非線性系統之模糊適應穩定廣義預測控制

(1)

ܧቢّր௚̝ሀቘዋᑕᘦؠᇃཌྷ࿰ീଠט

ૺฮ௨

઼ϲ๔ৈԫఙጯੰҽࣜ۩አրᓾर

ች୻Ѱ

઼ϲ̚Ꮈ̂ጯ࿪፟̍඀ጯրି଱

ၡ! ࢋ

ώ͛ўд൴णዋϡٺ˘পؠᙷݭܧቢّր௚̝ሀቘዋᑕᘦؠᇃཌྷ࿰ീଠ ט(fuzzy adaptive stable generalized predictive control)̝నࢍ͞ڱጯĄώ͞ڱඕ Ъ˘׍ѣણᇴҤീΑਕ̝Takagi-Sugeno ሀቘሀݭĂͽ̈́˘׍ѣ੠ᖸଠטّਕ

̈́ܲᙋᘦؠّ̝ᘦؠᇃཌྷ࿰ീଠטڱ݋҃ј̝າݭሀቘዋᑕڱ݋ĄགྷϤଠט׌

੼ޘܧቢّצଠវ۞࿪ཝሀᑢඕڍĂΞᑭរ෠၁ώ͛ٙ೩ଠטඉர׍ѣᐹள۞

ԩ̒ᕘ(disturbance rejection)ᄃనؠᕇ੠ᖸ(setpoint tracking)ඈপّĄ ᙯᔣෟĈሀቘሀݭăዋᑕଠטăᇃཌྷ࿰ീଠטĄ

FUZZY ADAPTIVE STABLE GENERALIZED PREDICTIVE CONTROL FOR A CLASS OF NONLINEAR SYSTEMS

Ya-Ling Chang

Department of Refrigeration and Air Conditioning National Chin-Yi Institute of Technology

Taichung, Taiwan 411, R.O.C.

Ching-Chih Tsai

Department of Electrical Engineering National Chung-Hsing University

Taichung, Taiwan 402, R.O.C.

Key Words: adaptive control, fuzzy modeling, generalized predictive control.

ABSTRACT

This paper develops methodologies for an adaptive stable generalized predictive control with fuzzy modeling for a class of nonlinear systems.

This new type of controller is composed of a fuzzy model with on-line parameter estimation, and a stable generalized predictive control with good tracking, guaranteed stability and disturbance rejection capability.

Numerical simulations for controlling two highly nonlinear processes are

described. These show the excellent disturbance rejection and setpoint

tracking performance of the proposed control method.

(2)

˘ă݈! ֏

ధктЍ࿪ă࿪̄ăྤੈăϠۏԫఙĂ؉Ѽඈ੼ࡊԫ யຽ̙ᕝ౹າᄃ̿৺Ăтңଠטᄦ඀ซ҃யϠᐹ։ݡኳ̝

யݡ˜ߏቁܲயຽјΑ۞ᙯᔣԫఙĄ൒҃Ăд఺ிк۞ҋ જ̼ᄦౄ̍඀྆·ϋ඾ధкኑᗔ۞ܧቢّଠטયᗟĂࡶͽ

็௚ჟቁ۞ቢّޙሀᄃր௚ଠטԫμޝᙱᒔ଀΄ˠ႕ຍ۞

ඕڍĄЯѩᑕϡܧቢّଠטă࿰ീଠטăᙷৠགྷშྮᄃሀ ቘଠטඈ͞ڱĂͽྋՙኑᗔ۞ܧቢّયᗟซ҃଀זٙᅮ۞

ଠטّਕ̏ߏϫ݈̍ຽ፟ጡᄃ඀Ԕଠטᅳા͔྆੓ᇃھᎸ

኷۞ࡁտࢦᕇ[1-3]Ą

д઼࡚ΐэߦҹ඿̂ጯϑᇇ(Zadeh)ି଱ٺ 1965 ѐ൴

ܑĶሀቘะЪķ۞ኢ͛ޢĂሀቘநኢᄃ׎ᑕϡٺ஘൴णĄ ᚶࡻ઼ࣖ೶̂ጯMamdani ି଱ٺ 1974 ѐĂјΑгซҖᄐ ঈ͔ᑜ۞ሀቘଠט၁រޢĂሀቘଠט˘ۡజᇃھгᑕϡྋ ՙ̍ຽࠧ۞ܧቢّր௚ଠטયᗟĄт̫ሀቘଠט۞நኢࡁ տ̏ు႙јሢĂͷ̏ѣ̙͌ඕၹԆፋ۞३ᚱ඾ү[4-8]Ąѩ γĂٺ1985 ѐ੓ĂሀቘទᏭ˵јΑгజϡͽޙϲೡࢗր௚

જၗҖࠎ̝ሀቘޙሀ(fuzzy modeling)͞ڱ [7,8]ĂJang [7]

ᄃͳ[8]ඈጯ۰࠰෠၁ሀቘր௚၆ٺኑᗔͷܧቢّצଠវ ޙሀ࠹༊ѣड़ΞҖĄдሀቘޙሀ۞ᑕϡ͞ࢬĂ̏ѣிкј Α۞९ּĂּтŠkrjanc ̈́ Matko дܧቢّሤϹೱצଠវ

۞९ּ˯ֹϡ Takagi-Sugeno ݭёሀቘޙሀᄃሀݭ࿰ീଠ ט(model predictive control)ඉரඈ͞ё଀ז࠹༊ᐹ෸̝ଠ ט ј ड़[9] Ą Mollov ඈ ࡁ տ ጯ ۰ ೩ ΍ ܧ ቢ ّ ඀ Ԕ ۞ Takagi-Sugeno ݭёሀቘሀݭᄃ࿰Ҥଠטጡ۞ЪјĂጱ࡭˘

࣎ᘦؠิّ۞ౕਫ਼ྮଠטր௚[10]ĄϤѩΞۢሀቘր௚၆ ٺޙϲܧቢّצଠវ̝ሀݭ׍ѣ၁ᅫΞҖ۞Αड़Ą

Ϥࡻ઼ͱߺ̂ጯClarke ඈጯ۰ٙ೩΍۞ᇃཌྷ࿰ീଠט ߏሀݭ࿰ീଠטԫఙ̚౵૱జֹϡ۞ԫఙ̝˘ĂТॡϤٺ ιߏܧ૱टٽజ˞ྋᄃྻზેҖ۞ڱ݋Ă߇д̍யຽࠧ̏

јࠎ౵צᝌܓ۞ሀݭ࿰Ҥଠט͞ڱ̝˘[11,12]ćͷ̏జј Αг၁Җдధк̍யຽࠧ۞ᑕϡ˯[1-3,13]Ă׎ᐹᕇࠎĈ(1)

׍ঐੵր௚ᘦၗᄱम̝ਕ˧Ă߇၆ٺሀݭ̙̽੨̈́Ϗሀݭ

̼̝જၗѣ˘ؠ඀ޘ̝ૻઉّć(2)ጾѣధк၁ᅫአፋ୊

ตĂਕአፋր௚Җࠎ֭Լචր௚ّਕć(3)׍ѣ࿰ീࣃࢍზ ΑਕĂ߇Ξΐ˯Ꮾ΍ˢࢨט̝ଠטયᗟĄᔵ൒ᇃཌྷ࿰ീଠ טጾѣధкᐹᕇĂҭιޝ͌ଣ੅ր௚ᘦؠّ̝યᗟĂ߇ Rossiter ඈˠٺ 1992 ѐ೩΍ᘦؠᇃཌྷ࿰ീଠטڱ݋[14]Ă

֭ྻϡᘦؠа଱ଠטጡ̝পّА྿זր௚̰ొ۞ᘦؠĂГ

ેҖᙷҬᇃཌྷ࿰ീଠט͞ڱĂᖣͽܲ঻ᄃ೩چྍڱ݋̝ᐹ

։পّĄ

ώ͛۞ϫ۞ߏ੫၆̍ຽࠧ૱֍۞ܧቢّր௚۞ଠט યᗟĂඕЪሀቘޙሀڱᄃRossiter ඈˠ[14]ٙ೩۞ᘦؠᇃཌྷ

࿰ീଠטඉர൴ण΍˘າዋᑕଠטڱ݋ĂͽЋဦྋՙ˯ࢗ

̝યᗟĄٙଳϡ̝͞ڱߏ੫၆צଠវߏ˘࣎ໂ׍ܧቢّপ

ّ۞ր௚Ă׎ፆү̙Ԋࢨٺ̈ቑಛ̰ĂӀϡሀቘޙሀ[7-10]

̈́ΐᝋ౵̈π͞Ҥീڱ[15]၆ր௚ፋវઇԊొቢّ̼Ăซ

҃଀זצଠវБાّቢّሀݭć׎Ѩ૟ٙ଀ሀݭણᇴ΃ˢ ᘦؠᇃཌྷ࿰ീଠטጡ̚ĂԆјଠטనࢍĂֹଠטր௚ّਕ

྿јٙഇ୕̝ଠטّਕĄώ͛۞׎ι̰ट௡ᖐໄࢗт˭Ĉ ௐ˟༼ࠎ࠹ᙯᇴጯؠཌྷĂௐˬ༼ྎႽౘࢗтңޙၹܧቢّ

ր௚̝ሀቘሀݭᄃણᇴҤീ۞͞ڱĂͽ̈́ௐα༼ӈඕЪሀ ቘሀݭᄃᘦؠᇃཌྷ࿰ീଠטڱ݋൴ण΍˘າ͞ڱĶሀቘዋ ᑕᘦؠᇃཌྷ࿰ീଠטķĂௐ̣༼݋Ӏϡ̙Тצଠវ̝࿪ཝሀ ᑢඕڍᄲځٙ೩ڱ݋۞ΞҖّᄃّਕĂௐ̱༼ࠎඕኢĄ

˟ă࠹ᙯᇴጯؠཌྷ

ؠཌྷ1 ឰ F ޽ϯٙѣ z

^-1

̝

n-1 Ѩкีё f

(

z

)=

f ₀

+

f ₁ z ⁻ ¹

) 1 1 − ( −

+

−

+L

f _n z ⁿ

۞Шณ۩มĂͷᏴፄ[1,z

^-1

,…,z

^-(n-1)

] ઇι۞ૄغะЪć҃ɚ{f(z)}ܑϯ f(z)۞Ѩᇴ n-1Ă

[ f f f n ] ^T

f

=

₀ ₁

L

₋ ₁

ࠎ

f(z)ᙯٺૄغะЪٙј R ⁿ

۞ ळ ᇾ Җ Ш ณ Ă ݋ ؠ ཌྷ P:

f

→

f

(

z

) Ă

f z f

( )→ :

P

^-1

Ą

ؠཌྷ2 ၆ٺ f(z)ᛳٺ FĂឰ Cfܑϯਫ਼୊৏ੱͷᛳٺ

R ^n×n

Ă

݋ؠཌྷ[C

_f

]

_ij

=

f _i ₋ _j

Ă׎̚Ă

f _k

ࠎ

f(z)˯ z ^-k

ี۞ܼᇴĄ ӈĂ













=













=













=

−

0 1 2 1

0 1 2

0 1 0

0 3 2 1

3 0

1 2

2 1 0 1

1 2 1 0

3 2 1

3 3 3 2 3 1 3

2 3 2 2 2 1 2

1 3 1 2 1 1 1

0 0 0

f f f

f

f f f

f f

n n

n n n

n n n n n n

n n

n n n

f

L O O M M

M O L L L

M M M M M

L L L L

M M M M M

L L L

C

ؠந1 ၆ٺ f(z)ăh(z)ᛳٺ FĂЯкีё׍ࢷڱϹೱّĂ߇ Ξ଀ਫ਼୊৏ੱᙯܼ

C _f C _h

=C

_h C _f

Ą

ؠཌྷ3 ৏ੱϤਫ਼୊৏ੱ C

f

݈ɢҖၹј۰Ăؠཌྷࠎ

Γ f

ćਫ਼

୊৏ੱ

C _f

౺ዶ۞ޢ

n-ɢҖٙၹј৏ੱĂؠཌྷࠎ Μ

Ą

_f

ࡶ

e i

΃ܑд

R ⁿ

˯ௐ

i ࣎ᇾ໤ૄغШณĂ݋ Γ f

̈́

Μ f

Ξؠཌྷт˭Ĉ

(3)

Γ _f

=

C _f

[

e ₁

,

e ₂

,K,

e _µ

]Ă

Μ _f

=

C _f

[

e _µ + ₁

,K,

e _n

]

! ! ӈĂ

















=

−

0 1 2

1 1 3

2 0 2

1 0 1 0

0 0 0

0 0

f f

f

f f

f

f f

f f f f

n n n

n

n n

n f

L L

O L L

M O O M

M M M

L L

M M M O O M M

L L L

L L L L

µ µ µ µ

C µ

ؠཌྷ4 ၆ٺ f(z)ᛳٺ FĂឰ H

f

ܑϯ႔ҹ৏ੱͷᛳٺ

R ^n×n

Ă

݋ؠཌྷ[H

_f

]

_ij

=

f _i ₋ ₁ ₊ _j

Ă׎̚Ăf

k

ࠎ

f(z)˯ z ^-k

ี۞

ܼᇴĄ ӈĂ













=













=













=

−

− +

−

+

−

+

− +

−

+

− +

−

+

− +

−

+

− +

−

0 0

0

0 0

0

1 1 2

1 2

1 1 2 2 2 1

2 2 3 2 1

1 3

2 1 2

1 1 3

1 2 1 1 1

1 3 3 1 3 2 1 3 1 1 3

1 2 3 1 2 2 1 2 1 1 2

1 1 3 1 1 2 1 1 1 1 1

L L

M M N M M

L L

M M N M M

L L

L M M M M M

L L L

n

n n

n n n

n

n n n

n

n n

n n n

n n

n n n

f

f f

f

f f f

f

f f f

f

f f f

f

f f f

f

f f

H

ˬăሀቘሀݭ̝ᇴጯሀݭᄃણᇴҤീ

ሀቘޙሀ̝ϫ۞дԱವ˘ཏజ׹ؼؠཌྷ̝ણᇴĂ൒ޢ ϡ׎ޙၹIF-THEN ሀቘఢ݋Ăֹ֭ѩఢ݋ཏ׍ೡࢗצଠវ Ꮾ΍ˢតᇴҖࠎਕ˧Ą˘ਠֽᄲĂሀቘఢ݋׍ѣ݈Іొણ ᇴ̈́ޢІొણᇴĂѩڱߏϡ݈ІొણᇴֽౘࢗצଠវᏮ΍

ˢតᇴมᐖၗܧቢّߍडᙯܼĂޢІొણᇴ݋ࠎ Takagi- Sugeno (T-S)ݭё۞ሀቘሀݭĂӀϡ׍ؼᏵ۞а଱Ꮾ΍ˢੈ

ֽཱིϼ౴ր௚જၗҖࠎĂͽԆјѩሀቘሀݭĄ

ώ༼Ϻྻϡᇾ໤̼ᓁ׶ඈٺ1 ̝ࣧந૟ఢ݋̶ྋĂซ

ֹ҃ր௚ᜩᑕ̶јᇴડ๴Ă֭ેҖ౵̈π͞Ҥീڱ၆Ч࣎

̄ఢ݋ઇણᇴҤീ(ӈր௚ፋវઇԊొቢّ̼)Ă౵ޢ૟̄

ఢ݋Ꮾ΍ࢷ˯׎ΐᝋπӮ̝ᝋࢦࣃ଀ᓁវᏮ΍ᜩᑕĂтѩ ӈ଀זצଠវБાّ̝ቢّሀݭĄ׎ྎ௟͞ڱтЬٙࢗĄ

1. ሀቘሀݭ̝ᇴጯሀݭ

૱֍۞ಏᏮˢಏᏮ΍ܧቢّր௚ሀݭߛၹࠎĈ

)) 1 (

, ), ( ), 1 ( , ), 1 ( ), ( ( ) 1 (

+

−

+

−

= +

m k u

k u n k y k

y k y k

y F

L L

(1)

дё(1)Ă

y

(

k

),

y

(

k

−1),L,

y

(

k

−

n

+1)̈́

u

(

k

),

u

(

k

−1),

…,

u

(

k

− m+1)ЧҾܑϯր௚ؼᏵ۞Ꮾ΍̈́ᏮˢੈཱིĂF ܑ ܧቢّבᇴĄ

ЯѩĂሀቘޙሀ̝ϫ۞ĂӈߏࢋͽTakagi-Sugeno(T-S) ݭёሀቘሀݭֽܕҬѩܧቢّבᇴ

Fć҃ T-S ݭёௐ j ࣎

ఢ݋ΞᆷјĈ

) ...

( then

is and...and is

if :

1 1 1

j N

j N N j j

x x f y

A x A

x R

= (2)

дё(2)̚Ăx

i

,i=1,…,N ߏᏮˢੈཱིĂA

i j

ߏ

x i

дௐ

j ࣎ఢ݋

˯۞ᕩᛳבᇴĂϺΞܑࠎ

x i

Ꮾˢ۩ม۞̄ะЪٕሀቘቑ ಛĂy ࠎᏮ΍ੈཱིĂ̈́ f

^j

дT-S ݭё̝ఢ݋˯఼૱ࠎᏮˢ

ੈཱི۞ቢّבᇴĄ

ͽฟਫ਼ྮՎลᜩᑕڱ(ͅᑕѡቢڱ)଀ր௚ໄரߛၹĂ

֭ޙၹё(2)̝ T-S ݭёሀቘሀݭĂт˭Ĉ

K i k u b

k u b k y k

y A k y R

i

i i

,..., 1 , ) 1 (

) ( ) ( a ) 1 ( then is ) ( if :

1

0

=

− +

+

=

+ (3)

ё(3)̚а଱ؼᏵ۞Ꮾ΍ˢੈཱི y(k)ău(k)̈́ u(k-1)ࠎሀቘր

௚ᏮˢតᇴĂy(k+1)ࠎᏮ΍តᇴĂ҃ A

i ,i=1,…,K ߏᏮˢត

ᇴ

y(k)дௐ i ࣎ఢ݋˯۞ᕩᛳבᇴćͷѩ఍઄ؠᕩᛳבᇴ

࣎ᇴߏඈٺఢ݋࣎ᇴ

KĂᏮˢតᇴ̝ᕩᛳבᇴϺืӣᄏٙ

ѣౕਫ਼ྮր௚۞ፆүቑಛĂтѩ̖Ξ௑Ъሀቘր௚̝ԆБ

ّّኳ[8]Ą

݋T-S ݭёሀቘሀݭᓁᏮ΍ӈࠎĈ

)) ) 1 (

) ( ) ( a ))(

( ( ( ) 1 (

1 1 0

− +

∑ +

= +

=

k u b

k u b k y k k

y

i K

i β i ϕ i i

(4)

׎̚Ăᇾ໤̼႕֖඀ޘ

∑

=

= K

i A

A i

k y

k k y

i i

1

( ( )) )) ( )) (

( (

µ ϕ µ

β

(5)

ଂё(4)ΞۢĂሀݭᓁᏮ΍޽ϯࠎЧఢ݋Ꮾ΍ࢷ˯׎ᕩᛳ඀

ޘ۞ΐᝋπӮࣃ̝ᓁ׶ćΩĂ

ϕ

(k)ܑਫ਼ᕩጡߏϤր௚Ꮾ΍

ˢੈཱིٙ௡јĄͷё(5)̝ٙѣఢ݋ᇾ໤̼႕֖඀ޘٙะЪ јШณݭёĂт˭Ĉ

[ _k ]

T

=

β ₁ β ₂

...

β

(6)

(4)

֭૟ё(6)΃ˢё(4)̚ĂΞ଀ሀቘሀݭᓁᏮ΍̝ШณݭёĂ ࠎĈ

) 1

~ ( )

~ ( ) ( a~

) 1 ( ) ( )

( ) 1 (

1 0

− + +

=

− +

+

= +

k u b k u b k y

k u k u k

y k

y

β

^T

a β

^T

b β

^T

b

(7)

׎̚

[ ]

[ K ]

T T K T K

b b b

1 12 11

0 02 01

2 1

...

a ...

a a

=

1 0

b b a

(8)

тѩĂё(7)ӈࠎྍր௚̝БાّቢّሀݭĄ

2. ሀቘሀݭ̝݈ІొણᇴҤീ

ଳϡ[7-9]̝͞ڱĂֹϡˬ֎ԛᕩᛳבᇴĂ֭གྷဘྏ᏾

ᄱڱטؠఢ݋ᇴĂ૟Ꮾ΍ቑಛࡗரઇӮ̶ĂͽԆјሀቘఢ

݋݈Іొᕩᛳבᇴ

A _i

ણᇴ̝నؠĄ

3. ሀቘሀݭ̝ޢІొણᇴҤീ

ॲፂณീٙ଀ր௚Ꮾ΍ˢྤफ़Ăֶሀݭ̼ჟৠ૟ྤफ़

৏ੱ

ɕ

̶ྋј

K ࣎̄৏ੱ ɕ 1

,

ɕ 2

,…,

ɕ _K

Ă൒ޢГᖣϤ౵̈

π͞Ҥീڱ၆Տ˘ఢ݋̝ણᇴ࣎ҾઇҤീĂͽ࡭ٺ଀זྵ

ָ۞T-S ݭё̝ሀቘሀݭޢІొણᇴ aăb₀̈́

b

₁Ą

ࢵАĂႊზڱ̝ሀቘሀݭؠࠎё(7)Ă఺֭྆઄ؠᇾ໤

̼႕֖඀ޘߏᄃॡมѣᙯ۞Ă݋ጱ࡭ё(9)јϲć׎̚ĂI

΃ܑᇾ໤ಏҜШณĄ 1 ) ( )

( = =

∑

k ^T k I

i β i β

(9)

൒ޢЯё(9)ඈٺ 1Ă߇Ξ૟׎ࢷдё(7)۞ν͘ᙝĂய Ϡт˭ԛёĈ

) 1 ( ) (

) ( ) ( ) ( ) ( ) 1 ( ) (

− +

+

= +

k u k

k u k k y k k

y k

T

T T

T

1

0

b

b a

I

β

β β

β

(10)

Ϻӈ

)) 1 ( ) (

) ( ) ( ) ( )a ( ( ) 1 ( ) (

1 1 0

1

− +

∑ +

∑ + =

=

k u b k

k u b k k y k k

y k

i i K

i i i i i

K

i i

β

β β

β

(11)

˫Ξ̶౷јё(12) K ࣎ቢّ͞඀ёĂՏ࣎͞඀ёܑϯՏ࣎

ఢ݋၆ٺሀቘሀݭᓁᏮ΍̝ણᄃ඀ޘĄ

) 1 ( ) ( ) (

) ( ) ( )a ( ) 1 ( ) ( :

) 1 ( ) ( ) (

) ( ) ( )a ( ) 1 ( ) ( :

) 1 ( ) ( ) (

) ( ) ( )a ( ) 1 ( ) ( :

1 0 12 2

02 2 2

2 2 2

11 1

01 1 1 1 1 1

− +

+

= +

− +

+

= +

− +

+

= +

k u b k k u

b k k y k k

y k R

k u b k k u

b k k y k k

y k R

k u b k k u

b k k y k k

y k R

K K

K K K

β β β

β

β β β

β

β β β

β

M

(12)

ࠎ˞଀זё(12)̚ሀቘሀݭޢІొણᇴ a

i

ăb

0i

̈́

b 1i

,

i=1,…,KĂ݋Տ˘࣎ఢ݋۞ਫ਼ᕩጡ ϕ i

(k)૟జֹϡĂ׎ؠཌྷт

˭Ĉ

[ ]

K i

k u k k u k k y k

k _i _i _i

i

,..., 1

) 1 ( ) ( ) ( ) ( ) ( ) ( ) (

=

−

=

β β β

ϕ

(13)

ͷ੫၆ٙѣᏮ΍ˢྤफ़၆Ăё(13)˯۞Տ˘࣎ఢ݋ਫ਼ᕩጡ

˫Ξ௡јт˭̝ਫ਼ᕩ৏ੱ

ɕ _i

Ĉ













−

=

) 1 ( ) ( ) ( ) ( ) ( ) (

(1) ) 2 ( (2) ) 2 ( ) 2 ( ) 2 (

(0) ) 1 ( (1) ) 1 ( (1) (1)

N u N N u N N y N

u u

y

u u

y

i i

i

i i

i

i i

i

β β

β

β β

β

β β

β

M M

Ψ

M (14)

дё(14)̚ĂN ΃ܑᏮ΍ˢྤफ़၆࣎ᇴćѩ఍ࠎ႕֖ᝥҾ

඀Ԕ̝̚፬ᐽ୧Іࢎؠё(15)୧ІĂᖣͽՙؠྍඊਫ਼ᕩጡ

ྤफ़Ξӎΐˢਫ਼ᕩ৏ੱ̚Ăซ҃଀זѣड़۞ਫ਼ᕩ৏ੱĄ(׎

̚ĂɚܑࠎᗔੈҤീត̼ณ)

N k

i

(

k

)>

δ

=1,...,

β

(15)

дѩĂϺؠཌྷ࠹၆ᑕٺௐ

R ⁱ

ఢ݋̝Ꮾ΍ࠎĈ )

1 ( ) ( ) 1

(

k

+ =

k y k

+

y ⁱ β _i

(16)

֭૟׎ඕЪјт˭̝Ꮾ΍ྤफ़ШณĈ













+

=

) 1 ( ) (

(3) (2)

(2) (1)

N y N

y y

i i i i

β β β

Y

M (17)

ତ඾Ăଂё(12)ăё(13)̈́ё(16)̚ĂӈΞ଀ௐ R

ⁱ

ఢ݋̝ሀ ቘሀݭ৏ੱԛёĂࠎĈ

i

i k i k

y

( +1)=

ϕ

( )

θ

(18)

׎̚

T [ i i i ]

i

= a

b ₀ b ₁ θ

౵ޢĂϤё(16)̈́ё(18)̝Шณ৏ੱԛё̚ĂેҖ౵̈π͞

Ҥീڱ଀זௐ

R ⁱ

ఢ݋̝ሀቘሀݭޢІొ౵ָણᇴĈ

i T i T i i

i

=(

Ψ Ψ

)

⁻ ¹ Ψ Y

θ

(19)

дё(19)่̚Ր଀ௐ R

ⁱ

ఢ݋̝ޢІొ౵ָણᇴćТᇹ͘ڱ ӈΞ૟ٙѣఢ݋̝ޢІొৌ၁ણᇴՐ΍Ă֭ၹјё(8)̝ણ ᇴШณ

aăb

0̈́

b

1ćϺ଀זё(7)̝ր௚Бાّ۞ቢّॡត ሀݭણᇴĂт˭Ĉ

1 0

b b a

)) ( ( )) ( (

)) ( ( )) ( ( a~

1 0

k k

b

k k

b

k k

~ T

T

ϕ ϕ

β β β

=

(20)

(5)

ГѨࢦࢗĂྍБાّ۞ቢّॡតሀݭࠎĈ

) 1 ( ) ( ) ( a ) 1

(

k

+ =

~ y k

+

b 0 u k

+

b 1 u k

−

y _m _m ^~ ^~

(21)

αăሀቘዋᑕᘦؠᇃཌྷ࿰ീଠט

ሀቘዋᑕᘦؠᇃཌྷ࿰ീଠטߏඕЪ݈ࢗ۞ሀቘޙሀ

͞ڱᄃዋᑕᘦؠᇃཌྷ࿰ീଠט҃யϠ̝າڱ݋ćྍڱ݋̝

࿰ീᏮ΍ࣃ̈́ଠטఢ݋ӈֶፂT-S ݭё̝Бાّቢّॡត ሀቘሀݭٙᒔ଀Ą

׎ଠטඉரࠎĈௐ˘ՎĂӀϡ˯ࢗሀቘሀݭ̼̈́ᅍਫ਼

౵̈π͞Ҥീڱඈ଀ѩր௚ሀݭણᇴĂГ૟׎΃ˢଠטጡ

̚ćௐ˟ՎĂֹϡᘦؠа଱ଠטጡপֽّ྿זր௚̰ొ۞

ᘦؠĂ֭ᑕϡྍপֽّᖎౕ̼ਫ਼ྮᖼೱፆү(ӈણ҂ੈཱི c ၆ր௚Ꮾ΍

y ̈́ଠטᆧณŔu ม̝ᙯܼ)ć౵ޢĂેҖᘦؠ

ᇃཌྷ࿰ീଠט˯Ꮾ΍࿰ീ̈́ͽϏֽણ҂ੈཱིࠎ၆෪̝౵ָ

̼ՎូĂͷࢦኑѩˬՎូд˭˘࣎פᇹॡมĄ఺྆ࢋপҾ

ૻአĂዋᑕᘦؠᇃཌྷ࿰ീଠט̝౵ָ̼၆෪ᔵߏ੫၆Ϗֽ

ણ҂ੈཱི҃ܧϏֽଠטᆧณઇనࢍĂҭଂણ҂ੈཱི

c ၆ଠ

טᆧณŔu มᖎಏ۞ౕਫ਼ྮᖼೱᙯܼۢĂྍඉரࠎมତ၆ ϏֽଠטᆧณઇనࢍĄ

1. ᘦؠ۞а଱ਫ਼ྮనࢍ

ࠎ˞Հዋآ̍ຽࠧᑕϡĂଳϡ׍᎕̶ጡ̝ሀݭĂ૟ё (7)׌ᙝࢷ˯ ∆(z)Ă֭੫၆ଠטᆧณ଀ր௚ᖼொבᇴĈ

) ( ) a(

) ( ) ( ) ( )

(

z B z A z z ¹ b z z z

g

= =

⁻

∆ (22)

дё(22)̚ĂB(z)̈́ A(z)՟ѣВТ̙ᘦؠ۞࿬ᕇćz

^-1

ܑր௚

g(z) ׍ ˘ ࣎ ؼ Ᏽ ᗓ ೸ ॡ ม ࣎ ᇴ Ă ͷ

a(

z

)=1−a~

z ^- ¹

ă

1 1

0

~

) ~

(

z

=

b

+

b z ⁻

b

̈́∆(

z

)=1

-z ⁻ ¹

Ą

൒ޢྋBezout ޮඈё 1 ) ( ) ( ) ( )

(

z X z

+

A z Y z

=

B

(23)

дѩྋBezout ޮඈё̝ЯĂߏιࡶࠎր௚ౕਫ਼ྮপّ͞඀

ё݋Ξܲᙋր௚ᘦؠć׎ຐڱᙷҬሢۢ۞ໂᕇщཉڱĂӈ ր௚ౕਫ਼ྮপّ͞඀ёඈٺԓ୕۞ౕਫ਼ྮপّ͞඀ё (1)Ă݋Ξܲᙋໂᕇ̙ົརдಏҜ๪γซ҃ܲᙋր௚ᘦؠĄ ͷтڍ

X(z)̈́ Y(z)ࠎ Bezout ޮඈё۞ྋĂQ(z)ܑϯ˘ֱᘦ

ؠ ۞ ᖼ ொ ב ᇴ ٕ ѣ ࢨ Ѩ ᇴ ̝ к ี ё Ă ݋ υ х д

) ( ) ( ) ( )

(

z Y z -B z Q z

M

= ̈́

N

(

z

)=

X

(

z

)+

A

(

z

)

Q

(

z

) Ϻ ࠎ Bezout ޮඈё۞ྋĂ߇଀ဦ 1 ̝ણᇴ̼ᘦؠଠטጡĈ

) ( ) ( )

~(

z M z N z

K

= (24)

Ϥဦ1 ଀˭ЕᙯܼĈ(Ꮾ΍ቑಛ nŸa(z)Ѩᇴ k)

) )] ( ( ) ( [ )]

( ) ( [ 1

)]

( ) ( [ ) ( )

( )

(

B z

z A z M z N z B

z A z M z B z

c z

y

=

= + (25)

=

N(z) = X(z) + A(z)Q(z) +

−

B(z) A(z) 1

M(z)

1

Y(z)−B(z)Q(z) ∆u(k+1) y(k+1)

c(k+1)

ဦ1 ׍ᘦؠਫ਼ྮΑਕ̝ଠטր௚͞๴ဦ

) )] ( ( ) ( [ )]

( ) ( [ 1

) ( 1 )

( )

(

A z

z A z M z N z B

z M z

c z

u

=

= +

∆ (26)

ଂ˯׌ё଀ր௚ౕਫ਼ྮপّ͞඀ёĈ

1

) ( ) ( ) ( ) (

)]

( ) ( ) ( )[

( )]

( ) ( ) ( )[

(

) ( ) ( ) ( ) (

=

+

=

+ +

−

=

+

z X z B z Y z A

z Q z A z X z B z Q z B z Y z A

z N z B z M z A

(27)

д ё(27) Ă Ξ࠻ ΍ ѩ ڱ ݋ ᒔ଀ ˘ ᘦ ؠ ̝ ր௚ Ă ֭ གྷ Ϥ Sylvester ͞඀ёྋ଀ͧ A(z) ลᇴ͌˘ล۞઱˘ྋ M(z)̈́

N(z)Ăͷଯ଀ਫ਼୊৏ੱᙯܼт˭Ĉ(ё̚ĂI _n

ࠎ˘

n ลಏҜ

৏ੱĄ)

n M N A

b C z C C I

C

−1 + = (28)

2. ࿰ീ͞඀ё

Ϥဦ1 ଀Ĉ

ր௚ሀݭ

A

(

z

)

y

(

k

+1)=

b

(

z

)∆

u

(

k

) (29)

ଠט͞඀ё

) 1 ( ) ( ) (

) ( ) ( ) ( ) ( ) (

1

+

=

∆

− N z y k -z

k c

k y z -N k c k u z

M

(30)

၆ٺ

n ՎШ݈Ꮾ΍ቑಛֽᄲගࠎĈ

ր௚ሀݭ

H y

r

C y

r

H u

r

C u

r

f b p f b

p A

A

+ = ∆ + ∆ (31)

ଠט͞඀ё

N f p z N z

f M p M

-

y C y

c H

C u H u

r r r

r r

1

0

− 1 −

∆ +

∆

=

⁽³²⁾

дё(31)ăё(32)̚

H _A

,H

_b

,H

_M

,H

_z-1N

ࠎ

R ^{n× n}

۞႔ҹ৏ੱ

C A

,C

b

,C

M

,C

z-1N

ࠎ

R ^{n× n}

۞ਫ਼୊৏ੱ

̈́

R ⁿ

ҖШณ

p T f T

T p T

f T

n k u , , k u

1 - n k u , , k u , k u

1 - n k c , , k c

1 n k y , , k y , k y

n k y , , k y

)]

( ) 1 ( [

)]

( ) 1 ( ) ( [

)]

( ) ( [

)]

( ) 1 ( ) ( [

)]

( ) 1 ( [

0

−

∆

−

∆

=

∆

+

∆ +

∆

=

∆

+

=

+

−

=

+ +

=

r L r L r L r L r L

u

c

y

(6)

Ӏϡؠந1 ̝ C

f C h

=C

h C f

ᄃё(28)ᙯܼĂ૟ё(31)νࢷ

z

⁻¹

N

C

ᄃё(32)νࢷ

C ࠹ΐޢ଀ ur _A

∆

f

֭΃ˢё(31)ፋந ޢĂ଀

ϏֽᏮ΍Шณ

y

r

C c

r

p y

r

p u

r

p p

f

=

b ₀

−

₁

−

₂

∆ (33) ё(33)̚

b M M b

A N M

b z

H C H C p

−

=

+

= −

2

1

¹

(34)

3. ᘦؠᇃཌྷ࿰ീଠט

ᘦؠᇃཌྷ࿰ീଠט̚၆෹࿅ଠטቑಛ̝ଠטณ࠰ෛ

үؠࣃĂซֹ҃఺ֱϏֽଠטᆧณ࠰ࠎ࿬ćᙷҬ͞ڱĂዋ ᑕᘦؠᇃཌྷ࿰ീଠט၆ଠט໚ᐝ۞ણ҂ੈཱི

c Ϻనѣણ҂

ቑಛɢĂι૟

R ⁿ

ણ҂ੈཱིШณ cr

0

̶౷ј݈

R ^µ

Шณ

cr ᄃޢ R ^n-µ

Шณc Ăӈ

^∞













+ +

− +

+

=













=

∞

) (

) 1 (

) (

c

0 1 - n k c

k c

M r M

r

µ µ c

c

(35)

݈۰ࠎдᇶၗॡ౵̼̈۞ણ҂ੈཱིĂߏ่ਕజనࢍ۞

ొЊć҃ޢ۰ҖࠎົഈШι۞ᘦၗࣃજүĂΞֶ၁ᅫࢋՐ

࿰АઇՙؠĂ˘ਠֽᄲĂдᘦၗॡĂᏮ΍ੈཱི

y ૟Ᏺೈన

ؠᕇ

r ۞࢖ྫ r

r=[

r

(

k

+1)

,

L

, r

(

k

+

n

)]

^T

ĂͷϤဦ1 ̚࠻΍

ણ҂ੈཱི

c Ϻ૟Ᏺೈ N(z)y ۞ੈཱིҖซĂ߇ΞᏴፄ













+ +

















=

∞

=

) (

) 1 (

0 0

0 n k r

k r

N N

N N N

x x

N

M M

L L

L

O M O M O O M

M O O M O M O

L L

L

M M M O O M O M

L L L L

r

µ µ

µ

r C c

















=

0 0

N N N

N N

N N N

N

x N x

L L

L

O M O M O O M

M O O M O M O

L L

L

M M M O O M O M

L L L L

M M M M M O O M

L L L L L

µ µ

µ

C

(36)

дё(36)̚Ă

C ܑϯ C _N N

౵ޢ

n-ɢЕ৏ੱć˘ਠдՎลన

ؠᕇੈཱི

r ˯ĂਕͽಏҜ৏ੱࢷ˯ N(z)ۡ߹ᆧৈ̝౵ޢ n-

ɢЕ৏ੱֽ΃ആĄ

ͽણ҂ቑಛɢઇ҂ᇋĂдё(33)˯ֶ Γ

f

̈́

Μ f

ؠཌྷ૟ cr

0

Шณઇ̶౷ޢ଀ё(37)Ꮾ΍࿰ീШณćଠטᆧณ࿰ീШณ

͞ࢬߏ૟ё(33)΃ˢҌё(31)̚ፋநĂГ૟ cr

0

Шณઇ̶౷

఍நޢ଀ё(38)Ăඕڍт˭Ĉ

Ꮾ΍࿰ീШณ

y

r

Γ c

r

y p y

r

p u

r

p p

f

=

b

+

^∞

−

₁

−

₂

∆ (37) ଠטᆧณ࿰ീШณ

u

r

Γ c

r

u p y

r

p u

r

p p A

f

= +∆ − − ∆

∆

^∞ ₃ ₄

(38)

дё(37)̈́ё(38)̚

N b M z A

N A z N A z

b M M b

A N M

b z

H C H C p

1 1 1 1

4 3 2 1

−

+

=

−

=

+

=

(39)

ͷ

y ^∞ = M _b c ^∞

Ă∆

u ^∞

=

M _A c ^∞

(40)

׎̚Ă

y ^∞

̈́ u∆

^∞

࠰ࠎඕЪ

Μ f

ࢷ᎕̝ีĂ߇׎݈ɢ̮࣎৵

࠰ࠎ࿬Ă҃౺ዶ̮৵ࠎ΁ࣇ࿰ീ̝ᘦၗҖࠎĄ

ዋᑕᘦؠᇃཌྷ࿰ീଠט̝౵ָ̼ඉரĂߏͽϏֽણ҂

ੈཱིΝ౵ّ̼̈ਕ޽ᇾֽԆјଠטࠎϫ۞Ă׎ّਕ޽ᇾт

˭Ĉ

y u

r

r r ² r ²

f σ f

J

= − + ∆ (41)

૟నؠᕇ࢖ྫ

rr Ăё(37)Ꮾ΍࿰ീШณ yr _f

̈́ё(38)ଠטᆧ ณ࿰ീШณ∆

ur f

΃ˢё(41)ّ̝ਕ޽ᇾ J ̚Ă൒ޢેҖ౵

ָ̼ඉர΄∂

J

∂

cr

=0Ă֭૟ଯጱඕڍፋநޢ଀౵ָ̼۞ዋ ᑕᘦؠᇃཌྷ࿰ീଠט̝ଠטఢ݋Ĉ

) ( ) ( ) ( ) ( )

(

z u k c k N z y k

M

∆ = − (42)

ё̚

p r

r

p y

r

p u

r

T p u p T y T

k r

c

( )= + + ∆ (43)

ͷ

) Γ (

] [

) Γ (

] Γ

Γ [

) ( ] [

4 1 2

1 3 1 1

1 1 1

p p

Γ Γ Γ Γ Γ p

p p

Γ Γ Γ p

C M Γ

C M Γ Γ Γ Γ Γ Γ p

u y r

T A b T

A - T A T b

b T T

T A b T

A - T A T b

T b T

A N T A

b N b T b T A - T A T b

b T T

e e -

- e

σ σ

σ

+ +

=

+ +

=

+

=

(44)

҃ё(43)ϺΞଯ଀

非線性系統之模糊適應穩定廣義預測控制

ૺฮ௨

઼ϲ๔ৈԫఙጯੰҽࣜ۩አրᓾर

ች୻Ѱ

઼ϲ̚Ꮈ̂ጯ࿪፟̍඀ጯրି଱

ၡ! ࢋ

FUZZY ADAPTIVE STABLE GENERALIZED PREDICTIVE CONTROL FOR A CLASS OF NONLINEAR SYSTEMS

Ya-Ling Chang

Department of Refrigeration and Air Conditioning National Chin-Yi Institute of Technology

Taichung, Taiwan 411, R.O.C.

Ching-Chih Tsai

Department of Electrical Engineering National Chung-Hsing University

Taichung, Taiwan 402, R.O.C.

Key Words: adaptive control, fuzzy modeling, generalized predictive control.

ABSTRACT

This paper develops methodologies for an adaptive stable generalized predictive control with fuzzy modeling for a class of nonlinear systems.

This new type of controller is composed of a fuzzy model with on-line parameter estimation, and a stable generalized predictive control with good tracking, guaranteed stability and disturbance rejection capability.

Numerical simulations for controlling two highly nonlinear processes are

described. These show the excellent disturbance rejection and setpoint

tracking performance of the proposed control method.

˘ă݈! ֏

˟ă࠹ᙯᇴጯؠཌྷ

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