Section 10–5 / State Observers 777
Transfer Function of Minimum-Order Observer-Based Controller. In the
minimum-order observer equation given by Equation (10–89):
define, similar to the case of the derivation of Equation (10–90),
Then, the following three equations define the minimum-order oberver:
(10–101)
(10–102)
(10–103)
Since Equation (10–103) can be rewritten as
(10–104)
by substituting Equation (10–104) into Equation (10–101), we obtain
(10–105)
Define
Then Equations (10–105) and (10–104) can be written as
(10–106)
(10–107)
Equations (10–106) and (10–107) define the minimum-order observer-based controller.
By considering uas the output and –yas the input,U(s)can be written as
Since the input to the observer controller is –Y(s),rather than Y(s),the transfer function
of the observer controller is
(10–108)
This transfer function can be easily obtained by using the following MATLAB statement:
[num,den] = ss2tf(Atilde, Btilde, -Ctilde, -Dtilde) (10–109)
U(s)
- Y(s)
=
num
den
=-CC
As I-A
B
- 1
B
+D
D
=-CC
As I-A
B
- 1
B
+D
D[-Y(s)]
U(s)=CC
As I-A
B-^1 B
+D
DY(s)
u =C
H+D
y
H=A
H+ B
y
D
=-AKa+Kb KeB
C
=-Kb
B
=Bˆ -FˆAKa+Kb KeB
A
=Aˆ -FˆKb
=AAˆ -FˆKbBH+CBˆ -FˆAKa+Kb KeBDy
H=AˆH+Bˆy+FˆC-KbH-AKa+Kb KeByD
=-KbH
- AKa+Kb KeBy
u =-Kx =-CKa KbDB
y
xb
R =-Ka y-Kb xb
u =-Kx
H=xb-Ke y
H=AˆH+Bˆy+Fˆu
Fˆ =Bb-Ke Ba
Bˆ =AˆKe+Aba-Ke Aaa
Aˆ =Abb-Ke Aab