u
6
7
~
2( + 10) 0 0
;
+ 10 2( + 10)
6
7
s
s
s
s
s
P
P
P
y
w
y
p
y
u
6
7
4
5
;s
?
?
?
+ 10
?
s
with feedback perturbation set
= 5 5]
(10.40)
;
:
10.3 PERTURBATION FEEDBACK FORM
229
The closed-loop transfer matrices are given by
2
2
~ =
2
;
(10.41)
H
2
q
w
( + 10)S
10T
10T
s
s
s
;
s
;
2
3
s
+ 10S
~ =
s
6
7
(10.42)
6
3
7
H
z
p
4
5
s
10T
s
;
= 2
1
s
(10.43)
H
2
q
p
100T ; + 10:
s
;
s
(The reader worried that these transfer matrices may be unstable should recall
that the interpolation conditions of section 7.2.5 require that (10) = 0 similar
T
conditions guarantee that these transfer matrices are proper, and have no pole at
= 0.)
s
Relative Uncertainty in P0
The plant perturbation set (10.10) can be expressed in the perturbation feedback
form shown in gure 10.14, for which
2
rel err 0
3
2
~~
~
~ 3
?
?
?
W
P
?
P
P
P
z
z
z
w
p
u
0
~
= 6
7
?
?
?
?
(10.44)
6
7
4
5
P
P
P
q
w
q
p
q
u
0 0 0
0
1
~
4
5
P
P
P
y
w
y
p
y
u
rel err 0
?
?
?
;W
P
?
and
=
1
(10.45)
f
j
k
k
g
:
1
In this case the feedback perturbations are normalized unknown-but-bounded trans-
fer functions, that cause the transfer function 0 to become (1 + rel err ) 0.
P
W
P
For this perturbation feedback form, the transfer matrices ~, ~ , and
H
H
H
q
w
z
q
p
p
are given by
~ =
0
0
(10.46)
;T
;T
=P
T
=P
H
q
w
~ =
rel err 0
W
P
S
(10.47)
H
z
p
rel err
;W
T
=
rel err
(10.48)
H
;W
T
:
q
p
Saturating Actuators
We consider the perturbed plant set (10.15) which consists of the single nonlinear
system nonlin . This can be expressed in perturbation feedback form by express-
fP
g
230
CHAPTER 10 ROBUSTNESS SPECIFICATIONS VIA GAIN BOUNDS
proc
p
sensor
n
y
n
r
+
+
+
+
q
r
q
r
r
y
u
0
P
+
r
+
+
;
rel err
W
q
p
A perturbation feedback form for the plant perturbation
Figure
10.14
set (10.10).
ing each saturator as a straight signal path perturbed by a dead-zone nonlinearity,
( ) =
( )
Dz
a
a
;
Sat
a
as shown in gure 10.15.
~
~
w
z
+
(
lin
P
.
q
r
;
.
+
. q
r
.
y
u
;
8
9
>
>
<
.
=
q
.
p
>
.
>
:
nonlin
The nonlinear system shown in gure 10.5 is redrawn as
Figure
10.15
the nominal plant, which is LTI, connected to nonlin, which is a dead-zone
nonlinearity.
This perturbation feedback form is described by the augmented plant
2
lin
lin
lin 3
2
~~
~
~ 3
P
;P
P
P
P
P
z
w
z
p
z
u
z
u
z
u
z
w
~
= 6 0
0
7
(10.49)
4
5
P
P
P
I
4
5
q
w
q
p
q
u
~
lin
lin
lin
P
P
P
y
w
y
p
y
u
P
;P
P
y
w
y
u
y
u
10.4 SMALL GAIN METHOD FOR ROBUST STABILITY
231
and = nonlin , where = nonlin( ) is de ned by
f
g
p
q
( ) =
( ( ) ) = 1 ...
(10.50)
p
t
S
Dz
q
t
=S
i
n
:
i
i
i
i
u
In this case the feedback perturbation nonlin is a memoryless nonlinearity.
For this perturbation feedback form, the transfer matrices ~, ~ , and
H
H
H
q
w
z
q
p
p
are given by
~ = (
lin ) 1 lin
;
(10.51)
H
K
I
;
P
K
P
q
w
y
u
y
w
~ =
lin(
lin) 1
(10.52)
;
H
;P
I
;
K
P
z
p
z
u
y
u
=
lin(
lin) 1
;
(10.53)
H
;K
P
I
;
K
P
:
q
p
y
u
y
u
10.4
Small Gain Method for Robust Stability
10.4.1
A Convex Inner Approximation
We consider a perturbed plant set that is given by a perturbation feedback form.
P
Suppose that the norm gn is a gain (see chapter 5). Let denote the maximum
k
k
M
gain of the possible feedback perturbations, i.e.,
= sup
gn
(10.54)
M
k
k
:
2
is thus a measure of how \big" the feedback perturbations can be.
M
Then from the small gain theorem described in section 5.4.2 (equations (5.29{
5.31) with 1 = and 2 =
) we know that if
H
H
H
q
p
gn
1
(10.55)
kH
k
M
<
q
p
then we have for all
,
2
(
) 1
M
;
I
;
H
gn
q
p
1
gn :
;
M
kH
k
q
p
From (10.23) we therefore have
pert( )
~~
~ gn
~ gn
M
kH
k
kH
k
z
p
q
w
for all
(10.56)
H
;
H
gn
2
:
z
w
1
gn
;
M
kH
k
q
p
We will refer to the closed-loop convex speci cation (10.55) as the small gain condi-
tion (for the perturbation feedback form and gain used). (10.55) and (10.56) are a
precise statement of the idea expressed in section 10.3.1: the closed-loop system will
be robust if the three closed-loop transfer matrices ~ , ~, and
are \small
H
H
H
z
p
q
w
q
p
enough".
232
CHAPTER 10 ROBUSTNESS SPECIFICATIONS VIA GAIN BOUNDS
It follows that the closed-loop convex speci cation on given by
H
gn 1
(10.57)
kH
k
<
=
M
q
p
~ gn
(10.58)
kH
k
<
1
z
p
~ gn
(10.59)
kH
k
<
1
q
w
~~ gn
(10.60)
kH
k
<
1
z
w
implies that
pert gn
for all
(10.61)
kH
k
<
1
2
i.e., the robustness speci cation formed from the perturbed plant set and the
P
speci cation
gn
holds. If the gain
gn is nite only for stable transfer
kH
k
<
1
k
k
matrices, then the speci cation (10.57{10.60) implies that pert is stable, and thus
H
the speci cation (10.57{10.60) is stronger than the speci cation of robust stability.
In this case, we may think of the speci cation (10.57{10.60) as a closed-loop convex
speci cation that guarantees robust stability.
As a more speci c example, the RMS gain (
norm)
is nite only for
H
k
k
1
1
stable transfer matrices, so the speci cation
1 , along with stability
kH
k
<
=
M
q
p
1
of ~ , ~, and ~~ (which is usually implied by internal stability), guarantees
H
H
H
z
p
q
w
z
w
that the robust stability speci cation rob stab holds for .
D
H
The speci cation (10.57{10.60) can be used to form a convex inner approxi-
mation of a robust generalized stability speci cation, for various generalizations of
internal stability (see section 7.5), by using other gains. As an example, consider
the -shifted
norm, which is nite if and only if the poles of its argument have
a
H
1
real parts less than . If the speci cation (10.57{10.60) holds for this norm, then
;a
we may conclude that the feedback perturbations cannot cause the poles of the
closed-loop system to have real parts equal to or exceeding . (We comment that
;a
changing the gain used will generally change , and will give a di erent speci ca-
M
tion.)Since the small gain condition (10.55) depends only on , the largest gain of
M
the feedback perturbations, it follows that the conclusion (10.56) actually holds for
a set of feedback perturbations that may be larger than :
sgt =
gn
f
j
k
k
M
g
:
By using di erent perturbation feedback forms of a perturbed plant set, and
di erent gains that are nite only for stable transfer matrices, the small gain condi-
tion (10.55) can be used to form di erent convex inner approximations of the robust
stability speci cation.
10.4.2
An Extrapolated First Order Bound
It is interesting to compare (10.56) to a corresponding bound that is based on a
rst order di erential analysis. Since
(
) 1
;
I
;
H
'
q
p
10.4 SMALL GAIN METHOD FOR ROBUST STABILITY
233
(recall that means equals, to rst order in ), the rst order variation in the
'