This is shown in gure 7.10.
The requirement that the closed-loop transfer matrix from to be zero is
v
e
satis ed because the observer state error,
^, is uncontrollable from , and
x
;
x
v
therefore the transfer matrix from to
^ is zero. The transfer matrix from
v
x
;
x
v
to is y times this last transfer matrix, and so is zero.
e
C
Applying the modi ed controller paradigm to the estimated-state-feedback con-
troller yields the observer-based controller shown in gure 7.10. The observer-based
controller is just an estimated-state-feedback controller, with the output prediction
error processed through a stable transfer matrix and added to the actuator signal
Q
before the observer tap.
In fact, this augmentation is such that the modi ed controller paradigm yields
every controller that stabilizes the plant. Every stabilizing controller can be real-
ized (likely nonminimally) as an observer-based controller for some choice of stable
transfer matrix .
Q
From the observer-based controller we can form simple state-space equations for
the parametrization of all controllers that stabilize the plant, and all closed-loop
transfer matrices achieved by controllers that stabilize the plant.
The state-space equations for the augmented nominal controller are, from (7.29{
7.31), _^=( P u sfb est y)^+ est + u
(7.32)
x
A
;
B
K
;
L
C
x
L
y
B
v
=
sfb^ +
(7.33)
u
;K
x
v
=
y^
(7.34)
e
y
;
C
x :
The state-space equations for the closed-loop system with the augmented con-
troller are then found by eliminating and from (7.32{7.34) and the plant equa-
u
y
tions (2.19{2.21) of section 2.5:
_ = P
u sfb^ + w + u
x
A
x
;
B
K
x
B
w
B
v
_^ = est y + ( P u sfb est y)^ + est yw + u
x
L
C
x
A
;
B
K
;
L
C
x
L
D
w
B
v
= z
zu sfb^ + zw + zu
z
C
x
;
D
K
x
D
w
D
v
= y
y^ + yw
e
C
x
;
C
x
D
w
:
The transfer matrices 1, 2, and 3 can therefore be realized as
T
T
T
1( )
2( )
1
T
s
T
s
3( )
0
= T(
T); T + T
(7.35)
C
sI
;
A
B
D
T
s
where
T =
P
u sfb
A
;B
K
A
est y
P
u sfb
est y
L
C
A
;
B
K
;
L
C
T =
w
u
B
B
B
est yw
u
L
D
B
7.5 SOME GENERALIZATIONS OF CLOSED-LOOP STABILITY
165
T =
z
zu sfb
C
;D
K
C
y
y
C
;C
T =
zw
zu
D
D
D
yw
0
:
D
If has state-space realization
Q
_Q = Q Q + Q
(7.36)
x
A
x
B
e
= Q Q + Q
(7.37)
v
C
x
D
e
then a state-space realization of the observer-based controller can be found by elim-
inating and from the augmented controller equations (7.32{7.34) and the
e
v
Q
realization (7.36{7.37):
_^ = ( P u sfb est y u Q y)^
x
A
;
B
K
;
L
C
;
B
D
C
x
+ u Q Q + ( est + u Q)
(7.38)
B
C
x
L
B
D
y
_Q = Q y^ + Q Q + Q
(7.39)
x
;B
C
x
A
x
B
y
= ( sfb + Q y)^ + Q Q + Q
(7.40)
u
;
K
D
C
x
C
x
D
y
so that
( ) = K(
K) 1
;
K + K
(7.41)
K
s
C
sI
;
A
B
D
where
K =
P
u sfb
est y
u Q y
u Q
A
;
B
K
;
L
C
;
B
D
C
B
C
A
Q y
Q
;B
C
A
K =
est + u Q
L
B
D
B
Q
B
K =
sfb
Q y
Q
;K
;
D
C
C
C
K = Q
D
D
:
Some algebra veri es that the closed-loop transfer matrix
given by (2.27) of
H
section 2.5 does indeed equal 1 + 2 3.
T
T
QT
7.5
Some Generalizations of Closed-Loop Stability
So far, our discussion in this chapter has been built around the notion of a stable
transfer function, i.e., a transfer function for which each pole satis es
0. We
p
<p
<
saw in chapter 5 that stability is equivalent to several other important properties of a
transfer function, e.g., niteness of its peak or RMS gain. In fact, the material in this
chapter can be adapted to various generalized notions of stability. The references
discuss these ideas in a general setting we will describe a speci c example in more
detail.
166