where ( ) is a non-decreasing frequency-dependent magnitude bound.
M
!
8.2.5
Worst Case Peak Regulation
If we model the disturbance as unknown-but-bounded, say, d
d, and
kw
k
M
1
require that the worst case peak deviation of the commanded variable due to d is
w
less than max pk reg, i.e.,
M
cd d
max pk reg whenever
d
d
kH
w
k
M
kw
k
M
1
1
then we can specify the peak-gain bound
pk dis =
cd pk gn
max pk reg
d
H
fH
j
kH
k
M
=
M
g
:
8.3
Actuator Effort
In any control system the size of the actuator signals must be limited, i.e.,
act
act
kuk
M
for some appropriate norm
act and limit act. Reasons include:
k
k
M
Actuator heating. Large actuator signals may cause excessive heating, which
will damage or cause wear to the system. Such constraints can often be
expressed in terms of an RMS norm of , possibly with weights (see sec-
u
tion 4.2.2).
Saturation or overload. Exceeding absolute limits on actuator signals may
damage an actuator, or cause the plant to be a poor model of the system
P
to be controlled. These speci cations can be expressed in terms of a scaled or
weighted peak norm of .u
Power, fuel, or resource use. Large actuator signals may be associated with
excessive power consumption or fuel or resource use. These speci cations are
often expressed in terms of a scaled or weighted average-absolute norm of .u
Mechanical or other wear. Excessively rapid changes in the actuator signal
may cause undesirable stresses or excessive wear. These constraints may be
expressed in terms of slew rate, acceleration, or jerk norms of (see sec-
u
tion 4.2.8).
These limits on the size of can be enforced by limiting in an appropriate way
u
the size of ac and ad, the closed-loop transfer matrices from the command and
H
H
disturbance signals to the actuator. For example, if the command signal is modeled
as a stochastic process with a given power spectral density, then a weighted 2
H
norm-bound on ac will guarantee a maximum RMS actuator e ort due to the
H
8.4 COMBINED EFFECT OF DISTURBANCES AND COMMANDS
191
command signal. If the command signal is modeled as unknown-but-bounded, and
the peak of the actuator signal must be limited, then the actuator e ort speci cation
is a limit on the peak gain of ac. These speci cations are analogous to many we
H
have already encountered in this chapter.
We mention one simple but important distinction between a limit on the size of
and the associated limit on the size of ac (or ad). We will assume for simplicity
u
H
H
that the command and actuator signals are scalar, and the disturbance is negligible.
Suppose that we have the constraint
1
(8.18)
kuk
1
on our actuator signal (perhaps, an ampli er driving a DC motor saturates), and
our command signal is constant for long periods of time, and occasionally changes
abruptly to a new set-point value between 1 and 1. We can ensure that (8.18)
;
holds for all such command signals with the closed-loop convex design speci cation
2 ac pk step 1
(8.19)
kH
k
:
This speci cation ensures that even with the worst case full-scale set-point changes,
from 1 to 1 and vice versa, the peak of the actuator signal will not exceed one.
;
By linearity, the speci cation (8.19) ensures that a set-point change from 0 6 to
;
:
0 4 will yield an actuator signal with
0 6. Roughly speaking, for such a
:
kuk
:
1
set-point change we are only making use of 60% of our allowable actuator signal
size this may exact a cost in, say, the time required for the commanded variable to
converge to within 0 01 of the nal value 0 4.
:
:
This is illustrated in gure 8.14, which shows two command signals and the
associated actuator signals in a control system that satis es the speci cation (8.19).
The command signal in gure 8.14(a) is one of the worst case, full-scale set-
w
point changes, and causes the actuator signal , shown in gure 8.14(b), to nearly
u
saturate. The command ~ in gure 8.14(c), however, results in the actuator signal
w
in gure 8.14(d), which uses only 48% of the allowable actuator capability.
8.4
Combined Effect of Disturbances and Commands
So far we have treated command inputs and disturbances separately the speci ca-
tions we have seen constrain the behavior of the closed-loop system when one, but
not both, of these exogenous inputs acts. As a simple example, assume that the
system has a single command input, a single disturbance, and a single actuator, so
that c = d = a = 1. Consider the two speci cations
n
n
n
env =
min( )
( ) max( ) for all
0
H
fH
j
s
t
s
t
s
t
t
g
rms act =
ad 2 1
H
fH
j
kH
k
g
:
The rst speci cation requires that the step response from c to c lie inside the
w
z
envelope given by min and max, as in gure 8.5. This means that the commanded
s
s
192