signal , and we must include and in the regulated variables vector , in which
w
u
y
z
case the four transfer matrices (7.11{7.14) appear as submatrices of the closed-loop
transfer matrix . Internal stability is then expressed as the speci cation that
H
these entries of be stable, which we mentioned in chapter 6 (section 6.4.1) is an
H
a ne speci cation.
It seems clear that any sensible set of design speci cations should limit the e ect
of sensor and process noise on and . Indeed, a sensible set of speci cations will
u
y
constrain these four transfer matrices more tightly than merely requiring stability
some norm of these transfer matrices will be constrained. So sensible sets of spec-
i cations will generally be strictly tighter than internal stability internal stability
will be a redundant speci cation. We will see examples of this in chapter 12: for
the LQG and
problems, niteness of the objective will imply internal stability,
H
1
provided the objectives satisfy certain sensibility requirements. See the Notes and
References at the end of this chapter.
7.2.3
Closed-loop Affineness of Internal Stability
We now consider the speci cation that is the closed-loop transfer matrix achieved
H
by a controller that stabilizes the plant:
=
(
) 1
;
stable =
zw + zu
yu
yw
H
P
P
K
I
;
P
K
P
(7.15)
H
H
for some that stabilizes
:
K
P
Of course, this is a stronger speci cation than realizability.
Like rlzbl, stable is also a ne: if and ~ each stabilize , then for each
H
H
K
K
P
, the controller
given by (7.10) also stabilizes . In the example of
2
R
K
P
section 7.1.1, the controllers (a) and (b) each stabilize the plant. Hence the ve
K
K
controllers that realize the ve step responses shown in gure 7.2 also stabilize the
plant.
We can establish that stable is a ne by direct calculation. Suppose that con-
H
trollers and ~ each stabilize the plant, yielding closed-loop transfer matrices
K
K
H
and ~, respectively. We substitute the controller
given in (7.10) into the four
H
K
transfer matrices (7.11{7.14), and after some algebra we nd that
(
yu ) 1
(
) 1
) ~(
) 1
;
;
;
yu =
yu
yu + (1
yu ~
yu
K
I
;
P
K
P
K
I
;
P
K
P
;
K
I
;
P
K
P
(
yu ) 1 =
(
) 1
1
;
+ (1 ) ~(
);
;
yu
yu ~
K
I
;
P
K
K
I
;
P
K
;
K
I
;
P
K
(
yu ) 1
) 1
1
;
)(
);
;
yu = (
yu
yu + (1
yu ~
yu
I
;
P
K
P
I
;
P
K
P
;
I
;
P
K
P
(
yu ) 1 = (
) 1 + (1 )(
) 1
;
;
;
yu
yu ~
I
;
P
K
I
;
P
K
;
I
;
P
K
:
Thus, the four transfer matrices (7.11{7.14) achieved by
are a ne combinations
K
of those achieved by and ~. Since the right-hand sides of these equations are all
K
K
stable, the left-hand sides are stable, and therefore
stabilizes .
K
P
We can use the same device that we used to simplify our description of rlzbl.
H
The four transfer matrices (7.11{7.14) can be expressed in terms of the transfer
7.2 INTERNAL STABILITY
155
matrix given in (7.7):
R
(
yu ) 1
;
yu =
yu
(7.16)
K
I
;
P
K
P
R P
(
yu ) 1
;
=
(7.17)
K
I
;
P
K
R
(
yu ) 1
+
)
;
yu = (
yu
yu
(7.18)
I
;
P
K
P
I
P
R
P
(
yu ) 1 = +
(7.19)
;
yu
I
;
P
K
I
P
R :
Hence we have
8
yu
9
R P
>
>
>
>
<
=
stable =
zw + zu yw
R
are stable
(7.20)
H
P
P
R P
+ yu
:
I
P
R
>
>
>
>
:
( + yu ) yu
I
P
R
P
We will nd this description useful.
7.2.4
Internal Stability for a Stable Plant
If the plant is stable, then in particular yu is stable. It follows that if is stable,
P
P
R
then so are yu, + yu , and ( + yu ) yu. Hence we have
R P
I
P
R
I
P
R
P
stable = zw + zu yw
stable
(7.21)
H
fP
P
R P
j
R
g
:
This is just our description of rlzbl, with the additional constraint that be stable.
H
R
Given any stable , the controller that stabilizes and yields a closed-loop
R
P
transfer matrix = zw + zu yw is
H
P
P
R P
= ( + yu) 1
(7.22)
;
K
I
R P
R :
Conversely, every controller that stabilizes can be expressed by (7.22) for some
P
stable .
R
7.2.5
Internal Stability via Interpolation Conditions
For the classical SASS 1-DOF control system (see section 2.3.2), the speci cation of
internal stability can be expressed in terms of the I/O transfer function , although
T
the speci cation is not as simple as stability of alone (recall our motivating
T
example). We have already noted that a closed-loop transfer matrix
that is
H
realizable has the form
=
0(1
)
P
;
T
;T
T
H
0
0
;T
;T
=P
T
=P
=
0 0 0
0
1
1
P
0 0 0 +
;P
;
(7.23)
T
1
1 0 1 0
;
;
=P
=P
where is the I/O transfer function. We will describe stable as the set of transfer
T
H
matrices of the form (7.23), where satis es some additional conditions.
T
156