لﺮﺘﻨﮐ يﺎﻫ ﻢﺘﺴﯿﺳ ﯽﺣاﺮﻃ و ﻞﯿﻠﺤﺗ ﺖﻟﺎﺣ يﺎﻀﻓ هزﻮﺣ رد هﺮﯿﻐﺘﻣﺪﻨﭼ
ﻖﯾﺪﺻ ﯽﮐﺎﺧ ﯽﻠﻋ لﺮﺘﻨﮐ هوﺮﮔ -
ﺮﻬﻣ 1399
2
ﻪﻣﺪﻘﻣ •
ﻢﯿﻫﺎﻔﻣ ﻪﯾﺎﭘ
: لﺮﺘﻨﮐ يﺮﯾﺬﭘ
و ﺖﯾور يﺮﯾﺬﭘ
لﺮﺘﻨﮐ يﺮﯾﺬﭘ
ﯽﻌﺑﺎﺗ )
ﯽﺟوﺮﺧ (
ﻪﯾﺮﻈﻧ ﻖﻘﺤﺗ
ﺶﻫﺎﮐ ﻪﺒﺗﺮﻣ
تﻻدﺎﻌﻣ يﺎﻀﻓ
ﺖﻟﺎﺣ
ﻪﻠﭘﻮﮐد يزﺎﺳ
ﻢﺘﺴﯿﺳ يﺎﻫ
هﺮﯿﻐﺘﻣﺪﻨﭼ ﺎﺑ
ﮏﺑﺪﯿﻓ ﺣ
ﺖﻟﺎ
ﻢﯿﻫﺎﻔﻣ ﻪﯾﺎﭘ
: لﺮﺘﻨﮐ يﺮﯾﺬﭘ
و ﺖﯾور ﺬﭘ
يﺮﯾ
( ) ( ) ( )
( ) ( ) ( )
, ,
rank , rank
n n n m l n
x t A x t Bu t y t Cx t Du t
A R B R C R
B m C l
•
× × ×
= +
= +
∈ ∈ ∈
= =
طﺮﺷ مزﻻ
و ﯽﻓﺎﮐ لﺮﺘﻨﮐ
يﺮﯾﺬﭘ ﺴﯿﺳ
ﻢﺘ :
1
n n nm
c B A B A − B R ×
Φ = ∈
ﺲﯾﺪﻧا يﺎﻫ
لﺮﺘﻨﮐ يﺮﯾﺬﭘ
ﺴﯿﺳ ﻢﺘ
:
4
1 1
1 1 1
1
1
1
1
( 2, , )Controllability indices
we have,
, max ( 1, , )Controllability index
(1 ) Partial Controllability Matrix
i
i i i
m
i c i
i i
q c
i
b A b A b
i m
b A b A b
n i m
B A B A B q n
µ
µ
µ ν
µ µ
µ
−
−
=
−
→
=
→
= = =
Φ = ≤ ≤
∑
لﺎﺜﻣ ﮏﯾ •
6
لﺎﺜﻣ ﮏﯾ •
طﺮﺷ مزﻻ
و ﯽﻓﺎﮐ ﺖﯾور
يﺮﯾﺬﭘ ﺘﺴﯿﺳ
ﻢ.
ﺲﯾﺪﻧا يﺎﻫ
ﺖﯾور يﺮﯾﺬﭘ
ﺘﺴﯿﺳ ﻢ.
ﺲﯾﺮﺗﺎﻣ ﺖﯾور
يﺮﯾﺬﭘ ﯽﯾﺰﺟ
ﻢﺘﺴﯿﺳ .
ﻢﺘﺴﯿﺳ يﺎﻫ
لﺮﺘﻨﮐ ﺮﯾﺬﭘ
و ﺖﯾور ﺮﯾﺬﭘ
.
فﺬﺣ ﺐﻄﻗ
- ﺮﻔﺻ ﻢﺘﺴﯿﺳرد
يﺎﻫ لﺮﺘﻨﮐ
:
1 1
(sI − A )− B or C sI( − A )−
8
لﺮﺘﻨﮐ • يﺮﯾﺬﭘ
و ﺖﯾور يﺮﯾﺬﭘ
رد ﻒﯿﺻﻮﺗ ﺲﯾﺮﺗﺎﻣ
ﺳ ﻢﺘﺴﯿ
( ) ( ) ( ) ( )
1
P( )
( ) ( ) ( ) ( ) ( )
P s Q s
s R s W s
G s R s P − s Q s W s
= −
⇒ = +
Let, ( ) 0 ( ) ( ) ( ) ( ) ( ) ( )
L s
P s L s P s Q s L s Q s
≠
=
=
( ) ( ) 1( ) ( ) ( ) G s R s P − s Q s W s
⇒ = +
ﺎﯾآ ﺶﻫﺎﮐ ﻪﺒﺗﺮﻣ
يا رد ﻞﯿﮑﺸﺗ ﺲﯾﺮﺗﺎﻣ
ﻊﺑﺎﺗ ﻞﯾﺪﺒﺗ
خر هداد ا
؟ﺖﺳ
( ) 0
( ) ( ) ( ) ( ) ( ) ( ) D s
P s P s D s R s R s D s
≠
=
=
( ) ( ) 1( ) ( ) ( ) G s R s P − s Q s W s
⇒ = +
ﻪﺑ رﻮﻃ ﻪﺑﺎﺸﻣ
،يا ﺮﮔا
:
ﺎﯾآ ﺶﻫﺎﮐ ﻪﺒﺗﺮﻣ
يا رد ﻞﯿﮑﺸﺗ ﺲﯾﺮﺗﺎﻣ
ﻊﺑﺎﺗ ﻞﯾﺪﺒﺗ
خر هداد ا
؟ﺖﺳ
10
( ) 0 i.d.z ( ) 0 o.d.z L s
D s
= ⇒
= ⇒
{ } { } { }
{ } { } { }
Let,
i.d.z, o.d.z, and
o.d.z after removing all the i.d.z, then i.o.d.z
i i
i
i i i
β γ
θ
δ γ θ
= =
=
= − =
ﺪﻨﭼ ﻒﯾﺮﻌﺗ
:
لﺎﺜﻣ ﮏﯾ •
12
لﺎﺜﻣ ﮏﯾ •
لﺎﺜﻣ ﮏﯾ •
14
{ }
{ } { } { { } { } }
{ }
( ) 0 =System matrix poles Let,
,
Then, =TFN matrix poles
i
i i i i i
i
P s α
η α β γ δ
η
= ⇒
= − −
ﺐﻄﻗ ﻢﺘﺴﯿﺳ
:
16
لﺎﺜﻣ ﮏﯾ •
لﺮﺘﻨﮐ يﺮﯾﺬﭘ
ﯽﻌﺑﺎﺗ )
ﯽﺟوﺮﺧ (
ﻒﯾﺮﻌﺗ لﺮﺘﻨﮐ
يﺮﯾﺬﭘ ﯽﺟوﺮﺧ
طﺮﺷ مزﻻ
و ﯽﻓﺎﮐ لﺮﺘﻨﮐ
يﺮﯾﺬﭘ ﯽﺟوﺮﺧ
1 n
o C B CA B CA − B D
Φ =
ﻒﯾﺮﻌﺗ لﺮﺘﻨﮐ
يﺮﯾﺬﭘ ﯽﻌﺑﺎﺗ
( ) TFN Matrix with inputs and outputs
G s m l
18
( ) ( ) ( )
( ) s.t. R ( ) ( ) ? Y s G s U s
U s Y s Y s
=
∃ =
ﻂﯾاﺮﺷ لﺮﺘﻨﮐ
يﺮﯾﺬﭘ ﯽﻌﺑﺎﺗ
:
1. inputs outputs 2. rank ( )
m l
G s l
≥
=
لاﻮﺳ ﯽﻠﺻا
:
ﺎﺑ هدروآﺮﺑ نﺪﺷ
ﻂﯾاﺮﺷ ﻻﺎﺑ
:
( ) T ( )[ ( ) T ( )] 1 R ( ) U s =G s G s G s − Y s
لﺎﺜﻣ ﮏﯾ •
20
22
ﻢﯿﻫﺎﻔﻣ ﻪﯾﺎﭘ
: ﻖﻘﺤﺗ ﯽﻣ
لﺎﻤﯿﻧ و
ﻖﻘﺤﺗ يﺎﻫ
ﮑﯿﻧﻮﻧﺎﮐ لﺎ
ﻖﻘﺤﺗ ﯽﻣﺮﯿﻏ
لﺎﻤﯿﻧ و
ﺶﻫﺎﮐ ﻪﺒﺗﺮﻣ
ﻖﻘﺤﺗ تﺮﺒﻠﯿﮔ
ﻪﯾﺮﻈﻧ • ﻖﻘﺤﺗ
رد ﻢﺘﺴﯿﺳ يﺎﻫ
هﺮﯿﻐﺘﻣﺪﻨﭼ
ﻖﻘﺤﺗ يﺎﻫ
ﯽﻣﺮﯿﻏ لﺎﻤﯿﻧ
ﺎﯾ ﻢﯿﻘﺘﺴﻣ
[
1]
( ) ( ) m ( )
G s = g s g s
ﺶﯾﺎﻤﻧ ﯽﻧﻮﺘﺳ
) ﺎﯾ ﻪﺑ رﻮﻃ ﻪﺑﺎﺸﻣ
( ﯽﻔﯾدر ﺲﯾﺮﺗﺎﻣ
ﺑﺎﺗ ﻊ ﻞﯾﺪﺒﺗ :
1 2
1 2
1 1
( )
j j
j
j j
j
ij ij ij
ij j j
s s
g s
s s
δ δ
δ
δ δ
δ
β β β
α α
− −
−
+ + +
= + + +
1
1
j
j
ij ij ij
ij δ δ
β β β
β
−
=
j
j j
αδ
α
24
ﻖﻘﺤﺗ لﺎﮑﯿﻧﻮﻧﺎﮐ
لﺮﺘﻨﮐ ﺮﯾﺬﭘ
:
1 2
1 2
1 1
( )
j j
j
j j
j
ij ij ij
ij j j
s s
g s
s s
δ δ
δ
δ δ
δ
β β β
α α
− −
−
+ + +
= + + +
1 1
0 1 0 0
0 0 1 , 0
j j 1
j j
j j j
ij T ij
A b
c
δ δ
α α α
β
−
= =
− − −
=
ﺐﯿﮐﺮﺗ ﻖﻘﺤﺗ
يﺎﻫ لﺎﮑﯿﻧﻮﻧﺎﮐ
لﺮﺘﻨﮐ ﺮﯾﺬﭘ
ياﺮﺑ ﻢﺘﺴﯿﺳ
ﭼ هﺮﯿﻐﺘﻣﺪﻨ :
1 1
2 2
11 12 1
21 22 2
0 0 0 0
0 0 0 0
,
0 0 0 0
c c
m m
m m c
A b
A b
A B
A b
c c c
c c c
C
c c c
= =
=
26
لﺎﺜﻣ ﮏﯾ •
28
يﺮﻄﻗ ﻖﻘﺤﺗ • )
تﺮﺒﻠﯿﮔ (
ﺖﻟﺎﺣ يﺎﻀﻓ
لوا ﺖﻟﺎﺣ :
ﻘﺣ يراﺮﮑﺗ ﺮﯿﻏ يﺎﻫ ﺐﻄﻗ ﺎﺑ ﻞﯾﺪﺒﺗ ﻊﺑﺎﺗ ﺲﯾﺮﺗﺎﻣ ﯽﻘﯿ
ﻪﮐ ﺪﯿﻨﮐ ضﺮﻓ :
{
1, , n}
,Bn , n ,a C
di g
λ λ
DΛ =
زا ترﻮﺻ ﺚﯿﻤﺳا
ﮏﻣ نﻼﯿﻣ ﻪﺒﺳﺎﺤﻣ
ﯽﻣ دﻮﺷ
. lim ( )
D s G s
= →∞
ﻢﯾراد
{
1}
1
( ) , ,
n
k
n n
k k
n
G s C diag s s B D G D
λ λ
s=
λ
= − − + = +
∑
−
هﺪﻧﺎﻣ يﺎﻫ
ﺲﯾﺮﺗﺎﻣ ﻊﺑﺎﺗ
ﻞﯾﺪﺒﺗ ﺪﻨﺘﺴﻫ
lim ( ) ( ) :
and
k
k k
s
T
k nk nk
G s G s
G c b
λ λ
= → −
= يدورو و ﯽﺟوﺮﺧ ﺲﯾﺮﺗﺎﻣ ﻒﯾدر و نﻮﺘﺳ ﻦﯿﻣاk
30
لﺎﺜﻣ ﮏﯾ •
مود ﺖﻟﺎﺣ :
ﯽﻘﯿﻘﺣ يراﺮﮑﺗ يﺎﻫ ﺐﻄﻗ ﺎﺑ ﻞﯾﺪﺒﺗ ﻊﺑﺎﺗ ﺲﯾﺮﺗﺎﻣ
ﺒﺗ ﻊﺑﺎﺗ ﺲﯾﺮﺗﺎﻣ ﻪﮐ ﺪﯿﻨﮐ ضﺮﻓ ﻪﻟﺎﺴﻣ نﺪﺷ ﺮﺗ هدﺎﺳ ياﺮﺑ ﮏﯾ ﻞﯾﺪ
دﺪﻌﺗ ﺎﺑ رﺮﮑﻣ يﺎﻫ ﺐﻄﻗ 3
دراد .
ﺪﻫد ﯽﻣ ﯽﯾﺰﺟ يﺎﻫﺮﺴﮐ ﻂﺴﺑ :
( 1 )3 1 ( 1 )2 2 ( 1 ) 3
( )
G s M M M
s λ s λ s λ
= + +
− − −
1 1
1
2 2
( ) rank M r rank M r
M
=
=
و
{ }
{ }
{ }
1
1 1 2
1
1 ( 1)
, ,
, , , , ,
l lr
l lr l r lr
b b
b b b + b
32
ﻪﺘﮑﻧ :
1 2 3
r ≤ ≤ ≤r r m
ﺎﺑ ﻒﯾﺮﻌﺗ يﺎﻫرادﺮﺑ
ﯽﻧﻮﺘﺳ
،ﺐﺳﺎﻨﻣ ﯾراد
ﻢ:
1 1
1 1 1
1
1 1 1 1
1 11 1 12 2 1
1 2
11 12 1
1
1
( ) [ ( ) ]
l l r lr
l
l l l
r r r
lr
l l T l l T
r r r r
M c b c b c b b
c c c b C B
b
C M B B B
−= + + +
=
⇒ =
ﻪﺑ رﻮﻃ ﻪﺑﺎﺸﻣ
:
2
2
3
3
1 2
2 21 22 2
1 2
3 31 32 3
l l r
lr l l r
lr
b
M c c c b
b b
M c c c b
b
=
=
34
ﮏﯾ • لﺎﺜﻣ
[ ] [ ]
2 2
2
1 2
2 2
1 1 1
1
2 2
1 0
1 1 ( 1)
( ) 1 ( )
1 1 1
( 1)
0 ( 1)
1 1
1 1 1 1
( ) ( ) 0
1 1
( 1) ( 1) ( 1) ( 1)
1 0
( ) 2 , and 1 1 1 1
0 1
( ) 2 , and the number of Jorda
G s M s s
s
s
G s M M G s
s s s s
r M r M
r M r
M
+
= + − ⇒ =
+
⇒ = + + + ⇒ = + − + +
⇒ = = = + −
⇒ = =
n blocks of order 2 is 2 and 1 is 0.
1 1 0 0 0 0
0 1 0 0 1 1
0 0 1 1 0 0
x x u
−
−
= +
−
36
ﮏﯾ • لﺎﺜﻣ
[ ]
2
2
1 2
2
2
1 1 1
1 1
1 0
( 1) ( 2)
( 1) ( 2)
( ) ( )
1 1
( 1) ( 2) 0
1 1 1
( ) ( 1) ( 1) ( 2)
1 0 0 0 0 1
1 1 1
( ) ( 1) 0 0 ( 1) 1 0 ( 2) 0 1
( ) 1 , and 1 1 0 one Jo 0
s s
s s
G s M s
s s s
G s M M M
s s s
G s s s s
r M r M
+ +
+ +
= ⇒ =
+ +
⇒ = + +
+ + +
⇒ = + + + + +
⇒ = = =
[ ]
1
2 2 2 1
2
rdan block of order 2.
( ) 1 , and 0 1 0 zero Jordan block of order 1.
1
r M r M r r
M
⇒ = = = − =
[ ]
( ) 1, and 1 0 1 1
1 1 0 0 0
0 1 0 1 0
0 0 2 0 1
1 0 1 0 1 1
r M M
x x u
y x
•
⇒ = =
−
= − +
−
=
38
ﮏﯾ • لﺎﺜﻣ
3 2 3 2
4
3 2 2 3
1 2 3 4
4 3 2
4 3
2
1 1 2
( ) 1 1.5 1 1 1.5 2
9 1 1 2
1 1 1 1
( )
1 1 2 0 0 0
1 1
( ) 1 1 2 1.5 1 1.5
1 1 2 1 0 1
1 0 1 1 0 1
1 1
0 0 0 0 0 0
9 1 0 1 0 1
s s s s
G s s s s
s s s s s s s
G s M M M M
s s s s
G s s s
s s
− + − + −
= + + − −
− − + − + − −
⇒ = + + +
−
⇒ = − + −
− − −
− −
+ +
− −
[ ]
1 1 1 1
1
2 2 2
2
2 1
1 1 2 1
1 1 2 ( ) 1 , and 1 1 1 2 one Jordan block of order 4.
1 1 2 1
0 0 0 0 0
1 1 2
1.5 1 1.5 ( ) 2 , and 1 0.5
1 0 1
1 0 1 0 1
and one J
M r M r M
M r M r M
M r r
−
= − ⇒ = = = −
−
−
= − −− ⇒ = = = − − −
− =
1
3 2 3 3
3
3 2
1
ordan block of order 3.
1 0 1 0 0 1 1 1 2
0 0 0 ( ) 3 , and 0 0 0 1 0 1
9 1 0 1 5 3 1 0 1
and one Jordan block of order 2.
1 0 1
M M
M r r M
M r r
M M
− −
= ⇒ = = = − −
− − − −
− =
−
= =
40
0 1 0 0 0 0 0
0 1 0 0 0 0 0
0 1 0 0 0
0 1 1 2
0 1 0 0 0 0
0 1 0 0 0
0 1 0 1
0 0 1 0 0 0
0 1 0 1
1 0 0 0 0 0 0 1 1
1 1 0 0 0.5 0 0 0 0
1 0 1 0 1 5 1 3 0
x x u
y x
•
−
= +
− −
−
−
= −
− −
The Plant is
Controllable but
Unobservable
ﺶﻫﺎﮐ ﻪﺒﺗﺮﻣ
تﻻدﺎﻌﻣ يﺎﻀﻓ
ﺖﻟﺎﺣ
تﻻدﺎﻌﻣ • يﺎﻀﻓ
ﺖﻟﺎﺣ ﺮﯿﻏ
ﯽﻣ لﺎﻤﯿﻧ
تﻻدﺎﻌﻣ • يﺎﻀﻓ
ﺖﻟﺎﺣ ﯽﻣ
لﺎﻤﯿﻧ
42
ﺶﻫﺎﮐ ﻪﺒﺗﺮﻣ
ﻢﺘﺴﯿﺳ يﺎﻫ
ﯽﻣﺮﯿﻏ لﺎﻤﯿﻧ
ﻖﻘﺤﺗ لﺎﮑﯿﻧﻮﻧﺎﮐ
لﺮﺘﻨﮐ ﺮﯾﺬﭘ
:
1 1 1
2 2 2
1 2
0 0
0 P( )
0 0
( )
m m m
m
sI A B
sI A B
s
sI A B
C C C D s
δ
δ
δ
−
−
=
−
− − −
ﯽﻣ ناﻮﺗ
ﺎﺑ ياﺮﺟا تﺎﯿﻠﻤﻋ
ﺐﺳﺎﻨﻣ يﺎﻫﺮﻔﺻ
ﻪﻠﭘﻮﮐد ﯽﺟوﺮﺧ
ار :دﺮﮐ فﺬﺣ و ﯽﯾﺎﺳﺎﻨﺷ
12
P( ) 0
0 ( )
u u u
o o o
o
sI A A B
s sI A B
C D s
−
= −
−
Pmin ( )
( )
o o o
o
sI A B
s C D s
−
= −
44
ﻪﺘﮑﻧ ود :
ﻧ لﺮﺘﻨﮐ يﺎﻫ ﻢﺘﺴﯿﺳ ياﺮﺑ ﻪﺑﺎﺸﻣ ﺪﻧور ﺮﯾﺬﭘﺎ
يور ﺮﺑ ﻢﺘﯾرﻮﮕﻟا لﺎﻤﻋا :
0 A B N C
=
لﺎﺜﻣ ﮏﯾ •
46
ﺶﻫﺎﮐ ﻪﺒﺗﺮﻣ
تﻻدﺎﻌﻣ يﺎﻀﻓ
ﺖﻟﺎﺣ ﯽﻣ
لﺎﻤﯿﻧ
1 11 1 12 2 1
x A x A x B u x A x A x B u
= + +
⇒ = + +
48
ود شور لواﺪﺘﻣ
:
شور شﺮﺑ
شور هﺪﻧﺎﻣ
يراﺬﮔ
شور شﺮﺑ
ﺖﻟﺎﺣ يﺮﻄﻗ
:
50
لﺎﺜﻣ :
ﻦﯿﺑرﻮﺗ يزﺎﮔ
2 2ₓ ﺎﺑ 12 ﺮﯿﻐﺘﻣ ﺖﻟﺎﺣ
ﺦﺳﺎﭘ يﺎﻫ
ﯽﺟوﺮﺧ :
0 5 10 15 20 25 30
0 0.5 1 1.5 2 2.5 3
y 1
time (sec)
0 5 10 15 20 25 30
-4 -2 0 2 4
y 1
time (sec)
2 3 4
10 15
52
ﺮﯾدﺎﻘﻣ ﯽﯾﺎﻨﺜﺘﺳا
:
10-3 10-2 10-1 100 101 102
-50 -40 -30 -20 -10 0 10 20 30
Frequency (rad\sec)
Singular Values (dB)
شور هﺪﻧﺎﻣ
يراﺬﮔ
ﮏﯾ ﻪﺘﺳد
ﺮﯿﻐﺘﻣ زا
ﻪﺘﺳد يﺎﻫﺮﯿﻐﺘﻣ
ﺖﻟﺎﺣ ﺮﮕﯾد
ﯾﺮﺳ ﻊ ﺪﻧﺮﺗ .
54
لﺎﺜﻣ :
ﻦﯿﺑرﻮﺗ يزﺎﮔ
2 2ₓ ﺎﺑ 12 ﺮﯿﻐﺘﻣ
ﺖﻟﺎﺣ و
ﺶﻫﺎﮐ :ﯽﺟوﺮﺧ يﺎﻫ ﺦﺳﺎﭘ .يراﺬﮔ هﺪﻧﺎﻣ شور ﻪﺑ ﻪﺒﺗﺮﻣ
0 5 10 15 20 25 30
0 0.5 1 1.5 2 2.5 3
y 1
time (sec)
0 5 10 15 20 25 30
0 1 2 3 4
y1
time (sec)
0 5 10 15 20 25 30
0 1 2 3 4
y 2
time (sec)
0 5 10 15 20 25 30
0 5 10 15
y 2
time (sec)
ﺮﯾدﺎﻘﻣ ﯽﯾﺎﻨﺜﺘﺳا
:
10-3 10-2 10-1 100 101 102
-50 -40 -30 -20 -10 0 10 20 30
Frequency (rad\sec)
Singular Values (dB)
56
بﺎﺨﺘﻧا ﻪﺒﺗﺮﻣ
لﺪﻣ ﯽﮑﯿﻣﺎﻨﯾد
ﻖﻘﺤﺗ يﺎﻫ
ﺲﻧﻻﺎﺑ هﺪﺷ
نﺎﯿﻣاﺮﮔ يﺎﻫ
لﺮﺘﻨﮐ يﺮﯾﺬﭘ
و ﺖﯾور يﺮﯾﺬﭘ
:
شﺮﺑ ﺲﻧﻻﺎﺑ
هﺪﺷ
هﺪﻧﺎﻣ يراﺬﮔ
ﺲﻧﻻﺎﺑ هﺪﺷ
58
لﺎﺜﻣ :
ﻦﯿﺑرﻮﺗ يزﺎﮔ
2 2ₓ ﺎﺑ 12 ﺮﯿﻐﺘﻣ ﺖﻟﺎﺣ
و ﺶﻫﺎﮐ ﻪﺒﺗﺮﻣ
ﻪﺑ
شور :ﯽﺟوﺮﺧ يﺎﻫ ﺦﺳﺎﭘ .هﺪﺷ ﺲﻧﻻﺎﺑ يراﺬﮔ هﺪﻧﺎﻣ و شﺮﺑ
0 5 10 15 20 25 30
-1 0 1 2 3
y 1
time (sec)
0 5 10 15 20 25 30
-1 0 1 2 3 4
y 1
time (sec)
0 5 10 15 20 25 30
-1 0 1 2 3 4
y 2
time (sec)
0 5 10 15 20 25 30
-5 0 5 10 15
y 2
time (sec)
ﺮﯾدﺎﻘﻣ ﯽﯾﺎﻨﺜﺘﺳا
:
-3 -2 -1 0 1 2
-50 -40 -30 -20 -10 0 10 20 30
Singular Values (dB)
