1 5
:
:
1
1
0 5
0 5
:
:
()
)
t
0
( 0
t
u
w
0 5
0 5
;
:
;
:
1
1
;
;
1 5
1 5
;
:
0
2
4
6
8
10 ; : 0
2
4
6
8
10
t
t
(a)
(b)
1 5
1 5
:
:
1
1
0 5
0 5
) :
:
)
~(t 0
~( 0
t
u
w
0 5
0 5
;
:
;
:
1
1
;
;
1 5
1 5
;
:
0
2
4
6
8
10 ; : 0
2
4
6
8
10
t
t
(c)
(d)
The specication (8.19) ensures that the actuator signal mag-
Figure
8.14
nitude will not exceed one during set-point changes in the range 1 to 1.
;
The input in (a) shows a set-point change that drives the actuator sig-
w
nal, shown in (b), close to its limit. Because of linearity, smaller set-point
changes will result in smaller actuator eort: the set-point change ~ in (c)
w
produces the actuator signal ~ in (d), which only uses 48% of the available
u
actuator eort.
8.4 COMBINED EFFECT OF DISTURBANCES AND COMMANDS
193
variable c will lie in the given envelope when a step input is applied to c and the
z
w
disturbance input is zero. The second speci cation requires that the RMS value of
the actuator be less than one when white noise is applied to d and the command
u
w
input is zero. If a unit step command input is applied to c and a white noise is
w
applied to d simultaneously, the response at c is simply the sum of the responses
w
z
with the inputs acting separately. It is quite likely that this c would not lie in the
z
speci ed envelope, because of the e ect of the disturbance d similarly the RMS
w
value of would probably exceed one because of the constant component of the
u
actuator signal that is due to the command.
This phenomenon is a basic consequence of linearity. These separate speci ca-
tions often su ce in practice since each regulated variable may be mostly dependent
on either the command or disturbances. For example, in a given system the distur-
bance may be small, so the component of the actuator signal due to the disturbance
(i.e., ad d) may be much smaller than the component due to the command (i.e.,
H
w
ac c) therefore an actuator e ort speci cation that limits the size of ac will
H
w
H
probably acceptably limit the size of , even though it \ignores" the e ect of the
u
disturbance.
We can also describe this phenomenon from a more general viewpoint. Each
speci cation we have considered so far is a speci cation on some submatrix of H
that does not contain all of its columns, and therefore considers the e ects of only a
subset of the exogenous inputs. In contrast, a speci cation on a submatrix of that
H
contains all of its columns will consider the e ects of all of the exogenous inputs,
acting simultaneously. For example, the RMS actuator e ort speci cation rms act
H
involves only the submatrix ad, and does not consider the e ect of the command
H
on the RMS value of the actuator signal. On the other hand, the speci cation
q
rms act cmb =
ad 22 + ac(0)2 1
H
H
kH
k
H
which is a speci cation on the bigger submatrix ac ad] of , correctly guarantees
H
H
H
that the RMS value of will not exceed one when the command is a unit step and
u
the disturbance is a white noise (and, we should add, etc = 0).
w
This discussion suggests that a general actuator e ort speci cation should really
limit the size of the transfer matrix ac ad]. Limiting the sizes of ac and ad
H
H
H
H
separately will, of course, limit the size of ac ad] this corresponds to a prior
H
H
allocation of actuator eort between regulation and command-following tasks.
In cases where di erent types of models for the commands and disturbances are
used (or indeed, di erent types of models for di erent components of either), it
can be di cult or cumbersome to formulate a sensible speci cation on the bigger
submatrix of . Returning to our example, let us form a speci cation on the
H
response of c that considers both a unit step at c (a particular signal) and a
z
w
white noise at d (a stochastic process). A possible form for such a speci cation
w
194