Odi Boy P H Dosen Pembimbing : Prof. Dr. Ir. Achmad Jazidie, M.Eng.

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Odi Boy P H 2206100194

Dosen Pembimbing :

Prof. Dr. Ir. Achmad Jazidie, M.Eng.

DASAR TEORI

KESIMPULAN SIMULASI

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LATAR BELAKANG PERMASALAHAN

TUJUAN

•Pendulum terbalik adalah sistem yang

nonlinear dan tidak stabil

1

• Semakin berkembangnya sistem

kontrol tracking menggunakan

model fuzzy Takagi-Sugeno

2

• Performansi tracking H

_∞

menggunakan

pendekatan LMI semakin berkembang

3

(3)

•Memaksa kereta bergerak

mengikuti sinyal referensi yang

diberikan dengan tetap

mempertahankan batang

pendulum tetap dalam posisi

tegak

LATAR BELAKANG PERMASALAHAN

(4)

• Desain kontroler fuzzy

berbasis model

Takagi-Sugeno menggunakan

kriteria performansi

tracking H

_∞

LATAR BELAKANG PERMASALAHAN TUJUAN

(5)

FUZZY TAKAGI-SUGENO PERFORMANSI TRACKING H_∞ LINEAR MATRIX INEQUALITY SISTEM KONTROL MODEL REFERENSI PLANT INVERTED PENDULUM

MODEL FISIK

motor DC l

pusat massa sistem sumbu rotasi

(6)

MODEL MATEMATIS

3 1

x

x

4 2

x

x

2 2 4 2 2 2 2 4 3

sin

)

sin

(

cos

)

sin

(

x

l

J

x

f

x

g

x

l

x

T

F

a

x



c p 2 2 4 2 2 2 4 2 4

sin

)

sin

(

cos

x

l

J

x

f

x

g

x

T

F

x

l

x



c p

l

m

)

(

a

l

2

J

FUZZY TAKAGI-SUGENO PERFORMANSI TRACKING H_∞ LINEAR MATRIX SISTEM KONTROL MODEL REFERENSI PLANT INVERTED PENDULUM

(7)

)

(

)

(

)

(

t

A

x

t

B

r

t

x



_r _r _r _r FUZZY TAKAGI-SUGENO PERFORMANSI TRACKING H_∞ LINEAR MATRIX INEQUALITY SISTEM KONTROL MODEL REFERENSI PLANT INVERTED PENDULUM

(8)

MODEL FUZZY TAKAGI-SUGENO

3 ATURAN

u

B

x

A

x



₁ ₁

x

C

y

₁

Aturan plant ke-1:

If x

₂

is F

₁

(-0.174 rad)

then

Aturan plant ke-2:

If x

₂

is F

₂

(0 rad)

then

x



A

2

x

B

2

u

x

C

y

₂

)

(

)

(

)

(

t

K

₁

x

t

x

t

u

_r

)

(

)

(

)

(

t

K

₂

x

t

x

t

u

_r

Aturan kontroler ke-1:

If x

₂

is F

₁

(-0.174 rad )

then

Aturan kontroler ke-2:

If x

₂

is F

₂

(0 rad))

then

x

C

y

₃

Aturan plant ke-3:

If x

₂

is F

₃

(+0.174 rad)

then

Aturan kontroler ke-3:

If x

₂

is F

₃

(+0.174 rad)

then

u

B

x

A

x



₃ ₃

u

(

t

)

K

₃

x

_r

(

t

)

x

(

t

)

(9)

FUNGSI KEANGGOTAN

-1.50 -1 -0.5 0 0.5 1 1.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x2(radian) G ra d e o f M F F1 F2 F3 FUZZY TAKAGI-SUGENO PERFORMANSI TRACKING H_∞ LINEAR MATRIX INEQUALITY SISTEM KONTROL MODEL REFERENSI PLANT INVERTED PENDULUM

x

F

5 .

1 ,

0

5 .

1

0 ,

4 .

1

5 .

1

0

5 .

1 ,

5 .

1

5 .

1

5 .

1 ,

0

2

(10)

z y u d 2 y C T T x Ax Bu Dd x z x Qx u Ru  u Kx

u = sinyal kontrol

y = variabel terukur

d = disturbance

z = output performansi

(11)

Gain L₂ dikatakan mengalami pelemahan dengan faktor sebesar γ jika:

2 0 0 0 2 0 2

)

(

)

(

)

(

)

(

dt

d

dt

Ru

u

Qx

x

dt

t

d

dt

t

z

T T T FUZZY TAKAGI-SUGENO PERFORMANSI TRACKING H_∞ LINEAR MATRIX INEQUALITY SISTEM KONTROL MODEL REFERENSI PLANT INVERTED PENDULUM

(12)

2 0 0

)

(

~

)

(

~

)

(

)

(

)

(

)

(

f f t T t r T r

dt

t

w

t

w

dt

t

x

t

x

Q

t

x

t

x

)

(

~

)

(

~

)

(

)

(

)

(

)

(

0 2 0

t

w

t

w

dt

t

x

t

x

Q

t

x

t

x

f f t T t r T r T

t

r

t

w

t

v

t

w

~

(

)

(

),

(

),

(

)

adalah waktu batas kontrol

Q matriks pembobot definit positif ρ level atenuasi yang didefinisikan

f t FUZZY TAKAGI-SUGENO PERFORMANSI TRACKING H_∞ LINEAR MATRIX SISTEM KONTROL MODEL REFERENSI PLANT INVERTED PENDULUM

(13)

)]

(

)

(

))[

(

))

(

)]

(

)

(

))[

(

)

(

1 1 1

_h

_z

_t

_A

_x

_t

_B

_u

_t

t

z

t

u

B

t

x

A

t

z

t

x

_i L i i i L i i i L i i i



)]

(

))[

(

))

(

)]

(

))[

(

)

(

1 1 1

_h

_z

_t

_C

_x

_t

t

z

t

x

C

t

z

t

y

L i i i L i i L i i i FUZZY TAKAGI-SUGENO PERFORMANSI TRACKING H_∞ LINEAR MATRIX INEQUALITY SISTEM KONTROL MODEL REFERENSI PLANT INVERTED PENDULUM

(14)

g j j ij i

z

t

F

z

t

1

))

(

))

(

L i i i i

t

z

t

z

t

z

h

1

))

(

))

(

))

(

L j r j j L j j L j r j j

t

x

t

x

K

t

z

h

t

z

t

x

t

x

K

t

z

t

u

1 1 1

)]

(

)

(

[

))[

(

))

(

)]

(

)

(

ˆ

[

))[

(

)

(

(15)

)]

(

)

(

[

))

(

))

(

)

(

1 1

t

w

t

x

A

t

z

h

t

z

h

t

x

L i ij L j j i



)

(

)

(

)

(

t

x

t

x

t

x

r r j i j i i ij

A

K

B

K

B

A

0 )

(

)

(

)

(

t

r

t

w

t

w

(16)

0

1

2

P

Q

A

P

A

_ijT _ij

Q

22 11

0

0 P

P

(17)

0

22 21 12 11

F

Q

P

A

P

A

F

Q

K

B

P

F

Q

P

K

B

A

P

K

B

A

F

r T r j i T j i i T j i i 22 22 2 22 22 22 11 21 12 11 11 2 11 11 11

1

1 )

(

)

(

(18)

0

2 22 22 22 21 12 11

I

P

H

Q

A

P

A

H

Q

K

B

P

H

Q

P

K

B

A

P

K

B

A

H

r T r j i T j i i T j i i 22 22 22 11 21 12 11 11 2 11 11 11

1 )

(

)

(

Q

P

A

P

A

F

_rT _r ₂₂ ₂₂ 2 22 22 22

1

(19)

0

1 )

(

0

1 )

(

~

0

1 )

(

)

(

1 11 11 2 11 11 11 11 2 11 11 11 1 11 11 11 11 2 11 11

Q

W

I

Y

B

Y

B

W

A

W

Q

W

I

Y

B

Y

B

W

A

W

K

Y

P

W

Q

P

K

B

A

P

K

B

A

T j i j i i T i T j i j i i T i j j j i i T j i i FUZZY TAKAGI-SUGENO PERFORMANSI TRACKING H_∞ LINEAR MATRIX INEQUALITY SISTEM KONTROL MODEL REFERENSI PLANT INVERTED PENDULUM

(20)

0 ,

0 subject to

min

22 22 11 11 2 , ₂₂ 11 T T P P

P

0

2 22 22 22 21 12 11

I

P

H

(21)

PLOT GRAFIK HASIL DESAIN MODEL SIMULINK

LINEARISASI PLANT

0079

.

0

0 7691

.

14

0 0001

.

0

0 2366

.

0

1

0

1

0

1

A

2173

.

1 8266

.

0

1

B

0079

.

0

0 0319

.

15

0 0001

.

0

0 2524

.

0

1

0

1

0

2

A

2370

.

1 8272

.

0

2

B

0079

.

0

0 7691

.

14

0 0001

.

0

0 2366

.

0

1

0

1

0

3

A

2173

.

1 8266

.

0

3

B

(22)

LINEARISASI PLANT

5

6

0

1

0

5

6

0

1

0

r

A

8 .

3

0

9 .

2

0

r

B

(23)

LINEARISASI PLANT

55

0

55

0

55

0

55

10

5

Q

0.0043

0.0044

-0.0170

0.0037

-0.0044

-0.0047

0.0175

-0.0041

0.0170

0.0175

-0.0693

0.0148

-0.0037

-0.0041

0.0148

-0.0046

11

P

1.2008

1.2148

0.1102

-0.0205

-1.2148

2.0394

0.0194

-0.0588

0.1102

-0.0194

-1.0087

1.1566

0.0205

-0.0588

1.1566

2.1897

22

P

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LINEARISASI PLANT

79.0635

-71.2274

315.3671

-56.9462

1

K

78.9962

-71.2851

315.2746

-57.0520

2

K

79.0635

-71.2274

315.3671

-56.9462

3

K

GAIN KONTROLER

35 .

0 LEVEL PELEMAHAN

(25)

LINEARISASI PLANT

0 5 10 15 20 25 30 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 t(s) x 1 (m ) referensi kontroler fuzzy

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LINEARISASI PLANT

0 5 10 15 20 25 30 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 t(s) x 1 (m e te r) sinyal referensi rho=1 rho=0.75 rho=0.55 rho=0.35 rho=0.15 PLOT GRAFIK HASIL DESAIN MODEL SIMULINK

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0 5 10 15 20 25 30 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 t(s) x 2 (r a d ) VIDEO EKSPERIMEN MODEL SIMULINK PLOT GRAFIK