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In modern plants, process variable (PV) measurement signals are
typically scaled to engineering units before they are displayed on the control room
HMI computer screen or archived for storage by a process data historian. This is done
for good reasons.
When operations staff walk through the plant, the assorted field gauges
display the local measurements in engineering units to show that a vessel is operating,
for example, at a pressure of 25 psig (1.7 barg) and a temperature of 140 oC (284 oF).
It makes sense, then, that the computer screens in the control room
display the set point (SP) and PV values in these same familiar engineering
units because:
| • |
It helps the operations staff
translate their knowledge and intuition from their field experience over to the
abstract world of crowded HMI computer displays. |
| • |
Familiar units will
facilitate the instinctive reactions and rapid decision making that
prevents an unusual occurrence from escalating into a crisis
situation. |
| • |
The process was originally designed in
engineering units, so this is how the plant documentation will list
the operating
specifications. |
Controlguru.com Articles Compute Kc With Units
Like a control room display, the Controlguru.com e-book
presents PV values in engineering
units. In most articles, these PVs are used directly
in tuning correlations to compute controller gains, Kc. As a result, the
Kc values also carry engineering units.
The benefit of this approach is that controller gain maintains the
intuitive familiarity that engineering units provide. The difficulty is that
commercial controllers are normally configured to use a dimensionless Kc (or
dimensionless
proportional band, PB).
To address this issue, we explore below how to convert a Kc with engineering
units into the standard dimensionless (%/%) form.
The conversion formula presented at the end of this article is reasonably
straightforward to use. But it is derived from several subtle concepts that
might benefit from explanation. Thus, we begin with a background
discussion on units and scaling, and work our way toward our Kc conversion
formula goal.
From Analog Sensor to Digital Signal
There are many ways to measure a process variable and move the signal
into the digital world for use in a computer based control system. Below is
a simplified sketch of one approach (click for a large view).
Other operations in the
pathway from sensor to control system not shown in the simplified sketch might include a
transducer, an amplifier, a transmitter, a scaling element, a linearizing
element, a signal filter, a multiplexer, and more.
The central issue for this discussion is that the PV signal arrives
at the computers and controllers
in a raw digital form. The continuous analog PV measurement has been
quantized (broken into) a range of discrete increments or digital
integer "counts" by an A/D (analog
to digital) converter.
More counts dividing the span of a measurement
signal increases the
resolution of the measurement when expressed as a digital value. The ranges
offered by most vendors result from the
computer binary 2n
form where n is the number of bits of resolution used by the A/D
converter.
|
Example: a 12 bit A/D converter digitizes an analog
signal into 212 = 4096 discrete increments normally expressed to
range from 0 to 4095 counts. |
|
A 13 bit A/D converter digitizes an analog signal into 213
= 8192 discrete increments normally expressed to range from 0 to 8191
counts. |
|
A 14 bit A/D converter digitizes an analog signal into 214
= 16384 discrete increments normally expressed to range from 0 to 16383
counts. |
Example: if a 4 to 20 mA (milliamp) analog
signal range is
digitized by a 12 bit A/D converter into 0 to 4095 counts, then the resolution is:
(20
–
4 mA) ÷ 4095 counts = 0.00391 mA/count |
A signal of 7 mA from an analog range of 4
to 20 mA changes to digital counts
from the 12 bit A/D converter as:
(7
–
4 mA) ÷ 0.00391 mA/count = 767 counts |
A signal of 1250 counts from a 12 bit A/D
converter corresponds to an input signal of 8.89 mA from an analog
range of 4 to 20 mA as:
4 mA + (1250 counts)∙(0.00391 mA/count) = 8.89 mA |
Scaling the Digital PV Signal to Engineering Units for Display
During the configuration phase of a control project, the minimum and maximum (or zero and span) of the PV
measurement must be entered. These values are used to scale the digital PV signal to engineering units for display and
storage.
| Example:
if a temperature range of 100 oC to 500 oC is digitized into
0 to 8191
counts by a 13 bit A/D converter, the signal is scaled
for display and storage by setting the minimum digital value of 0 counts = 100 oC, and
maximum digital value of 8191 counts = 500 oC |
Each digital count from the 13
bit A/D converter gives a resolution of:
(500
–
100 oC)
÷ 8191 counts
= 0.0488 oC/count |
A signal of 175 oC from an analog range of 100 oC
to 500 oC changes to digital counts
from the 13 bit A/D converter as:
(175
–
100 oC) ÷ 0.0488 oC/count =
1537 counts |
A signal
of 1250 counts from the 13 bit A/D converter corresponds to an
input signal of 161 oC
from an analog range of 100 oC to 500 oC as:
100 oC + (1250 counts)∙(0.0488 oC/count) = 161 oC |
As discussed at the top of
this article, the intuition and field knowledge of the
operations staff is maintained by
using engineering units in control room displays and when storing data
to a historian.
For this same reason, modern control
software uses engineering units when passing
variables between the function blocks used for calculations and
decision-making. Calculation and decision functions are easier to
understand, document and debug when the logic is written using
floating point
values in common engineering units.
Scaling the Digital PV Signal for Use by the PID Controller
Most commercial PID controllers use a
controller
gain, Kc (or
proportional band, PB) that is expressed as a standard dimensionless %/%.
| Note:
Controller gain in commercial controllers is often said to be
unitless or dimensionless, but Kc actually has units of (% of CO
signal)/(% of PV signal). In a precise mathematical world, these
units do not cancel, though there is little harm in speaking as
though they do.
|
Prior to executing the PID controller
calculation, the PV signal must be scaled to a standard 0% to 100% to match
the "dimensionless" Kc. This happens every loop sample time,
T, regardless of whether we are measuring temperature,
pressure, flow, or any other process variable.
To perform this scaling, the minimum and
maximum PV values in engineering units corresponding to the 0% to 100%
standard PV range must be entered during setup and loop configuration.
| Example: if a temperature
range of 100 oC to 500 oC is digitized into
0 to 8191 counts by a 13
bit A/D converter,
the signal is scaled for the PID control calculation by setting the minimum digital value
of 0 counts = 0%, and
the maximum digital value of 8191 counts = 100%. |
Each digital count from the 13
bit A/D converter gives a resolution of:
(100
–
0%) ÷ 8191 counts = 0.0122%/count |
A signal of 1537 counts
(175 oC) from a 13 bit A/D converter would translate to a
signal of 18.75% as:
0% + (1537)∙(0.0122%/value) = 18.75% |
A signal
of 1250 counts (161 oC) from a 13 bit A/D converter would translate to a
signal of 15.25% as:
0% + (1250)∙(0.0122%/value) = 15.25% |
Control Output is 0% to 100%
The controller output (CO) from commercial controllers normally default
to a 0% to 100% digital signal as well.
Digital to analog (D/A) converters begin the transition of moving the digital CO values
into the appropriate electrical current and voltage required by the valve, pump or other final control element
(FCE) in the loop.
| Note: While CO commonly defaults to
a 0% to 100% signal, this may not be appropriate when
implementing the
outer primary controller in a cascade. The outer primary CO1
becomes the set point of the inner secondary controller, and signal scaling
must match.
For example, if SP2 is in engineering units, the CO1 signal must be
scaled accordingly. |
Care Required When Using Engineering Units For Controller Tuning
It is quite common to analyze and design controllers using data retrieved
from our process historian or captured from our computer display. Just as with the articles
in this e-book, this means the computed Kc values will likely be scaled in
engineering units.
The sketch below
highlights (click for a large view)
that scaling from engineering units to a standard
0% to 100% range used in commercial controllers requires careful attention
to detail.
The conversion of PV in engineering units to a standard 0% to 100% range
requires knowledge of the maximum and minimum PV values in engineering
units. These are the same values that are entered into our PID
controller software function block during setup and loop configuration. The general conversion formula is:
where:
PVmax
= maximum PV value in
engineering units
PVmin
= minimum PV value in
engineering units
PV = current PV value in engineering
units
Example: a temperature
signal ranges from 100 oC to 500 oC and we
seek to scale it to a range of 0% to 100% for use in a PID
controller. We
set:
PVmin = 100 oC and
PVmax = 500 oC |
A temperature of 175 oC
converts to a standard 0% to 100% range as:
[(175 –
100 oC)
÷ (500 –
100 oC)]∙(100 –
0%) = 18.75% |
A temperature of 161 oC
converts to a standard 0% to 100% range as:
[(161 –
100 oC) ÷ (500 –
100 oC)]∙(100 –
0%) = 15.25% |
Applying Conversion to Controller Gain, Kc
The discussion to this point provides the basis for the formula
used to convert Kc from engineering units into dimensionless (%/%):
| Example:
the moderate Kc value in our
P-Only control of the heat exchanger study is Kc = –
0.7 %/ oC.
For this process, PVmax = 250 oC and PVmin = 0 oC |
Kc =
(–
0.7 %/ oC)∙[(250
– 0 oC)
÷ (100 –
0%)]
=
–
1.75 %/%
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Final Thoughts
Textbooks are full of rule-of-thumb guidelines for estimating initial Kc
values for a controller depending on whether, for example, it is a flow
loop, a temperature loop or a liquid level loop. While we have great
reservations with such a "guess and test" approach to tuning, it is
important to recognize that such rules are based on a Kc that is expressed
in a dimensionless (%/%) form.
Return to the
Table of Contents to learn more.
Copyright © 2008 by Douglas J. Cooper. All Rights Reserved.
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