= (
) 1
~(
~) 1
;
;
B
K
I
;
P
K
P
;
K
I
;
P
K
P
y
u
y
u
y
u
y
u
= ~(
~) 1;
C
K
I
;
P
K
y
u
= (
) 1 ~
~ 1
;
(
);
D
K
I
;
P
K
;
K
I
;
P
K
:
y
u
y
u
The special form of
given in (7.10) is called a bilinear or linear fractional de-
K
pendence on . We will encounter this form again in section 7.2.6.
7.1.1
An Example
We consider the standard plant example of section 2.4. The step responses from the
reference input to p and for each of the controllers (a) and (b) of section 2.4
r
y
u
K
K
are shown in gures 7.2(a) and (b). Since rlzbl is a ne, we conclude that every
H
transfer matrix on the line passing through (a) and (b),
H
H
(a) + (1 ) (b)
H
;
H
2
R
can be realized as the closed-loop transfer matrix of our standard plant with some
controller. Figures 7.2(c) and (d) show the step responses from to p and of ve
r
y
u
of the transfer matrices on this line.
The average of the two closed-loop transfer matrices ( = 0 5) is realized by
:
the controller 0 5, which can be found from (7.10). Even though both (a) and
K
K
:
(b) are 3rd order, 0 5 turns out to be 9th order. From this fact we draw two
K
K
:
conclusions. First, the speci cation that be the transfer matrix achieved by a
H
controller of order no greater then ,
K
n
=
+
(
) 1
;
rlzbl =
H
P
P
K
I
;
P
K
P
z
w
z
u
y
u
y
w
H
H
n
for some with order( )
K
K
n
is not in general convex (whereas the speci cation rlzbl, which puts no limit on the
H
order of , is convex). Our second observation is that the controller 0 5, which
K
K
:
yields a closed-loop transfer matrix that is the average of the closed-loop transfer
matrices achieved by the controllers (a) and (b), would not have been found by
K
K
varying the parameters (e.g., numerator and denominator coe cients) in (a) and
K
(b).
K
7.2
Internal Stability
We remind the reader that a transfer function is proper if it has at least as many
poles as nite zeros, or equivalently, if it has a state-space realization a transfer
function is stable if it is proper and all its poles have negative real part nally, a
transfer matrix is stable if all of its entries are stable. We noted in section 6.4.1
that these are ane constraints on a transfer function or transfer matrix.
7.2 INTERNAL STABILITY
151
1:8
20
( )
1:6
a
13
s
(t)
1:4
( )b
10
;
;
1:2
23
s
(t)
1
;
;
0:8
0
@
I
@
( )
@
I
@
b
0:6
13
( )
a
s
(t)
10
;
23
0:4
s
(t)
0:20
20
;
0:2
;
0
1
2
3
4
5
0
1
2
3
4
5
t
t
(a)
(b)
1:8
20
=
;0:3
1:6
=
1:3
=
0
1:4
=
1
10
1:2
() 1
() 0
t
t
13 0:8
0:6
23
s
s
H
Y
10
@
I
H
;
=
0:5
0:4
@
=
0:5
0:2
@
I
@
@
I
=
0
H
Y
H
@
=
1
0
20
=
;0:3
;
H
Y
H
=
1:3
0:2
;
0
1
2
3
4
5
0
1
2
3
4
5
t
t
(c)
(d)
(a) shows the closed-loop step responses from to p for the
Figure
7.2
r
y
standard example with the two controllers (a) and (b). (b) shows the step
K
K
responses from to . In (c) and (d) the step responses corresponding to
r
u
ve dierent values of are shown. Each of these step responses is achieved
by some controller.
7.2.1
A Motivating Example
Consider our standard example SASS 1-DOF control system described in section 2.4,
with the controller
( ) = 36 + 33s
K
s
10
:
;
s
This controller yields the closed-loop I/O transfer function
( ) =
33 + 36
+ 36
s
T
s
3 + 10 2 + 33 + 36 = 33s
( + 3)2( + 4)
s
s
s
s
s
which is a stable lowpass lter. Thus, we will have p
provided the reference
y
r
signal does not change too rapidly the controller yields good tracking of slowly
r
K
152