Stability
In this chapter we consider the design specications of realizability and internal
(closed-loop) stability. The central result is that the set of closed-loop transfer
matrices realizable with controllers that stabilize the plant is ane and readily
described. This description is referred to as the parametrization of closed-loop
transfer matrices achieved by stabilizing controllers.
7.1
Realizability
An important constraint on the transfer matrix
is that it should be the closed-
H
2
H
loop transfer matrix achieved by some controller , in other words, should have
K
H
the form
+
(
) 1
for some . We will refer to this constraint
;
P
P
K
I
;
P
K
P
K
z
w
z
u
y
u
y
w
as realizability:
rlzbl =
=
+
(
) 1
for some
(7.1)
;
H
H
H
P
P
K
I
;
P
K
P
K
:
z
w
z
u
y
u
y
w
We can think of rlzbl as expressing the dependencies among the various closed-loop
H
transfer functions that are entries of .
H
As an example, consider the classical SASS 1-DOF control system described in
section 2.3.2. The closed-loop transfer matrix is given by
2
0
0
0
3
P
P
K
P
K
;
= 6 1 + 0
1 + 0
1 + 0 7
P
K
P
K
P
K
6
7
H
6
0
7
P
K
K
K
4
5
;
1 + 0
;
1 + 0
1 + 0
P
K
P
K
P
K
=
0(1
)
P
;
T
;T
T
0
0
;T
;T
=P
T
=P
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