Given one controller nom that stabilizes the plant, we can construct a large fam-
K
ily of controllers that stabilize the plant, just as the formula (7.10) shows how to
construct a one-parameter family of controllers that stabilize the plant.
The construction proceeds as follows:
We modify or augment the nominal controller nom so that it produces an
K
auxiliary output signal (of the same size as ) and accepts an auxiliary input
e
y
signal (of the same size as ) as shown in gure 7.4. This augmentation is
v
u
done in such a way that the closed-loop transfer matrix from to is zero
v
e
while the open-loop controller transfer matrix from to remains nom.
y
u
K
We connect a stable u
y transfer matrix
from to as shown in
n
n
Q
e
v
gure 7.5, and collect nom and together to form a new controller, .
K
Q
K
w
z
P
y
u
nom
K
v
e
ev = 0 closed loop
H
The nominal controller nom is augmented to produce a signal
Figure
7.4
K
and accept a signal . The closed-loop transfer function from to is 0.
e
v
v
e
The intuition is that should also stabilize , since the system we added
K
P
Q
to nom is stable and \sees no feedback", and thus cannot destabilize our system.
K
However, can change the closed-loop transfer matrix . To see how a ects the
Q
H
Q
closed-loop transfer matrix , we de ne the following transfer matrices in gure 7.4:
H
1 is the closed-loop transfer matrix from to ,
U
w
z
2 is the closed-loop transfer matrix from to ,
U
v
z
3 is the closed-loop transfer matrix from to .
U
w
e
7.3 MODIFIED CONTROLLER PARADIGM
159
w
z
P
y
u
nom
K
Q
v
e
K
Modication of nominal controller nom with a stable transfer
Figure
7.5
K
matrix .
Q
Since the transfer matrix from to in gure 7.4 is zero, we can redraw gure 7.5
v
e
as gure 7.6. Figure 7.6 can then be redrawn as gure 7.7, which makes it clear
that the closed-loop transfer matrix resulting from our modi ed controller is
H
K
simply
= 1 + 2 3
(7.27)
H
U
U
QU
which must be stable because , 1, 2 and 3 are all stable.
U
U
U
Q
It can be seen from (7.27) that as varies over all stable transfer matrices,
Q
H
sweeps out the following a ne set of closed-loop transfer matrices:
mcp =
1 + 2
3
stable
H
fU
U
QU
j
Q
g
:
Of course, mcp
stable. This means that a (possibly incomplete) family of
H
H
stabilizing controllers can be generated from the (augmented) nominal controller
using this modied controller paradigm.
If the augmentation of the nominal controller is done properly, then the modi ed
controller paradigm yields every controller that stabilizes the plant , in other
P
words, mcp = stable. In this case, 1, 2, and 3 can be used as 1, 2, and 3
H
H
U
U
U
T
T
T
in the free parameter representation of stable given in equation (7.25).
H
160