¶ p ckc c c ÷¿ ³S’ , Z
™ ³S™¢°. 0†t kz(t)k /Âù #ŸY ½¶ /ÂY ÆX¢°.
1 2+ 3
0
2
1 (1
1)
é (2.50) ú Ã8, ¡ò \׺½ ½¶šêõ mŸ æ©t™ õ ¾3 zA©b ¢°™
,ú N ½ °.
2.3
½X èqªŽ
zGý CqŸ  ú ’ Ÿ æ©t V = 500 ft=sec; h = 10000 ft A\ ½eü
±\× F-16 ¨WŸ‚ °üY ù ; ; <ºú s}°.
c = 2:659Æ ; c = 0Æ ; c = 0Æ
0 t 1 (sec)
c = 10Æ ;
c = 0Æ ; c = 50Æ
1 t 10 (sec)
c = 0Æ ; c = 0Æ
10 t 20 (sec)
c =
2Æ ;
,ù pull-up, push-over ®  Ïô @¦ý Ÿôú y¢°.
A 2.1 ‚ 0† yÛ ¢ <ºú f Ÿ æ©t !n = 12; !n = 3; = 0:8 3ò
xÍèßð Í× command lter PÌü}°.
1
xd1 (s)
xc1 (s)
2
= (s + ! )(s !+n 2!n ! s + ! )
n
n
n
1
1
2
2
2
1
2
2
1
(2.52)
2
zGº½™ k = 3; k = 8 ¿ zA •L, 8³ù A\\× T = 1478:5 lbf õ ”¿
î“ •°.
” 2.2 ™ æ <ºˆ:Y zGº½‚ ¢ èqªŽ @Yõ #Í6°. ”‚t >
xù <ºˆ:ú y¢°. ”‚t Ã" CKý Cqèßð‚ :Ìý ¨WŸ™ ‹ü
„  <ºú Í 0† L ÷6, ˜y¤ù Is ù ¡ò©æ 4‚t <º
ú 8[ L üú N ½ °.
1
2
22
2.3
½X èqªŽ
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
Ÿªú 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 èqªŽ
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
Ÿªú 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 èqªŽ
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™ ¨WŸ èßð‚ Ì ™ ½ˆ# W³G½ ÝÝìú L² “ ML
CqŸõ zG •°. ¨WŸ ô:"ù üx́ 6, ¨WŸ Ïô‚™ Š “ ]ý: Ç\ Ä! 3 Ì Ÿ :[‚ ¨WŸ AÝ¢ ½¡: ?Ýú u™ ,ù Í
q²Ï Æ°. Z¢, two-timescale —ú Ì¢ üxÍ ¨WŸ CqŸ E͂™ W³G
½ ÝÝ쁂 © Cqèßð ÝKA©•°™ , uü}°. 0†t ¨WŸ C
qŸõ zG£ :‚™ ½ˆ „ ÝÝì –³ú L² ™ , |Å °. $‚t™ :
ÿCqŸªú êE翂 ÿÌ Š ¢ –³ú ™èř CqŸõ zG¢°.
2
8
W³G½ ?Ý ¡ò‚ ¢ –³
3.1
$‚t™ šê Ïô“Aé‚ Â¢ ?Ý ¡òõ L²¢°. é (2.18) ‚t A
ý f (; ; p; q; r); f a (; ); g (; ) ‚ ¢ ÝÝìú L² Š, 8Aý ¥½ú 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 é‚t = B [u u^] ¿ Aü}÷6, ,ù èßð ?Ý ¡ò‚ © Œf ™ ¨
°. Œ, f (; ; p; q; r); f a (; ); g (; ) ‚ ¢ ?Ý ¡ò‚ ©t, KA ’<YA
‚t òb’d’ ¥½ yۂ zT ¨ 8 ý°™ ,ú Ý£ ½ °.
2
2
2
2
2
3.2
êEç¿ uS
:ÿCqŸ¿ PÌ Ÿ æ¢ êEç¿ uSõ U_Ã. ” 3.1 ‚ #Í6 3-layer
êEç¿ù sq• ³ xÆnn 2 RN ‚  Š, °ü êÒ ynn 2 RN ú ;³¢°.
1
ynni
=
N2
X
j =1
"
wij N1
X
k=1
3
vjk xÆnnk + vj
!
+ wi
#
i = 1; 2; : : : ; N3
(3.4)
ŠŸt vjk ™ ¥«® ¥« W (layer) P @êL, wij ™ ¥«® ƒ¥« W P
@ê6, ™ bias õ y¢°. ”ýL Ni ™ i¥« W K‚ ™ ‘¢ ½õ y 6, () ™ °üY vÇü™ sigmoid activation ¥½õ #Í6°.
(z ) =
1
1+e
z
(3.5)
êEç¿ ;³ [Gõ #Í4™ é (3.4) ™ °üY ³¢ ±µÍ׿ vÇþ
½ °.
ynn = W T ~(V T xnn )
(3.6)
3
êEç¿ú Ì¢ üxÍ :ÿ ü±Cq
1
1
xÆnn1
xÆnn
.
.
.
.
.
.
”
N3 ; V
1
6w1 w11 w12
6
w21 w22
2
1
.
.
.
.
.
.
ynnN3
êEç¿ uS
N ; xnn 2 RN ® ~ : RN
3.1
2
4 .
..
ynn1
ynn2
N2
2 RN +1
2 +1
= 666w
N2
.
.
.
xÆnnN1
WT
1
2
2
ŠŸt W 2 RN
Aü}°.
29
..
.
..
.
3-layer
1 +1
2
3
7
7
777 ;
.. 5
.
2
7! RN +1 ™ °üY
2
2
6v1 v11 v12
VT
6
= 666v
4 .
..
v21 v22
2
..
.
..
.
3
7
7
777
.. 5
.
xnn = [1; xÆnn1 ; xÆnn2 ; : : : ; xÆnnN1 ]T
~(z ) = [1; (z1 ); (z2 ); : : : ; (zN2 )]T
¢ êEç¿ù ù§W ½, N ?Û& ¾8 ¥½õ Ù ™ AÝê¿
vÇ£ ½ °™ , N²K °. Œ, °ü Aýõ T¢°.
Aý
: RN 7! RN š¥½® N > 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¿
Š º½‚ U ™ Ä!¢ üxÍ ¥½õ :ù º½õ Ì Š AÝ& vÇ£ ½ .
30
3.3
CqŸ zG „ KA ©u
°™ ,ú y 6, êEç¿ Š Ûb‚ ÿÌüL ™ $ |Å¢ î°.
3.3
CqŸ zG „ KA ©u
:ÿCqŸªù èßð ;³ [Gõ Ì Š "A¢ º½õ ìè÷¿ 8A
¥÷¿, èßð \×# ½Ù ÞE ºÜ Ì†ê ’Æ¢  ú uú ½ êÀ Cq
èßð uSõ ºÜèř CqŸª°. Ÿªù 8A ™ º½‚ 0† ¾3 ”?
:ÿCqŸª (direct adaptive control) Y ? :ÿCqŸª (indirect adaptive control) ¿ uÛ
£ ½ °. ? :ÿCqŸªù èß𠺽õ ìè÷¿ 8A¢ °ü, 8Aý ìC º½†L A L CqŸ º½õ @A ™ Ÿª6, ”? :ÿCqŸ
ªù ”? CqŸ º½õ ìè÷¿ 8A ™ Ÿª°. Ÿªú Ì Š Cq
Ÿõ zG£ :‚™ Cqèßðú èßð‚# CqŸ º½¿ º½Ü ™ YA
j¥ý°. 0†t Ÿªù ³¢ xÍèßðú |î÷¿ Œ;ü}°.
Æ f[‚t™ üxÍ :ÿCqŸª (nonparametetric nonlinear adaptive control) ú Ì
Š CqŸõ zG¢°. Ÿªù èßð ;³ [Gõ Ì Š "A¢ ¥½ –P
(function approximator) ¿ üxÍ ô:?Ýú vÇ ™ Ÿª°. CqŸõ zG£ : Cq
èßðú º½Ü “ MŸ :[‚, Ÿªù Ä!¢ üxÍ èßð Cq‚ Ì :¦ °. X 3.1 <‚t™ W³G½ ÝÝ쁂 © ¨ 8 ý°™ ,ú ÕL, 3.2 <‚
t™ êEç¿ ¥½õ AÝ& vÇ£ ½ °™ ,ú Õ°. Æ f[‚t™ ¥
½ –P¿ êEç¿ú zA Š, êEç¿ ;³ ÝÝ쁂 © Œf ™ ¨
ú \© êÀ êEç¿ º½õ ºÜèÇ°. Ÿªù èßð ]ý: º
½ I¨ ¥½ –P º½õ 8A¢°™ Q8‚t ”? :ÿCqŸª¿ Ûêþ ½
ê “, Ɗ: ”? :ÿCqŸªY™ µý Cqèßðú º½Ü Š vÇ “
M™°™ " °.
3
êEç¿ú Ì¢ üxÍ :ÿ ü±Cq
31
ŸÆ A
3.3.1
:ÿCqŸõ zG ™ YA‚t™ 2 $‚t PÌ¢ A 2.1 2.5 ® ¥Í °üY ù
Aú Ì¢°.
A êEç¿ ³÷¿™ xnn = [xd; x ; x ]T ú PÌ 6 ³ù qE N ‚ Â
Š é ú TèÇ°
A é ú T ™ \: @ê™ î¢ 6 ” \¢©æ WM ; VM ú NL
°
3.1
1
(3.7)
3.2
1
,
2
.
(3.7)
,
.
kW kF WM ; kV kF VM
(3.8)
ŠŸt kkF ™ ±µ Frobenius normy ú y¢°.
3.3.2
CqŸ zG
Aý 3.1 ‚ © êEç¿ W³G½ ÝÝ쁂 © Œf ™ ¨ú AÝ& vÇ
êÀ ™ \: @ê U&¢°. ” # CqŸõ zG ™ YA‚t™ ÝÝì
‚ © Œf ™ ¨ ‚ [¢ AÃõ uú ½ {Ÿ :[‚, Aý 3.1 ú T ™ \:
@ê W; V õ GS£ ½ {°. 0†t, Cqèßð‚™ é (3.7) ú T ™ \:
^ ; V^ ú Ì 6, KAú Ã$£ ½ êÀ zGý :ÿªY‚ ©
W; V ‚ ¢ 8A W
^ ; V^ ú ºÜèÇ°.
W
CqŸ‚™ 8A W^ ; V^ PÌüt¿ \: @ê® 8A ò‚ ¢ –
³ Œf¢°. °ü ÃSAý™ KA ©uYA‚t \: @ê® 8A ò
‚ © Œf ™ –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
CqŸ 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)
ŠŸt 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 ‚  Š 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 Ÿu÷¿ 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  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)
0†t 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
ŠŸt c ™ j \½õ a 6, kAxk kAkF kxk †™ "ú Ì •°.
C, é (3.10) ‚ æ éú  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‚ © Œf ™ êEç¿ ;³¡ò é (3.9) ¿
vÇü6, ” ÆÙ w ¾Ÿ é (3.11) ¿ C¢ý°™ ,ú Êu°.
°ü ÃSAý™ CqŸ‚ 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
CqŸ zG „ KA ©u
(kx k + kx k)
1
(3.20)
2
kv max fC3 ; C4 g
(3.21)
ŠŸt ù j \½L 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 Š 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  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) ‚  Š 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)
0†t é (3.22) ’<ü}°.
X‚t tÁ¢ AY ÃSAýõ Ÿ#¿ :ÿCqŸõ 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°üYA 8 èßð©®êE翍@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 é‚t 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)
ŠŸt ; w ; v ™ j \½¿ ? zGº½° Z¢ ¡ò ½¶¢G™ k ; k ; ú S<¥÷¿ ¾Ÿ 4¿ C¢£ ½ °
’< °üY ù òb’d’ ¥½õ A .
.
,
1
2
.
.
V = 12 z1T z1 + 12 z2T z2 + 21
w
h
i
~ ~ + 1 tr V~ T V~
2v
h
i
tr W T W
(3.30)
¥½õ Ïô“Aé (2.25), (2.26) Y :ÿªY [Gé (3.28), (3.29) ‚ ©t yÛ¢ ù
é (2.42) õ Ì Š °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 é‚t 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
CqŸ zG „ KA ©u
é (3.32) õ é (3.31) ‚  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
æ‚t = B [u u] ¿ Aü}°. æ é‚ é (3.9), (3.28), (3.29) õ  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 œ©•°.
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 é‚t 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 4Z~
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)
0†t k 4C > 0 ú T êÀ k ; õ zA 8, æ é P¥« ¨‚ j¥ý ±µ
positive denite 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 fw ; v g
(3.38)
õ T êÀ õ xØ 8 /[:÷
V_ 2V + C5
(3.39)
æ é‚t V > C Æ : V_ < 0 t¿, z ; z ; Z~F ™ î¢ 6 °ü š¦ 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 ‚ ët¿, k ; k õ S< Š ½¶©æõ ¿ 3
½ °.
æ Aý™ W³G½ ?݂ ÝÝì ™ E͂ êEç¿ú Ì Š ; ; <º
ú 8[£ ½ ™ CqŸõ zG ™ YAú #Í6°. CqŸõ :Ì 8 <º8[ ¡
ò „ êEç¿ @ê ¡ò D ¿ ½¶ 6, ¡ò ¾Ÿõ zG S<£ ½
°™ Pìú Êu°.
Ɗ:÷¿ :ÿCqªY :Ìý èßðù ½ˆ, unmodeled dynamics „ èß𠺽 ô ºÜ‚ © ÝKA©— ½ °™ , uü}°. ¢ Ç\ù ” ق
0†t parameter drift, high-gain instability, fast adaptation, high-frequency instability ¿ uÛý
1
2
3
1
1
2
1
2
38
3.3
CqŸ zG „ KA ©u
°. Æ f[‚t™ ¢ ³>ú •Ä Ÿ æ Š :ÿªY (robust adaptive law) ¢ [ê -modication Ÿªú Ì •°. é (3.29), (3.28) ‚t w W^ ; v V^ ¨ù
-modication Ÿª‚ © 8 ý ¨÷¿t èßð ?Ý ¡ò‚ © :ÿªY º½ W^ ; V^ ŒS ™ ,ú ““ ™ ‹£ú 6, ™ ” –³ú S< ™ zGº½°.
-modication Ÿª ½‚ switching-, -modication Ÿª #ú :Ì£ ½ ÷6 :Ìý Ÿ
ª‚ 0†t ½¶—Y KA ’< YA SšB µ†•°.
20
20, 21
 ©u
3.3.3
òb’d’¥½õÌ ŠX<‚tzGýCqŸ‚©¡ò\׺½ z = [zT ; zT ]T
® êEç¿ @ê 8A¡ò Z~ ©æõ îê .
Aý é
¿ vÇü™ èßð‚t \׺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)
ŠŸt = 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  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 é‚t N ½ " kza(t)k ™ #Ÿ kza(t )k ‚t ½¶ C ÷¿ ³S’ , Z™
³S™¢°. 0†t kza(t)k /Âù #ŸY ½¶ /ÂY ÆX¢°.
0
2
3
1
5
2
é (3.40) ú Ã8, ¡ò \׺½ ½¶šêõ mŸ æ©t™ õ ¾3 zA©b ¢°™
,ú N ½ °.
3.4
½X èqªŽ
zGý CqŸ  ú ’ Ÿ æ©t ½X èqªŽú ½± •°. #Ÿ \׺
½ „ <ºˆ: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 ” ¾Ÿ™ v 3.1 ‚ Aýüq °.
” 3.2 ™ v 3.1 ‚ Aýý W³G½ ?Ý ¡ò U& ™ EÍ èqªŽ @Y
õ #Í6°. ”‚t >xù <ºˆ:ú y 6, ¡ >xù 2 $‚t îê¢ backstepping
Ÿªú Ì¢ CqŸ èqªŽ @Y°. ìxù $‚t îê¢ :ÿCqŸõ PÌ
¢ EÍ èqªŽ @Y°. ”‚t Ã" W³G½ ÝÝ쁂 ¢ –³ú L²
“ ML CqŸõ zG •ú :‚™, W³G½ ?Ý ¡ò¿ © Cqèßð  9 ý°™ ,ú N ½ °. Š8‚ êEç¿Y :ÿCqŸõ @¦¢ üxÍ :ÿCqŸ
™ W³G½ ?Ý ¡ò‚ ©  9 { Ÿuˆ: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 èqªŽ
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 èqªŽ
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 èqªŽ
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™ ½ˆ‚# W³G½ ÝÝ쁂 ¢ –³ú ™èş æ Š êEç¿
ú Ì¢ :ÿCqŸªú PÌ •°. $‚t™ üxÍ CqŸªú Ì Š ½
ˆ „ ÝÝ쁂 &¢ ü± Cqèßðú zG¢°.
3
4.1
CqŸ zG „ KA ©u
CqŸªù ÝÝì \½, è, Z™ \׺½ ¥½¿ vÇü™ "A¢ –‹
4‚ š©°L A L, ” –‹ 4‚t ’Æ¢  ú uú ½ ™ LAý Í× C
qŸõ zG ™ Ÿª°. 0†t CqŸªú Ì Š CqŸõ zG£ :‚™ ÝÝ
ì „ ½ˆ‚ ¢ –³ ¾Ÿõ yý @A©b ¢°. ÝÝ쁂 ¢ –³ CqŸõ
zG£ : 8A¢ ©æõ «q#™ E͂™ Cqèßð  „ KAú Ã$£ ½ {°.
:ÿCqŸ èßð ;³ [Gõ Ì Š CqŸ º½õ { ºÜèÅê
À zGü™ ,Y™ µý, CqŸ™ Cqèßð uSõ ºÜèœ M™°.
¨WŸ Cq[C‚t™ ÏÌ\× š¦‚  Š H1 ® ù xÍ CqŸª÷
¿ zGý Cqú ¨WŸ \׺½‚ 0† à Š PÌ ™ ߀‰ (gain
scheduling) Ÿª Gý PÌü}°. ® µý Æ f[‚t™ ÷L[¶ 10 @Yõ Ì Š üxÍ ü±CqŸõ zG •°. Æ f[‚t zG¢ Cq¨ù W³G½ Ý
Ý쁂 © Œf ™ ¨ú \© êÀ ºÜ ™ " °.
46
4
üxÍ ü±Cq
47
ŸÆ A
4.1.1
CqŸõ zG ™ YA‚t™ 2 $‚t PÌ¢ A 2.1 2.5 ® ¥Í W³G½
ÝÝì „ ½ˆ‚ © Œf ™ ¨ ¾Ÿõ C¢ ™ °ü Aú Ì¢°.
A W³G½ ÝÝì „ ½ˆ‚ © Œf ™ ¨ i ¾Ÿ™ 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)
CqŸ zG
°ü ÃSAý™ ÝÝ쁂 © Œf ™ ¨ú \© êÀ ºÜ ™ Cq¨ 
—ú #Í46 KA ’<YA‚ Ìý°.
ÃSAý 2 R ; ; ' 2 R ‚  Š °ü Ù#é ¢°
3
4.1
’<
.
+
.
k k '
k k + ' '
k k 3 '3 '
k k3 + 3 '3
; ' > 0 t¿
k k
k k + ' 1
¢°. 0†t éú Ì 8 é (4.2) Tý°™ ,ú Ö3 ÃÆ ½ °.
é (4.3) ú ’< Ÿ 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
CqŸ zG „ KA ©u
W³G½ ÝÝì „ ½ˆ‚ © Œf ™ ¨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)
X‚t tÁ¢ AY ÃSAýõ Ÿ#¿ CqŸõ zG¢ @Yõ Aý 8 °ü Aý®
°.
Aý é
¿ vÇü™ èßð‚t Cq³ u °üY Aü8 è
ß𠩙
°
(4.4), (4.5)
4.1
,
exponentially attractive
u = B2 1
ŠŸt 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)
ŠŸt = 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 òb’d’ ¥½õ L² .
.
V = 12 z1T z1 + 12 z2T z2
(4.12)
é (4.4), (4.7) ú é (4.12) ‚  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  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
CqŸ zG „ KA ©u
æ é‚ é (4.6) ú  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
ŠŸt 0 , h (; )u + °. æ é‚t h (; )u õ y“¨÷¿ s Š 0 ú A
•™Ú, h (; )u ™ ]ý:÷¿ ¨WŸ Cq8‚ © Œf ™ * Ûú y Ÿ :[
‚ ” ¾Ÿ 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)
0†t èß𠩙 exponentially attractive °.
æ Aý™ W³G½ ?݂ ÝÝì ™ E͂ üxÍ CqŸªú :Ì Š
; ; <ºú 8[£ ½ ™ CqŸõ zG ™ YAú #Í6°. Z¢, ÝÝì „ ½ˆ
‚ © Œf ™ ¨ ¾Ÿõ yý 8A£ ½ ú :, CqŸõ :Ì 8 <º8[ ¡ò
0 ÷¿ ½¶¢°™ ,ú Êu°. ¢`, æ Aý‚t™ °üY ù A 8 üq P
Ìü}°.
4
üxÍ ü±Cq
51
A °ü <Cõ T ™ \½ m; m; m > 0 U&¢° jj m; jj m; jj m õ T ™ ? ; ; ® ‚  Š g (; ; ; )+ g a (; ) ™
°
.
4.2
1
invertible
1
.
ÃSAý 2.1 ‚t N ½ " g (; ; ; ) ™ invertible °. Z¢, g a (; ) ™ šê
p; q; r ‚ © Œf ™ W³G½ *Û÷¿t, v ‚ Aýý ,¤ g (; ; ; ) ‚ ü©
” ¾Ÿ Í Ÿ :[‚ g (; ; ; ) + g a (; ) ¨‚t –³ Í °. 0†t æ
® ù Aú Ì£ ½ °.
1
1
1
1
4.1.3
1
 ©u
òb’d’¥½õÌ Š, X<‚tzGýCqŸ‚©¡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™ èßð‚t ¡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  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¶šêõ mŸ æ©t™ ; õ ¾3 zA©b ¢°
™ ,ú N ½ °.
52
4.2
4.2
½X èqªŽ
½X èqªŽ
zGý CqŸ  ú ’ Ÿ æ©t ½X èqªŽú ½± •°. #Ÿ \׺½
„ <ºˆ:ù 2.3< 4ÌY ÆX¢°. zGº½™ k = 3; k = 8; = 0:05; = 0:1 ¿ z
A •°. W³G½ ?Ý ¡ò™ ¿ f •L, ” ¾Ÿ™ v 4.1 ‚ Aýüq °.
” 4.1 ù v 4.1 ‚ Aýý W³G½ ?Ý ¡ò U& ™ EÍ èqªŽ @Y
õ #Í6°. ”‚t >xù <ºˆ:ú y 6 ¡ >xù 2$‚t îê¢ backstepping
Ÿªú Ì¢ CqŸ èqªŽ @Y°. ìxù $‚t îê¢ CqŸõ PÌ
¢ EÍ èqªŽ @Y°. ”‚t Ã" CqŸª‚ ¢ Cqèßðù W³
G½ ?Ý ¡ò U& ™ EÍ‚ê  9 { Ÿuˆ: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 èqªŽ
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 èqªŽ
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 èqªŽ
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 ¨WŸ CqŸõ zG ™ “ªú CK L, CKý zG“ª‚
¢ ¨WŸ èßð KAú ½¡:÷¿ ©u •°. "&, üxÍ :ÿCqÁY üx
Í CqÁú :Ì Š ?Ý ¡ò „ ½ˆ‚ © &¢ "ú #Í4™ ¨WŸ C
qŸ zG“ªú Cè •°. Z¢, Æ f[‚t CK¢ CqŸ zGŸªú F-16 üxÍ ¨W
Ÿ ?݂ :Ì Š ½X èqªŽú ½±¥÷¿, ”  ú *’ •°. ” @Yõ A
ý 8 °üY °.
1.
2.
¨WŸ W³G½ ?Ý ¡ò „ ½ˆ U& “ M™°L A L, backstepping
Ÿª‚ © üxÍ ¨WŸ CqŸõ zG ™ “ªú CK •°. Ÿªù ŸU
‚ üxÍ ¨WŸ CqŸ‚ PÌüqµÏ ˆÞxÍܟªY µý, W³G½ Lò y
Û¨ú PÌ “ M÷6 nonminimum phase system ‚ê :Ì£ ½ °™ " °.
Z¢, /–‚ PÌü™ timescale separation Aú Ì¢ ¨WŸ CqŸ zGŸªY µ
ý KAú ’< Ÿ æ© üÇì: Aú PÌ “ M™°. Œ, Æ f[‚t CK
¢ üxÍ ¨WŸ CqŸ zGŸªù ¨WŸ ô:"ú :<& ßÌ¢ CqŸ zG
Ÿª6, CqŸ zGŸª :Ìý ¨WŸ èßð KAú ½¡:÷¿ ’<£
½ °™ $> °.
üxÍ :ÿCqÁY êEç¿ú Ì Š ¨WŸ W³G½ ?Ý ¡ò „ ½
ˆ‚ © &¢ üxÍ ¨WŸ CqŸõ zG ™ “ªú CK •°. Ÿªù
êEç¿ universal approximation "ú Ì Š êEç¿ ½ˆ „ ?Ý ¡
ò‚ © Œf ™ ¨ú AÝ& vÇ£ ½ °L A L, êEç¿ ;³ ½
59
60
ˆ „ ?Ý ¡ò‚ © Œf ™ ¨ú \© êÀ êEç¿ @êõ :ÿC
qªY‚ © ºÜèř Ÿª°. W³G½ ?Ý ¡ò „ ½ˆ U& ™ E
͂ê <º8[ ¡ò uniformly ultimately bounded °™ ,ú ’< •°.
3.
üxÍ CqÁú Ì Š ¨WŸ W³G½ ?Ý ¡ò „ ½ˆ‚ © &
¢ üxÍ ¨WŸ CqŸõ zG ™ “ªú CK •°. Ÿªù ½ˆ „ ?Ý
¡ò‚ © Œf ™ ¨ ¾Ÿ "A¢ ©æ K‚ j¥ý°L A L, ” –‹ K
‚t &¢ "ú #Í4™ CqŸõ zG ™ Ÿª°. W³G½ ?Ý ¡ò „
½ˆ U& ™ E͂ê <º8[ ¡ò exponentially attractive °™ ,ú ’< •°.
°üY ù u ³ó ½±üqb ¢°L fý°. Æ f[‚t üxÍ ¨WŸ ?Ý
CqŸõ zG£ :‚™ ¨WŸ Ïô“Aé üxÍ ¨ú \© (cancellation) L KA
ú æ¢ xÍ Cq¨ú 8 •°. Ïô“Aé ¨ù ¨WŸ ô‹¡: "ú #Í4
L ÷t¿ üxÍ ¨ KAú ©X™ Å÷¿ Ì “™ M™°. 0†t üxÍ ¨ú
\© ™ Í× CqŸõ zG Ÿ ð™, \׺½ ºÜ‚ 0ô W³G½ ºÜ "
ú ßÌ $# °ô Í× òb’d’ ¥½õ Ì Š ¨WŸ ]ý: "ú ?Û&
ßÌ ™ CqŸõ zG ™ “ªú u©b £ ,°. Z¢, Æ f[‚t™ :ÿCq® CqŸªú Ì Š W³G½ ¡ò‚ © &¢ "ú #Í4™ CqŸõ zG •°. Cqèßð &‚ –³ú yX™ ř¿™ ÝÝì ½‚ unmodeled dynamics
°. ¨WŸ E͂™ “Ÿ „ ôŸ ô‹¡: "‚ ©  JÜþ ½
÷t¿, ¢ –³‚ © &¢ "ú #Í4™ CqŸõ 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.
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62
[8] Schumacher, C. and Khargonekar, P. P., Missile Autopilot Designs Using
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Appendix A
F-16
¨WŸ ?Ý
[?/!
A.1
F-16
¨WŸ ôLA _vG (body xed axis) ‚ ¢ [?/!™ °üY °.
Ix = 9496 slug ft2
Iy = 55814 slug ft2
Iz = 63100 slug ft2
Ixz = 982 slug ft2
Ii ™ [?/!¿ uý ¨÷¿t v A.1 ‚ Aýüq °.
A.2
W³G½ ?Ý
WŸ‹¡: *Y ?/!™ °üY ‹ü, ˜y¤, šê, Cq³‚ ¢ ¥
½¿ vèý°. F-16 ¨WŸ W³G½ ?Ýù ÷L[¶ 19 Òõ Ì •÷6, Æ f
[‚t™ Cq³ W³G½ ?݂ 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
¨WŸ ?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
¨WŸ ?Ý
67
v
A.1
[?/!¿ uý ¨ 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
òb’d’ KA Á
Appendix B
A
°ü <Cõ T ™ \½ B; d > 0 ® T = T (Æ) > 0 U& 8 yۓAé
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<CõT ™\½ ; d > 0 ® = (Æ) U& 8yۓAé 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 ‚  Š V (x) °ü [Géú T ™ òb’d’ ¥½†L 8
,
c1 kxk2 V (x) c2 kxk2
(B.3)
@V
f (t; x) c3 kxk2 + e t
@x
(B.4)
èß𠩙 °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 c2c 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
ŠŸt c ; c ; c ; ; ™ j \½°
’< òb’d’ ¥½ è‚ Â¢ 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 c2c 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 c2c e
1 3
2
t0 h
2
e
(t t0 )
2
e
c t t )i
0
c
3
2 2(
CqŸ „ Ïô“Aé yÛ
Appendix C
C.1
2, 3
2, 3
$‚t PÌý CqŸ 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 é‚t 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 CqŸ yÛ
$‚t PÌý CqŸ 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 é‚t 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
=
=
=
=
=
=
@g1 (; ; ; ) + g1a (; )
R2 (x1 ; x3 )
@
@g (; ; ; ) + g1a (; )
R2 (x1 ; x3 ) 1
R2 (x1 ; x3 )
@
@g (; ; ; ) + g1a (; )
R2 (x1 ; x3 ) 1
R2 (x1 ; x3 )
@
@g (; ; ; ) + g1a (; )
R2 (x1 ; x3 ) 1
R2 (x1 ; x3 )
@
@f1 (; ) @f1g (; ; ; ) @v1
k1 I33
+ @x
@x1
@x1
1
@f1g (; ; ; )
@x3
R2 (x1 ; x3 )
2 R31 ù é (4.10) ‚t °üY zGü}°.
v1 =
:, @x@v
1
1
1 k1 k2
Æ
k1 k3 + 3 '3 1
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
2
1 k1 k2 @Æ1
k1 k3 + 3 '3 @x1
3
2
1
1
3
C
CqŸ „ Ïô“Aé yÛ
C.3
73
Ïô“Aé yÛ
Ïô“Aé 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 ]
Ïô“Aé 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 coefcients 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