Instead of stability, we consider the requirement that each pole of a transfer
function should satisfy
0 1 and
. We will call such transfer
<p
;
:
j=pj
;<p
functions -stable ( stands for generalized). In classical terminology, -stability
G
G
G
guarantees a stability degree and a minimum damping ratio for a transfer function,
as illustrated in gure 7.11.
( )
=
p
( )
<
p
;
;
= 0 1
p
;
:
A transfer function is -stable if its poles lie in the region
Figure
7.11
G
to the left. In classical control terminology, such transfer functions ha
p
ve a
stability degree of at least 0 1 and a damping ratio of at least 1 2. All
:
=
of the results in this chapter can be adapted to this generalized notion of
stability.
We say that a controller
-stabilizes the plant if every entry of the four
K
G
P
transfer matrices (7.11{7.14) is -stable (c.f. de nition 7.1). It is not hard to show
G
that -stable, the speci cation that the closed-loop transfer matrix is achievable
H
G
by a -stabilizing controller, is a ne in fact, we have
G
8
yu
9
R P
>
>
>
>
<
=
-stable =
zw + zu yw
R
are -stable
H
P
P
R P
G
G
+ yu
I
P
R
>
>
>
>
:
( + yu ) yu
I
P
R
P
which is just (7.20), with \ -stable" substituted for \stable".
G
For the SASS 1-DOF control system, the speci cation -stable can be expressed
H
G
in terms of the interpolation conditions described in section 7.2.5, with the following
modi cations: condition (1) becomes \ is -stable", and the list of poles and zeros
T
G
of 0 must be expanded to include any poles and zeros that are -unstable, i.e., lie
P
G
in the right-hand region in gure 7.11.
7.5 SOME GENERALIZATIONS OF CLOSED-LOOP STABILITY
167
There is a free parameter representation of -stable:
H
G
-stable = 1 + 2 3
is a -stable u y transfer matrix
H
fT
T
QT
j
Q
G
n
n
g
G
which is (7.25) with \ -stable" substituted for \stable". This free parameter rep-
G
resentation can be developed from state-space equations exactly as in section 7.4,
provided the state-feedback and estimator gains are chosen such that P
u sfb
A
;
B
K
and P
est y are -stable, i.e., their eigenvalues lie in the left-hand region of
A
;
L
C
G
gure 7.11.
168