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外文资料与翻译
PID Contro
l
Introduction
The
PID
controller
is
the
most
common
form
of
feedback.
It
was
an
essential
element
of
early
governors
and
it
became
the
standard
tool
when
process
control
emerged in the
1940s. In process control today, more than 95% of
the control loops
are of PID type, most
loops are actually PI control. PID controllers are
today found in
all areas where control
is used. The controllers come in many different
forms. There
are standalone systems in
boxes for one or a few loops, which are
manufactured by
the hundred thousands
yearly. PID control is an important ingredient of
a distributed
control
system.
The
controllers
are
also
embedded
in
many
special
purpose
control
systems. PID control is often combined
with logic, sequential functions, selectors, and
simple function blocks to build the
complicated automation systems used for energy
production, transportation, and
manufacturing. Many sophisticated control
strategies,
such
as
model
predictive
control,
are
also
organized
hierarchically.
PID
control
is
used
at
the
lowest
level;
the
multivariable
controller
gives
the
set
points
to
the
controllers at the lower level. The PID
controller can thus be said
to be the
“bread and
butter of control
engineering. It is an important component in every
co
ntrol engineer’s
tool box.
PID controllers have survived many
changes in technology, from mechanics and
pneumatics
to
microprocessors
via
electronic
tubes,
transistors,
integrated
circuits.
The
microprocessor
has
had
a
dramatic
influence
the
PID
controller.
Practically
all
PID
controllers
made
today
are
based
on
microprocessors.
This
has
given
opportunities
to
provide
additional
features
like
automatic
tuning,
gain
scheduling,
and continuous adaptation.
6.2 The Algorithm
We
will
start
by
summarizing
the
key
features
of
the
PID
controller.
The
“textbook” version of the PID algorithm
is described by:
t
?
1
de
?
t
?
?
?
?<
/p>
6.1
u
?
t
?
?
K<
/p>
e
?
t
?
?
e
?
?
?
d
?
?
T
d
?
?
dt
?
T
i
0
?
?
where
y
is the measured process
variable,
r
the reference
variable,
u
is the control
signal and
e
is
the control error
（
e
=
y
sp
?
y
）
. The reference
variable is often called
1
/
14
the set
point. The control signal is thus a sum of three
terms: the P-term
（
which is
proportional to the
error
）
, the I-term
（
which is proportional to
the integral of the
error
）
, and the
D-term
（
which is
proportional to the derivative of the
error
）
. The
controller parameters are proportional
gain
K
, integral time
T
i
, and
derivative time
T
d
.
The
integral,
proportional
and
derivative
part
can
be
interpreted
as
control
actions
based
on
the
past,
the
present
and
the
future
as
is
illustrated
in
Figure
2.2.
The
derivative
part
can
also
be
interpreted
as
prediction
by
linear
extrapolation
as
is
illustrated
in
Figure
2.2.
The
action
of
the
different
terms
can
be
illustrated
by
the
following figures which
show the response to step changes in the reference
value in a
typical case.
Effects of Proportional, Integral and
Derivative Action
Proportional control
is illustrated in Figure 6.1. The controller is
given by D6.1E
with
T
i
=
?
and
T
d
=0. The figure
shows that there is always a steady state error in
proportional
control. The
error will decrease with
increasing
gain, but
the tendency
towards oscillation will also increase.
Figure 6.2 illustrates the effects of
adding integral. It follows from D6.1E that the
strength of integral action increases
with decreasing integral time
T
i
. The figure shows
that the steady state error disappears
when integral action is used. Compare with the
discussion
of
the
“magic
of
integral
action”
in
Se
ction
2.2.
The
tendency
for
oscillation
also
increases
with
decreasing
T
i
. The properties of
derivative action are
illustrated in
Figure 6.3.
Figure 6.3 illustrates the
effects
of
adding derivative
action. The parameters
K
and
T
i
are
chosen
so
that
the
closed
loop
system
is
oscillatory.
Damping
increases
with
increasing
derivative
time,
but
decreases
again
when
derivative
time
becomes
too
large. Recall that derivative action can be
interpreted as providing prediction by
linear extrapolation over the time
T
d
. Using this
interpretation it is easy to understand
that derivative action does not help if
the prediction time T
d
is
too large. In Figure 6.3
the period of
oscillation is about 6 s for the system without
derivative Chapter 6. PID
Control
2
/
14
Figure 6.1
Figure 6.2
Derivative actions cease to
be effective when
T
d
is
larger than a 1 s (one sixth of
the
period). Also notice that the period of
oscillation increases when derivative time is
increased.
A
Perspective
There is much
more to PID than is revealed by
（
6.1
）
.
A faithful implementation
of
the
equation
will
actually
not
result
in
a
good
controller.
To
obtain
a
good
PID
controller it is also necessary to
consider
。
3
/
14
Figure
6.3
??
Noise filtering and high
frequency roll off
??
Set
point weighting and 2 DOF
??
Windup
??
Tuning
??
Computer
implementation
?
In
the
case
of
the
PID
controller
these
issues
emerged
organically
as
the
technology
developed
but
they
are
actually
important
in
the
implementation
of
all
controllers. Many of
these questions are closely related to fundamental
properties of
feedback, some of them
have been discussed earlier in the book.
6.3 Filtering and Set Point
Weighting
Differentiation is
always sensitive to noise. This is clearly seen
from the transfer
function
G
(
s
)
=
s
of a differentiator which
goes to infinity for large
s
. The following
example is also illuminating.
y
?
t
?
?
sin
t
?
n
?
t
?
?<
/p>
sin
t
?
a<
/p>
n
sin
?
n<
/p>
t
where the noise
is sinusoidal noise with frequency w. The
derivative of the signal
is
dy
?
t
?
?
cos
t
?
n
?
t
?
?
cos
t
?
a
n
cos
?
n
t
dt
The
signal to noise ratio for the original signal is
1/
a
n
but the signal to noise ratio
of the differentiated signal is
w/
a
n
. This ratio
can be arbitrarily high if w is large.
In a practical controller with
derivative action it is there for necessary to
limit the
high
frequency
gain
of
the
derivative
term.
This
can
be
done
by
implementing
the
derivative term as
4
/
14
D
?
?
1
?<
/p>
s
T
d
N
s
KT
d
6.2
instead of
D
=
sT
d
Y
. The
approximation given by
(6.2) can be
interpreted as the
ideal derivative
sT
d
filtered by a first-order system with
the time constant
T
d
< br>/
N
. The
approximation
acts
as
a
derivative
for
low-frequency
signal
components.
The
gain,
however,
is
limited
to
KN
.
This
means
that
high-
frequency
measurement
noise
is
amplified at most by a
factor
KN
. Typical values of
N
are 8 to 20.
Further limitation of the high-
frequency gain
The
transfer
function
from
measurement
y
to
controller
output
u
of
a
PID
controller
with the approximate derivative is
?
s
KT
d
?
1
?
?
C
?
S
?
< br>?
?
K
1
?
?
?
S
T
I
1
?
s
T
d
N
?
?
?
This controller has
constant gain
lim
C
?
s
?
?
?
K
?
1
?
N
?
s
?
?
at
high frequencies. It follows from the discussion
on robustness against process
variations in Section 5.5 that it is
highly desirable to roll off the controller gain
at high
frequencies. This can be
achieved by additional
low pass
filtering of the control signal by
F
?
s
?
?
?
1
?
s
T
f
?
1
n
where
T
f
is
the filter time constant and
n
is the order of the filter. The choice
of
T
f
is a
compromise between filtering capacity and
performance. The value of
T
f
can be
coupled
to
the controller time constants
in
the same way as for the
derivative filter
above. If the
derivative time is used,
T
f
=
T
d
/
N
is a suitable choice. If the controller
is
only PI,
T
f
=
Ti
/
N
may be suitable.
The controller can also be implemented
as
?
?
1
C
?
s
?
?<
/p>
?
K
?
1
?
?
s
T
d
?
?
s
?
T
i
?
?
?
1
?
s
T
d
1
N<
/p>
?
2
6.3
This structure has the advantage that
we can develop the design methods for an
ideal
PID
controller
and
use
an
iterative
design
procedure.
The
controller
is
first
designed for the
process
P
(
s
).
The design gives the controller parameter
T
d
. An ideal
controller for the process
P
(
s
)/(1+
sT
d
/
N
)
2
is then designed giving a
new value of
T
d
5
/
14
etc.
Such
a
procedure
will
also
give
a
clear
picture
of
the
tradeoff
between
performance and
filtering.
Set Point
Weighting
When using the
control law given by
（
6.1
）
it follows that a step change in the
reference
signal
will
result
in
an
impulse
in
the control
signal.
This
is
often
highly
undesirable there for
derivative action is frequently not applied to the
reference signal.
This problem can be
avoided by filtering the reference value before
feeding it to the
controller.
Another
possibility
is
to
let
proportional
action
act
only
on
part
of
the
reference signal. This is called set
point weighting. A PID controller given by
（
6.1
）
then becomes
t
?
1
?
dr
?
t
?
dy
?
t
?
?
?
?
?
u
?
< br>t
?
?
K
?
br
?
t
?
?
y
?
t<
/p>
?
?
e
?
d
?
?
c
?
6.4
?
?
T
d
?
?
?<
/p>
?
dt
?
?
?
dt
T
i
0
?
where
b
and
c
are
additional parameter. The integral term must be
based on error
feedback
to
ensure
the
desired
steady
state.
The
controller
given
by
D6.4E
has
a
structure with two degrees
of freedom because the signal path from
y
to
u
is different
from that from
r
to
u
. The transfer function
from
r
to
u
is
?
?
U
?
s
?
1
?
?
c
r
?
s
?
?
K
b
?
?
< br>cs
T
d
?
6.5
?
?
R
?
s
?
s
T
i
?
?
Time
t
Figure 6.4
Response to a
step in the reference for systems with different
set
point weights
b
= 0 dashed,
b
= 0
?
5 full and
b
=1
?
0
dash dotted. The process has the
transfer function
P
（
s
）
=1/
（
s
+1
）
3
and the controller parameters are
k
= 3,
k
i
=
1
?
5
and
k
d
=
1
?
5.
and the
transfer function from
y
to
u
is
6
/
14
?
?
U
?
s<
/p>
?
1
?
c
y
?
s
?
?
K
?
1
?
?
s
T
d
?
6.6
?
s
?
R
?
s<
/p>
?
T
i
?
?
Set
point weighting
is
thus
a special case of
controllers having two degrees
of
freedom.
The system obtained
with the controller
（
6.4
）
respond to load disturbances and
measurement
noise
in
the
same
way
as
the
controller
（
6.1
）
.
The
response
to
reference
values
can
be
modified
by
the
parameters
b
and
c
.
This
is
illustrated
in
Figure 6.4, which shows the response of
a PID controller to set-point changes, load
disturbances,
and
measurement
errors
for
different
values
of
b
.
The
figure
shows
clearly the effect of changing
b
. The overshoot for set-
point changes is smallest for
b
= 0, which is the case where the
reference is only introduced in the integral term,
and
increases with increasing
b
.
The parameter
c
is normally zero to avoid
large transients in the control signal
due to sudden changes in the set-point.
6.4 Different
Parameterizations
The PID
algorithm given by Equation
（
6.1
）
can be represented by
the transfer
function
?
?
1
G
?
s
?
?
K
?
1
?
?
s
T
d
?
6.7
?
s
?
T
i
?<
/p>
?
K
?
K
?
T
6.8 <
/p>
?
?
T
?
T
?
i
i
d
T
?
T
?
?
T
?
i
i
d
6.9
T
d
?
T
?
T
?
T
?
?
T
?
i
< br>d
i
d
An
interacting
controller
of
the
form
Equation
D6.8E
that
corresponds
to
a
non-
interacting controller can be found only if
T
?
?
4
< br>T
?
i
d
The parameters are then given by
K
K
?
?
1
?
1
?
< br>4
T
d
2
?
T
i
?
i
T
?
?
T
2
i
i
?
1
?
1
?
4
T
d
< br>T
?
6.10
7
/
14