PPT - State Feedback Controller Design - Electrical Engineering (EE) PDF Download

 Page 1


1
State Feedback Controller Design
§State Feedback and Stabilization
§Full-Order Observer Design
§Separation Principle
§Reduced-Order Observer
Page 2


1
State Feedback Controller Design
§State Feedback and Stabilization
§Full-Order Observer Design
§Separation Principle
§Reduced-Order Observer
2
0 0
) ( , x t x Bu Ax x = + = &
( ) ( ) ( ) ( ) ( )
n
BF A I l l l l l l l - - - = - - L
2 1
det
Fx u - =
( )x BF A
BFx Ax Bu Ax x
- =
- = + =
&
Stabilization by State Feedback: Regulator Case
Given ,n , i
i
L 1 , = l
Controllable
Û
) (A,B
There exists a state feedback matrix, F, such that
Plant:
State Feedback Law:
Closed-Loop System:
Theorem
State Feedback and Stabilization
Page 3


1
State Feedback Controller Design
§State Feedback and Stabilization
§Full-Order Observer Design
§Separation Principle
§Reduced-Order Observer
2
0 0
) ( , x t x Bu Ax x = + = &
( ) ( ) ( ) ( ) ( )
n
BF A I l l l l l l l - - - = - - L
2 1
det
Fx u - =
( )x BF A
BFx Ax Bu Ax x
- =
- = + =
&
Stabilization by State Feedback: Regulator Case
Given ,n , i
i
L 1 , = l
Controllable
Û
) (A,B
There exists a state feedback matrix, F, such that
Plant:
State Feedback Law:
Closed-Loop System:
Theorem
State Feedback and Stabilization
3
State Feedback System  (Regulator Case)
C
A
B
D
+
+
+
+
F
ò
x
u
x
&
y
Page 4


1
State Feedback Controller Design
§State Feedback and Stabilization
§Full-Order Observer Design
§Separation Principle
§Reduced-Order Observer
2
0 0
) ( , x t x Bu Ax x = + = &
( ) ( ) ( ) ( ) ( )
n
BF A I l l l l l l l - - - = - - L
2 1
det
Fx u - =
( )x BF A
BFx Ax Bu Ax x
- =
- = + =
&
Stabilization by State Feedback: Regulator Case
Given ,n , i
i
L 1 , = l
Controllable
Û
) (A,B
There exists a state feedback matrix, F, such that
Plant:
State Feedback Law:
Closed-Loop System:
Theorem
State Feedback and Stabilization
3
State Feedback System  (Regulator Case)
C
A
B
D
+
+
+
+
F
ò
x
u
x
&
y
4
( )( ) ( )
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ë
é
+ - + - + -
= -
n n
c c
f a f a f a
F B A
L
M O M M
L
L
2 2 1 1
0 0 0
0 1 0
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ë
é
=
1
0
0
M
c
B
[ ]x f f f Fx u
n
- - - = - = L
2 1
State Feedback Design in Controllable Form
( ) ( ) ( ) ( )
1 1
1
det f a s f a s F B A sI
n
n n
n
c c
+ + + + + = - -
-
L
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ë
é
- - -
=
-
n
n
c
a a a
I
A
L
M
2 1
1
0
0
(8.1)
Page 5


1
State Feedback Controller Design
§State Feedback and Stabilization
§Full-Order Observer Design
§Separation Principle
§Reduced-Order Observer
2
0 0
) ( , x t x Bu Ax x = + = &
( ) ( ) ( ) ( ) ( )
n
BF A I l l l l l l l - - - = - - L
2 1
det
Fx u - =
( )x BF A
BFx Ax Bu Ax x
- =
- = + =
&
Stabilization by State Feedback: Regulator Case
Given ,n , i
i
L 1 , = l
Controllable
Û
) (A,B
There exists a state feedback matrix, F, such that
Plant:
State Feedback Law:
Closed-Loop System:
Theorem
State Feedback and Stabilization
3
State Feedback System  (Regulator Case)
C
A
B
D
+
+
+
+
F
ò
x
u
x
&
y
4
( )( ) ( )
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ë
é
+ - + - + -
= -
n n
c c
f a f a f a
F B A
L
M O M M
L
L
2 2 1 1
0 0 0
0 1 0
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ë
é
=
1
0
0
M
c
B
[ ]x f f f Fx u
n
- - - = - = L
2 1
State Feedback Design in Controllable Form
( ) ( ) ( ) ( )
1 1
1
det f a s f a s F B A sI
n
n n
n
c c
+ + + + + = - -
-
L
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ë
é
- - -
=
-
n
n
c
a a a
I
A
L
M
2 1
1
0
0
(8.1)
5
( )
1
1
1
c
n
cn
n
n
i
i
a s a s s + + + = -
-
=
Õ
L l
1 1 1
1 1 1
a a f
a a f
a a f
c
n cn n
n cn n
- =
- =
- =
- - -
M
[ ]
n cn c c
a a a a a a F - - - = L
2 2 1 1
Comparing (8.1) and (8.2), we have
Suppose the desired characteristic polynomial
(8.2)
(8.3)