ඕЪѷീԫఙ̈́ዋᑕڱనࢍ˘࣎Ξតඕၹଠטጡ
ޙᎸ
ϖ྿ԫఙጯੰ̄ր
ၡ! ࢋ
ώ͛ͽՂܠᏚ͈ᘦؠّநኢࠎૄᖂĂඕЪѷീ̈́ዋᑕڱĂనࢍ˘
າ۞ΞតඕၹଠטጡĂͽֹߙ˘ᙷ۞צᕘրͽ྿јᘦؠଠט۞ϫ۞Ą͛
̚ᖣϤѷീԫఙֽҤീצଠրٙዎצז۞Ϗ̝ۢᕘજณĂТॡͽ˘࣎צ ዋᑕڱٙଠט۞ᆧৈีֽྃᐺдֹϡྍѷീԫఙॡٙΞਕயϠ۞Ҥീᄱ मĂซ҃ܲᙋٙనࢍ۞ଠטጡਕֹܳצଠրซˢАٙఢထ۞ึπࢬĂ֭
Яֹ҃Ꮾᜩᑕѣૻิ۞ᘦؠপّĄޢ၆˘࣎ಏᕚրซҖ̶ژᄃሀ ᑢĂͽᙋځٙ೩۞ଠט͞ڱߏΞҖ۞Ą
ᙯᔣෟĈΞតඕၹଠטăึπࢬăѷീăዋᑕڱĄ
A VARIABLE STRUCTURE CONTROLLER DESIGN BASED ON GREY PREDICTION AND ADAPTIVE LAWS
Chien-Hsing Chou
Department of Electronic Engineering Yungta Institute of Technology and Commerce
Pingtung, Taiwan 909, R.O.C.
Key Words: variable structure control, sliding hyperplane, grey prediction, adaptive law.
ABSTRACT
Based on the Lyapunov stability theorem, a novel variable structure controller incorporated with a grey prediction scheme and an adaptive law is presented for a class of perturbed systems to achieve stability control. By using the grey prediction scheme, we can directly predict the unknown perturbations. The adaptive component is used to compensate for the uncertain prediction errors between the real value and the predicted value.
It is also shown that the proposed control scheme ensures robust stability after the controlled system enters a pre-designed sliding hyperplane. Finally, a pendulum system is given to demonstrate the feasibility of the proposed control scheme.
˘ă݈! ֏
“Ξតඕၹଠט”(variable structure control)[1]ߏ˘ᖎ ಏ֭ѣૻิ(robust)পّ۞ଠט͞ڱĄдצଠրዎצѣ ኜ к ᕘ જ (perturbations) ۞ ଐ ڶ ˭ Ă ּ т Ĉ ણ ᇴ ۞ ត ̼ (parameter variations)ăրሀё۞̙ቁؠ(model uncer-
tainties)̈́γొ۞̒ᕘ(external disturbances)ඈĂᑕϡѩඈଠ טጡĂΞֹצଠրԣిͷѣड़г྿јְАٙనࢍăఢ ထр˞۞જၗᜩᑕҖࠎ(dynamic response performance)Ă֭
ͷдצଠրซˢึሀၗ(sliding mode)̝ޢĂӈΞ၆Ч
̽੨ё(matching)[2]۞ᕘજѣૻิ۞ͅᑕপّ[3,4]Ą็
˯ĂΞតඕၹଠטጡ۞ᑕϡ݈ᗟĂߏࢋਕАҖۢր
ٙዎצז۞ᕘજᇴࣃ̝̂̈ࠎң[5-7]ĂҭϤٺ˯۞Ч ёᕘજ֭ܧߏ˘ј̙ត۞ܫཱིĂιࣇــົᐌᒖဩᄃॡ ม҃ѣٙតજ(time varying)Ăٙͽ˘ਠ၆Ξតඕၹଠט҃
֏Ăдଠטጡ۞నࢍᄃፆү̝݈ĂӈۢଉᕘજٙΞਕயϠ
۞͞ёᄃតજ۞ቑಛĂߏ˘ีᅲࠎᚑॾ۞୧ІࢨטĄЯѩĂ ၆Ξតඕၹଠטጡ۞݈ࢨטซҖ࣒ϒᄃԼචߏ˘ีࣃ
ΐͽࡁտ۞ኝᗟĄ
ѷրநኢ(grey system theory)[8]˜ߏ˘ྵາ۞ր
நኢĂιߏд 1982 ѐϤ઼̂̚ౙ۞ጯ۰ዒჸᐷିٙ
೩Ąιᄃ็நኢࢋ۞मҾдٺѷրநኢᄮ ࠎĂՏ˘ඊٙפ۞ྤफ़࠹̢มӮѣࡶ̒۞ᙯᓑّ
(relevancy) х д Ă ̙ ᑕ Ϊ ߏ ༊ ј ࠹ ̢ ϲ ۞ (independent)ಏ৷ᐌ፟(random)ྤफ़ֽ࠻ޞĄܕѐֽྍந ኢ̏జᇃھгᑕϡдЧ̙Т۞ᅳા̚Ă̚ѣᙯٺ
ീ(prediction)͞ࢬ۞ᑕϡՀᒔధкˠ۞ڦຍ[9-11]Ąѷ
ീ۞ԫఙᄃԔĂࢋ˜ߏӀϡ͌ณ̏ۢ۞ྤफ़Ăགྷ ᇴፂந(operation)ăᇴፂϠј(generation)ĂϠј۞ᇴ ፂޙϲѷҒሀݭ(grey model)ĂГӀϡ̈π͞ڱ(least square technique) ᒔ פ ሀ ݭ ۞ ણ ᇴ (coefficient) ҃ ̝ [12]ĄϤٺٙᅮࢋ۞ྤफ़ณࠤ͌Ăٙᅮྻϡ۞ᇴጯሀёᖎ ಏĂ̙ᅮࢋᖳಱ۞ࢍۢᙊĂՀ̙ᅮࢋᝥҾצଠր۞
ลᇴ(order)ĂΪࢋዋ༊۞ၡפٙᅮࢋ۞ྤफ़֭ΐͽࡶ̒۞
ፆү̈́ྻზӈΞĄЯѩѷീܧ૱ዋЪϡֽଵੵ݈Ξ តඕၹଠטጡдᑕϡॡ็˯۞˘ֱࢨטĄ
ᑕϡѷീٺΞតඕၹଠטጡॡĂ༊Ξ၆хѣ̙ቁؠ Я৵۞צᕘրઇീଠט۞ड़ਕĂͽԼච็Ξតඕ ၹଠטጡ۞˘ֱࢨטĄҭϤٺീڱѺ̶̝Ѻгჟ
ᄱĂٕкٕ͌ົѣֱధ۞ᄱमхдĂֱ҃ᄱम۞̂̈˵
̙ߏдְАಶΞΐͽפ۞Ąٙͽֶ็۞͞ڱĂనؠ
˘࣎ѣྵ̂۞̷ೱଠטᆧৈ(switching control gain)۞ྃ
ᐺี(compensation)ĂͽഇਕҹڇྍีᄱमٙΞਕֽ۞ᇆ ᜩĂߏ˘ۡତͷѣड़۞͞ڱĂҭѩᓝ๕υົᆧΐᏜ۞
ଠטΑதঐਈ֭Тॡΐᆐᝫજ(chattering)۞ன෪Ąࠎ˞ਕ ࢫҲѩ̙υࢋ۞࿅ณ(excessive)ଠטᏮˢĂΐˢ˘࣎ਕዋ ॡгአፋᄃ࣒ϒྃᐺี̂̈۞ዋᑕّ(adaptive)ᆧৈĂߏྋ ՙયᗟ۞р͞ڱ[5,7,13]Ą
дώኢ͛ٙ̚ࢋࡁտăଣ۞̰टĂࢋ˜੫၆Ξត ඕၹଠטጡ็˯۞ࢨט୧ІĈۢᕘજณ۞ֹ̂̈̈́ϡ
࿅ณ۞ྃᐺᆧৈĂΐͽ࣒ϒᄃԼචĄࢵА͔ϡѷീԫ ఙֽҤീצଠրᕘજณ۞̂̈Ăͽೲੵдֹϡྍᙷଠט ጡॡυืࢋۢٙዎצז۞ᕘજ̂̈ቑಛࠎң۞ࢋՐĂ
֭ͽ˘࣎צዋᑕڱٙአଠ۞ᆧৈֽྃᐺă࣒ϒдֹϡྍ
ѷീԫఙᄃԔॡٙΞਕயϠ۞ᄱमĂซֹ҃ፋ࣎צ ଠ ր ѣ Ӯ ̹ г ࣃ ࠎ ѣ ࢨ (uniformly ultimately bounded)۞ّኳ[14,15]ĄޢགྷϤቑּ۞ሀᑢᄃ̶ژֽ
ᙋځӍඈٙ೩۞ଠט͞ڱĂੵ˞ਕྋੵ݈۞ࢨט୧І
҃྿јଠט̝ϫ۞γĂ֭ͷ̪ቁܲፋ࣎צଠրѣૻิ
۞ᘦؠ(robust stability)পّĄ
˟ăր̝ೡᄃన
ώ͛ଣ˘ѣ̙ቁؠᕘજЯ৵۞צᕘրĂ
ᇴጯೡёт˭ٙϯĈ
) (
) ( )) ( ( ) ( )) ( ( ) (
u , x , v
u x , B B x x , A A x
t
t t t
t t
+
∆ + +
∆ +
=
&
(1)
̚x(t)∈Rnࠎր۞ېၗតᇴ(state variable)ШณĂ֭ͷ
న Ч ࣎ ې ၗ ត ᇴ ̝ ᇴ ࣃ ̂ ̈ Ξ ͽ གྷ Ϥ ณ ീ ҃ ז (measurable) ću(t)∈Rmࠎ ଠ ט Ꮾ ˢ ܫ ཱི (control input signal) Ą A∈ Rn×n ࠎ ̏ ۢ ۞ ې ၗ ੱ (state matrix) Ă
n×m
∈ R
B ࠎ̏ۢ۞႕৩(full rank)Ꮾˢੱ(input matrix)ͷ (A, B)ѣΞଠטّ(controllability)Ą
ࠎ˞ֹր(1)ਕૉ྿јᘦؠଠט̝ϫ۞ĂӍඈన
ྍצଠր۞ᖐߛၹЪͽ˭۞ࢨט୧ІĄ
(˘) ∆ A(⋅,⋅)ă∆ B(⋅,⋅)ᄃv(⋅,⋅,⋅)̶Ҿܑ̂̈Ϗۢҭᇴࣃࠎ ѣࢨ۞ాᜈё(unknown but bounded continuous)ણᇴ
۞ត̼ăሀё۞̙ቁؠăγొ۞̒ᕘٕր۞ܧቢّ
ొЊ(nonlinearity) [16]Ą
(˟) хд࠹၆ᑕዋ༊ჯޘ(dimension)ăϏۢҭాᜈ۞ב ᇴ f(⋅,⋅,⋅)Ă ֹ ˭ Е ۞ ̽ ੨ ё ୧ І ਕ ૉ ј ϲ [1-7, 11,13]Ĉ
) , , (
) ( ) , ( ) ( ) , ( ) , , (
u x v
u x B x x A u x Bf
t
t t t t t
+
∆ +
∆
= (2)
ѣ˞˯Е۞୧І̝ޢĂତ(2)ёˢ(1)ё֭གྷዋ༊۞ፋ நĂΞт˭۞րજၗܑϯёĈ
f B u B x A
x&= + + (3)
ώቔኢ͛ٙࢋࡁտăଣ۞ϫᇾĂдٺ̙ᅮࢋְА
ۢᕘજณ̂̈۞݈ᗟ˭ĂඕЪѷീԫఙ̈́ዋᑕڱĂ నࢍ˘࣎ѣૻิপّ۞ΞតඕၹଠטጡĂͽֹܳт(3)
ٙϯ۞צᕘצଠրĂͽ྿јᘦؠଠט۞ϫ۞Ą
ˬăึπࢬ͞ё
дనࢍΞតඕၹଠטጡॡâਠΞ̶ࠎ̂ՎូĄࢵ
А ಶ ߏ ၆ צ ଠ ր న ࢍ ˘ ࣎ ዋ ༊ ۞ ึ π ࢬ (sliding hyperplane)Ąдώኢ͛̚Ăٙᅮࢋ۞ึπࢬ͞ёజన
ࢍࠎĈ
∫ + ⋅
−
= t d
t) 0( )
( Cx C A BK x τ
s ăs(t)∈Rm (4)
ё̚C∈Rm×nߏ˘࣎૱ܼᇴ႕৩ੱĂᏴפ۞ֶፂߏֹ
)
(CB−1ͽхдࠎࣧĂ҃૱ܼᇴੱK∈Rm×n۞Ᏼפ
ֶፂĂߏͽֹੱ(A+BK)̝পᇈࣃ(eigenvalue)Бᇴळ རдνΗኑπࢬ(left-half complex plane)˯ࠎࣧć˵ಶߏ
ֹ˭Е۞̙ඈёĈ
0
<
)]
+ ( [λ A BK
Re (5)
ਕૉјϲࠎࣧĄ༊ੱC ̈́ K ̶Ҿజֶ˯۞͞ڱ
ٙᏴפ̝ޢĂГགྷϤዋ༊۞ଠטጡٙጱ͔ĂΞֹт(3) ёٙϯ۞צଠրĂజᜭඉҌٙనࢍ۞ึπࢬ˯Ă֭
ͷҋѩ̝ޢዸдྍπࢬ˯Ăซ҃யϠٙഇ۞જၗᜩ ᑕҖࠎĄ
நኢ˯Ă༊צଠրซˢึπࢬ̝ޢĂΞጱ˘
࣎т˭ٙϯ۞ඈड़ଠטጡu (equivalent controller)[1]Ĉ eq f
Kx
ueq= − (6)
ྍඈड़ଠטጡ(6)ˢ(3)ё̚[17]֭གྷ࿅˘ֱ̼ᖎՎូ۞
நޢĂྍצଠր۞ඈड़ౕਫ਼ྮ(closed-loop)જၗ͞ё
ົѣт˭̝ԛёĈ ) ( ) ( )
(t A BK xt x& = +
Ϥ˯ёΞг࠻âόצଠրజጱ͔ซˢึπࢬ
̝ޢĂྍր۞જၗᜩᑕҖࠎ̙Гצז̙ቁؠᕘજี۞
ᇆᜩĄЯѩĂዋ༊ͷϒቁгᏴፄ˘࣎Ξͽ႕֖(5)ё۞ੱ
KĂГགྷଠטڱ(6)۞ጱ͔ĂӈΞܲᙋፋ࣎צଠրߏ႙ ܕᘦؠ۞(asymptotical stable)ć˵ಶߏᄲΞͽֹצଠր྿
јᘦؠଠט۞ϫ۞Ą
дԆјึπࢬ͞ё۞నࢍ̝ޢĂତ˭ֽ۞̍үӈ ߏଯጱ˘࣎ዋ༊۞ଠטጡĂͽഇдྍଠטጡ۞ጱ͔̝
˭Ăਕૉᒔٙనࢍ۞ଠטड़ڍĄдѩ͔ϡѷրநኢ
ֽ൴णӍඈٙᅮࢋ۞ଠטጡĄ
αăѷീ
ѷրநኢߏ˘າͷໂ൴णሕ˧۞ଠט͞ڱĂΪ
ࢋචΐӀϡٙᒔ۞ѣࢨྤफ़ĂѷീӈΞ၆ኑᗔͷੈि
̙Ԇፋ۞ր൴೭ീ۞ਕ˧Ăࢋ۞ჟលӈдٺԱ
ᇆᜩր۞ણᇴĂ֭ޙϲᇴጯᙯܼёĄ
ྻϡѷീԫఙॡĂࢵАӈߏࢋפ˘ඊࣧؕ۞ྤफ़ ᇴЕ(original data sequence)Ĉ
{ (0)( )| =1,2, , }
(0) d k k N
d = L
̚N ܑྤफ़۞࣎ᇴͷࠎ˘ѣࢨ۞ᇴࣃĄତ၆ྤफ़
ᇴ ፂ d(0)ઇ ˘ Ѩ ΐ Ϡ ј ྻ ზ (accumulated generated operation)ĂӈΞᒔт˭ٙϯ۞ΐϠјᇴЕd(1)Ĉ
{ (1)( )| =1,2, , }
(1) d k k N
d = L
҃ࣧؕᇴЕd(0)ᄃΐϠјᇴЕd(1)̝ᙯܼࠎĈ
∑=
= k
i
i d k d
1 (0)
(1)( ) ()
னдϤͅШ۞៍ᕇֽ࠻Ăࡶ၆d(1)ઇഴڱྻზĂѣт
˭۞ᙯܼёĈ
1) ( ) ( )
( (1) (1)
(0) k =d k −d k−
d
ѩӈܑϯd(0)ߏd(1)۞ഴϠјྻზ(inverse accumulated generation operation)ᇴЕĄѣ˞˯Е۞ྤफ़ᇴፂޢĂତ
ӈΞޙϲ˘࣎ GM(1,1) ሀݭĂְ҃၁˯ιಶߏ˘࣎˘ล
̶͞ё(first-order differential equation)Ĉ
[ ] a d c
t d d
d + ⋅ (1)=
(1)
(7)
̚a Ⴭࠎ൴णܼᇴ(developing coefficient)Ăc ჍࠎѷҒᏮ ˢ(grey input)ĄࠎዋቁгଯҤᇴፂd(1)۞൴णᔌ๕ĂΞ Ӏϡ̏ۢ۞ྤफ़ᇴፂ̈́̈π͞ڱ(least squares method)ֽ
פણᇴa ̈́ c ۞ᇴࣃ[12,18]Ă҃ٙᅮ۞ྻზё̈́Ч࠹
ᙯ۞ણᇴࣃ̶Ҿт˭ٙϯĈ
(N ⋅N ) ⋅N ⋅D
=
− T
B T B
c B
a 1
(8a)
( )
( )
( )
+
−
−
+
−
+
−
=
1 ) ( 1) 2 (
1
1 ) 3 ( ) 2 2 ( 1
1 ) 2 ( ) 1 2 ( 1
(1) (1)
(1) (1)
(1) (1)
N d N d
d d
d d
B
M M
N (8b)
[d(0)(2) d(0)(3) L d(0)(N)]T
=
D (8c)
̚N Ⴭࠎΐੱ(accumulated matrix)ĂD ჍࠎܼᇴШB ณ(coefficient vector)Ąдనୁؕ୧І(initial condition)ࠎ
) 1 ( ) 0 ( (0)
(1) d
d = ۞ଐڶ˭Ăྍ˘ล̶͞ё(7)۞ᇴҜԛ ё۞ྋΞܑࠎĈ
a e c a d c
k
d ⋅ ak+
(0)− −
= 1) +
( (1) ˆ(1)
ତ၆ᇴЕdˆ(1)ઇഴϠјྻზĂܮΞᒔт˭ٙϯ۞
ഴϠјᇴЕdˆ(0)Ĉ
( )ea d ac e ak
k
d ⋅ −
−
⋅
−
=
+1) 1 (1)
( (0)
ˆ(0)
N
k=2,3,L, (9)
ѩёӈࠎࣧؕྤफ़ᇴፂ۞ѷീ̳ёĄٙͽĂӀϡ̏ۢ۞
ྤफ़ᇴፂĂགྷѷീ̳ё(8)ă(9)۞ྻზĂӈΞീ˭˘
ඊྤफ़۞ᇴࣃĄдώቔኢ͛̚ĂӍඈᝑг(iteratively)
ֹϡາᒔ۞̣ඊྤफ़ᇴፂĂֽീ˭˘ඊӈன۞
ྤफ़ᇴࣃĄ
̣ăΞតඕၹଠטጡ۞నࢍ
˘ਠ҃֏Ăࠎ˞ֹึ୧ІsT(t s)&(t)<0ͽјϲĂ дፆүΞតඕၹଠטጡ̝݈ӈ̏ۢᕘજٙΞਕត̼۞ቑ
ಛĂ˜ߏ˘ІЪͼ็۞న݈೩ĄҭϤٺצଠրώ
֗ඕၹ̈́ٙᒖဩ۞ኑᗔޘֹĂְАӈ̏ۢրᕘજ ณ̂̈ࠎң۞న݈೩Ăــڱд၁ᅫ۞ଠטրٙ̚
ቁϲĄࠎ˞ਕҹڇѩࢨטĂӍඈଳϡѷീԫఙֽྋ ՙ̝Ą
ࢵАĂޙϲ˘࣎т˭۞ᇾր(nominal system)Ĉ )
( ) ( )
(t t t
nominal Ax Bu
x& = +
˯ёᄃ(3)ё࠹ഴޢĂΞĈ ) ( ) ( )
( )
(t xt xnominal t Bf t
d ∆ & − & = (10)
Ϥٺ݈̏నր۞ېၗតᇴ x(t)ΞགྷϤณീ҃זͷ րੱAăB ࠎ̏ۢĂ߇ӍඈΞӀϡ(10)ё֭གྷϤཝפ ᇹԔ۞ፆүᄃᐂĂ҃פྻზॡٙᅮࢋ۞̣ඊྤफ़ᇴ ፂ {d(t−5Ts),d(t−4Ts),d(t−3Ts),d(t−2Ts),d(t−Ts)} Ă
̚۞T ࠎ˘࣎ໂ̈۞ϒᇴĂܑീԔٙ̚నs
ؠ ۞ ീ ม ॡ ม Ą ତ ٙ פ ۞ ྤ फ़ ᇴ ፂ Е
{d(t−nTs),n=1,2,3,4,5}నࠎֹϡѷീԫఙॡٙᅮࢋ۞
ࣧؕྤफ़ᇴЕĂГӀϡٺ݈༼ٙ̚ౘ۞ѷീԔ̳̈́
ё(8a-c)ᄃ(9)Ăͽפᕘજี f(t)۞Ҥࣃfest(t)ć˵ಶ ߏᄲր۞ᕘજี f(t)ΞᖣϤӀϡ͌ᇴ̏ۢ۞ྤफ़ᇴ ፂ֭གྷѷീ۞ԫఙᄃԔ҃జീࠎ fest(t)Ąдѩࣃ
˘೩۞ߏĂώീڱ̙֭צטٺᕘજณ۞̂̈Ăٙͽ ώڱ၆ٺᕘજѣໂૻ۞ዋᑕਕ˧Ą
д ၁ ᅫ ۞ ྻ ϡ ҂ ᇋ ˯ Ă ې ၗ ጱ ב ᇴ x&(t−hTs) ă
{1,2,3,4,5}
∈
h ۞ৌ၁ᇴࣃϫ݈̪ڱۡତᒔĂҭߏ
) (t−hTs
xˆ& Ă˵ಶߏx&(t−hTs)۞ҤീࣃĂݒΞགྷϤ˘ֱ͞
ڱଂ̏ۢ۞ېၗࣃx(t−hTs)҃ז[19,20]Ąְ၁˯ĂЯࠎ
̏ѣ˘ֱܫཱིந۞͞ڱ(signal processing methods)Ăּ
т Ă Ӏ ϡ ࠹ Ҝ ᕭ گ ጡ (zero phase filter) ă ͐ প ա ਬ (Butterworth)ᕭگጡඈĂΞϡֽפ̏ۢېၗតᇴ۞ጱבᇴ ࣃĂٙͽפ x&(t−hTs)۞Ҥീࣃྵפx&(t)ֽᖎಏ [20]ĄѩγĂд͛ᚥ[21,22]̚Ϻኢ̈́ӍˠΞགྷϤણ҂ሀё
៍ീጡ(model reference observer)ֽࢦޙր۞ېၗតᇴͽ
̈́ېၗតᇴ۞ጱבᇴĄ͛ᚥ[23]ኢӀϡᏮˢᏮ៍ീ
ڱ(input-output observation)ͽᒔפր۞ېၗតᇴ۞ጱב ᇴࣃĄٙͽགྷϤͽ˯۞͛ᚥ͔̈́ᄲځΞͽۢĂ͛ٙ̚
ᅮ۞̣ඊᇴፂྤफ़ΞజЪந۞ᒔĂ҃ЯീٙΞਕயϠ
۞ᄱमĂӍඈనЪ˭Е۞୧І[19]Ĉ x
f
f− est ≤ f0+f1 (11)
̚ f ̈́0 f ̶ҾܑϏۢ۞ϒࣃ૱ᇴ(positive constant)Ą 1 дనࢍΞតඕၹଠטጡॡĂ఼૱ᅮࢋ͔ˢ˘࣎ѣ֖
ૉ̂ଠטᆧৈ۞̷ೱྃᐺีĂͽԺטϏۢ۞̙ቁؠีٙΞ ਕֽ۞̙։ᇆᜩĂซֹ҃ፋ࣎ଠטրЪՂܠᏚ
͈ᘦؠّநኢ(Lyapunov stability theorem)۞ࢋՐĄҭࢋ
ۢྍീᄱमณ۞̂̈˜ߏ˘І̙͉ΞਕԆј۞ЇચĂ˵
ಶߏᄲдώּ̚ࢋۢ f ̈́0 f ۞̂̈˜ߏ˘І࠹༊ӧᙱ1
۞ְĂЯѩֹϡྵ̂۞ଠטᆧৈߏڱᔖҺ۞Ą҃ྵ̂
̷ೱᆧৈ۞ྃᐺӈຍקྵк۞ਕঐਈᄃᚑࢦ۞ᝫજன ෪Ă҃ᝫજໂΞਕົ͔൴րϏሀё̼ొЊ۞ᐛᜩᑕ [20]Ăٙͽࠎ˞ഴ͌ߏีЯ৵ٙΞਕயϠ۞̙։ᇆᜩĂӍ ඈଳϡ࣎Ξត۞ዋᑕᆧৈ(adaptive gain)fˆ0(t)̈́fˆ1(t)
ֽአዋăᔌܕྍ࣎Ϗۢ۞૱ᇴf ̈́0 f Ă҃ྍዋᑕᆧৈ1 ᄃ၁ᅫ૱ᇴม۞मࣃؠࠎĈ
0 0 0(t) f (t) f
f = ˆ −
~ Ăf~1(t)= ˆf1(t)−f1
ტЪͽ˯۞ኢĂͽ˭ӈ೩ώ͛ٙనࢍ۞Ξតඕၹ ଠטጡĈ
[ˆ ˆ x ]
B C s s C B
s CB f
Kx u
) ( + ) ( )
(
) (
1 0 T 1
T T
1
t f t f
est g
−
−
−
−
−
=
(12)
̚g ࠎ˘࣎Ξϡֽአፋଠטჟޘ۞ϒࣃଠטᆧৈĂ
festࠎགྷϤ݈ѷീٙീ۞ᕘજณĄٙᅮࢋ۞ዋᑕڱ
ࠎĈ
[ ()]
)
( T 0
0 t f t
f s CB ˆ
ˆ& =α −θ
(13a)
[ ()]
)
( T 1
1t f t
f s CB x ˆ
ˆ& =β −η
(13b)
̚α, β, θ̈́η̶Ҿܑϒࣃ૱ᇴĄ
ତ˭ֽĂӍඈᙋځඕЪௐα༼ٙ̈́̚۞ѷീ
ԫఙ̈́ዋᑕڱ(13)҃ז۞Ξតඕၹଠטጡ(12)ĂΞֹ
т(3)ёٙϯ۞צଠրѣӮ̹гࣃࠎѣࢨ۞ّ
ኳ[14,15]ĄѩγĂࠎᔖҺٺ̶ژଯጱ࿅ٙ̚ΞਕயϠ۞
ኑᗔّĂТॡϺ̙εΝ˘ਠ̼۞পّĂనੱC ͽజዋ༊۞ᏴפĂͽֹC = ֽᖎ̼˘ֱႊზăଯጱ۞B I ԔĄ
ؠநĈ҂ᇋт(1)ёٙϯ۞צᕘրĂд႕֖݈͛ٙ̚೩
̈́۞Чีన̝ޢĂྍրΞଯႊј(3)ё̝ٙ
ϯĄࡶଳϡ(12)ё۞ଠטጡᄃ(13)ё۞ዋᑕڱ֭Т ॡҋѷീԔ̳̈́ё(8)ă(9)פրᕘજี f
۞ീࣃ festĂѩצଠրົѣӮ̹г
ѣࢨ۞ّኳĄ
ᙋځĈࢵАᏴؠ˘࣎ѣϒؠཌྷّ(positive definite)۞Ղܠ
Ꮪ͈͞ё
( 2 1 12)
0 1
2 T
1 f f
V ~ ~
s
s + − + −
⋅
= α β (14)
(4)ё၆ॡม t ̶֭ඕڍˢ(3)ё̝ޢĂΞ
f Kx u
s&= − + ĄѩॡГ(14)ё၆ॡม t ̶֭ˢ˯ϯ s&
۞ܑϯёĂΞ
) (
)
( 1 0 0 1 1 1
T f f f f
V& ~ ~& ~ ~&
f x K u
s − + + − + −
= α β (15)
ତ̶Ҿଠטጡ(12)ᄃዋᑕڱ(13)ˢ(15)ёĂགྷፋந ޢΞ
[ ]
[ ]
{ } { [ ]}
[ ]
[ ]
{ ()} {[ ()]}
) ( + ) ( )
(
) ( )
(
) ( + ) ( )
(
1 T
1 0 T 0
1 0 T 1 T
1 T
1 1 0
T 0 1
1 0 T 1 T
t f f
t f f
t f t f g
t f f
t f f
t f t f g
V
est est
x ˆ
~ s s ˆ
~
ˆ x s ˆ
s s f f s
x ˆ
~ s s ˆ
~
f x K ˆ x
s ˆ s s f x K s
η θ
η β
β θ
α α
− +
− +
− − −
=
− +
− +
− − − − +
=
−
−
−
& −
னд(11)ёˢ˯ё֭གྷซ˘Վ۞̼ᖎޢĂΞጱ˭Е
۞ᙯܼё
( ) ( )
( ) ([ )] ( ) ([ )]
( ) ( )
( ) ( )
) (
4 4 2
2
2
2 1 2 0 2 1 1 2 0 0 2
1 1 1 0 0 0 2
1 1 1 0
0 0
1 0 1
0 2
t g
f f f
f f
f g
f f f f f f g
f f f f
f f
f f f
f g
V
Γ s
~ s ~
ˆ ˆ ˆ
s ˆ
x ˆ s s ˆ
ˆ ˆ
ˆ x s ˆ x s
s
−
−
=
+ + +
− +
−
−
=
−
−
−
−
−
=
+
−
− + +
−
− +
+
− + +
−
≤
η θ η
θ
η θ
η θ
&
̚ () ( 2) ( 2) 2 4 12 4 0
2 1 1 2 0
0 f f f f f
f
t =θ ~ + +η ~+ −θ −η
Γ Ą༊
≠0
θ ăη≠0ॡĂ΄Γ(t)=0ĂΞд~f0(t) f~1(t)
− πࢬ˯
ထ˘࣎ፚԛ۞ѡቢĄॲፂ[14,15]ٙ೩۞ؠநΞۢĂ ፋ࣎צଠրѣӮ̹гࣃࠎѣࢨ۞ّኳĄ
ොĈ
(˘) ଂ݈ᙋځ۞ՎូᄃඕڍΞۢĂ༊ዋᑕ૱ᇴజᏴؠࠎ
=0
=η
θ ॡĂጱΓ(t)=0Ă֤ᆃдs(t)≠0ॡĂົ
ѣV&<0Ąѩӈܑϯ༊t→∞ॡĂs(t)ᔌܕٺĂ
҃s(t)ᔌܕٺ۞ຍཌྷӈܑϯצଠրజᜭඉҌึ
πࢬ˯Ą˘όրซˢึπࢬ̝ޢĂтТдௐ ˬ༼ٙ̚ઇ۞ኢĂրົѣ႙ܕᘦؠ۞ّኳĄ (˟) ༊Ᏼؠθ ≠0ăη≠0ॡĂࡶᆧৈᄱम(~f0(t),f~1(t))
۞ ᇴࣃརдፚѡቢΓ(t)=0̝γॡĂܑΓ(t)>0Ăѩ ॡ୮ႷયгΞۢV&<0Ąࡶᆧৈᄱम(f~0(t),~f1(t))
۞ᇴࣃརдፚѡቢΓ(t)=0̝̰ॡĂܑΓ(t)<0Ă
֤ᆃд˭ЕડมĈ
s(t) <[−Γ(t) g]12 (16)
ົזV&>0Ą
ტЪ(˘)ᄃ(˟)۰۞̶ژඕڍΞۢĂώ ଠטր
ົ ѣ ˘ ࣎ ۞ ќ ᑦ ቑ ಛ (ultimate bound)
[ () ]12
)
(t Γ t g
s = − Ą
གྷϤ(16)ё݈̈́۞ᄲځΞͽۢĂѣ˘ֱΞҖ۞͞
ڱΞϡֽᆧซώଠט۞ჟޘĄۡ̚ତ۞͞ڱӈߏአ
̂ଠטᆧৈ૱ᇴg ۞ᇴࣃ̂̈ĄځពгĂֹϡྵ̂ᇴࣃ۞
ଠטᆧৈ gĂົѣྵ̈۞ќᑦડมĄᔘѣಶߏଳϡྵ̈
۞ീมॡมT Ăಶ˘ਠ۞གྷរ˯ֽᄲĂֹϡྵ̈۞s
ീมົѣྵָ۞ീჟޘć˵ಶߏᄲΞֹ f ᄃ festม۞मࣃត̈Ă߇҃Ξֹќᑦቑಛᒺ̈ĄΩ˘ߏ
mg Y
X T
ဦ 1 צଠ̝ಏᕚր (l1 l0=0.5Ăg l0=10Ăml02=1)
ഴ̈ዋᑕ૱ᇴθ̈́η۞ᇴࣃĄϤΓ(t)۞ܑϯёΞ࠻Ă༊θ
̈́η႙႙ត̈ॡĂΓ(t)ϺТॡត̈ĂҋгΞֹќᑦቑಛ Ϻᐌ̝ᒺ̈Ąޢ༊θ=η=0ĂΓ(t)=0Ăѩॡࡶs≠0Ă
V&<0Ăѩӈܑϯրѣ႙ܕᘦؠ۞ّኳĄ҃ࣃ
ڦຍ۞ߏĂѩ͞ёົࢫҲ(13)ё۞ዋᑕਕ˧ĂЯࠎ
≠0
s ጱ(13)ёܑٙϯ۞ዋᑕᆧৈfˆ0(t)̈́fˆ1(t)႙႙г ᆧ̂ĂޢΞਕౄјεଠ۞ᚑࢦޢڍĄٙͽనࢍ۰υื
дଠטჟޘᄃዋᑕਕ˧˯ઇᝋᏊפĂͽഇਕЪଠט ጡనࢍ۞ࢋՐĄ
̱ăཝሀᑢቑּ
дώ༼̚ĂӍඈ၆˘࣎ᕚܜࠎl(φ)۞ಏᕚրซҖ
Ҝཉଠט۞ሀᑢᄃ̶ژĂͽរᙋώଠט͞ڱ۞ّਕĄྍ
ಏᕚր۞ඕၹтဦ 1 ٙϯĂ֭ѣт˭۞̶ܑϯё[5]Ĉ
{ }
) )cos(
( ) (
)]
cos(
1 )[
( 10 )]
cos(
5 0 1 )[
sin(
5
0 2
φ φ
φ φ
φ φ φ
φ
t v
T sin
. .
+
∆
+ +
− +
= &
&&
̚Ϗۢ۞̙ቁؠี̈́̒ᕘี̶ҾࠎĈ ]2
2 ) [cos(
25 0 )
( = +
∆φ . φ Ăv(t)=2cos(3t)
ॲ ፂ ր ۞ প ّ Ă ې ၗ ត ᇴ Ш ณ Ξ జ ؠ ཌྷ ࠎ
[1 2]T=[ ]φ φ&T
∆ x x
x Ă҃ଠטᏮˢࠎu(t)=TĄགྷϤ
˯ٙઇ۞నؠĂྍಏᕚր۞ېၗܑ͞ϯёΞజጱ
ࠎĈ
[ ( ) (, , )]
1 0 ) ( 0 0
1 0 )
(t xt ut f t xu
x +
+
=
&
̚
{ }
{0105[[11 0cos(5cos()]sin()]sin() )[1} (( ))] ()()cos(( ) )
) , , (
1 1
2 2 1 1
1 1
1 1
x t v x x x x
x t u x x
x f
+
∆ +
+
∆
∆
− + +
−
=
⋅
⋅
⋅
. .
ࠎԆјᘦؠଠט۞ϫ۞ĂΞតඕၹଠטጡٙᅮࢋ۞ึ
π ࢬ ͞ ёs(t)ӈ ֶ ( 4 ) ё ֽ న ࢍ Ă ̚ ֭ Ᏼ ؠ
[ ]0 1
=
C ̈́
−
= −
7 12
0
K 0 ĄϤٙᏴפăޙၹ۞ੱ K φ
l(φ)=l0+ l1cos(φ)
2 1 0 -1 -2
5
0 10
states 1:
2:
sec.
ဦ 2 צଠր̝ېၗྫဦ (1Ĉx1(t)Ă2Ĉx2(t))
0.3 0.2 0.1 0.0
-0.10 5 10
sliding
sec.
ဦ 3 ึπࢬ͞ёs(t)۞ྫဦ
20
10
0
-100 5 10
controller
sec.
ဦ 4 ଠטᏮˢܫཱི
20 10 0 -10
-200 5 10
controller
sec.
ဦ 5 ็Ξតඕၹଠטጡ۞Ꮾˢܫཱི
ΞͽۢRe[λ(A+BK)]<0јϲĄତĂֶ(12)ёనࢍ
ଠטᏮˢܫཱིu(t)֭̚నؠg=1Ąଠטጡٙ̚ᅮࢋ۞ዋ ᑕڱ˜Ϥ(13)ёٙޙϲĂዋᑕ૱ᇴ̶Ҿజᖎಏгన ؠࠎα=β=1ăθ=η=1Ă҃ festߏӀϡᑢᇾր
20
10
0
-10
4
0 8 12 16
controller
sec.
ဦ 6 ˘ਠ PID ଠטጡ۞Ꮾˢܫཱི
60 30 0 -30
-600 4 8 12 16
controller
sec.
ဦ 7 ̙ቁؠ۞ᕘજี̶Ҿдௐˬࡋॡᆧ̂ࢺГٺௐ
̱ࡋॡᆧ̂ˬࢺ۞ଐڶ˭ĂࠎԆјᘦؠଠט̝ϫ۞
ٙᅮࢋ۞ଠטᏮˢܫཱི
1:
0.010 2:
0.005
0.000
4 8 12 16
x(t) norms
sec.
ဦ 8 ѣ̙Т۞ଠטᆧৈg ॡĂx(t) ̂̈۞ͧྵဦ(1Ĉ
=1
g Ă2Ĉg=5)
nominal u
+
=
1 0 0 0
1
0 x
x& ĂགྷϤཝפᇹԔ۞ፆүᄃ
ᐂ֭ඕЪ͛ٙ̚ኢ۞ѷീԔᄃ(8)ă(9)ёٙᒔפ۞
ᕘ જ ീ ࣃ Ą ޢ న צ ଠ ր ۞ ୁ ؕ ୧ І ̶ Ҿ ࠎ
[ 2 0]T
(0)=π
x ̈́[fˆ0(0) fˆ1(0)] =[0 0]Ă֭ͽ .0010 ࡋ ࠎࢍზ፟ሀᑢॡࢍზᄃפᇹ۞มॡมĂٙሀᑢ۞ඕڍ̶
Ҿणϯٺဦ 2 Ҍဦ 14Ą
ဦ 2 ٙणϯ۞ߏצଠր۞ېၗྫဦĂϤဦ̚Ξ࠻
צଠրѣ։р۞ଠטඕڍĄဦ 3 ٙणϯ۞ߏึπ ࢬ͞ёs(t)۞ᇴࣃྫဦĂϤဦ̚Ξ࠻ྍึ͞ё
̝ᇴࣃ̂̈Ăܲд˘࣎ໂ̈۞ቑಛ̝̰Ąဦ 4 णϯ˞ࠎ
