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外文资料与翻译
PID Contro
l
6.1
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
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
u
?
t
?
?
K
?
e
?
t
?
?
?
e
?
?
?
d
?
< br>?
T
?
T
i
0
?
d
d
e
?
t
?
?<
/p>
?
dt
?
?
6.1
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
?
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
?
d
t
?
cos
t
?
n
?
t
?<
/p>
?
cos
t
?<
/p>
a
n
cos
?<
/p>
n
t
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
?
?
KT
1
?
s
T
s
d
d
N
6.2
instead
of
D
=
s
T
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
/
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
?
1
C
?
S
?
?
?
K
?
1
?
?
S
T
?
?
I
s
KT
1
?
s
T
d
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
?
< br>s
?
?
1
?
1
?
s
T
f
?
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
?
?
?
K
?
1
?
?
s
T
?
s
T
i
?
?
?
d<
/p>
?
?
1
?
1
?
s
T
d
N
?
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
?
dr
?
t
?
dy
?
t
?
?
?
?
u
?
t
< br>?
?
K
br
?
t
?
?
y
?
t
?
?<
/p>
c
?
?
?
?
e
?
?
?
d
?
?
T
d
?
?
dt
dt
?
?
?
T
i
0
?
?
6.4
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
?
R
?<
/p>
s
?
?
c
r
?
s
?
?
?
1
K
?
b
?
?
cs
T
?
s
T
i
?
?
?
d
?
?
6.5
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
U
?
s
?
?
1
?
c
y
?
s
?
?
K
?
1
< br>?
?
s
T
?
R
?
s
?
s
T
i
?
?
?
d
?
?
6.6
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
?
< br>?
K
?
1
?
?
s
T
?
s
T
i
?
?
?
d
?
?
6.7
K
?
K
?
T
6.8 <
/p>
?
?
T
?
T
?
i
i
d
T
i
?
T
?
?
T
?
6.9
i
d
T
d
?
T
?
T
?
T
?
?
T
?
i
d
< br>i
d
An
interacting
controller
of
the
form
Equation
D6.8E
that
corresponds
to
a
non-
interacting controller can be found only if
T
?
The parameters
are then given by
K
?
< br>?
K
i
?
4
T
?
d
?
1
?
2
i
1
?
4
T
d
T
?
i
T
?
i
?
T
2
?
1
?
1
?
4
T
d
< br>T
?
6.10
i