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中英文互译
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 control
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: <
/p>
t
?
1
de
?
t
?
?
?
6.1
?
?
u
?
< br>t
?
?
K
?
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 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
Section
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
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
。
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
< br>?
t
?
?
sin
t
?
n
?
t
?
?
s
in
t
?
a
n
sin
?
n
t
where the noise is
sinusoidal noise with frequency w. The derivative
of the signal is
dy
?
< br>t
?
?
cos
< br>t
?
n
?
t
?
?
cos
t
?
a
n
c
os
?
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
D
?
?
1
?
s
T
< br>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
?<
/p>
s
KT
d
?
1
?
?
C
?
S
?
?
?
K
1
< br>?
?
?
S
T
I
1
?
s
T
d
N
?
?
?
This controller
has constant gain
lim
C
?
s
?
?
< br>?
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
?
< br>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
=<
/p>
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>
P
(
s
)/(1
+
sT
d
/
N
)
2
is then
designed giving a new value of
T
d
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<
/p>
?
dr
?
t
?
dy
?
t
?
?
?
?
?
u
?
t
?
?
K
?
br
?
t
?
?
y
?
t
?
?
e
?
d
?
?
c
?
6.4
?
?
T
d
?
?
?
?
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
?
?<
/p>
c
r
?
s
?
?
K
b
?
?
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
）<
/p>
and the controller parameters are
k
= 3,
k
i
=
1
and the transfer function from
y
to
u
is
3
5 and
k
d
=
1
5.
?
?
U
?
s
?
1<
/p>
?
c
y
?
s
?
?
K
?
1
?
?
s
T
d
?
6.6
?
s
?
R
?
s
?
T
i
?
< br>?
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
< br>1
?
?
s
T
d
?
6.7
?
s
?
T
i
?
?
K
?
K
?
T
?
?
T
?
T
?
i
i
i
d
6.8
T
?
T
?
?
T
?
i
d
6.9
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