where is the classical closed-loop I/O transfer function. Hence if is the closed-
T
H
loop transfer matrix realized by some controller , i.e.,
rlzbl, then we must
K
H
2
H
have
12 = 21
(7.2)
H
H
0 22 = 21
(7.3)
P
H
H
11
0 21 = 0
(7.4)
H
;
P
H
P
13 =
12
(7.5)
H
;H
23 =
22
(7.6)
H
;H
:
It is not hard to show that the ve explicit speci cations on given in (7.2{7.6)
H
are not just implied by realizability they are equivalent to it. Roughly speaking,
we have only one transfer function that we can design, , whereas the closed-loop
K
transfer matrix contains six transfer functions. The ve constraints (7.2{7.6)
H
among these six transfer functions make up the missing degrees of freedom.
The speci cations (7.2{7.6) are ane, so at least for the classical SASS 1-DOF
controller, realizability is an a ne speci cation. In fact, we will see that in the
general case, rlzbl is a ne. Thus, realizability is a closed-loop convex speci cation.
H
To establish that rlzbl is a ne in the general case, we use a simple trick that
H
replaces the inverse appearing in (7.1) with a simpler expression. Given any n n
u
y
transfer matrix , we de ne the
transfer matrix by
K
n
n
R
u
y
= (
) 1
;
(7.7)
R
K
I
;
P
K
:
y
u
This correspondence is one-to-one: given any
transfer matrix , the
n
n
R
n
n
u
y
u
y
transfer matrix given by
K
= ( +
) 1
(7.8)
;
K
I
R P
R
y
u
makes sense and satis es (7.7).
Hence we can express the realizability speci cation as
rlzbl =
=
+
for some
(7.9)
H
fH
j
H
P
P
R P
n
n
R g
:
z
w
z
u
y
w
u
y
This form of rlzbl can be given a simple interpretation, which is shown in gure 7.1.
H
The transfer matrix can be thought of as the \controller" that would realize the
R
closed-loop transfer matrix if there were no feedback through our plant, i.e.,
=
H
P
y
u
0 (see gure 7.1(b)). Our trick above is the observation that we can reconstruct
the controller that has the same e ect on the true plant as the controller that
K
R
operates on the plant with
set to zero. Variations on this simple trick will
P
y
u
appear again later in this chapter.
From (7.9) we can establish that rlzbl is a ne. Suppose that
~
rlzbl.
H
H
H
2
H
Then there are two
transfer matrices and ~ such that
n
n
R
R
u
y
=
+
H
P
P
R P
z
w
z
u
y
w
~ =
+
~
H
P
P
R
P
:
z
w
z
u
y
w
7.1 REALIZABILITY
149
+
q
r
w
z
P
z
w
+
+
r
q
P
P
K
y
w
z
u
+
P
y
u
R
(a)
+
q
r
w
z
P
z
w
+
P
P
R
y
w
z
u
(b)
The closed-loop transfer matrix can be realized by the feed-
Figure
7.1
H
back system (a) for some if and only if it can be realized by the system
K
(b) for some transfer matrix . In (b) there is no feedback.
R
Let
. We must show that the transfer matrix
=
+ (1
) ~ is also
;
H
2
R
H
H
realizable as the closed-loop transfer matrix of our plant with some controller. We
note that = +
H
P
P
R
P
z
w
z
u
y
w
where
=
+ (1 ) ~
R
R
;
R
:
This shows that
rlzbl.
H
2
H
We can nd the controller
that realizes the closed-loop transfer matrix
K
H
using the formula (7.8) with . If and ~ are controllers that yield the closed-
R
K
K
loop transfer matrices and ~, respectively, the controller that realizes the closed-
H
H
loop transfer matrix
is not
+ (1 ) ~ it is
H
K
;
K
= ( + ) 1( + )
(7.10)
;
K
A
B
C
D
where = + ~(
~) 1;
A
I
K
I
;
P
K
P
y
u
y
u
150