ﭼﻜﻴﺪﻩ ﻣﻘﺪﻣﻪ ﻃﺮﺍﺣﻲ ﮐﻨﺘﺮﻝ ﮐﻨﻨﺪﻩ ﺑﺮﺍﻱ ﺳﻴﺴﺘﻢ ﻫﺎﻱ ﻏﻴﺮ

٠

ﺐﻟﺎﻄﻣ ﺖﺳﺮﻬﻓ :

ﻩﺪﻴﻜﭼ

...

...

...

...

...

. ۱

ﻪﻣﺪﻘﻣ

...

...

...

...

...

. ۲

ﻲﻄﺧﺮﻴﻏ ﻱﺎﻫ ﻢﺘﺴﻴﺳ ﻱﺍﺮﺑ ﻩﺪﻨﻨﮐ ﻝﺮﺘﻨﮐ ﻲﺣﺍﺮﻃ

...

...

...

۴

ﺵﻭﺭ FEEDBACK LINEARIZATION

...

...

...

...

٤

ﺵﻭﺭ LINEARIZATION

INPUT-OUTPUT

...

...

...

...

٦

ﺵﻭﺭ LINEARIZATION

INPUT STATE

...

...

...

...

٨

ﻢﺘﺴﻴﺳ ﻲﺿﺎﻳﺭ ﻝﺪﻣ

...

...

...

...

...

۱۰

ﺵﻭﺭ ﻲﺳﺭﺮﺑ FEEDBACK LINEARIZATION

ﻲﺸﻨﮐﺍﻭ ﻲﺸﺧﺮﭼ ﻝﻭﺪﻧﺎﭘ ﻱﻭﺭ ﺮﺑ

...

۱۱

ﻱﺯﺎﺳ ﻪﻴﺒﺷ ﻭ ﺞﻳﺎﺘﻧ :

...

...

...

...

...

۱۴

ﻪﺠﻴﺘﻧ ﻱﺮﻴﮔ :

...

...

...

...

...

۲۱

١

ﻩﺪﻴﻜﭼ :

ﻝﺮﺘﻨﻛ ﻭ ﻞﻴﻠﺤﺗ ﻩﮊﻭﺮﭘ ﻦﻳﺍ ﺭﺩ ﻳ

ﺳ ﮏ ﻴ ﺸﺧﺮﭼ ﻝﻭﺪﻧﺎﭘ ﻢﺘﺴ ﻲ

ﺸﻨﮐﺍﻭ

١ﻲ ﺳﺭﺮﺑ ﺩﺭﻮﻣ ﻲ

ﻪـﺘﻓﺮﮔ ﺭﺍﺮـﻗ

ﺳﺍ ﺖ . ﺵﻭﺭ ﺯﺍ ﻢﺘﺴﻴﺳ ﻦﻳﺍ ﻝﺮﺘﻨﻛ ﻱﺍﺮﺑ Feedback linearization

ﺷ ﻩﺩﺎﻔﺘﺳﺍ ﺖـﺳﺍ ﻩﺪ

. ﻱﺮﻴﮔﺭﺎـﻜﺑ

ﻲﻣ ﻥﺎﺸﻧ ﺵﻭﺭ ﻦﻳﺍ ﻪﻛ ﺪﻫﺩ

ﻲﻣ ﻲﺑﻮﺨﺑ ﺵﻭﺭ ﻦﻳﺍ ﺍﻮﺗ

ﺪﻧ ﻩﺪﺷ ﺺﺨﺸﻣ ﻩﺩﻭﺪﺤﻣ ﺭﺩ ﺪـﻨﻛ ﺭﺍﺪﻳﺎﭘ ﺍﺭ ﻢﺘﺴﻴﺳ

.

ﺭﺍﺰـﻓﺍ ﻡﺮـﻧ ﺭﺩ ﻱﺮﺗﻮﻴﭙﻣﺎـﻛ ﻱﺯﺎـﺳ ﻪﻴﺒـﺷ ﺯﺍ ﻩﺩﺎﻔﺘـﺳﺍ ﺎـﺑ ﺵﻭﺭ ﻦـﻳﺍ ﻱﺮﻴﮔﺭﺎـﻜﺑ ﺯﺍ ﻞﺻﺎﺣ ﺞﻳﺎﺘﻧ ﻥﺎﻳﺎﭘ ﺭﺩ ﻤﻴﺳ ﺖﺳﺍ ﻩﺪﻣﺁ ﻚﻨﻴﻟﻮ .

Reaction Wheel Pendulum١

٢

ﻪﻣﺪﻘﻣ :

ﺧﺮﭼ ﻝﻭﺪﻧﺎﭘ ﺸ

ﻲ ﺸﻨﮐﺍﻭ ﻲ ﻪﮐ ﻪﻧﻮﮕﻧﺎﻤﻫ ﻞﮑﺷ ﺭﺩ

۱ ﺖﺳﺍ ﻩﺪﺷ ﻩﺩﺍﺩ ﻥﺎﺸﻧ ﺖﺳﺍ ﻲﮑﻳﺰﻴﻓ ﻝﻭﺪﻧﺎﭘ ﮏﻳ

ﺖـﺳﺍ ﻩﺪﺷ ﻞﺼﺘﻣ ﻥﺁ ﻱﺎﻬﺘﻧﺍ ﻪﺑ ﻥﺭﺎﻘﺘﻣ ﮏﺴﻳﺩ ﮏﻳ ﻪﮐ .

ﺎـﺑ ﻱﺯﺍﻮـﻣ ﺖـﻬﺟ ﺭﺩ ﺪـﻧﺍﻮﺗ ﻲـﻣ ﮏـﺴﻳﺩ ﻦـﻳﺍ

ﻪﻧﺍﺩﺍﺯﺁ ﺕﺭﻮﺼﺑ ﻝﻭﺪﻧﺎﭘ ﺶﺧﺮﭼ ﺪﺧﺮﭽﺑ

. ﺭﻮـﺗﻮﻣ ﮏـﻳ ﻂﺳﻮﺗ ﮏﺴﻳﺩ ـﻳﺮﺤﺗDC

ﺭﻭﺎﺘـﺸﮔ ﻭ ﺩﻮـﺷ ﻲـﻣ ﮏ

ﺍﺰﺠﻣ ﻱ ﻟﻮﺗ ﻴ ﺍ ﻪﻈﺤﻟ ﺏﺎﺘﺷ ﻂﺳﻮﺗ ﻩﺪﺷ ﺪ ﻱ

ﺩﻳ ﻣ ﮏﺴ ﻲ ﺪﻧﺍﻮﺗ ﺳ ﻝﺮﺘﻨﮐ ﺖﻬﺟ ﻴ

ﻢﺘـﺴ ﺭﺍﺮـﻗ ﻩﺩﺎﻔﺘـﺳﺍ ﺩﺭﻮـﻣ

ﮔﻴ ﺩﺮ .

ﻞﮑﺷ ۱ - ﺸﺧﺮﭼ ﻝﻭﺪﻧﺎﭘ ﻲ

ﺸﻨﮐﺍﻭ ﻲ

ﺍﺮﺑ ﻩﺪﺷ ﻩﺩﺎﻔﺘﺳﺍ ﻪﻟﺎﻘﻣ ﺭﺩ ﻱ

ﺍﻳ ﻟﺮﺘﻨﮐ ﺵﻭﺭ ﻭﺩ ﻩﮊﻭﺮﭘ ﻦ ﻲ

ﺍﺮﺑ ﻱ ﺍ ﻝﺮﺘﻨﮐ ﻳ

ﺳ ﻦ ﻴ ﺭﺎﮐ ﻪﺑ ﻢﺘﺴ

ﺖﺳﺍ ﻩﺪﺷ ﻩﺩﺮﺑ .

ﺵﻭﺭ

Swingup Control

ﻭ

Feedback Linearization

. ﺭﺩ ﻪﮐ ﺭﻮﻄﻧﺎﻤﻫ

ﺎﻫ ﺶﺨﺑ ﻱ

ﺪﻌﺑ ﻱ

ﺵﻭﺭ ﺪﺷ ﺪﻫﺍﻮﺧ ﺺﺨﺸﻣ ﻢﻫ ﻪﻟﺎﻘﻣ

Feedback Linearization

ﺎﻬﻨﺗ

ﻩﺩﻭﺪﺤﻣ ﺭﺩ

2

1 <π

ﺍﻳ

q

ﺳ ﻦ ﻴ ﺩﺮﮐ ﺪﻫﺍﻮﺧ ﻝﺮﺘﻨﮐ ﺍﺭ ﻢﺘﺴ .

ﺍﺮﺑﺎﻨﺑ ﻳ ﺍﺮﺑ ﻪﻃﻮﺑﺮﻣ ﻪﻟﺎﻘﻣ ﺭﺩ ﻦ ﻱ

ﺳ ﻥﺪﻧﺎﺳﺭ ﻴ

ﺍ ﻪﺑ ﻢﺘﺴ ﻳ

ﺵﻭﺭ ﺯﺍ ﻩﺩﻭﺪﺤﻣ ﻦ

Swingup Control

ﻩﺪﺷ ﻩﺩﺎﻔﺘﺳﺍ ﺖﺳﺍ

.

ﻳ

ﻨﻌ

ﻲ

ﺭﺩ

٣

ﺳ ﻝﺮﺘﻨﮐ ﻊﻗﺍﻭ ﻴ

ﻩﺩﻭﺪﺤﻣ ﺭﺩ ﻢﺘﺴ

2

1 ≥π

ﻨﭽﻤﻫ ﻭ

q

ﻴ ﻝﻭﺪﻧﺎﭘ ﻥﺩﺮﺑ ﻻﺎﺑ ﻦ ﻪﻴﺣﺎﻧ ﻦﻳﺍ ﺯﺍ

ﻟﺮﺘﻨﮐ ﺵﻭﺭ ﻂﺳﻮﺗ ﻲ

Swingup Control

ﻣ ﻡﺎﺠﻧﺍ ﻲ ﺩﻮﺷ .

ﺍ ﻡﺎﺠﻧﺍ ﺯﺍ ﻑﺪﻫ ﻪﮐ ﺎﺠﻧﺁ ﺯﺍ ﻳ

ﺳﺭﺮﺑ ﻩﮊﻭﺮﭘ ﻦ ﻲ

ﺵﻭﺭ

Feedback Linearization

ﻭﺭ ﺮﺑ ﻱ ﺍﻳ ﻦ

ﺳ ﻴ ﻢﺘﺴ ﺍ ﺭﺩ ﺍﺬﻟ ﺖﺳﺍ ﻳ

ﺍ ﺎﻬﻨﺗ ﺵﺭﺍﺰﮔ ﻦ ﻳ

ﺳﺭﺮﺑ ﺯﺍ ﻭ ﻩﺩﺍﺩ ﻪﺟﻮﺗ ﺩﺭﻮﻣ ﺍﺭ ﺵﻭﺭ ﻦ ﻲ

ﺵﻭﺭ

ﻟﺮﺘﻨﮐ ﻲ

Swingup Control

ﻣ ﺮﻈﻨﻓﺮﺻ ﻲ

ﺩﻮﺷ .

ﺍ ﺭﺩ ﻳ ﺳﺭﺮﺑ ﻪﺑ ﺍﺪﺘﺑﺍ ﺵﺭﺍﺰﮔ ﻦ ﻲ

ﻠﮐ ﻲ ﺎﻫ ﻩﺪﻨﻨﮐ ﻝﺮﺘﻨﮐ ﻱ

ﻏ ﻴ ﻄﺧﺮ ﻲ , ﺵﻭﺭ

Feedback

Linearization

ﺎﻫ ﺶﺨﺑ ﻭ ﻱ

ﻫﺍﻮﺧ ﻥﺁ ﻒﻠﺘﺨﻣ ﻴ

ﺖﺧﺍﺩﺮﭘ ﻢ .

ﺳﺭﺮﺑ ﻪﺑ ﺲﭙﺳ ﻲ

ﺭ ﻝﺪﻣ ﻳ ﺿﺎ ﻲ

ﺳ ﻴ ﻣ ﻢﺘﺴ ﻲ ﭘ ﺯﺍﺩﺮ ﻳ ﺵﻭﺭ ﻭ ﻢ

Feedback Linearization

ﻭﺭ ﺮﺑ ﺍﺭ ﻱ ﺍﻳ ﺳ ﻦ ﻴ ﻣ ﻝﺎﻤﻋﺍ ﻢﺘﺴ ﻲ

ﻨﮐ ﻴ ﻢ . ﻧ ﺎﻬﺘﻧﺍ ﺭﺩ ﻴ

ﺎﺘﻧ ﺰ ﻳ ﺒﺷ ﻭ ﺞ ﻴ

ﺯﺎﺳ ﻪ ﻱ ﺎﻫ ﻱ ﺍ ﻝﺎﻤﻋﺍ ﺯﺍ ﻞﺻﺎﺣ ﻳ

ﻭﺭ ﺮﺑ ﺵﻭﺭ ﻦ ﻱ

ﺳ ﻴ ﺍﺭ ﻢﺘﺴ

ﻣ ﺭﺍﺮﻗ ﻪﺟﻮﺗ ﺩﺭﻮﻣ ﻲ

ﻫﺩ ﻴ ﻢ .

٤

ﻲﻄﺧﺮﻴﻏ ﻱﺎﻫ ﻢﺘﺴﻴﺳ ﻱﺍﺮﺑ ﻩﺪﻨﻨﮐ ﻝﺮﺘﻨﮐ ﻲﺣﺍﺮﻃ :

ﺘﻨﮐ ﺯﺍ ﻩﺩﺎﻔﺘﺳﺍ ﻩﺪﻤﻋ ﻞﻳﻻﺩ ﺯﺍ ﻲﮑﻳ

ﻦﻳﺍ ﻪﮐ ﺖﺳﺍ ﻲﻌﻴﺳﻭ ﻩﺩﻭﺪﺤﻣ ﻲﻄﺧﺮﻴﻏ ﻱﺎﻫ ﻩﺪﻨﻨﮐ ﻝﺮ

ﻝﺮﺘﻨﮐ ﻦﻳﺍ ﻪﮐ ﺖﺳﺍ ﻲﻳﻻﺎﺑ ﺭﺎﻴﺴﺑ ﺖﻗﺩ ﻦﻴﻨﭽﻤﻫ ﻭ ﺪﻨﮐ ﻞﻤﻋ ﺪﻧﺍﻮﺗ ﻲﻣ ﻩﺪﻨﻨﮐ ﻝﺮﺘﻨﮐ ﺩﺭﺍﺩ ﻩﺪﻨﻨﮐ .

ﻪﻟﺎﺴﻣ ﻲﮑﻳ ﺪﻳﺁ ﻲﻣ ﺶﻴﭘ ﻪﻟﺎﺴﻣ ﻭﺩ ﺎﻫ ﻩﺪﻨﻨﮐ ﻝﺮﺘﻨﮐ ﻉﻮﻧ ﻦﻳﺍ ﻲﺣﺍﺮﻃ ﺭﺩ

Regulation

ﻭ

ﻪﻟﺎﺴﻣ ﻱﺮﮕﻳﺩ

Traking

ﺖﺳﺍ .

ﻲﻄﺧﺮﻴﻏ ﻲﮑﻴﻣﺎﻨﻳﺩ ﻢﺘﺴﻴﺳ

) ), ( ), (

(x t u t t f

x&=

ﺪﻳﺮﻴﮕﺑ ﺮﻈﻧ ﺭﺩ ﺍﺭ .

ﻪﻟﺎﺴﻣ ﺭﺩ

Regulation

ﻝﺮﺘﻨﮐ ﻥﻮﻧﺎﻗ ﺪﻳﺎﺑ ﻪﻴﻟﻭﺍ ﻂﻳﺍﺮﺷ ﺮﮔﺍ ﻪﮐ ﻢﻳﺭﻭﺁ ﺖﺳﺩ ﻪﺑ ﻱﺭﻮﻃ ﺍﺭ u

ﻝﻮﺒﻗ ﻞﺑﺎﻗ ﻱﺎﻀﻓ ﺯﺍ ﺎﺠﮐ ﺮﻫ ﺭﺩ ﻕﻮﻓ ﻢﺘﺴﻴﺳ )

(

Ω

ﻪﺑ ﺍﺭ ﻢﺘﻴﺳ ﻝﺮﺘﻨﮐ ﻥﻮﻧﺎﻗ ﻦﻳﺍ ﻩﺎﮕﻧﺁ ﺪﺷﺎﺑ

ﺪﻧﺎﺳﺮﺑ ﻝﺩﺎﻌﺗ ﻪﻄﻘﻧ .

ﻥﺎﻣﺯ ﺭﺩ ﻲﻨﻌﻳ

∞

→

ﺖﻟﺎﺣ ﻪﺑ

t

=0

ﺪﺳﺮﺑ

x

.

ﻪﻟﺎﺴﻣ ﺭﺩ

Trakin g

ﻝﺮﺘﻨﮐ ﻥﻮﻧﺎﻗ ﻥﺩﺭﻭﺁ ﺖﺳﺪﺑ ﻑﺪﻫ ﻂﻳﺍﺮﺷ ﺮﻫ ﺯﺍ ﻪﮐ ﻱﺭﻮﻃ ﺖﺳﺍ u

ﻱﺎﻀﻓ ﺭﺩ ﻪﻴﻟﻭﺍ ﻱﺎﻄﺧ ﻢﻴﻨﮐ ﻉﻭﺮﺷ

Ω

Traking

ﻲﻨﻌﻳ

) ( ) (t y t y − _d

ﻞﻴﻣ ﺮﻔﺻ ﺖﻤﺳ ﻪﺑ

ﻱﺎﻫ ﺖﻟﺎﺣ ﻦﻴﻨﭽﻤﻫ ﻭ ﺪﻨﮐ ﺪﻧﺎﻤﺑ ﺩﻭﺪﺤﻣ ﺰﻴﻧ ﻢﺘﺴﻴﺳ

.

ﺵﻭﺭ ﻲﻄﺧﺮﻴﻏ ﻱﺎﻫ ﻩﺪﻨﻨﮐ ﻝﺮﺘﻨﮐ ﻲﺣﺍﺮﻃ ﺭﺩ ﻱﺩﺮﺑﺭﺎﮐ ﺭﺎﻴﺴﺑ ﻱﺎﻬﺷﻭﺭ ﺯﺍ ﻲﮑﻳ

Feedback

linearization

ﺖﺳﺍ .

ﺵﻭﺭ Feedback linearization :

ﺮﻴﻏ ﺎﻫﺎﮔ ﺎﻳ ﻭ ﻲﻄﺧ ﻢﺘﺴﻴﺳ ﮏﻳ ﻪﺑ ﺍﺭ ﻲﻄﺧﺮﻴﻏ ﮏﻴﻣﺎﻨﻳﺩ ﺎﺑ ﻢﺘﺴﻴﺳ ﺍﺪﺘﺑﺍ ﺵﻭﺭ ﻦﻳﺍ ﺭﺩ ﺵﻭﺭ ﻂﺳﻮﺗ ﺲﭙﺳ ﻭ ﻩﺩﺮﮐ ﻞﻳﺪﺒﺗ ﻲﻄﺧ ﺎﺘﺳﺍ ﻱﺎﻫ

ﻲﻣ ﻝﺮﺘﻨﮐ ﺍﺭ ﻥﺁ ﻲﻄﺧ ﺩﺭﺍﺪﻧ ﻛ

ﺪﻨﻨ . ﺭﺩ

ﺖﻬﺟ ﻲﻄﺧ ﻩﺪﻨﻨﮐ ﻝﺮﺘﻨﮐ ﮏﻳ ﺏﺎﺨﺘﻧﺍ ﻭ ﻲﻄﺧﺮﻴﻏ ﺮﺛﺍ ﻑﺬﺣ ﻑﺪﻫ ﺵﻭﺭ ﻦﻳﺍ ﺭﺩ ﻊﻗﺍﻭ

ﺖﺳﺍ ﻱﺯﺎﺳ ﺭﺍﺪﻳﺎﭘ

.

٥

ﺖﺳﺍ ﺖﻤﺴﻗ ﻭﺩ ﻞﻣﺎﺷ ﺵﻭﺭ ﻦﻳﺍ .

۱ ( Linearization

Input State

۲ ( Linearization

Input-Output

ﺵﻭﺭ ﺭﺩ

Linearization Input-Output

ﻃ ﻪﻟﺎﺴﻣ ﻪﻟﺎﺴﻣ ﻲﺣﺍﺮ

Traking

ﺭﺩ ﺎﻣﺍ ﺖﺳﺍ

ﺵﻭﺭ

Linearization Input Stat

e

ﻪﻟﺎﺴﻣ ﻲﺣﺍﺮﻃ ﻪﻟﺎﺴﻣ

Regulation

ﺖﺳﺍ .

ﻝﺮﺘﻨﮐ ﻥﻮﻧﺎﻗ ﻪﮐ ﺖﺳﺍ ﻦﻳﺍ ﻑﺪﻫ ﺵﻭﺭ ﻭﺩ ﺮﻫ ﺭﺩ ﻪﺘﺳﺍﻮﺧ ﻪﮐ ﻢﻳﺭﻭﺁ ﺖﺳﺩ ﻪﺑ ﻱﺭﻮﻃ ﺍﺭ u

ﺩﺯﺎﺳ ﻩﺩﺭﻭﺁﺮﺑ ﺍﺭ ﺮﻈﻧ ﺩﺭﻮﻣ ﻪﻟﺎﺴﻣ ﻱﺎﻫ .

ﻝﺮﺘﻨﮐ ﻥﻮﻧﺎﻗ ﻲﺣﺍﺮﻃ ﻪﻟﺎﺴﻣ ﻲﻠﮐ ﺖﻟﺎﺣ ﺭﺩ ﻢﺘﺴﻴﺳ ﻱﺍﺮﺑ u

ﻡﺮﻓ ﻪﺑ ﻱﺩﻭﺭﻭ ﮏﺗ ﻲﻄﺧﺮﻴﻏ

) ), ( ), ( ( )

(t f x t u t t

x& =

ﺖﺳﺍ ﻪﻠﺣﺮﻣ ﻭﺩ ﻞﻣﺎﺷ .

۱ ( ﻲﻄﺧﺮﻴﻏ ﺖﻟﺎﺣ ﻞﻳﺪﺒﺗ ﻥﺩﺭﻭﺁ ﺖﺳﺩ ﻪﺑ )

ﻲﻨﻌﻳ ﺯﺍ ﻲﻄﺧﺮﻴﻏ ﻲﻌﺑﺎﺗ x

ﺪﺷﺎﺑ z ( ﻪﺑ

ﺕﺭﻮﺻ

) (x w Z =

ﻝﺮﺘﻨﮐ ﻥﻮﻧﺎﻗ ﻥﺩﺭﻭﺁ ﺖﺳﺩ ﻪﺑ ﺲﭙﺳ ﻭ ﺮﻴﻏ ﻲﻌﺑﺎﺗ ﺩﻮﺧ ﻪﮐ u

ﺯﺍ ﻲﻄﺧ ﻭ x

ﺖﺳﺍ v )

) , (x v g u=

( ﻪﺑ

ﻢﺘﺴﻴﺳ ﻪﺑ ﻲﻄﺧ ﺮﻴﻏ ﻢﺘﺴﻴﺳ ﻪﮐ ﻱﺭﻮﻃ

ﻲﻄﺧ

BV AZ

Z& = +

ﺩﻮﺷ ﻞﻳﺪﺒﺗ .

۲ ( ﻲﺣﺍﺮﻃ ﻱﺍﺮﺑ ﻲﻄﺧ ﺩﺭﺍﺪﻧﺎﺘﺳﺍ ﻱﺎﻫﺪﺘﻣ ﺯﺍ ﻲﮑﻳ ﺯﺍ ﻩﺩﺎﻔﺘﺳﺍ . v

ﺍ ﻪﻣﺍﺩﺍ ﺭﺩ ﻳ

ﻟﺮﺘﻨﮐ ﺵﻭﺭ ﻭﺩ ﻦ ﻲ

ﺳﺭﺮﺑ ﺩﺭﻮﻣ ﻪﺻﻼﺧ ﺭﻮﻃ ﻪﺑ ﺍﺭ ﻲ

ﻣ ﺭﺍﺮﻗ ﻲ ﻫﺩ ﻴ ﻢ .

٦

ﺵﻭﺭ Linearization Input-Output

:

ﺘﺴﻴﺳ ﺕﻻﺩﺎﻌﻣ ﺪﻴﻨﮐ ﺽﺮﻓ ﺪﺷﺎﺑ ﺮﻳﺯ ﺕﺭﻮﺻ ﻪﺑ ﺎﻣ ﻢ

:

) ۱ (

) x ( h y

u ) x ( g ) x ( f x

=

+

& =

) ۲ (

v ) x ( ) x ( u =α +β

ﻪﮐ ﻢﻳﺮﻴﮔ ﻲﻣ ﻪﺠﻴﺘﻧ ﻻﺎﺑ ﺕﻻﺩﺎﻌﻣ ﺯﺍ ﻪﮐ

) ۳ (

) x ( h y

v ) x ( ) x ( g ) x ( ) x ( g ) x ( f x

=

β + α +

& =

ﻢﻴﻧﺍﻮﺗ ﻲﻣ ﺎﻣ ﺕﻻﺩﺎﻌﻣ ﻦﻳﺍ ﺯﺍ ﻩﺩﺎﻔﺘﺳﺍ ﺎﺑ input-output

ﻱﺯﺎـﺳ ﻩﺩﺎﻴﭘ ﻢﺘﺴﻴﺳ ﻱﻭﺭ ﺮﺑ ﺍﺭ

ﻢﻴﻨﮐ . ﻝﺮﺘﻨﮐ ﻥﻮﻧﺎﻗ ﻪﮐ ﺖﺳﺍ ﻦﻳﺍ ﺎﻣ ﻑﺪﻫ ﻢﻴـﻨﮐ ﺪﻴﻟﻮﺗ ﺍﺭ u

. ﻢﻴﻧﺍﻮـﺘﺑ ﺪـﻳﺎﺑ ﺎـﻣ ﻲـﻨﻌﻳ u

ﺴﻴﺳ ﺎﺗ ﻢﻴﻨﮐ ﻝﺎﻤﻋﺍ ﻢﺘﺴﻴﺳ ﻪﺑ ﺍﺭ ﺐﺳﺎﻨﻣ ﺩﻮﺷ ﺭﺍﺪﻳﺎﭘ ﻢﺘ

.

ﺪﻳﺎﺑ ﺍﺪﺘﺑﺍ ﺭﺎﮐ ﻦﻳﺍ ﻱﺍﺮﺑ ﻖﻳﺮﻃ ﺯﺍ ﻭ ﻢﻳﺯﺎﺴﺑ ﺍﺭ v

ﻪـﻟﺩﺎﻌﻣ ﻪـﺑ ﻪﺟﻮﺗ ﺎﺑ ﻭ v )

۲ ( ﻢﻳﺯﺎـﺴﺑ ﺍﺭ u

ﻞﮑﺷ ﺕﺭﻮﺻ ﻪﺑ ﻲﻟﺮﺘﻨﮐ ﻞﮑﺷ

۲ - ﺖﺳﺍ .

ﻞﮑﺷ ﺭﺩ ۲

- ﮏـﻴﻣﺎﻨﻳﺩ ﻱﺯﺎﺳ ﺭﺍﺪﻳﺎﭘ ﻱﺍﺮﺑ ﻲﺟﺭﺎﺧ ﻪﻘﻠﺣ ﻭ ﻥﺩﺮﮐ ﻲﻄﺧ ﻱﺍﺮﺑ ﻲﻠﺧﺍﺩ ﻪﻘﻠﺣ

ﺩﻮﺷ ﻲﻣ ﻪﺘﻓﺮﮔ ﺭﺎﮑﺑ ﻪﺘﺴﺑ ﻪﻘﻠﺣ

.

ﻪﻠﺌﺴﻣ ﺭﺩ I-O-L

ﺎﺑ ﺎﻣ ﻢﻳﺭﺍﺩ ﺭﺎﮐ ﻭ ﺮﺳ ﻥﺩﺮﮐ ﻝﺎﺒﻧﺩ ﻪﻠﺌﺴﻣ .

ﻪـﮐ ﺖـﺳﺍ ﻦـﻳﺍ ﻑﺪـﻫ ﻲﻨﻌﻳ

ﺪﻨﮐ ﻝﺎﺒﻧﺩ ﺍﺭ ﺎﻣ ﺮﻈﻧ ﺩﺭﻮﻣ ﻱﺩﻭﺭﻭ ﻢﺘﺴﻴﺳ ﻲﺟﻭﺮﺧ .

٧

ﻪﻠﺌﺴﻣ ﺭﺩ I-O-L

ﻦﻴﺑ ﻢﻴﻘﺘﺴﻣ ﻪﻄﺑﺍﺭ ﺍﺪﺘﺑﺍ ﻪﮐ ﻢﻴﻨﮐ ﻲﻣ ﻲﻌﺳ ﺎﻣ ﻭ y

ﻢﻴﻨﮐ ﺍﺪﻴﭘ u .

ﺭﺩ

ﻢﻴﻨﮐ ﺏﺎﺴﺣ ﺰﻴﻧ ﺍﺭ ﻲﺒﺴﻧ ﻪﺟﺭﺩ ﺪﻳﺎﺑ ﻪﻠﺣﺮﻣ ﻦﻳﺍ .

ﺎﻌﻓﺩ ﺩﺍﺪﻌﺗ ﺯﺍ ﺖﺳﺍ ﺕﺭﺎﺒﻋ ﻲﺒﺴﻧ ﻪﺟﺭﺩ ﺕ

ﺯﺍ ﻱﺮﻴﮔ ﻖﺘﺸﻣ ﻦﻴﺑ ﻢﻴﻘﺘﺴﻣ ﻪﻄﺑﺍﺭ ﻪﺑ ﺎﺗ y

ﻭ y ﻢﻴﺳﺮﺑ u . ﻪﺟﺭﺩ ﺯﺍ ﻲﺒﺴﻧ ﻪﺟﺭﺩ ﺮﮔﺍ

ﻪﺟﺭﺩ ﺎﺑ ﻲﺒﺴﻧ ﻪﺟﺭﺩ ﺮﮔﺍ ﻭ ﻢﻴﻨﮐ ﻲﺳﺭﺮﺑ ﺰﻴﻧ ﺍﺭ ﺮﻔﺻ ﮏﻴﻣﺎﻨﻳﺩ ﺪﻳﺎﺑ ﺪﺷﺎﺑ ﺮﺘﻤﮐ ﻢﺘﺴﻴﺳ ﻩﺎﮕﻧﺁ ﺪﺷﺎﺑ ﻲﮑﻳ ﻢﺘﺴﻴﺳ I-O-L

ﻪﺑ Input state ﺕﺭﻮﺻ ﻦﻳﺍ ﺭﺩ ﻪﮐ ﺩﻮﺷ ﻲﻣ ﻞﻳﺪﺒﺗ

ﻭ ﻱﺮﻳﺬﭘ ﻝﺮﺘﻨﮐ ﻁﺮﺷ

involutively ﺍﺭ

ﻢﻴﻨﮐ ﻲﺳﺭﺮﺑ ﺪﻳﺎﺑ ﺰﻴﻧ .

ﺭﺩ ﻪﮐ ﺪﻴﻨﮐ ﺽﺮﻓ ﻦﻴﺑ ﻢﻴﻘﺘﺴﻣ ﻪﻄﺑﺍﺭ ﻪﺑ ﻱﺮﻴﮕﻘﺘﺸﻣ ﺭﺎﺑ n

ﻭ y ﺮﺑ u ﻪﺑ ﻢﻴﺳﺮﺑ ﻢﺘﺴﻴﺳ

ﺮﻳﺯ ﺕﺭﻮﺻ :

u ) x ( a f y

) n

( = ₁ +

) ۴ (

ﻢﻴﻨﮐ ﻲﻣ ﻞﻤﻋ ﺮﻳﺯ ﺕﺭﻮﺻ ﻪﺑ ﺕﺭﻮﺼﻨﻳﺍ ﺭﺩ :

ﻞﮑﺷ ٢

ﮫﻠﯿﺳﻮﺑ يزﺎﺳ ﻲﻄﺧ :

Feedbacklinearization

٨

) ۵ (

0 5

5 4 1 5

3 5

2 5

1 5

1 2

1

1 1

1 2

1

= + +

+

−

−

=

−

= +

−

−

−

−

=

−

= +

=

−

−

−

) n ( ) n ( n

) n ( n )

n (

d ) n ( ) n (

d ) n (

e e k ...

e k e k )

) f v )( x ( u a )

v u ) x ( a f )

e k ...

e k e k y v )

v e y y )

&

&

ﻣ ﻩﺪﻫﺎﺸﻣ ﻪﮐ ﺭﻮﻄﻧﺎﻤﻫ ﻲ

ﻝﻮﻣﺮﻓ ﻪﺑ ﻪﺟﻮﺗ ﺎﺑ ﺩﻮﺷ )

۲ - ۵ ( ﺍﺪﺘﺑﺍ ﺎﻣ ﺲﭙﺳ ﻭ ﻢﻳﺯﺎﺳ ﻲﻣ ﺍﺭ v

ﻱﻭﺭ ﺯﺍ ﻢﻴﻧﺍﻮﺗ ﻲﻣ v

ﺪﻳﺁ ﻲﻣ ﺖﺳﺪﺑ ﺎﻣ ﻲﻟﺮﺘﻨﮐ ﻱﺩﻭﺭﻭ ﺐﻴﺗﺮﺗ ﻦﻳﺍ ﻪﺑ ﻭ ﻢﻳﺭﻭﺁ ﺖﺳﺪﺑ ﺍﺭ u .

ﻝﻮﻣﺮﻓ ﻪﺑ ﻪﺟﻮﺗ ﺎﺑ )

۵ - ۵ ( ﺳﺭﺩ ﺏﺎﺨﺘﻧﺍ ﺎﺑ ﻲﻣ ﺐﻳﺍﺮﺿ ﺖ

ﻪﺑ ﺍﺭ ﻢﺘﺴﻴﺳ ﻱﺎﻄﺧ ﻱﺎﻬﺒﻄﻗ ﻢﻴﻧﺍﻮﺗ

ﻢﻴﻨﮐ ﺮﻔﺻ ﺍﺭ ﺎﻄﺧ ﺐﻴﺗﺮﺗ ﻦﻳﺍ ﻪﺑ ﻭ ﻢﻳﺮﺒﺑ ﻢﻴﻫﺍﻮﺨﺑ ﻪﮐ ﻪﺤﻔﺻ ﻱﺎﺠﮐ ﺮﻫ .

ﺍ ﺭﺩ ﻳ ﺕﻻﺩﺎﻌﻣ ﻦ

f1

ﺪﻳﺁ ﻲﻣ ﺖﺳﺪﺑ ﻢﺘﺴﻴﺳ ﻱﺎﻬﺘﻟﺎﺣ ﻱﻭﺭ ﺯﺍ .

ﺵﻭﺭ Linearization Input State

:

ﺍ ﺭﺩ ﻳ ﺣﺍﺮﻃ ﻪﻟﺎﺴﻣ ﺵﻭﺭ ﻦ ﻲ

ﻪﻟﺎﺴﻣ Regulation ﺖﺳﺍ

. ﺍﺮﺑ ﻱ ﺍ ﻝﺎـﻤﻋﺍ ـﻳ

ﻪـﺑ ﺵﻭﺭ ﻦ ـﻳ

ﮏ

ﺳ ﻴ ﻏ ﻢﺘﺴ ﻴ ﻄﺧ ﺮ ﻲ ﺎﺑﻳ ﺘﺴ ﻲ ﺳ ﻥﺁ ﻴ ﺍﺭﺍﺩ ﻢﺘﺴ ﻱ

ﺍﺮـﺷ ﻳ ـﺻﺎﺧ ﻂ ﻲ

ﺪـﺷﺎﺑ . ﺳ ﻊـﻗﺍﻭ ﺭﺩ ﻴ

ﻤﺘـﺴ ﻲ

Linearizable Input State

ﻝﺮﺘﻨﮐ ﻪﮐ ﺖﺳﺍ ﺬﭘ

ﻳ ﻭ ﺮ involutive ﺪـﺷﺎﺑ

. ﻪـﻣﺍﺩﺍ ﺭﺩ

ﺍﻳ ﺍﺮﺷ ﻦ ﻳ ﻠﮐ ﺪﻧﻭﺭ ﻭ ﻂ ﻲ

ﺍ ﻝﺎﻤﻋﺍ ﻳ ﺳﺭﺮﺑ ﺩﺭﻮﻣ ﺵﻭﺭ ﻦ ﻲ

ﻣ ﺭﺍﺮﻗ ﻲ ﮔ ﻴ ﺩﺮ .

۱ ( ﺍﺮـــﺑ ﻱ ـــﺳﺭﺮﺑ ﻲ ﺍﺮـــﺷ ﻳ ﺮﮐﺬـــﻟﺍ ﻕﻮـــﻓ ﻂ ﻪـــﻋﻮﻤﺠﻣ ﺍﺪـــﺘﺑﺍ

vector fields

{

^{g adfg ... adnf−1g}

}

ﮑﺸﺗ ﺍﺭ ﻴ ﻣ ﻞ ﻲ ﻫﺩ ﻴ ﻢ . ﺍ ﺮﮔﺍ ﻳ ﻪﻋﻮﻤﺠﻣ ﻦ ﻪﺘﺴﺑﺍﻭ ﺮﻴﻏ

ﻄﺧ ﻲ

ﺪـــﺷﺎﺑ , ﺳ ﻴ ﺬـــﭘ ﻝﺮـــﺘﻨﮐ ﻢﺘـــﺴ ﻳ

ﺖـــﺳﺍ ﺮ .

ـــﻨﭽﻤﻫ ﻴ

ﻪـــﻋﻮﻤﺠﻣ ﻦ vector fields

٩

{

^{g adfg ... adnf−2g }}

ﮑﺸﺗﺍﺭ ﻴ ﻣ ﻞ ﻲ ﻫﺩ ﻴ ﻢ . ﺍ ﺮﮔﺍ ﻳ ﻏ ﻪﻋﻮﻤﺠﻣ ﻦ ﻴ

ـﻄﺧ ﻪﺘـﺴﺑﺍﻭ ﺮ ﻲ

ﺳ ﺪﺷﺎﺑ ﻴ ﺍﺭ ﻢﺘﺴ involutive

ﻮﮔ ﻳ ﺪﻨ . ﺍﺮﺑﺎـﻨﺑ ﻳ ﺍ ﺮـﮔﺍ ﻦ ـﻳ

ﺳ ﺩﻮـﺑ ﺭﺍﺮـﻗﺮﺑ ﻁﺮـﺷﻭﺩ ﻦ ﻴ

ﻢﺘـﺴ

Linearizable Input State

ﻣ ﻭ ﺖﺳﺍ ﻲ

ﺍ ﻥﺍﻮﺗ ﻳ ﺍ ﺮﺑ ﺍﺭ ﺵﻭﺭ ﻦ ـﻳ

ﺳ ﻦ ﻴ ﻝﺎـﻤﻋﺍ ﻢﺘـﺴ

ﺩﺮﮐ . ﻧﻭﺭ ﻪﻣﺍﺩﺍ ﺪ

ﺍ ﻝﺎﻤﻋﺍ ﻳ ﺯ ﺕﺭﻮﺻ ﻪﺑ ﺵﻭﺭ ﻦ ﻳ

ﺖﺳﺍ ﺮ .

۲ ( ﺍﺮﺷ ﺮﮔﺍ ﻳ ﻄ ﻲ ﻟﻭﺍ ﺩﻮﺑ ﺭﺍﺮﻗﺮﺑ ﺪﺷ ﺮﮐﺫ ﻕﻮﻓ ﺭﺩ ﻪﮐ ﻴ

ﺖﻟﺎﺣ ﻦ

z1

ﻭﺭ ﻪﺑ ﻪﺟﻮﺗ ﺎﺑ ﺍﺭ ﺯ ﻂﺑﺍ

ﻳ ﻪﺑ ﺮ

ﻣ ﺖﺳﺩ ﻲ ﺭﻭﺁ ﻳ ﻢ .



 



≠

=

=

=

−

−

0

0 ...

1

1 1

1

1

2

z L

z L

z L z L

g ad

g g ad

ad g

n f

n f f

( ۶ )

ﺍ ﻪﺑ ﻻﺎﺑ ﻂﺑﺍﻭﺭ ﻊﻗﺍﻭ ﺭﺩ ﻳ

ﻟﺩ ﻦ ﻴ ﺍ ﺭﺩ ﻪﮐ ﺖﺳﺍ ﻞ ﻳ

ﺒﺴﻧ ﻪﺟﺭﺩ ﺖﻟﺎﺣ ﻦ ﻲ

ﺮﺑﺍﺮﺑ ﺖﺳﺍn .

۳ ( ﺪﺒﺗ ﻳ ﺖﻟﺎﺣ ﻞ ﺕﺭﻮﺻ ﻪﺑz

] ...

[ )

(x z₁ L z₁ L ¹z₁

z = _f ⁿ_f⁻

ﺩﻮﺑ ﺪﻫﺍﻮﺧ .

۴ ( ﻪﻄﺑﺍﺭ ﻪﺑ ﻪﺟﻮﺗ ﺎﺑ

v x x

u =α( )+β( )

ﺩﺎﻘﻣ ﻳ ﺮ

) α(x

ﻭ

) β(x

ﺯ ﺕﺭﻮﺻ ﻪﺑ ﻳ

ﻣ ﺖﺳﺩ ﻪﺑ ﺮ ﻲ

ﺁﻳ ﺪ .

1 1

1 1

1

) 1 (

) (

z L x L

z L L

z x L

n f g

n f g

n f

−

−

=

= − β

α

) ۷ (

١٠

ﻝﺪﻣ ﺭﻳ ﺿﺎ ﻲ ﻢﺘﺴﻴﺳ :

ﻲﺧﺮﭼ ﻝﻭﺪﻧﺎﭘ ﻲﮑﻴﻣﺎﻨﻳﺩ ﺕﻻﺩﺎﻌﻣ ﻥﺩﺭﻭﺁ ﺖﺳﺪﺑ ﻱﺍﺮﺑ ﻥﺎﺳﺁ ﻩﺍﺭ ﮏﻳ ﻲﺸﻨﻛﺍﻭ

ﻦﻳﺍ ﻪﮐ ﺖﺳﺍ ﻦﻳﺍ

ﺍ ﺭﺩ ﻢﻴﻨﮐ ﻝﺪﻣ ﻱﺩﺍﺯﺁ ﻪﺟﺭﺩ ﻭﺩ ﺕﺎﺑﺭ ﮏﻳ ﺪﻨﻧﺎﻤﺑ ﺍﺭ ﻢﺘﺴﻴﺳ ﻳ

ﻦ ﻭ ﻝﻭﺍ ﮏﻨﻴﻟ ﺍﺭ ﻝﻭﺪﻧﺎﭘ ﻥﺍﻮﺗ ﻲﻣ ﺖﻟﺎﺣ

ﺖﻓﺮﮔ ﺮﻈﻧ ﺭﺩ ﻡﻭﺩ ﮏﻨﻴﻟ ﺍﺭ ﮏﺴﻳﺩ ﺶﺧﺮﭼ .

ﻢﻴﻨﮐ ﻲﻣ ﺽﺮﻓ ﻪﻛ

ﺖﻬﺟ ﺭﺩ ﻱﺩﻮﻤﻋ ﺕﺭﻮﺼﺑ ﻝﻭﺪﻧﺎﭘ

ﺪﺧﺮﭼ ﻲﻣ ﺖﻋﺎﺳ ﻱﺎﻫ ﻪﺑﺮﻘﻋ .

ﺪﺷﺎﺑ ﻲﻣ ﺮﻳﺯ ﺕﺭﻮﺻ ﻪﺑ ﻢﺘﺴﻴﺳ ﻲﮑﻴﻣﺎﻨﻳﺩ ﺕﻻﺩﺎﻌﻣ ﺕﺎﻴﺿﺮﻓ ﻦﻳﺍ ﺎﺑ .

0 ) ( ₁

2 12 1

11q +d q + q =

d && && φ ( ۸ )

τ

= + 22 2 1

21q d q

d && && (۹)

q1 ﻝﻭﺪﻧﺎﭘ ﻪﻳﻭﺍﺯ ,

q2

ﻭ ﮏﺴﻳﺩ ﻪﻳﻭﺍﺯ ﺪﺷﺎﺑ ﻲﻣ ﻢﺘﺴﻴﺳ ﻱﺩﻭﺭﻭ ﻭ ﺭﻮﺗﻮﻣ ﺭﻭﺎﺘﺸﮔ τ

.

ﻂﺑﺍﻭﺭ ﺭﺩ ﻩﺪﺷ ﻩﺪﻣﺁ ﻱﺎﻫﺮﺘﻣﺍﺭﺎﭘ ۸

ﻭ ۹ ﺪﻨﻳﺁ ﻲﻣ ﺖﺳﺩ ﻪﺑ ﺮﻳﺯ ﺕﺭﻮﺼﺑ .

2 1 1

2 1 2 2 1

11 ml m l I I

d = c + + +

2 22 21

12 d d I

d = = =

φ⁽q1⁾=−mg^sin(q1^{) m =m1lc1+m2l1}

ﻞﮑﺷ ﺭﺩ ﻝﻮﻬﺠﻣ ﻱﺎﻫﺮﺘﻣﺍﺭﺎﭘ ﻕﻮﻓ ﻂﺑﺍﻭﺭ ﺭﺩ ۱

ﺪﻨﺘﺴﻫ ﺺﺨﺸﻣ .

ﻱﺎﻫﺮﻴﻐﺘﻣ ﻦﺘﻓﺮﮔ ﺮﻈﻧ ﺭﺩ ﺎﺑ ﺍﺭ ﻢﺘﺴﻴﺳ ﺯﺍ ﻪﺘﻓﺎﻳ ﺶﻫﺎﮐ ﻝﺪﻣ ﮏﻳ ﺎﺠﻨﻳﺍ ﺭﺩ ﺕﺭﻮﺼﺑ ﺖﻟﺎﺣ

1

1 q

x =

ﻭ

1

2 q

x = &

ﻭ

2

3 q

x = &

ﻢﻴﻨﮐ ﻲﻣ ﻒﻳﺮﻌﺗ .

ﻝﻭﺪﻧﺎﭘ ﺖﻴﻌﻗﻮﻣ ﻝﺮﺘﻨﮐ ﺎﻣ ﻑﺪﻫ ﻊﻗﺍﻭ ﺭﺩ ,

ﻭ ﻝﻭﺪﻧﺎﭘ ﺖﻋﺮﺳ

ﻢﻴﻨﮐ ﻲﻣ ﺮﻈﻨﻓﺮﺻ ﮏﺴﻳﺩ ﺖﻴﻌﻗﻮﻣ ﻝﺮﺘﻨﮐ ﺯﺍ ﻭ ﺖﺳﺍ ﮏﺴﻳﺩ ﺖﻋﺮﺳ .

ﻢﺘﺴﻴﺳ ﺖﻟﺎﺣ ﻱﺎﻫﺮﻴﻐﺘﻣ ﻒﻳﺮﻌﺗ ﺎﺑ

ﺣ ﺕﻻﺩﺎﻌﻣ ﻕﻮﻓ ﺕﺭﻮﺻ ﻪﺑ ﺪﻳﺁ ﻲﻣ ﺭﺩ ﺮﻳﺯ ﻂﺑﺍﻭﺭ ﺕﺭﻮﺻ ﻪﺑ ﻢﺘﺴﻴﺳ ﺖﻟﺎ

.

^{x&1 =x2} (۱۰)

^{φ τ}

D x d

D x d

) det det (

12 1

22

2 =− −

& (۱۱)

^{φ τ}

D x d

D x d

) det det (

11 1

21

3 = +

& ( ۱۲)

١١

ﻕﻮﻓ ﻂﺑﺍﻭﺭ ﺭﺩ ﻪﮐ

0 detD=d₁₁d₂₂ −d₁₂d₂₁ >

ﺖﺳﺍ .

ﺖﺷﻮﻧ ﺮﻳﺯ ﺕﺭﻮﺻ ﻪﺑ ﺍﺭ ﻕﻮﻓ ﻂﺑﺍﻭﺭ ﻥﺍﻮﺗ ﻲﻣ .

τ ) ( ) (x g x f

x&= + ( ۱۳)





















=





















−

=

D d

D x d

g D x

d D x d x x f

det det 0 ) ( ,

) det (

) det (

) (

11 12

1 21

1 22 2

φ

φ ( ۱۴)

ﺎﺑ ﺍﺭ ﻝﻭﺪﻧﺎﭘ ﺲﻧﻻﺎﺑ ﻝﺮﺘﻨﮐ ﻢﻴﻫﺍﻮﺧ ﻲﻣ ﻢﻳﺍ ﻩﺩﺭﻭﺁ ﺖﺳﺩ ﻪﺑ ﺍﺭ ﻢﺘﺴﻴﺳ ﻲﮑﻴﻣﺎﻨﻳﺩ ﺕﻻﺩﺎﻌﻣ ﻪﮐ ﻝﺎﺣ ﻭﺭ ﺯﺍ ﻩﺩﺎﻔﺘﺳﺍ ﺵ

Feedback linearization ﻢﻴﻫﺩ ﻡﺎﺠﻧﺍ

.

ﺳﺭﺮﺑ ﻲ ﺵﻭﺭ Feedback linearization ﻭﺭ ﺮﺑ

ﻱ ﺸﺧﺮﭼ ﻝﻭﺪﻧﺎﭘ ﻲ

ﺸﻨﮐﺍﻭ ﻲ :

ﺭﺍﺪﻳﺎﭘ ﻢﺘﺴﻴﺳ ﻪﮐ ﻢﻴﻨﮐ ﻲﺣﺍﺮﻃ ﻱﺭﻮﻃ ﻲﻄﺧ ﺮﻴﻏ ﻢﺘﺴﻴﺳ ﻦﻳﺍ ﻱﺍﺮﺑ ﻱﺍ ﻩﺪﻨﻨﮐ ﻝﺮﺘﻨﮐ ﻢﻴﻫﺍﻮﺧ ﻲﻣ ﺩﺩﺮﮔ . ﻪﻟﺎﺴﻣ ﺎﺠﻨﻳﺍ ﺭﺩ ﻊﻗﺍﻭ ﺭﺩ ,

ﺖﺳﺍ ﻱﺯﺎﺳ ﺭﺍﺪﻳﺎﭘ ﻪﻟﺎﺴﻣ .

ﻝﺮﺘﻨﮐ ﻥﻮﻧﺎﻗ ﺪﻳﺎﺑ ﺍﺭ u

ﻢﻳﺭﻭﺁ ﺖﺳﺪﺑ ﻱﺭﻮﻃ

ﻝﻮﺒﻗ ﻞﺑﺎﻗ ﻱﺎﻀﻓ ﺯﺍ ﺎﺠﮐ ﺮﻫ ﺭﺩ ﻢﺘﺴﻴﺳ ﻪﻴﻟﻭﺍ ﻂﻳﺍﺮﺷ ﺮﮔﺍ ﻪﮐ )

(Ω ﻢﺘﺴﻴﺳ ﻝﺮﺘﻨﮐ ﻥﻮﻧﺎﻗ ﻦﻳﺍ ﻩﺎﮕﻧﺁ ﺪﺷﺎﺑ

ﺪﻧﺎﺳﺮﺑ ﻝﺩﺎﻌﺗ ﻪﻄﻘﻧ ﻪﺑ ﺍﺭ .

ﺵﻭﺭ ﺵﺭﺍﺰﮔ ﻦﻳﺍ ﺭﺩ ﺎﻣ Input-Output Linearization

ﻢﻴﻨﮐ ﻲﻣ ﻝﺎﻤﻋﺍ ﻢﺘﺴﻴﺳ ﻱﻭﺭ ﺮﺑ ﺍﺭ .

ﺘﺴﻳﺎﺑ ﺍﺪﺘﺑﺍ ﺎﻣ ﺵﻭﺭ ﻦﻳﺍ ﺭﺩ ﻢﻳﺭﻭﺁ ﺖﺳﺪﺑ ﻢﺘﺴﻴﺳ ﻲﺟﻭﺮﺧ ﻭ ﻱﺩﻭﺭﻭ ﻦﻴﺑ ﻢﻴﻘﺘﺴﻣ ﻪﻄﺑﺍﺭ ﮏﻳ ﻲ

. ﺭﺎﮐ ﻦﻳﺍ

ﺯﺍ ﻱﺮﻴﮔ ﻖﺘﺸﻣ ﺎﺑ ﺍﺭ ﻢﻴﻫﺩ ﻲﻣ ﻡﺎﺠﻧﺍ y

. ﺯﺍ ﺭﺎﺑ ﺮﻫ ﻲﻨﻌﻳ ﻦﻴﺑ ﻢﻴﻘﺘﺴﻣ ﻪﻄﺑﺍﺭ ﻪﮑﻨﻳﺍ ﺎﺗ ﻢﻳﺮﻴﮔ ﻲﻣ ﻖﺘﺸﻣy

ﻭ u ﺪﻳﺁ ﺖﺳﺪﺑ y .

ﺲﭙﺳ ﻭ ﺪﻨﮐ ﻑﺬﺣ ﺍﺭ ﻲﻄﺧ ﺮﻴﻏ ﺖﻤﺴﻗ ﺮﺛﺍ ﻪﮐ ﻢﻴﻨﮐ ﻲﻣ ﻲﺣﺍﺮﻃ ﻱﺭﻮﻃ ﺍﺭ u

ﻦﻴﻨﭽﻤﻫ ﺪﺑ ﻱﺍ ﻪﻧﻮﮔ ﻪﺑ ﺍﺭ v

ﺩﺯﺎﺳ ﺭﺍﺪﻳﺎﭘ ﺍﺭ ﻢﺘﺴﻴﺳ ﻪﮐ ﻢﻳﺭﻭﺁ ﺖﺳ .

ﺖﺳﺍ ﻩﺪﺷ ﻒﻳﺮﻌﺗ ﺮﻳﺯ ﺕﺭﻮﺻ ﻪﺑ ﻲﺟﻭﺮﺧ ﺕﻻﺩﺎﻌﻣ ﻩﺪﺷ ﻩﺩﺍﺩ ﻢﺘﺴﻴﺳ ﻱﺍﺮﺑ .

3 12 2

) 11

(x d x d x

h

y= = + ( ۱۵) ﺕﺎﻘﺘﺸﻣ ﻝﺎﺣ ﺑﺍﺭ ﻪﮐ ﻢﻴﻫﺩ ﻲﻣ ﻪﻣﺍﺩﺍ ﻲﻳﺎﺟ ﺎﺗ ﻭ ﻩﺩﺮﮐ ﻪﺒﺳﺎﺤﻣ ﺍﺭy

ﻱﺩﻭﺭﻭ ﻭ ﻲﺟﻭﺮﺧ ﻦﻴﺑ ﻲﻤﻴﻘﺘﺴﻣ ﻪﻄ

ﺪﻳﺁ ﺖﺳﺩ ﻪﺑ ﻢﺘﺴﻴﺳ .

١٢ τ

h L h L

y= _f + _g ( ۱۶)

h L_f

ﻭ

h L_g

Lie derivative ﻱﺎﻫ

ﻪﺑ ﺖﺒﺴﻧ h ﻭf

ﺼﺑ ﻪﮐﺪﻨﺘﺴﻫ g ﺪﻨﻳﺁ ﻲﻣ ﺖﺳﺩ ﻪﺑ ﺮﻳﺯ ﺕﺭﻮ

.

[ ]

^{( ) sin( )}

) det (

) det (

0 _{1 1}

1 21

1 22 2

12

11 x mg x

D x d

D x d x d d

h

L_f =− =





















−

= φ

φ

φ (۱۷)

ﻭ

[ ]

⁰

det 0 det

11 12 2

12

11 =





















−

=

D d

D d x d d

h

L_g ( ۱۸)

ﻢﻳﺭﺍﺩ ﺪﻧﻭﺭ ﻦﻳﺍ ﻪﻣﺍﺩﺍ ﺎﺑ .

0 ,

0 ,

2 2

3 ) 3 (

2 2

≠ +

=

=

= +

=

h L L h

L L h L y

L L h

L L L h L y

f g f

g f

f g f

f g f

τ

τ

&& τ

) ۱۹ (

ﻭ

) det cos(

) sin(

) det cos(

) ( ) sin(

) cos(

1 2 21

1 1 2 22

2 2 1 3

2 1 2

x g Dm h d

L L

x D x

g d m x x g m h L

x x g m h L

f g f f

−

=

+

−

=

=

ﺖﺳﺍ ﻱﺩﻭﺭﻭ ﺎﺑ ﻢﻴﻘﺘﺴﻣ ﻪﻄﺑﺍﺭ ﺯﺍ ﺲﭘ ﻲﺟﻭﺮﺧ ﺕﺎﻘﺘﺸﻣ ﻪﺒﺗﺮﻣ ﺮﺑﺍﺮﺑ ﻲﺒﺴﻧ ﻪﺟﺭﺩ ﻪﮑﻨﻳﺍ ﻪﺑ ﻪﺟﻮﺗ ﺎﺑ ﺮﺑﺍﺮﺑ ﻲﺒﺴﻧ ﻪﺟﺭﺩ ﻦﻳﺍﺮﺑﺎﻨﺑ ۳

ﺪﺷﺎﺑ ﻲﻣ .

ﻪﻟﺎﺴﻣ ﺍﺬﻟ ﺖﺳﺍ ﺮﺑﺍﺮﺑ ﻢﺘﺴﻴﺳ ﻪﺟﺭﺩ ﺎﺑ ﻲﺒﺴﻧ ﻪﺟﺭﺩ ﻥﻮﭼ Input-

Output Linearization ﻪﻟﺎﺴﻣ ﮏﻳ ﻪﺑ

Input State Linearization ﻮﺷ ﻲﻣ ﻞﻳﺪﺒﺗ

ﺩ .

ﻢﺘﺴﻴﺳ ﻦﻳﺍ ﺖﺳﺍ ﻢﺘﺴﻴﺳ ﻪﺟﺭﺩ ﻥﺎﻤﻫ ﻲﺒﺴﻧ ﻪﺟﺭﺩ ﻪﮐ ﺎﺠﻧﺁ ﺯﺍ ﻦﻴﻨﭽﻤﻫ Zero Dynamic

ﺩﺭﺍﺪﻧ .

١٣

ﻲﺒﺴﻧ ﻪﺟﺭﺩ ﻱﺍﺭﺍﺩ ﻢﺘﺴﻴﺳ ﺲﭘ ۳

ﻲﺟﻭﺮﺧ ﻪﺑ ﺦﺳﺎﭘ ﺭﺩ

3 12 2

) 11

(x d x d x

h

y= = +

ﻪﮐ ﺍﺮﻳﺯ ﺖﺳﺍ

h L L_g ²_f

ﻪﻴﺣﺎﻧ ﺭﺩ

2

2 ¹ π

π < <

− q

ﺖﺳﺍ ﺮﻔﺻ ﺮﻴﻏ .

ﺳﺭﺮﺑ ﺎﺑ ﻲ ﺬﭘ ﻝﺮﺘﻨﮐ ﻁﺮﺷ ﻳﺮ

ﻱ ﻣ ﻩﺪﻫﺎﺸﻣ ﻲ

ﺍ ﻪﮐ ﺩﻮﺷ ﻳ

ﺳ ﻦ ﻴ ﺬﭘ ﻝﺮﺘﻨﮐ ﻢﺘﺴ ﻳ

ﺮﺗﺎﻣ ﻪﮐ ﺎﺠﻧﺁ ﺯﺍ ﻭ ﺖﺳﺍ ﺮ ﻳ

ﺲ

ﻳg ﺮﺗﺎﻣ ﮏ ﻳ ﺖﺳﺍ ﺖﺑﺎﺛ ﺲ involutive

ﻧﻴ ﻣ ﺰ ﻲ ﺪﺷﺎﺑ .

ﺍﺮﺑﺎﻨﺑ ﻳ ﺪﻳﺪﺟ ﺖﻟﺎﺣ ﻱﺎﻫﺮﻴﻐﺘﻣ ﻢﻴﻧﺍﻮﺗ ﻲﻣ ﻦ

3 2 1,ξ ,ξ ﻢﻴﻨﮐ ﻒﻳﺮﻌﺗ ﺮﻳﺯ ﺕﺭﻮﺼﺑ ﺍﺭ ξ

.

2 1 2

3

1 2

3 12 2 11 1

) cos(

) (

) sin(

) ( ) (

x x g m x h L

x g m x h L

x d x d x h

f f

=

=

=

=

+

=

=

ξ ξ ξ

) ۲۰ (

ﺖﺳﺍ ﺮﻳﺯ ﺕﺭﻮﺻ ﻪﺑ ﻢﺘﺴﻴﺳ ﺪﻳﺪﺟ ﺖﻟﺎﺣ ﺕﻻﺩﺎﻌﻣ ﻦﻳﺍﺮﺑﺎﻨﺑ :







+

=

=

=

τ ξ

ξ ξ

ξ ξ

h L L h L³_{f g} ²_f

3 3 2

2 1

&

&

&

) ۲۱ (

ﺵﻭﺭ ﺭﺩ

Feedback linearization ﺭ ﻢﺘﺴﻴﺳ ﻱﺩﻭﺭﻭ ﻢﻴﻫﺍﻮﺧ ﻲﻣ ﺎﻣ ﻊﻗﺍﻭ ﺭﺩ

ﻪﺑ ﻱﺍ ﻪﻧﻮﮔ ﻪﺑ ﺍ

ﺲﭙﺳ ﻭ ﺪﻨﮐ ﻞﻳﺪﺒﺗ ﻲﻄﺧ ﻡﺮﻓ ﻪﺑ ﻢﺘﺴﻴﺳ ﺖﻟﺎﺣ ﺕﻻﺩﺎﻌﻣ ﻪﮐ ﻢﻳﺭﻭﺁ ﺖﺳﺩ ﻢﻴﻨﮐ ﻦﻴﻌﺗ ﻱﺍ ﻪﻧﻮﮔ ﻪﺑ ﺍﺭ u

ﺪﻨﮐ ﺭﺍﺪﻳﺎﭘ ﺍﺭ ﻢﺘﺴﻴﺳ ﻪﮐ .

ﻪﻄﺑﺍﺭ ﻪﺑ ﻪﺟﻮﺗ ﺎﺑ ﻮﻴﺘﮐﺍﺭ ﻲﺧﺮﭼ ﻝﻭﺪﻧﺎﭘ ﻢﺘﺴﻴﺳ ﻱﺯﺎﺳ ﻲﻄﺧ ﻱﺍﺮﺑ ﻦﻳﺍﺮﺑﺎﻨﺑ ۲۱

, ﺮﻳﺯ ﺕﺭﻮﺻ ﻪﺑ ﺍﺭ τ

ﻢﻴﻨﮐ ﻲﻣ ﻒﻳﺮﻌﺗ .

)]

( )[

(

1 ₃

2 u L h x

x h L

L_{g f} − ^f

=

τ ( ۲۲)

ﻄﺧ ﺕﺭﻮﺻ ﻪﺑ ﻢﺘﺴﻴﺳ ﺪﻳﺪﺟ ﺖﻟﺎﺣ ﺕﻻﺩﺎﻌﻣ ﻦﻳﺍﺮﺑﺎﻨﺑ ﻲ

ﺪﻳﺁ ﻲﻣ ﺭﺩ .







=

=

=

3 u

3 2

2 1

ξ ξ ξ

ξ ξ

&

&

&

(۲۳)

ﺪﻳﺪﺟ ﻝﺮﺘﻨﮐ ﺮﻴﻐﺘﻣ ﺩﺮﮐ ﻒﻳﺮﻌﺗ ﺮﻳﺯ ﺕﺭﻮﺻ ﻪﺑ ﻥﺍﻮﺗ ﻲﻣ ﺍﺭ u

.

1 3 1 2 1

1ξ k ξ k ξ

k

u=− − − ( ۲۴)

١٤

ﮐ ﺖﺳﺍ ﻱﺭﺎﻴﺘﺧﺍ ﻪﺘﺴﺑ ﻪﻘﻠﺣ ﻱﺎﻫ ﺐﻄﻗ ﻥﺎﮑﻣ ﻪ .

ﻐﺗ ﺎﺑ ﻭ ﻴ ﺮ ﻣk ﻲ ﺩﺮﮐ ﺎﺠﺑﺎﺟ ﺍﺭ ﺎﻬﻧﺁ ﻥﺍﻮﺗ .

ﻪﻴﺣﺎﻧ ﺭﺩ ﺎﻬﻨﺗ ﻮﻴﺘﮐﺍﺭ ﻲﺧﺮﭼ ﻝﻭﺪﻧﺎﭘ ﻪﮐ ﻢﻳﺩﺍﺩ ﻥﺎﺸﻧ ﺎﻣ

2

1 <π

Feedback linearizable x

ﺖﺳﺍ

ﺎﭘ ﻭ ﻩﺩﺮﮐ ﻝﺮﺘﻨﮐ ﺍﺭ ﻢﺘﺴﻴﺳ ﺵﻭﺭ ﻦﻳﺍ ﺯﺍ ﻩﺩﺎﻔﺘﺳﺍ ﺎﺑ ﻢﻴﻧﺍﻮﺗ ﻲﻣ ﺎﻣ ﻪﻴﺣﺎﻧ ﻦﻳﺍ ﺭﺩ ﺎﻬﻨﺗ ﻭ ﻢﻴﻨﮐ ﺭﺍﺪﻳ

.

ﺎﺘﻧ ﻳ ﺒﺷ ﻭ ﺞ ﻴ ﺯﺎﺳ ﻪ ﻱ :

ﻪﺑ ﺞﻳﺎﺘﻧ ﻩﺪﻣﺁ ﺖﺳﺩ ﻢﺘﺴﻴﺳ ﻱﺯﺎﺳ ﻩﺩﺎﻴﭘ ﺭﺩ

ﺱﺎﺳﺍ ﺮﺑ ﻱﺎﻫﺮﺘﻣﺍﺭﺎﭘ

ﻪﺑ ﻝﻭﺪﺟ ﺭﺩ ﻩﺪﻣﺁ ﺖﺳﺩ ۱

ﻲﻣ ﺪﺷﺎﺑ .

ﻢﺘﺴﻴﺳ ﻱﺎﻫﺮﺘﻣﺍﺭﺎﭘ - ۱ ﻝﻭﺪﺟ

ﺩ ﺕﻻﺩﺎﻌﻣ ﻪﺑ ﻪﺟﻮﺗ ﺎﺑ ﻳ

ﻣﺎﻨ ﻴ ﮑ ﻲ ﭘﻴ ﺯﺎﺳ ﻩﺩﺎ ﻱ ﺳ ﻴ ﺯ ﺕﺭﻮﺻ ﻪﺑ ﻢﺘﺴ ﻳ

ﺖﺳﺍ ﺮ .

ﻞﻜﺷ ۳ - ﻩﺩﺎﻴﭘ ﻢﺘﺴﻴﺳ ﻱﺯﺎﺳ

١٥

ﺍ ﻪﺑ ﻪﺟﻮﺗ ﺎﺑ ﻳ

ﻤﻋﺍ ﺯﺍ ﻪﮑﻨ ﺵﻭﺭ ﻝﺎ

Swing up ﺳ ﻩﺭﺍﻮﻤﻫ ﺎﻣ ﺖﺳﺍ ﻩﺪﺷ ﺮﻈﻨﻓﺮﺻ

ﻴ ﺍ ﻩﺩﻭﺪﺤﻣ ﺭﺩ ﺍﺭ ﻢﺘﺴ ﻱ

ﺵﻭﺭ ﻪﮐ Feedback Linearizat ion

ﻣ ﺮﻈﻧ ﺭﺩ ﺖﺳﺍ ﻝﺎﻤﻋﺍ ﻞﺑﺎﻗ ﻲ

ﮔ ﻴﺮ ﻳ ﻢ . ﺎﺘﻧ ﻳ ﺫ ﺭﺩ ﻩﺪﺷ ﻩﺪﻣﺁ ﺞ ﻳ

ﻞ

ﺍﺮﺷ ﺱﺎﺳﺍﺮﺑ ﻳ

ﻟﻭﺍ ﻂ ﻴ ﺍﺮﺑ ﻪ ﻱ ﻌﻗﻮﻣ ﻴ ﻝﻭﺪﻧﺎﭘ ﺖ ﺖﺳﺍ ﻩﺪﻣﺁ ﺖﺳﺪﺑ

.

ﺎﺘﻧ ﻳ ﺍﺮﺑ ﺞ ﻱ ﻟﻭﺍ ﻁﺮﺷ ﻴ ﻪ

rad q₁ =1

ﺎﮕﻨﻫ ﻲﻨﻌﻳ ﻩﺪﻣﺁ ﺖﺳﺩ ﻪﺑ ﺩﻮﺷ ﺎﻫﺭ ﺖﻴﻌﻗﻮﻣ ﻦﻳﺍ ﺯﺍ ﺯﺍ ﻝﻭﺪﻧﺎﭘ ﻪﻛ ﻲﻣ

ﺍﺮﺿ ﻭ ﻳ ﺐ

1 3 ,

30 2 ,

2 .

1= k = k =

k

ﺯ ﺕﺭﻮﺻ ﻪﺑ ﻳ

ﺖﺳﺍ ﺮ .

ﻞﮑﺷ ۴ - ﻌﻗﻮﻣ ﻴ ﻝﻭﺪﻧﺎﭘ ﺖ ﻪﻴﻟﻭﺍ ﺖﻴﻌﻗﻮﻣ ﻱﺍﺯﺍ ﻪﺑ

rad q₁ =1

ﻣ ﻪﻈﺣﻼﻣ ﻪﮐ ﺭﻮﻄﻧﺎﻤﻫ ﻲ

ﻤﮐ ﻥﺎﻣﺯ ﺕﺪﻣ ﺯﺍ ﺲﭘ ﻝﻭﺪﻧﺎﭘ ﺩﻮﺷ ﻲ

ﻝﺩﺎﻌﺗ ﺖﻟﺎﺣ ﻪﺑ )

(x=0 ﻣ ﻲ ﺪﺳﺭ .

١٦

ﻞﮑﺷ ۵ - ﺩ ﺖﻋﺮﺳ ﻳ ﮏﺴ ﻪﻴﻟﻭﺍ ﺖﻴﻌﻗﻮﻣ ﻱﺍﺯﺍ ﻪﺑ

rad q₁ =1

ﻞﮑﺷ ۶ - ﺍﺮﺑ ﻝﻭﺪﻧﺎﭘ ﻪﺑ ﻩﺪﺷ ﻝﺎﻤﻋﺍ ﺭﻭﺎﺘﺸﮔ ﻱ

ﻌﻗﻮﻣ ﻴ ﻟﻭﺍ ﺖ ﻴ ﻪ

rad q₁ =1

١٧

ﻞﮑﺷ ﺭﺩ ۵

- ﺩ ﺖﻋﺮﺳ ﻪﺑ ﻁﻮﺑﺮﻣ ﺭﺍﺩﻮﻤﻧ ﻪﮐ ﻳ

ﻣ ﮏﺴ ﻲ ﺪﺷﺎﺑ , ﻣ ﻪﻈﺣﻼﻣ ﻲ

ﺩ ﺖﻋﺮﺳ ﻪﮐ ﺩﻮﺷ ﻳ

ﺯﺍ ﺲﭘ ﮏﺴ

ﻧﺎﺛ ﺪﻨﭼ ﺖﺷﺬﮔ ﻴ

ﺳﺭ ﻝﺩﺎﻌﺗ ﺖﻟﺎﺣ ﻪﺑ ﻪ ﻴ

ﻭ ﻩﺪ ﺘﺑﺎﺛ ﺭﺍﺪﻘﻣ ﻪﺑ ﻲ

ﻣ ﻲ ﺪﺳﺭ . ﻟﺎﻤﻋﺍ ﺭﻭﺎﺘﺸﮔ ﻥﻮﭼ ﻲ

ﻊﻗﺍﻭ ﺭﺩ

ﺍ ﻪﻈﺤﻟ ﺏﺎﺘﺷ ﻱ

ﺩﻳ ﮏﺴ ﻣ ﻲ ﺍﺮﺑﺎﻨﺑ ﺪﺷﺎﺑ ﻳ

ﺭﺩ ﻪﮐ ﺭﻮﻄﻧﺎﻤﻫ ﻦ ﻞﮑﺷ

۵ - ﺖﺑﺎﺛ ﺯﺍ ﺲﭘ ﺖﺳﺍ ﻩﺪﺷ ﻩﺩﺍﺩ ﻥﺎﺸﻧ

ﺩ ﺖﻋﺮﺳ ﻥﺪﺷ ﻳ

ﺩ ﺏﺎﺘﺷ ﻥﺪﺷ ﺮﻔﺻ ﻭ ﮏﺴ ﻳ

ﻣ ﺮﻔﺻ ﺭﻭﺎﺘﺸﮔ ﮏﺴ ﻲ

ﺩﻮﺷ .

ﺎﺘﻧ ﻪﻣﺍﺩﺍ ﺭﺩ ﻳ

ﺒﺷ ﺯﺍ ﻞﺻﺎﺣ ﺞ ﻴ

ﻪ ﺯﺎﺳ ﻱ ﺳ ﻴ ﺍﺮﺑ ﺍﺭ ﻢﺘﺴ ﻱ

ﻌﻗﻮﻣ ﻴ ﻟﻭﺍ ﺖ ﻴ ﻪ

rad q₁ =1.5

ﺐﻳﺍﺮﺿ ﻭ

1 3 ,

30 2 ,

2 .

1= k = k =

k

ﺳﺭﺮﺑ ﺩﺭﻮﻣ ﻲ

ﻣ ﺭﺍﺮﻗ ﻲ ﻫﺩ ﻴ ﻢ .

ﻞﮑﺷ ۷ - ﻌﻗﻮﻣ ﻴ ﺍﺯﺍ ﻪﺑ ﻝﻭﺪﻧﺎﭘ ﺖ ﻱ

ﺖﻴﻌﻗﻮﻣ ﻟﻭﺍ

ﻴ ﻪ

rad q₁ =1.5

ﻲﻣ ﻩﺪﻫﺎﺸﻣ

ﺖﺷﺬﮔ ﺯﺍ ﺲﭘ ﻝﻭﺪﻧﺎﭘ ﺖﻴﻌﻗﻮﻣ ﻪﻛ ﺩﻮﺷ

۳ ﺖﻴﻌﻗﻮﻣ ﺯﺍ ﻪﻴﻧﺎﺛ

rad q₁ =1.5

ﻝﺩﺎﻌﺗ ﺖﻟﺎﺣ ﻪﺑ

ﻲﻨﻌﻳ

1 =0

ﻲﻣ q

ﺪﺳﺭ .

ﻞﻜﺷ ﺭﺩ ۸

ﻪﺑ ﻭ ﻩﺪﺷ ﺖﺑﺎﺛ ﻲﻤﻛ ﻥﺎﻣﺯ ﺕﺪﻣ ﺯﺍ ﺲﭘ ﻚﺴﻳﺩ ﺖﻋﺮﺳ ﺮﻔﺻ ﺖﻠﻋ ﻪﺑ ﻢﻫ ﺭﻭﺎﺘﺸﮔ ﻥﺁ ﺐﺟﻮﻣ

ﻲﻣ ﺮﻔﺻ ﻚﺴﻳﺩ ﺭﻭﺎﺘﺸﮔ ﻥﺪﺷ ﺩﻮﺷ

.

١٨

ﻞﮑﺷ ۸ - ﺩ ﺖﻋﺮﺳ ﻳ ﺍﺯﺍ ﻪﺑ ﮏﺴ ﻱ

ﺖﻴﻌﻗﻮﻣ ﻟﻭﺍ

ﻴ ﻪ

rad q₁ =1.5

ﻞﮑﺷ ۹ - ﺍﺮﺑ ﻝﻭﺪﻧﺎﭘ ﻪﺑ ﻩﺪﺷ ﻝﺎﻤﻋﺍ ﺭﻭﺎﺘﺸﮔ ﻱ

ﻌﻗﻮﻣ ﻴ ﻟﻭﺍ ﺖ ﻴ ﻪ

rad q₁ =1.5

١٩

ﻪﻴﺒــﺷ ﺞﻳﺎــﺘﻧ ﺐﻳﺍﺮــﺿ ﻱﺍﺯﺍ ﻪــﺑ ﻢﺘــﺴﻴﺳ ﻱﺯﺎــﺳ

k1=.05 ﻭ

k2=100 ﻭ

k3=20 ﻪــﻴﻟﻭﺍ ﺖــﻴﻌﻗﻮﻣ ﻭ

rad q₁ =1.5

ﺖﺳﺍ ﻩﺪﻣﺁ ﻞﻳﺫ ﺭﺩ .

ﻞﮑﺷ ۱۰ - ﻪﻴﻟﻭﺍ ﺖﻴﻌﻗﻮﻣ ﻱﺍﺯﺍ ﻪﺑ ﻝﻭﺪﻧﺎﭘ ﺖﻴﻌﻗﻮﻣ rad

q₁ =1.5 ﺐﻳﺍﺮﺿ ﻭ

k1=.05 ﻭ

k2=100 ﻭ

k3=20 ﻱﺍﺮﺑ u

ﺸﻣ ﻪﻛ ﺭﻮﻄﻧﺎﻤﻫ ﻲﻣ ﻩﺪﻫﺎ

ﻪﺑ ﺞﻳﺎﺘﻧ ﺩﻮﺷ ﻪﺑ ﺖﺒﺴﻧ ﺖﻟﺎﺣ ﻦﻳﺍ ﺭﺩ ﻩﺪﻣﺁ ﺖﺳﺩ

ﻊﻳﺮـﺳ ﻞﺒﻗ ﺖﻟﺎﺣ ﺭﺩ ﻭ ﻩﺩﻮـﺑ ﺮـﺗ

ﺯﺍ ﺮﺘﻤﻛ ۱

ﻲﻣ ﻝﺩﺎﻌﺗ ﺖﻟﺎﺣ ﻪﺑ ﻪﻴﻧﺎﺛ ﺪﺳﺭ

. ﻞﻜﺷ ﻱﺍﺮﺑ ﻱﺎﻫ

۱۱ ﻭ ۱۲ ﻢﻳﺭﺍﺩ ﺍﺭ ﺪﻧﻭﺭ ﻦﻴﻤﻫ ﻢﻫ .

ﻪـﻜﻨﻳﺍ ﻲﻨﻌﻳ

ﺖﺳﺍ ﺮﺘﺸﻴﺑ ﻞﺒﻗ ﺕﻻﺎﺣ ﻪﺑ ﺖﺒﺴﻧ ﺕﻻﺎﺣ ﻦﻳﺍ ﻱﺍﺮﺑ ﻝﺩﺎﻌﺗ ﺖﻟﺎﺣ ﻪﺑ ﻥﺪﻴﺳﺭ ﺖﻋﺮﺳ .

٢٠

ﻞﮑﺷ ۱۱ - ﺳ ﻪﻴﻟﻭﺍ ﺖﻴﻌﻗﻮﻣ ﻱﺍﺯﺍ ﻪﺑ ﮏﺴﻳﺩ ﺖﻋﺮ rad

q₁=1.5 ﺐﻳﺍﺮﺿ ﻭ

k1=.05 ﻭ

k2=100 ﻭ

k3=20 ﻱﺍﺮﺑ u

ﻞﮑﺷ ۱۲ - ﻪﻴﻟﻭﺍ ﺖﻴﻌﻗﻮﻣ ﻱﺍﺮﺑ ﻝﻭﺪﻧﺎﭘ ﻪﺑ ﻩﺪﺷ ﻝﺎﻤﻋﺍ ﺭﻭﺎﺘﺸﮔ rad

q₁ =1.5 ﺐﻳﺍﺮﺿ ﻭ

k1=.05 ﻭ

k2=100 ﻭ

k3=20 ﻱﺍﺮﺑ u

٢١

ﻪﺠﻴﺘﻧ ﻱﺮﻴﮔ :

ﺵﻭﺭ ﺮـﺛﺍ ﻩﮊﻭﺮـﭘ ﻦﻳﺍ ﺭﺩ Feedback Linearization

ﻲـﺸﺧﺮﭼ ﻝﻭﺪـﻧﺎﭘ ﻢﺘـﺴﻴﺳ ﻚـﻳ ﻱﻭﺭ ﺮـﺑ ﺍﺭ

ﻢﻳﺩﺍﺩ ﺭﺍﺮﻗ ﻲﺳﺭﺮﺑ ﺩﺭﻮﻣ ﻲﺸﻨﻛﺍﻭ .

ﻱﺮـﺳ ﻚـﻳ ﻱﺍﺭﺍﺩ ﺵﻭﺭ ﻦـﻳﺍ ﻢﻳﺪـﻳﺩ ﻢـﻫ ﻩﮊﻭﺮـﭘ ﻦﻳﺍ ﺭﺩ ﻪﻛ ﺭﻮﻄﻧﺎﻤﻫ

ﺖﻳﺩﻭﺪﺤﻣ ﺖﺳﺍ ﺎﻫ

. ﻱﺍﺭﺍﺩ ﺖـﺳﺍ ﻦـﻜﻤﻣ ﺎـﻫ ﻢﺘـﺴﻴﺳ ﻱﻭﺭ ﺮـﺑ ﻝﺎـﻤﻋﺍ ﺖـﻬﺟ ﺵﻭﺭ ﻦـﻳﺍ ﻊﻗﺍﻭ ﺭﺩ ﻲﻨﻌﻳ

ﺖﻳﺩﻭﺪﺤﻣ ﻲﺧﺮﺑ ﻱﺍﺮﺑ ﺖﺳﺍ ﻦﻜﻤﻣ ﻲﺘﺣ ﺎﻳ ﻭ ﺪﺷﺎﺑ ﻲﻳﺎﻫ

ﻢﺘﺴﻴﺳ ﻪﺑ ﺍﺭ ﺵﻭﺭ ﻦﻳﺍ ﻥﺍﻮﺘﻧ ﺎﻫ ﺩﺮﺑ ﺭﺎﻛ

. ﻦـﻳﺍ ﺭﺩ

ﺵﻭﺭ ﻩﮊﻭﺮﭘ Feedback Linearization

ﺖـﻴﻌﻗﻮﻣ ﺯﺍ ﻲـﺼﺨﺸﻣ ﻩﺩﻭﺪـﺤﻣ ﺭﺩ ﺎﻬﻨﺗ ﻝﻭﺪـﻧﺎﭘ

ﻱﻭﺭ ﺮـﺑ

ﻪـﺑ ﺵﻭﺭ ﻦـﻳﺍ ﺯﺍ ﻩﺩﺎﻔﺘـﺳﺍ ﺎﺑ ﻢﺘﺴﻴﺳ ﻱﺯﺎﺳﺭﺍﺪﻳﺎﭘ ﺖﻬﺟ ﺭﺩ ﻲﺑﻮﺧ ﺞﻳﺎﺘﻧ ﻭ ﺩﻮﺑ ﻝﺎﻤﻋﺍ ﻞﺑﺎﻗ ﻢﺘﺴﻴﺳ ﺖـﺳﺩ

ﺪﻣﺁ . ﻲﻣ ﺍﺭ ﺵﻭﺭ ﻦﻳﺍ ﻪﺑ ﻥﺪﻴﺳﺭ ﻱﺎﺘﺳﺍﺭ ﺭﺩ ﻥﺍﻮﺗ

ﺵﻭﺭ ﺎﺑ ﺮﺘﻬﺑ ﺞﻳﺎﺘﻧ ﻱﺎـﻫ

Robust ﻭ

Adaptive ﺐـﻴﻛﺮﺗ

ﺩﺮﻛ