Introduction
Commercial controllers such as the PID series (proportional,
integral, derivative, and their combinations) are linear devices
within their normal operating range, i.e., within set limits of
input and output signals. Yet most fluid processes which they are
assigned to control are nonlinear to some degree, which means
that their gain in response to control action is subject to
change. In this case the controller cannot be optimally tuned
except at one specific operating condition. When the process load
or controller set point change, the tuning parameters may no
longer provide the optimum recovery from disturbances. In some
cases, the process gain can even vary with the magnitude of the
disturbance or the deviation of the controlled variable from set
point.
If the variation in process gain is less than about 50 percent
over the full operating range, the response penalty will not be
severe. However, the controller should be tuned at the operating
point where the gain is highest, so that the loop will be stable
at all times. For some loops, the gain variation is much greater
than this, in which case the controller gain is much lower than
optimum at normal production conditions, substantially
compromising its performance. Two examples are given here where
the gain variation was sufficient to cause control problems, and
was corrected by applying characterization.
Locating the Nonlinearity
Most nonlinear behavior is associated with the final
element—the control valve. Its characteristic is often
incorrectly chosen. But when the pressure drop across the valve
is variable—which is commonly the case—none of the
standard valve characteristics will deliver flow linearly with
controller output. Furthermore, some processes such as
liquid-liquid heat exchangers do not transfer heat in a linear
relationship with fluid flow, in which case temperature is not
linear with flow.
When the process nonlinearity is related to flow or process
load, the loop gain will be different for each value of load, and
therefore with each value of controller output. This is
manifested as variable damping as a function of controller
output. The loop could be lightly damped at high values of
controller output, or at low values, depending on the shape of
the nonlinear relationship. Quite often, the highest process gain
occurs at startup or standby conditions, requiring the controller
to be tuned there for stability; then it will not be sufficiently
responsive at normal production conditions.
It is also possible for the nonlinear behavior to be
associated with the controlled variable, but this is far less
common. The two applications most frequently encountered are flow
and pH loops. Nonlinear flow measurements are those where head or
differential pressure across a restriction is transmitted to the
controller. Modern dp transmitters and digital controllers can
extract the square root of the measured head, linearizing the
flow signal for orifices and nozzles; similar calculations can be
made to linearize flow signals from flumes and weirs.
With pH measurements, however, the problem is more complex.
The relationship between reagent delivery and pH is logarithmic,
capable of producing a gain variation over several orders of
magnitude. And it is not a simple relationship—the shape of
a curve is a function of the ionic species in solution and their
concentrations. Each process may have its own characteristic
curve, and even a family of curves, changing with time.
Measurement of oxidation-reduction potential and ionic species
other than hydrogen follow similar relationships.
If the variable process gain is associated with the controller
output, then a curve characterizer applied to the controller
output signal can correct the problem, by essentially modifying
the valve characteristic. For the head flowmeters described
above, characterization was applied to the flow measurement.
Similarly, for pH loops, characterization needs to be applied to
the measurement (and to the set point).
A typical titration curve for industrial wastewater is shown
in Fig. 1. The curve is produced by titrating a sample with
caustic in a laboratory, or by incrementally raising the dosage
of caustic delivered to the plant neutralization vessel under
constant load (a far slower and less-reliable procedure). The set
point for the pH controller is usually positioned in the region
of neutrality, where the curve has its steepest slope—for
this curve, the maximum slope is 75 for a pH range of 2-12.
Consequently, the controller gain must be adjusted for stability
there, to avoid limit-cycling, a constant-amplitude cycling which
can substantially increase reagent consumption.
Figure 2 describes the simulated response of a pH control loop
based on the titration curve of Fig. 1 (without nonlinear
compensation), following a step decrease in acid load to the
neutralization vessel. The vessel is simulated as being
well-mixed, and having a residence time of 20 minutes. The set
point (SP) is positioned at pH 7. Although the PI controller is
tuned for light damping at set point, as observed toward the end
of the response, recovery of the process variable (PV) from the
upset is very slow, and is followed by a large overshoot. Note
that the controller output (CO) moves very slowly toward its new
steady state, because the controller gain is very low (0.167).
For smaller load changes, the response will be better, and for
larger upsets, it will be worse.
The loop can be linearized by placing a complementary
nonlinear function in the path of the pH measurement and set
point—that function is shown in Fig. 3. In essence, this
characterizer converts pH values into equivalent concentration of
caustic in solution, linear with the delivery of caustic by the
controller.
Figure 4 repeats the step change in acid load with the
characterizer applied to the controller. The recovery is much
quicker and damping uniform—observe that the trajectory of
the controller output is representative of a linear loop. The
integrated error between PV and SP has been reduced by a factor
of two and the integrated absolute error by almost a factor of
three by the addition of the characterizer. The controller gain
has been increased to 7.7, but the loop is now more heavily
damped than before, and settles much more quickly.
Many other possible curves exist—some asymmetrical , as
well as the possibility that the set point may be positioned
somewhere other than in the center of the curve. For best
results, the characterizer should match the titration curve as
accurately as possible. This requires at least ten points on an
X-Y plot, uniformly distributed over the operating pH range (not
the reagent range). If the curve happens to be variable, then the
characterizer should be matched to the most nonlinear curve.
Compressor Control
Most compressors are fitted with a recirculation valve,
allowing some of the compressed gas to be returned to the suction
(after cooling) to control the capacity of the machine over a
wide flow range. Either the suction or discharge pressure is
controlled at the compressor, and the other pressure controlled
elsewhere or open to the atmosphere. The recirculation valve then
typically operates under constant pressure drop, in which case
recycled flow is linear with valve opening. The valve should
therefore have a linear characteristic.
In many compressor installations, however, an equal-percentage
valve has been incorrectly provided1. The gain of this
valve varies directly with the flow through it, as shown in Fig.
5. (Note that the valve is reverse-acting, so that it will open
on a signal failure.) Under no-load conditions, the valve must
recycle all the compressed flow, in which case it will approach
full opening, where its gain is highest. The pressure controller
must be tuned for stability here. As the load increases toward
the normal production conditions, the controller will close the
valve accordingly, to a point where its gain will be much lower
and therefore the response to upsets will be sluggish.
The response of a reciprocating compressor to a series of step
load changes is simulated in Fig. 6, with a linear and an
equal-percentage recirculation valve compared. As the load is
stepped from zero to 80 percent, the pressure transient with the
linear valve remains essentially the same. However, with the
equal-percentage valve, the pressure controller had to be tuned
at zero load, where the valve gain is highest. Subsequent steps
at higher loads show progressively deteriorating response, owing
to a reduction in loop gain.
A linear recirculation valve should therefore be installed
with each compressor. However, in the event that an
equal-percentage valve is already in place, the incorrect
characteristic can be corrected by inserting a complementary
characterizer in the controller-output path. The curve required
to linearize the valve of Fig. 5 appears in Fig. 7. Observe that
the valve curve is simply rotated 90 degrees to produce the
characterizer.
Fine Points
If the equal-percentage valve were direct-acting, the
characterizer curve would have to bend in the other direction,
i.e., lying above the diagonal. However, the characterizer should
always have a positive gain, so that the controller action
remains the same whether a characterizer is installed or not. The
same is true for pH characterizers. If the titration curve of
Fig. 1 were reversed, for example, by titrating a basic
wastewater with an acid reagent, the same characterizer would be
used as that shown in Fig. 2.
Remember that retuning of the controller is always required
after installing a characterizer, and always in the direction of
raising its gain. The gain of the pressure controller in the
simulation with the linear valve is over three times as high as
with the equal-percentage valve. Therefore control will always be
tighter after the characterizer is installed than before.
References
Shinskey, F. G., "Smoothing Out Compressor Control," Chem. Eng., Feb. 1999, pp. 127-130. |
