as shown in gure 7.9. so that the closed-loop transfer matrix is
Q
= 1 + 2 3
H
U
U
QU
:
Thus the set of closed-loop transfer matrices achievable by the modi ed controller
shown in gure 7.9 is
mcp = zw + zu
yw
stable
H
fP
P
QP
j
Q
g
:
The expression here for mcp is the same as the expression for stable in equa-
H
H
tion (7.21) in section 7.2.4. So in this case the modi ed controller paradigm gener-
ates all stabilizing controllers: any stabilizing controller for a stable plant can
K
P
be implemented with a suitable stable as shown in gure 7.9.
Q
The reader can also verify that the connection of with the augmented nominal
Q
controller yields = ( + yu) 1 |exactly the same formula as (7.22) with
;
K
I
QP
Q
Q
substituted for .
R
7.4
A State-Space Parametrization
A general method of applying the modi ed controller paradigm starts with a nom-
inal controller that is an estimated-state feedback. The estimated-state-feedback
controller is given by
=
sfb^
(7.28)
u
;K
x
where sfb is some appropriate matrix (the state-feedback gain) and ^ is an estimate
K
x
of the component of due to , governed by the observer equation
x
u
_^ = P ^ + u + est(
y^)
(7.29)
x
A
x
B
u
L
y
;
C
x
where est is some appropriate matrix (the estimator gain). The transfer matrix of
L
this controller is thus
nom( ) =
sfb(
P + u sfb + est y) 1
;
est
K
s
;K
sI
;
A
B
K
L
C
L
:
nom will stabilize
provided sfb and est are chosen such that P
u sfb
K
P
K
L
A
;
B
K
and P
est y are stable, which we assume in the sequel.
A
;
L
C
To augment this estimated-state-feedback nominal controller, we inject into
v
, before the observer tap, meaning that (7.28) is replaced by
u
=
sfb^ +
(7.30)
u
;K
x
v
and therefore the signal does not induce any observer error. For the signal we
v
e
take the output prediction error:
=
y^
(7.31)
e
y
;
C
x :
7.4 A STATE-SPACE PARAMETRIZATION
163
D
y
w
+
+
q
q
r
w
z
D
z
w
+
r
+
D
z
u
x
B
C
w
z
(
) 1
;
sI
;
A
P
+
q
r
y
u
C
B
y
u
+
P
nom
K
+
q
^
r
C
B
y
x
u
(
) 1
;
;
sI
;
A
P
est
L
sfb
K
;
r
+
q
v
e
Q
+
x
Q
C
Q
(
) 1
;
r
+
q
B
sI
;
A
Q
Q
D
Q
The modied controller paradigm as applied to a nominal
Figure
7.10
estimated-state-feedback controller nom. is added to the actuator signal
K
v
before the observer tap, and is the output prediction error. With the
u
e
stable realization added, the modied controller is called an observer-
Q
based controller.
164