, where is a set of transfer matrices of the appropriate
2
size.
P
w
z
q
p
y
u
pert( )
P
Each perturbed plant is equivalent to the nominal plant mod-
Figure
10.6
ied by the internal feedback .
We will call the feedback perturbation. The perturbed plant that results from
the feedback perturbation will be denoted pert( ), and will be called the
P
feedback perturbation set that corresponds to :
P
= pert( )
(10.16)
P
P
2
:
The symbol emphasizes its role in \changing" the plant into the perturbed
P
plant pert.
P
The input signal to the perturbation feedback, denoted , can be considered an
q
output signal of the plant . Similarly, the output signal from the perturbation
P
feedback, denoted , can be considered an input signal to the plant . Throughout
p
P
this chapter we will assume that the exogenous input signal and the regulated
w
output signal are augmented to contain and , respectively:
z
p
q
= ~w
= ~z
w
z
p
q
where ~ and ~ denote the original signals from gure 10.6. This is shown in g-
w
z
ure 10.7.
To call an exogenous input signal can be misleading, since this signal does
p
not originate \outside" the plant, like command inputs or disturbance signals, as
222
CHAPTER 10 ROBUSTNESS SPECIFICATIONS VIA GAIN BOUNDS
P
(
~
~ )
w
z
w
z
p
q
y
u
The plant showing as part of the exogenous input signal
Figure
10.7
P
p
and as part of the regulated output signal . By connecting the feedback
w
q
z
perturbation between and , we recover the perturbed plant pert().
q
p
P
the term exogenous implies. We can think of the signal as originating outside the
p
nominal plant, as in gure 10.6.
To describe a perturbation feedback form of a perturbed plant set , we give
P
the (augmented) plant transfer matrix
2
~~
~
~ 3
P
P
P
=
z
z
z
w
p
u
~
4
5
P
P
P
P
q
w
q
p
q
u
~
P
P
P
y
w
y
p
y
u
along with the set of perturbation feedbacks. Our original perturbed plant can
be expressed as
pert( ) =
~~
~
~
1
P
P
z
z
z
w
u
+ P p
(
);
~
(10.17)
P
P
P
~
I
;
P
:
q
w
q
u
q
p
P
P
P
y
w
y
u
y
p
The perturbation feedback form, i.e., the transfer matrix in (10.17) and the
P
set , is not uniquely determined by the perturbed plant set . This fact will be
P
important later.
When contains nonlinear or time-varying systems, the perturbation feedback
P
form consists of an LTI and a set of nonlinear or time-varying systems. Roughly
P
speaking, the feedback perturbation represents the extracted nonlinear or time-
varying part of the system. We will see an example of this later.
10.3.1
Perturbation Feedback Form: Closed-Loop
Suppose now that the controller is connected to the perturbed plant pert( ),
K
P
as shown in gures 10.8 and 10.9.
10.3 PERTURBATION FEEDBACK FORM
223
P
~
~
w
z
p
q
u
y
K
When the perturbed plant set is expressed in the perturbation
Figure
10.8
feedback form shown in gure 10.6, the perturbed closed-loop system can be
represented as the nominal plant , with the controller connected between
P
K
and as usual, and the perturbation feedback connected between and
y
u
q
.p
8
9
>
>
<
~
~
w
z
=
w
z
H
>
>
:
p
q
The perturbed closed-loop system can be represented as the
Figure
10.9
nominal closed-loop system with feedback connected from (a part of )
q
z
to (a part of ). Note the similarity to gure 2.2.
p
w
224
CHAPTER 10 ROBUSTNESS SPECIFICATIONS VIA GAIN BOUNDS
By substituting (10.17) into (2.7) we nd that the transfer matrix of the per-
turbed closed-loop system is
pert( ) = ~~ + ~ (
) 1
(10.18)
;
~
H
H
H
I
;
H
H
z
z
q
p
q
w
w
p
where
~~ = ~~ + ~ (
) 1
;
~
(10.19)
H
P
P
K
I
;
P
K
P
z
w
z
w
z
u
y
u
y
w
~ = ~ + ~ (
) 1
;
(10.20)
H
P
P
K
I
;
P
K
P
z
p
z
p
z
u
y
u
y
p
~ =
~ +
(
) 1
(10.21)
;
~
H
P
P
K
I
;
P
K
P
q
w
q
w
q
u
y
u
y
w
=
+
(
) 1
;
(10.22)
H
P
P
K
I
;
P
K
P
:
q
p
q
p
q
u
y
u
y
p
Note the similarities between gures 10.9 and 2.2, and the corresponding equa-
tions (10.18) and (2.7). Figure 2.2 and equation (2.7) show the e ect of connect-
ing the controller to the nominal plant to form the nominal closed-loop system
gure 10.9 and equation (10.18) show the e ect of connecting the feedback per-
turbation to the nominal closed-loop system to form the perturbed closed-loop
system.
We may interpret
pert( )
~~ = ~ (
) 1
(10.23)
;
~
H
;
H
H
I
;
H
H
z
z
q
p
q
w
w
p
as the change in the closed-loop transfer matrix that is caused by the feedback
perturbation . We have the following interpretations:
~~ is the closed-loop transfer matrix of the nominal system, before its ex-
H
z
w
ogenous input and regulated output were augmented with the signals and
p
.q
~ is the closed-loop transfer matrix from the original exogenous input signal
H
q
w
~ to . If ~ is \large", then so will be the signal that drives or excites
w
q
H
q
q
w
the feedback perturbation .
~ is the closed-loop transfer matrix from to the original regulated output
H
p
z
p
signal ~. If ~ is \large", then so will be the e ect on ~ of the signal , which
z
H
z
p
z
p
is generated by the feedback perturbation .
is the closed-loop transfer matrix from to . We can interpret
as
H
p
q
H
q
p
q
p
the feedback seen by , looking into the nominal closed-loop system.
Thus, if the three closed-loop transfer matrices ~ , ~, and
are all \small",
H
H
H
z
p
q
w
q
p
then our design will be \robust" to the perturbations, i.e., the change in the closed-
loop transfer matrix, which is given in (10.23), will also be \small". This vague idea
will be made more precise later in this chapter.
10.3 PERTURBATION FEEDBACK FORM
225
10.3.2
Examples of Perturbation Feedback Form
In this section will denote a transfer function that we have already given elsewhere.
?
In this way we emphasize the transfer functions that are directly relevant to the
perturbation feedback form.
Neglected Dynamics
Figure 10.10 shows one way to represent the perturbed plant set = cmplx
P
fP
g
described in section 10.2.1 in perturbation feedback form. In this block diagram,
the perturbation feedback acts as a switch:
= 0 yields the nominal plant
= turns on the perturbation, to yield the perturbed plant cmplx.
I
P
This perturbation feedback form is described by the augmented plant
2
(1)
3
err
0
?
?
?
P
?
2
~~
~
~ 3
0
0
6
7
?
?
?
?
P
P
P
z
w
z
p
z
u
6
7
~
=
std
std
6
7
(10.24)
4
5
0
0 0
0
0
0
P
P
P
P
P
6
7
q
w
q
p
q
u
~
std
(1)
std
6
0
1 0
err
0
0 7
P
P
P
4
5
P
P
P
y
w
y
p
y
u
(1)
err
(2)
err
?
?
?
;P
;P
?
where
(1)
1 25( 100) ( 100)2
80
err ( ) =
(2)
;
:
s=
;
s=
err ( ) = ;s=
P
s
1 + 1 25( 100) + ( 100)2
P
s
1 + 80
:
s=
s=
s=
and the feedback perturbation set
=
(10.25)
fI
g:
When the controller is connected, we have
std
~ =
0
P
S
;T
T
std
(10.26)
H
q
w
0
P
S
S
T
"
(1)
(2)
#
~ =
err
err
P
S
;P
T
(1)
(2)
(10.27)
H
std
std
z
p
err
0
err
0
;P
T
=P
;P
T
=P
"
(1)
(2) #
=
err
err
;P
T
;T
P
(1)
(2)
(10.28)
H
:
q
p
err
err
P
S
;T
P
Gain Margin: Perturbation Feedback Form 1
The perturbed plant set for the classical gain margin speci cation, given by P
in (10.4), can be expressed in the perturbation feedback form shown in gure 10.11.
226
CHAPTER 10 ROBUSTNESS SPECIFICATIONS VIA GAIN BOUNDS
proc
p
sensor
n
y
n
r
+
+
+
+
+
r
q
r
q
r
q
r
y
u
0
P
+
+
+
+
r
;
2
9
8
;1:25(s=100)
;
(s=100)
=
<