ﺗﺤﻘﻖ ﺗﺌﻮﺭ : ﭘﻨﺠﻢ ﻓﺼﻞ

ﻞﺼﻓ ﻢﺠﻨﭘ

: ﺭﻮﺌﺗ ﻖﻘﺤﺗ

ﻝﺎﻤﻴﻨﻴﻣ ﻖﻘﺤﺗ

1

1 1

( ) ( )

n n ( )

n n

G s Y s

s s s U s

b

a ^- a _- a

= =

+ + +L + ( )

G s ( )

D s

= b

(

^{sn +a1sn-1 + +L an-1s +an}

)

^{Y s( ) = bU s( )}

1

2 1

3 2

( ) ( )

( ) ( ) ( )

( ) ( ) ( )

x t y t

x t y t x t x t y t x t

ì =

ï = =

ïï = =

íï

& &

&& & x t&_n( ) = -a¹x t_n( ) - -L a_nx t¹( ) + bu t( )

Teacher: Khoshnam Shojaei

3 2

( 1)

( ) ⁿ ( ) 1( )

n n

x t y ^- t x _- t íï

ïï = =

î

&& &

M

&

[ ]

1 1

2 2

1 2 1

0 1 0 0 0

( ) ( )

0 0 1 0 0

( ) ( )

( )

0 0 0 1 0

( ) ( )

( ) 1 0 0 0 ( )

n n

n n n

x t x t

x t x t

u t

x t x t

y t x t

a a _- a _- a b

é ù é ù

é ù ê úé ù ê ú

ê ú ê úê ú ê ú

ê ú = ê úê ú+ ê ú

ê ú ê úê ú ê ú

ê ú ê úê ú ê ú

ë û êë- - - - úûë û ê úë û

=

& L

& L

M M M O M M

M M

& L

L L

ﻞﺼﻓ ﻢﺠﻨﭘ

: ﺭﻮﺌﺗ ﻖﻘﺤﺗ

ﻝﺎﻤﻴﻨﻴﻣ ﻖﻘﺤﺗ

1 1

0 1 1 1 1

1 1

0 1 1 1 1

( )

n n n

n n n n

n n n n

n n n n

s s s s s

G s d

s s s s s s

b b b b b b b

a a a a a a a

- -

- -

- -

- -

+ + + + + + +

= = +

+ + + + + + + +

L L

L L

( ) ( )

( ) G s N s

= D s

0 0

lim ( ) d s G s b

a

= ®¥ = ₁

1 1 ( )

ˆ ( ) s^{n n} s ⁿ Y s

G s b ^- + + b _- + b

= L =

s^®¥ a0 1 1

1

1 1

ˆ ( ) ( )

( )

n n

n n

n n

s s Y s

G s s s s U s

b b b

a a a

- -

-

+ + +

= =

+ L+ + + L

) ﻒﻟﺍ ( ﺮﻳﺬﭘ ﻩﺪﻫﺎﺸﻣ ﻡﺮﻓ :

ﺕﺭﻮﺻ ﻪﺑ ﺖﻟﺎﺣ ﺎﻫ ﺮﻴﻐﺘﻣ ﺏﺎﺨﺘﻧﺍ ﺎﺑ ،ﻪﻧﺎﻔﺳﺄﺘﻣ

(ⁿ 1), , , , y ^- K && &y y y ﻢﻴﺳﺭ ﻲﻤﻧ ﺮﻳﺯ ﻡﺮﻓ ﻪﺑ

( ) ( ) ( )

( ) ( ) ( )

x t Ax t Bu t y t Cx t Du t

= +

ìí = +

î

&

ﺖﺷﺍﺩ ﻢﻴﻫﺍﻮﺧ ﻪﮑﻠﺑ :

1 2 3

( ) ( )

( ) ( )

x t Ax t Bu

y t Cx t d u d u d u

= +

ìí = + + + +

î

&

& && L

ﻞﺼﻓ ﻢﺠﻨﭘ

: ﺭﻮﺌﺗ ﻖﻘﺤﺗ

ﺖﻟﺎﺣ ﺕﻻﺩﺎﻌﻣ ﺩﺭﺍﺪﻧﺎﺘﺳﺍ ﻡﺮﻓ ﻪﺑ ﻪﮐ ﻢﻴﻨﮐ ﻦﻴﻴﻌﺗ ﺍ ﻪﻧﻮﮔ ﻪﺑ ﺍﺭ ﺖﻟﺎﺣ ﺎﻫﺮﻴﻐﺘﻣ ﺪﻳﺎﺑ ،ﻦﻳﺍﺮﺑﺎﻨﺑ ﻢﻴﺳﺮﺑ

( ) ( ) ( ) :

( ) ( ) ( )

x t Ax t Bu t y t Cx t Du t

= +

ìí = + î

&

ﻞﺣ ﻩﺍﺭ :

1 1 2

1 1 1 2

1 1

ˆ ( ) ( )

1 ( )

n n n

n n n

n n n n

s s s s s s Y s

G s s s s s s s U s

b b b b b b

a a a a a

- - - - -

-

- - - -

+ + + + + +

= ´ = =

+ L+ + + + + +L

L L

1 1

1 1 1

( ) ^{n n} _{n n} ⁿ 1 _n ⁿ ( )

G s ⁼ s +a s ^- + +L a _- s +a ^´ s^{- =} +a s^- + +L a s^{- =} U s

(

1 1

)

¹

(

2 2

)

²

( )

( ) _{n n} ⁿ 0

Y s + a Y - bU s^- + a Y -b U s^- + +L a Y -b U s^- =

( ) ( ) ( )

1 1 1

1 1 2 2

( ) _{n n}

Y s = s^{- é}ë b U -a Y + s^- éë b U -a Y + +L s^- b U -a Y ùû^ùû

1( ) X s

1( ) Xn_- s

n( ) X s

ﻞﺼﻓ ﻢﺠﻨﭘ

: ﺭﻮﺌﺗ ﻖﻘﺤﺗ

( )

( )

( )

[ ]

1

1 1 1

1

1 2 2 2

1

2 1 1 1

1 1

( ) ( )

( ) ( )

( ) ( )

( )

n n

n n

n n

n n

X s s U Y X s

X s s U Y X s

X s s U Y X s

X s s U Y

b a

b a

b a

b a

-

- -

- -

-

- -

-

ì = éë - + ùû

ï = - +

ï éë ùû

ïí

ï = éë - + ùû

ïï = -

î

M

1 1 1

1 2 2 2

2 1 1 1

1

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

n n n

n n n

n n n

n n n

x t u t x t x t

x t u t x t x t

x t u t x t x t

x t u t x t

b a

b a

b a

b a

-

- -

- -

= - +

ìï = - +

ïïí

ï = - +

ï = -

ïî

&

&

M

&

&

1{ } L^- ·

[ ]

1 1

1 1

2 2

2

1 1

0 0 0

( ) ( )

1 0 0

( ) ( )

( )

0 1 0

( ) ( )

0 0 1

( ) 0 0 0 1 ( )

n n

n n

n

n n

x t x t

x t x t

u t

x t x t

y t x t

a b

a b

a

a b

- -

-

é - ù é ù

é ù ê - úé ù ê ú

ê ú ê úê ú ê ú

ê ú = ê - úê ú + ê ú

ê ú ê úê ú ê ú

ê ú ê úê ú ê ú

ë û êë - úûë û êë úû

=

& L

& L

L M

M M

M M M O M M

&

L L

ﻞﻣﺎﮐ ﻪﺒﺗﺭ ﻪﮐ ﺪﻳﺩ ﻢﻴﻫﺍﻮﺧ ،ﻢﻴﻫﺩ ﻞﻴﮑﺸﺗ ﺍﺭ ﺮﻳﺬﭘ ﻩﺪﻫﺎﺸﻣ ﺲﻳﺮﺗﺎﻣ ﻕﻮﻓ ﻖﻘﺤﺗ ﺍﺮﺑ ﺮﮔﺍ ﺖﺳﺍ .

ﺖﺳﺍ ﺮﻳﺬﭘ ﻩﺪﻫﺎﺸﻣ ﻩﺭﺍﻮﻤﻫ ﻕﻮﻓ ﻖﻘﺤﺗ ﻲﻨﻌﻳ .

ﻞﺼﻓ ﻢﺠﻨﭘ

: ﺭﻮﺌﺗ ﻖﻘﺤﺗ

) ﺏ ( ﺮﻳﺬﭘ ﻝﺮﺘﻨﮐ ﻡﺮﻓ :

1

1 1

1

1 1

( ) ( )

ˆ ( )

( ) ( )

n

n n

n n

n n

s s

N s Y s

G s D s s s s U s

b b b

a a a

-

- -

-

+ + +

= = =

+ L+ + +

L ( ) ( ) ( ) ( )

D s Y s = N s U s Y s( ) = N s D s U s( ) ^-1( ) ( ) ( ) V s

1

( ) ( ) ( )

( ) ( ) ( )

Y s N s V s V s D s U s^- ì =

í =

î

1( )

D s^{- Y s( )}

( )

U s V s( )

( ) N s ( )

U s G sˆ ( ) Y s( )

( ) ( ) ( ) D s V s =U s

1

2 1

3 2

( 1)

( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ⁿ ( ) ( )

x t v t

x t v t x t x t v t x t

x t v ^- t x t

ì =

ï = =

ïï = =

íï

ïï = =

& &

&& &

M

&

ﻞﺼﻓ ﻢﺠﻨﭘ

: ﺭﻮﺌﺗ ﻖﻘﺤﺗ

(

^{sn +a1sn-1 + +L an-1s +an}

)

^{V s( ) =U s( )}

1 2

1

( ) ( )

( ) ( )

( )

( ) ⁿ ( )

n

X s V s

X s sV s

X s

X s s V s^-

é ù é ù

ê ú ê ú

ê ú ê ú

= =

ê ú ê ú

ê ú ê ú

ë û

ë û

M M

( 1)

( ) ⁿ ( ) 1( )

n n

x t v ^- t x _- t

ï = =

î & ë ⁿ û ë û

1

1 1

( ) ( ) ( ) ( ) ( )

n n

n n

s V s = -a s V s^- - -L a _- sV s -a V s +U s

1 1 2 1 ( )

n n n n

x& = -a x - -L a _- x -a x +u t

1{ } L^- ·

(

^{1 1 1}

) ^[

^{1 2}

^]

1

( )

( ) ( ) ( ) ( ) ( )

( )

n

n n n

n

V s

Y s N s V s s s V s sV s

s V s

b ^- b _- b b b b

-

é ù

ê ú

ê ú

= = + + + =

ê ú

ê ú

ë û

L L

M

( ) ( )

Y s = CX s

ﻞﺼﻓ ﻢﺠﻨﭘ

: ﺭﻮﺌﺗ ﻖﻘﺤﺗ

[ ]

1 1

2 2

1 2 1

1 2 1

0 1 0 0 0

( ) ( )

0 0 1 0 0

( ) ( )

( )

0 0 0 1 0

( ) ( )

1

( ) ( )

n n

n n n

n n n

x t x t

x t x t

u t

x t x t

y t x t

a a a a

b b b b

- -

- -

é ù é ù

é ù ê ú é ù ê ú

ê ú ê ú ê ú ê ú

ê ú = ê ú ê ú + ê ú

ê ú ê ú ê ú ê ú

ê ú ê ú ê ú ê ú

ë û êë- - - - úû ë û ê úë û

=

& L

& L

M M M O M M

M M

& L

L

[

1 2 L 1

]

( ) _{n n n} ( )

y t = b b _- b _- L b x t

ﺖﺳﺍ ﻞﻣﺎﮐ ﻪﺒﺗﺭ ﻪﮐ ﺪﻳﺩ ﻢﻴﻫﺍﻮﺧ ،ﻢﻴﻫﺩ ﻞﻴﮑﺸﺗ ﺍﺭ ﺮﻳﺬﭘ ﻝﺮﺘﻨﮐ ﺲﻳﺮﺗﺎﻣ ﻕﻮﻓ ﻖﻘﺤﺗ ﺍﺮﺑ ﺮﮔﺍ

. .ﺖﺳﺍ ﺮﻳﺬﭘ ﻝﺮﺘﻨﮐ ﻩﺭﺍﻮﻤﻫ ﻕﻮﻓ ﻖﻘﺤﺗ ﻲﻨﻌﻳ

ﻞﺼﻓ ﻢﺠﻨﭘ

: ﺭﻮﺌﺗ ﻖﻘﺤﺗ

ﻥﺩﺮﺟ ﻖﻘﺤﺗ

ﻢﻴﻫﺩ ﻲﻣ ﻂﺴﺑ ﻲﺋﺰﺟ ﺎﻫﺮﺴﮐ ﻪﺑ ﺍﺭ ﻞﻳﺪﺒﺗ ﻊﺑﺎﺗ ،ﻖﻘﺤﺗ ﻦﻳﺍ ﺭﺩ:

.

ﻝﺎﺜﻣ :

( ) ( )( )

( ) ( )

3

1 2 3

13 3

11 12 2

3 2

( ) ( )

ˆ ( )

( )

Y s N s

G s U s s s s

e e

e e e

s s s

s s

l l l

l l l

l l

= =

- - -

= + + + +

- - -

- -

(s l1) (³ s l1)² s l¹ s l² s l³

= + + + +

- - -

- -

( ¹¹1)³ ( ¹²1)^{2 13 1 2 2 3 3}

( ) e ( ) e ( ) e ( ) e ( ) e ( )

Y s U s U s U s U s U s

s s s

s l s l l l l

= + + + +

- - -

- -

ﻞﺼﻓ ﻢﺠﻨﭘ

: ﺭﻮﺌﺗ ﻖﻘﺤﺗ

1

1

s-l e¹¹

1

1 s-l

1

1 s -l

e12

e y

( ¹¹1)³ ( ¹²1)^{2 13 1 2 2 3 3}

( ) e ( ) e ( ) e ( ) e ( ) e ( )

Y s U s U s U s U s U s

s s s

s l s l l l l

= + + + +

- - -

- -

3

1

s -l e³

2

1

s -l e²

13 S

e

u

y

b s a-

u x x ax bu& = +

ﻞﺼﻓ ﻢﺠﻨﭘ

: ﺭﻮﺌﺗ ﻖﻘﺤﺗ

1

1

1

s-l e¹¹

1

1 s -l

1

1 s -l

S

e12

e13

13 1 13

x& = l x +u

x13 x₁₂ x₁₁

x2

12 1 12 13

x& = l x + x x&11 = l1 11x + x12

2 2 2

x& = l x +u y

11 1 11

12 1 12

13 1 13

2 2 2

3 3 3

1 0 0 0 0

0 1 0 0 0

0 0 0 0 1

0 0 0 0 1

0 0 0 0 1

x x

x x

x x u

x x

x x

l l

l l

l

é ù é ù é ù é ù

ê ú ê ú ê ú ê ú

ê ú ê ú ê ú ê ú

ê ú ê= ú ê ú ê ú+

ê ú ê ú ê ú ê ú

ê ú ê ú ê ú ê ú

ê ú ê ú ê ú ê úë û

ë û ë û ë û

&

&

&

&

&

3

1

s -l e³

2

1

s -l x² e²

x3

&

3 3 3

x& = l x +u

u

ﻞﺼﻓ ﻢﺠﻨﭘ

: ﺭﻮﺌﺗ ﻖﻘﺤﺗ

ﮏﻳ ﻦﻳﺮﻤﺗ

ﺪﻴﺴﻳﻮﻨﺑ ﻖﻘﺤﺗ ﮏﻳ ﺮﻳﺯ ﻞﻳﺪﺒﺗ ﻊﺑﺍﻮﺗ ﺯﺍ ﮏﻳ ﺮﻫ ﺍﺮﺑ :

:

1 3 2

( ) 1

2 7 4

G s s

s s s

= +

+ + +

4 2

1( ) 4 3 2

5 2 7 4

s s

G s s s s s

= +

+ + + +

2 2

( ) 1

( 2) ( 2) G s s

s s

= +

+ +

ﻭﺩ ﻦﻳﺮﻤﺗ

ﺪﻴﺴﻳﻮﻨﺑ ﻥﺩﺮﺟ ﻖﻘﺤﺗ ﮏﻳ ﺮﻳﺯ ﻞﻳﺪﺒﺗ ﻊﺑﺎﺗ ﺍﺮﺑ :

: