2DOF Controller Design . ﺗﻮﺍﻥ ﺩﺭ ﻧﻈﺮ ﮔﺮﻓﺖ ﺳﺎﺧﺘﺎﺭ ﺩﻭ ﺩﺭﺟﻪ ﺁﺯﺍﺩﻱ ﻳ

(1)

2DOF Controller Design Ali Chaibakhsh

ﻲﻣ ﺮﻳﺯ ﺕﺭﻮﺻ ﻪﺑ ﺍﺭ ﻢﺘﺴﻴﺳ ﻚﻳ ﻱﺩﺍﺯﺁ ﻪﺟﺭﺩ ﻭﺩ ﺭﺎﺘﺧﺎﺳ ﺖﻓﺮﮔ ﺮﻈﻧ ﺭﺩ ﻥﺍﻮﺗ

.

ﺖﺷﺍﺩ ﻢﻴﻫﺍﻮﺧ ﺕﺭﻮﺻ ﻦﻳﺍ ﺭﺩ :

) ( ) ( ) ( )) ( ) ( ) (

( ) (

) (

1 1

s F s G s P s G s P s I

R s Y

PGFr PG

I PGu Y

−

+

=

⇒ +

=

ﺪﻳﺎﺑ ﻲﻤﺘﺴﻴﺳ ﻦﻴﻨﭼ ﻝﺮﺘﻨﻛ ﻱﺍﺮﺑ ﺶﻴﭘ

ﺯﺎﺴﻧﺍﺮﺒﺟ ﻭ F

ﺯﺎﺴﻧﺍﺮﺒﺟ ﻪﺑ ﻢﺘﺴﻴﺳ ﺦﺳﺎﭘ ﺎﺗ ﺩﻮﻤﻧ ﻲﺣﺍﺮﻃ ﺍﺭ G

ﺩﻮﺷ ﻚﻳﺩﺰﻧ ﺮﻈﻧ ﺩﺭﻮﻣ ﻢﺘﺴﻴﺳ ﺦﺳﺎﭘ .

ﻲﻣ ﺕﺭﻮﺻ ﻦﻳﺍ ﺭﺩ ﻡﺮﻓ ﺎﺗ ﺖﻓﺮﮔ ﺮﻈﻧ ﺭﺩ ﺍﺭ ﻲﺒﺳﺎﻨﻣ ﻞﻳﺪﺒﺗ ﻥﺍﻮﺗ

ﻡﺮﻓ ﺯﺍ ﺍﺭ ﻢﺘﺴﻴﺳ ﺍﺰﺠﻣ ﻱﺎﻬﻤﺘﺴﻴﺳ ﺕﺭﻮﺻ ﻪﺑ MIMO

ﺎﺗ ﺖﺷﻮﻧ MISO

ﻝﺮﺘﻨﻛ ﻢﺘﺴﻴﺳ ﻲﺣﺍﺮﻃ ﻥﺎﻜﻣﺍ

ﺍﺩ ﺩﻮﺟﻭ ﺮﻴﺧﺍ ﻡﺮﻓ ﻪﺑ ﺪﺷﺎﺑ ﻪﺘﺷ

.

ﺍﺭ ﻪﺘﺴﺑ ﻪﻘﻠﺣ ﻞﻳﺪﺒﺗ ﻊﺑﺎﺗ ﻪﻛ ﻲﺗﺭﻮﺻ ﺭﺩ )

(s ﺖﺷﺍﺩ ﻢﻴﻫﺍﻮﺧ ،ﻢﻴﻣﺎﻨﺑ T :

PGF T

PG I

s F s G s P s G s P I s T

= +

⇒

+

= ⁻

) (

) ( ) ( ) ( )) ( ) ( (

)

( ¹

ﺽﺮﻓ ﺎﺑ

≠0

، P

,....

2 , 1 1

1 ) (

=

−











= 

= +

ij i

Q p

GF T

G P

[

^G

] [

^GF ^BT

]

T B

P⁻¹ =∆+ ⇒ = ∆+ ⁻¹ −

(2)

ﻲﻣ ﺕﺭﻮﺻ ﻦﻳﺍ ﺭﺩ ﺑ ﻭﺩ ﻪﺑ ﺍﺭ ﻢﺘﺴﻴﺳ ﻥﺍﻮﺗ

ﺩﻮﻤﻧ ﻞﻳﺪﺒﺗ ﺭﺩ ﺍﺰﺠﻣ ﺶﺨ .

ﻞﻳﺪﺒﺗ ﻊﺑﺎﺗ ﻪﺑ ﻁﻮﺑﺮﻣ ﻝﻭﺍ ﺶﺨﺑ

ﻭ ﺏﻮﻠﻄﻣﺎﻧ ﻱﺩﻭﺭﻭ ﻪﺑ ﻁﻮﺑﺮﻣ ﻞﻳﺪﺒﺗ ﻊﺑﺎﺗ ﻪﻛ ﺮﮕﻳﺩ ﺶﺨﺑ ﻭ ﺪﺷﺎﺑ ﺏﻮﻠﻄﻣ ﺦﺳﺎﭘ ﺪﻳﺎﺑ ﻥﺁ ﺦﺳﺎﭘ ﻪﻛ ﻱﺮﻄﻗ ﺖﺳﺍ ﻢﺘﺴﻴﺳ ﺵﺎﺸﺘﻏﺍ .

ﻲﻣ ﺍﺭ ﻪﻄﺑﺍﺭ ﻦﻳﺍﺮﺑﺎﻨﺑ ﺩﻮﻤﻧ ﻲﺴﻳﻮﻧﺯﺎﺑ ﺮﻳﺯ ﺕﺭﻮﺻ ﻪﺑ ﻥﺍﻮﺗ

.

∑

≠ 

 



− 

=

= +

+

=

i

k ik

ki ij

ij i ii

ii i ii ij

ij ij ij ij

q c t

f g v

q g w q

c v w y

1

) (

ﻞﻳﺪﺒﺗ ﻊﺑﺎﺗ

ij iiv ﻞﻳﺪﺒﺗ ﻊﺑﺎﺗ w ﻭ ﻲﺟﻭﺮﺧ ﻦﻴﻣi

ﺖﺳﺍ ﻱﺩﻭﺭﻭ ﻦﻴﻣ j .

ﻞﻳﺪﺒﺗ ﻊﺑﺎﺗ

ij iic ﻞﻳﺪﺒﺗ ﻊﺑﺎﺗ w

ﺖﺳﺍ ﺏﻮﻠﻄﻣﺎﻧ ﻱﺩﻭﺭﻭ .

ﺮﻳﺯ ﺕﺭﻮﺻ ﻪﺑ ﺍﺭ ﻞﻳﺪﺒﺗ ﻊﺑﺎﺗ ﻲﻣ ﺮﻈﻧ ﺭﺩ

ﻢﻳﺮﻴﮔ . ﻚﻳ ﻭ ﻱﺩﻭﺭﻭ ﻭﺩ ﻢﺘﺴﻴﺳ ﻭﺩ ﻪﺑ ﺪﻳﺎﺑ ﻢﺘﺴﻴﺳ ﻦﻳﺍ

ﻞﻳﺪﺒﺗ ﻲﺟﻭﺮﺧ ﺩﺩﺮﮔ

.



 





+ +

= +

1 01

. 0

2 . 0 1

3 ) 1

( s s

s s

s s G

ﻞﻜﺷ ۱ : ﻝﻭﺍ ﻱﺩﻭﺭﻭ ﻪﺑ ﺖﺒﺴﻧ ﻪﻠﭘ ﺦﺳﺎﭘ

(3)

ﻞﻜﺷ ۲ : ﻱﺩﻭﺭﻭ ﻪﺑ ﺖﺒﺴﻧ ﻪﻠﭘ ﺦﺳﺎﭘ ﻡﻭﺩ

ﺖﺳﺍ ﺮﻳﺯ ﺕﺭﻮﺻ ﻪﺑ ﻢﺘﺴﻴﺳ ﺮﻔﺻ ﻭ ﺐﻄﻗ .

0.5575 -

: zero

3 -

: poles

ﺭﺩ ﻲﻟﺎﻘﺘﻧﺍ ﺮﻔﺻ ﭻﻴﻫ ﻱﺍﺭﺍﺩ ﻭ ﺭﺍﺪﻳﺎﭘ ﻢﺘﺴﻴﺳ ﻦﻳﺍ ﺖﺴﻴﻧ RHP

. ﺴﻴﺳ ﻦﻳﺍ ﺮﺑ ﻩﻭﻼﻋ ﻝﺮﺘﻨﻛ ﻢﺘ

ﺖﺳﺍ ﺮﻳﺬﭘ .

=0 (A) A,B))-rank rank(ctrb(

ﻦﻴﻨﭽﻤﻫ

=2 rank(C*B) ﺖﺳﺍ

.

ﻢﺘﺴﻴﺳ ﻪﻜﻨﻳﺍ ﻱﺍﺮﺑ ﻦﻴﻨﭽﻤﻫ Functionally Controllable

ﺪﻳﺎﺑ ﺪﺷﺎﺑ

≠0

ﻢﻳﺭﺍﺩ ،ﺪﺷﺎﺑ P

:

9 + s 6 + s

0.998 +

s P =1.79₂

ﺖﺷﺍﺩ ﻢﻴﻫﺍﻮﺧ ﻦﻳﺍﺮﺑﺎﻨﺑ :

















− =

994 2 368

6 79

1

9 15 7

994 2 368 6 79

1

09 0 06 9 01

6

994 2 368

6 79

1

8 1 2 10 2

6 994

2 368 6 79

1

9 15 7

2 2 3

1

. s + .

+ s .

s + +

s + s .

s + .

+ s .

. s - .

- s . - -s

. s + .

+ s .

. s - . - s . - -s .

s + .

+ s .

s + +

s + s P

(4)

ﻔﺘﺳﺍ ﻱﺍﺮﺑ

؛ﻲﻨﻌﻳ ،ﺪﺷﺎﺑ ﺭﺍﺮﻗﺮﺑ ﺰﺘﻳﻭﺮﻫ ﻪﺒﻠﻏ ﻁﺮﺷ ﺪﻳﺎﺑ ﺵﻭﺭ ﻦﻳﺍ ﺯﺍ ﻩﺩﺎ

1 2

002 0 21 0 .

.

2 2

12 12

22 11

s + + s

. s + .

+ s q q

q

q =

،ﻩﮋﻳﻭ ﺭﺍﺪﻘﻣ ﻢﺳﺭ ﺎﺑ

ﻞﻜﺷ ۳ : ﻩﮋﻳﻭ ﺮﻳﺩﺎﻘﻣ

12 12

22 11

. . q q

q

ﻲﻣ ﻩﺪﻫﺎﺸﻣ ﻪﭽﻧﺎﻨﭼ ﻲﻣ ﺪﺣﺍﻭ ﺯﺍ ﺮﺘﻜﭼﻮﻛ ﻩﺭﺍﻮﻤﻫ ﻥﺁ ﺭﺍﺪﻘﻣ ،ﺩﻮﺷ

ﺭﺍﺮﻗﺮﺑ ﺰﻴﺗﻭﺭﻮﻫ ﻪﺒﻠﻏ ﻁﺮﺷ ﻭ ﺪﺷﺎﺑ

ﻲﻣ ﺪﺷﺎﺑ .

ﻪﻛ ﻲﺗﺭﻮﺻ ﺭﺩ ﻲﻣ ﺪﺷﺎﺑ ﺏﻮﻠﻄﻣ ﻪﺘﺴﺑ ﺭﺍﺪﻣ ﻞﻳﺪﺒﺗ ﻊﺑﺎﺗ T

ﺖﺷﻮﻧ ﻥﺍﻮﺗ :



 



=



 

















+ +

22 2 21 2

12 1 11 1 22

21 12 11

2 22 21

12 1

111 1

1 1

f g f g

f g f g t

t t t q g

q g q q

ﻱﺩﻭﺭﻭ ﻱﺍﺮﺑ ﺕﺭﻮﺻ ﻦﻳﺍ ﺭﺩ

1ﻱﺎﻫ ﻭ r r2

ﻲﻣ ﺖﺷﻮﻧ ﻥﺍﻮﺗ :

(5)

2 22

21 22 22 12

2 22 22

1 11

12 11 12 22

1 11 12

2 22

21 22 21 11

2 22 21

1 11

12 11 11 21

1 11 11

1 1 1 1

g q

q q f t

g q t

g q

q q f t

g q t

g q

q q f t

g q t

g q

q q f t

g q t

+



 



−

=

+



 



−

=

+



 



−

=

+



 



−

=

) ( ∗

ﻲﻣ ﻩﺪﻫﺎﺸﻣ ﻪﭽﻧﺎﻨﭼ

ij،ﺩﻮﺷ ﺪﻧﺭﺍﺩ ﺩﻮﺟﻭ ﻕﻮﻓ ﻪﻄﺑﺍﺭ ﺭﺎﻬﭼ ﺖﻤﺳ ﻡﺩ ﺭﺩ ﺎﻫ t .

ﺍﺪﺘﺑﺍ ﺖﺳﺍ ﺮﺘﻬﺑ ﻦﻳﺍﺮﺑﺎﻨﺑ

ﻲﻣ ﺏﻮﻠﻄﻣ ﻥﺁ ﺦﺳﺎﭘ ﻪﻛ ﺍﺭ ﻲﻤﺘﺴﻴﺳ ﻪﻛ ﻢﻴﻳﺎﻤﻧ ﻲﺣﺍﺮﻃ ﻱﺭﻮﻃ ﺍﺭ ﺯﺎﺴﻧﺍﺮﺒﺟ ﺐﺒﺳ ﻭ ﻪﺘﻓﺮﮔ ﺮﻈﻧ ﺭﺩ ﺪﺷﺎﺑ

ﺩﺭﻭﺁ ﺩﻮﺟﻭ ﻪﺑ ﺍﺭ ﺏﻮﻠﻄﻣ ﺦﺳﺎﭘ .

















+ +

+

+ +

= +

25 7.07s

25 20

10 100

14.14s 100 )

(

2 2

s s

s s s

T

ﻢﺘﺴﻴﺳ ﻦﻳﺍ ﺏﻮﻠﻄﻣ ﺦﺳﺎﭘ ﺕﺭﻮﺻ ﻦﻳﺍ ﺭﺩ ﺩﻮﺑ ﺪﻫﺍﻮﺧ ﺮﻳﺯ ﺕﺭﻮﺻ ﻪﺑ

.

ﻞﻜﺷ ۴ : ﻪﻠﭘ ﺦﺳﺎﭘ ﻝﻭﺍ ﻱﺩﻭﺭﻭ ﻪﺑ ﺖﺒﺴﻧ T

(6)

ﻞﻜﺷ ۵ : ﻪﻠﭘ ﺦﺳﺎﭘ ﻱﺩﻭﺭﻭ ﻪﺑ ﺖﺒﺴﻧ T

ﻡﻭﺩ

ﻂﺑﺍﺭ ﺭﺩ ﻥﺩﺍﺩ ﺭﺍﺮﻗ ﻭ ﻩﺪﻣﺁ ﺖﺳﺪﺑ ﺏﻮﻠﻄﻣ ﺭﺍﺪﻘﻣ ﺯﺍ ﻩﺩﺎﻔﺘﺳﺍ ﺎﺑ

∗) ( ﺖﺷﺍﺩ ﻢﻴﻫﺍﻮﺧ :

T(2,2))))

* Q(2,2) -

Q(2,2)

* (F(2,2)

* (2,1) Q(2,2))/(Q

* T(1,2) + Q(2,1)

* (2,2) minreal((T

= g2

T(1,1))))

* Q(1,1) - Q(1,1)

* (F(1,1)

* (1,2) Q(1,1))/(Q

* T(2,1) + Q(1,2)

* (1,1) minreal((T

= g1

ﻪﺑ ﻪﺟﻮﺗ ﺎﺑ gi

ﻫ ﻩﺩﺎﺳ ﺎﺑ ﻩﺪﻣﺁ ﺖﺳﺪﺑ ﻱﺎ ﺮﻟﺮﺘﻨﻛ ﻚﻳ ﻡﺮﻓ ﻪﺑ ﺍﺭ ﻥﺁ ﻪﻓﺎﺿﺍ ﻱﺎﻫﺮﻔﺻ ﻭ ﺐﻄﻗ ﻑﺬﺣ ﻭ ﻱﺯﺎﺳ

ﻲﻣ ﻞﻳﺪﺒﺗ PID ﻢﻴﻨﻛ

.

0.001 + s 0.0313 +

s 0.4478 +

s

0.0776 +

s 0.2446 +

s 1.194

g ₃ ₂

2 1 =

z =

-0.6807 + 1.4024i -0.6807 - 1.4024i p =

-0.4286 + 0.7012i -0.4286 - 0.7012i -0.3109

k = 5.1847

(7)

ﻳ ﻞﻳﺪﺒﺗ ﻊﺑﺎﺗ ﻦﻳﺍ ﺮﻟﺮﺘﻨﻛ ﻚ

ﺎﺑ PID ﻭﺩ ﺖﺳﺍ ﻪﻓﺎﺿﺍ ﻲﻣﻮﻫﻮﻣ ﺐﻄﻗ .

ﻦﻴﻨﭽﻤﻫ

202 1

03 24 96

64 53

51 ²

2 s + .

. s + .

+ s g = .

z =

-0.6303 + 0.2629i -0.6303 - 0.2629i p = -1.2017 k = 51.5345

ﺮﻟﺮﺘﻨﻛ ﻪﻛ

g2

ﺮﻟﺮﺘﻨﻛ ﻚﻳ ﺖﺳﺍ PID

.

ﺮﻟﺮﺘﻨﻛ ﻪﻛ ﻲﺗﺭﻮﺻ ﺭﺩ ﺕﺭﻮﺻ ﻪﺑ



 



= 

2 1

0 0 g G g

ﻦﻳﺍ ﺭﺩ ﺎﺑ ﻞﻳﺪﺒﺗ ﻊﺑﺍﻮﺗ ،ﺕﺭﻮﺻ ﺪﺷ ﺪﻫﺍﻮﺧ ﻞﻳﺪﺒﺗ ﺮﻳﺯ ﺕﺭﻮﺻ ﻪﺑ ﻢﺘﺴﻴﺳ

.

0.2959 +

s 1.536 + s 3.095 + s 2.854 + s

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - :

#2

0.2572 +

s 1.157 + s 1.799 + s 0.9893

0 :

#1

output...

to 2 input from function Transfer

0 :

#2

13.2 + s 46.47 + s 58.55 + s 43.19 + s 23.06 + s 6.168 + s

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - :

#1

12.57 + s 29.6 + s 17.81 + s 9.281

output...

Y

2 3

4

2 3

4 5

6

2 3

ij =

ﻞﻳﺪﺒﺗ ﻊﺑﺎﺗ ﻦﻴﻨﭽﻤﻫ ﺩﻮﺑ ﺪﻫﺍﻮﺧ ﺮﻳﺯ ﺕﺭﻮﺻ ﻪﺑ ﺏﻮﻠﻄﻣﺎﻧ ﻱﺩﻭﺭﻭ

.

(8)

2.959 + s 12.7 + s 19.78 + s 11.85 + s

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - :

#2

s 0.0003866 +

s 0.03911 +

s 0.04517 +

s 0.01072

264.1 + s 678.4 + s 539.1 + s 383.2 + s 121.3 + s 25.17 + s

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - :

#1

s 0.126 + s 1.237 + s 3.925 + s 5.28 + s 4.368 + s

output...

2.959 + s 12.7 + s 19.78 + s 11.85 + s

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - :

#2

s 0.0003866 +

s 0.03911 +

s 0.04517 +

s 0.01072

264.1 + s 678.4 + s 539.1 + s 383.2 + s 121.3 + s 25.17 + s

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - :

#1

s 0.126 + s 1.237 + s 3.925 + s 5.28 + s 4.368 + s

output...

2 3

4

2 3

4

2 3

4 5

6

2 3

4 5

6

2 3

4

2 3

4

2 3

4 5

6

2 3

4 5

6

ﻞﻜﺷ ۶ : ﻪﻠﭘ ﺦﺳﺎﭘ ﺎﺑ ﻪﺘﺴﺑ ﻪﻘﻠﺣ

ﻝﻭﺍ ﻱﺩﻭﺭﻭ ﻪﺑ ﺖﺒﺴﻧ

(9)

ﻞﻜﺷ ۷ : ﻪﻠﭘ ﺦﺳﺎﭘ ﻱﺩﻭﺭﻭ ﻪﺑ ﺖﺒﺴﻧ ﺎﺑ ﻪﺘﺴﺑ ﻪﻘﻠﺣ

ﻡﻭﺩ

ﻞﻜﺷ ۸ : ﻪﻠﭘ ﺦﺳﺎﭘ ﻝﻭﺍ ﺏﻮﻠﻄﻣﺎﻧ ﻱﺩﻭﺭﻭ ﻪﺑ ﺖﺒﺴﻧ ﺎﺑ ﻪﺘﺴﺑ ﻪﻘﻠﺣ

(10)

ﻞﻜﺷ ۹ : ﻪﻠﭘ ﺦﺳﺎﭘ ﻡﻭﺩ ﺏﻮﻠﻄﻣﺎﻧ ﻱﺩﻭﺭﻭ ﻪﺑ ﺖﺒﺴﻧ ﺎﺑ ﻪﺘﺴﺑ ﻪﻘﻠﺣ

ﻲﻣ ﻩﺪﻫﺎﺸﻣ ﻪﭽﻧﺎﻨﭼ

ﻱﺩﻭﺭﻭ ﺮﺛﺍ ﻭ ﻩﺩﻮﻤﻧ ﺍﺰﺠﻣ ﺍﺭ ﻢﺘﺴﻴﺳ ﻲﺑﻮﺧ ﻪﺑ ﺮﻟﺮﺘﻨﻛ ﻦﻳﺍ ﺩﻮﺷ ﺯﺍ ﺍﺭ ﺏﻮﻠﻄﻣﺎﻧ ﻱﺎﻫ

ﻲﻣ ﻦﻴﺑ ﺩﺮﺑ

. ﻦﻴﻨﭽﻤﻫ Tracking

ﺷ ﻡﺎﺠﻧﺍ ﻲﺑﻮﺨﺑ ﻲﺟﻭﺮﺧ ﻭ ﻩﺪ

ﺩﻮﺧ ﺏﻮﻠﻄﻣ ﺭﺍﺪﻘﻣ ﻪﺑ ﺖﻋﺮﺳ ﻪﺑ ﺎﻫ

ﻲﻣ ﺪﻨﺳﺭ . ﻡﻭﺎﻘﻣ ﻝﺮﺘﻨﻛ ﻱﺎﻬﺷﻭﺭ ﺭﺩ ﻭ ﻩﺩﻮﺑ ﺮﺛﻮﻣ ﺭﺎﻴﺴﺑ ﺵﻭﺭ ﻦﻳﺍ ﺯﺍ ﻩﺩﺎﻔﺘﺳﺍ ﺭﺍﺮﻗ ﻩﺩﺎﻔﺘﺳﺍ ﺩﺭﻮﻣ QFT

ﻲﻣ ﺩﺮﻴﮔ .

Ali ChaibakhshDigitally signed by Ali Chaibakhsh DN: cn=Ali Chaibakhsh, c=IR Date: 2007.06.04 20:23:53 Z Signature

Not Verified