The last chapter concluded with the development of the standard (also called the “textbook,”
“classical,” or “ideal”) form of the PID controller. Equation 4-5 presented it as:
(5-1) Although it is desirable to have a thorough understanding of both the rationale and the behav-ior of this standard form, in real applications one often finds the need for subtle deviations from the standard form. Indeed, if one attempted to purchase exactly that form, or choose it from a microprocessor-based system vendor’s library of control algorithms, frustration would be the likely result. The reason: few manufacturers offer the standard form of the PID; most offer one or more variations of it, called “configuration options.”
Some of the modifications are offered by only a few manufacturers, or even only one. After all, each manufacturer claims to have features that will be an advantage to users of that product.
On the other hand, all manufacturers offer many of the modifications in forms that are at least similar to each other. In this chapter we describe some of the more common modifications available, as well as their application or purpose. Several manufacturers’ software control forms are reviewed at the end of the chapter. These particular manufacturers were selected to illustrate a range of features; their inclusion here should not be interpreted as an endorsement of their products.
Because of the power and flexibility of the microprocessor, users will encounter most of the modifications, or optional features, to standard PID control in a digital-based system rather than in an analog system. Nevertheless, for clarity of presentation, we will refer to analog ele-ments, gains, integrators, derivative units, as we did in the previous chapter, rather than rely on difference equations which would be a more exact representation of the digital form. Before the end of the chapter, however, we will consider the subject of the digital implementation of PID control.
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A step change in set point can be a rather harsh disturbance for a control loop. Several modifi-cations are available that “soften” the effect of such a change: derivative mode on measure-ment, proportional mode on measuremeasure-ment, linear combination of inputs to modes, and set point ramping.
Derivative Mode on Measurement
In control loops where the standard form of the PID (Equation 5-1 and Figure 4-13) is used, a set point change causes a large, but short-duration “spike” in the controller output. This is due to the derivative mode’s response to the very rapid change in error. In addition to the derivative spike, there is the proportional response, followed by a gradual change caused by the integral response, until the measurement again achieves equilibrium with the new set point (see Figure 5-1a).
Consider the derivative spike in the controller output. Most likely, the final control element will not be fast enough to respond totally to this rapid change. Furthermore, the process itself will act as a filter and prevent much of this signal from reaching the measurement value. Even so, this spike on the output signal is unwanted because it is probably undesirable to move the valve this rapidly. For instance, we may not wish to cause a thermal shock to heat-exchanger tubes, or we may not wish to upset a reactor catalyst bed.
One of the reasons for adding the derivative mode to the controller was to improve the response of the control loop to a load upset. This can be achieved by making the derivative unit responsive to measurement only, rather than to the error. Doing so will avoid the derivative spike caused by a set point change. A block diagram of this configuration option is shown in Figure 5-2; the response to a set point change is shown in Figure 5-1b. This modification is represented by the following equation:
(5-2)
Note that the sign of the derivative contribution to the controller output has been changed from
“+” to “–” in Figure 5-2. Since the derivative contribution must always act to oppose the direc-tion of modirec-tion of the measurement, the derivative contribudirec-tion must be negative on a load increase for a reverse-acting controller. If Figure 5-2 depicted a direct-acting controller, the sign of the derivative contribution would be changed to “+” and the signs at the summation point for set point and measurement would be reversed. Each figure in this chapter uses a reverse-acting controller as the basis of illustration.
Suppose two controllers are mounted side by side, one with the standard form of the PID, and the other identical except for the derivative mode on measurement rather than on error. If the controllers were controlling identical processes and they were identically tuned, the response
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be the action that follows a set point change: the controller with the standard PID would cause a spike in the controller output whereas the other controller would not.
Figure 5-1. Response of Process Variable and Controller Output to a Set Point Change 352&(669$5,$%/(
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Equipment manufacturers often provide a user configuration option called “derivative on error” or “derivative on measurement,” especially in microprocessor-based control systems.
Almost always, the better choice from an application viewpoint is “derivative on measure-ment.” Some manufacturers do not give the user this choice; rather, they provide derivative on measurement as the only option. This is not necessarily a deficiency in a controller, however, since “derivative on measurement” is the preferred choice for almost all applications.
A rare instance in which derivative on error would be preferred is when the derivative mode is used in the secondary controller of a cascade loop. (Cascade control is discussed in chapter 9.) For instance, in controlling an exothermic chemical reactor, the set point of a jacket water-tem-perature controller may be set by the output of the primary reactor temwater-tem-perature controller. Both the primary and the secondary may utilize the derivative mode to provide the fast response and stabilization that the exothermic process requires. In this case, however, the primary controller will not make abrupt changes in the secondary controller set point. For this reason, derivative response to changes in error, whether caused by set point change, measurement change, or both, is acceptable.
Proportional Mode on Measurement
By placing the derivative mode on the measurement, we have eliminated the derivative spike caused by a step change in set point. We still have, however, a step change in controller output, as shown in Figure 5-lb. This is called the proportional response or “proportional kick.” Even this may be a more abrupt change to the process than is desirable. The proportional response to a set point change can also be eliminated by making the proportional mode, as well as the derivative mode, responsive only to the measurement signal rather than to the error. This is shown in Figure 5-3 and mathematically by Equation 5-3. This leaves only the integral mode acting on the error. The loop response to a set point change is shown in Figure 5-4.
(5-3) Figure 5-2. Derivative on Measurement
3
Note that in Figure 5-3 the sign of the proportional contribution to the controller output has been changed from “+” to “–” to be consistent with the reverse-acting controller depicted there. For a direct-acting controller, both the sign of the proportional contribution to the output and the signs at the error summation point would be reversed.
Suppose that two controllers are mounted side by side, one with the standard form of PID and the other identical except that the proportional and derivative modes are on measurement rather than error. If the controllers were controlling identical processes and if both controllers were identically tuned, the response to a load upset would be identical. On a set point change, the output of the controller with P and D on measurement will exhibit neither the derivative spike nor the proportional kick. Instead, the integral mode of the controller will cause the con-troller output to begin a gradual change until the measurement achieves equilibrium with the new set point.
Because of the absences of the initial rapid forcing of the process caused by the proportional response, this controller would probably not drive the measurement to the new set point as quickly as would a standard PID, or even a PID with the derivative on measurement modifica-tion. However, this is not necessarily a fault in this controller, since the purpose in using this modification is to be gentler on the process. Because of the slower response to a set point change, this modification should not be used in the secondary controller in a cascade loop.
(Cascade control is discussed in chapter 9.)
Linear Combination of Inputs to Modes
The configuration choices represented by the block diagrams in Figures 4-13, 5-2, and 5-3 can be expressed in a comprehensive equation as follows:
(5-4) Figure 5-3. Proportional and Derivative Modes on Measurement
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where: e = error (xSP – x), for reverse-acting controller,
x = measurement,
xSP = set point,
β = proportional mode selection factor, β = 1 for proportional on error,
β = 0 for proportional on measurement,
= derivative mode selection factor,
= 1 for derivative on error,
= 0 for derivative on measurement.
The definitions here for β and indicate that they can have values of only 0 or 1. This pro-vides the option of having proportional and derivative modes that are wholly responsive to either the error or the measurement. Suppose that the restrictions on β and are relaxed so that either or both may take on any value between 0 and 1, that is:
In this case, one could obtain a linear combination of proportional and derivative modes on error or measurement. (One manufacturer refers to this as a “two degree of freedom” control-ler.) This variation is probably most useful when only an intermediate value for β is chosen, since will usually be left at 0 (providing derivative on measurement). When the value for β is intermediate, the controller can be tuned with a higher value of gain, thus providing a better response to a disturbance. There will also be some proportional action taken on a set point change. As a result, the time available to achieve a new set point will be greater than it would be if all of the proportional action was on measurement.
Figure 5-4. Process Variable and Controller Output Response to a Set Point Change with Both Proportional and Derivative Modes on Measurement
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Functionality that is equivalent to the combination of proportional mode on error or measure-ment can be obtained by placing a lead-lag filter on the set point. This filter should take the fol-lowing form:
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where β is the proportional mode selection factor, as earlier, and TI is the integral time. One manufacturer refers to this as “set point filtering.”
For the same reason that the proportional-on-measurement configuration option should not be used as the secondary controller in cascade loops, this configuration option should also be avoided for that application.
Set Point Ramping
Some controllers permit a step set point change to be made into a new target value. The actual set point used by the controller, however, is changed gradually (“ramped”) from its present value to the target value at a specified rate of change. This will result in a more gradual change in the process variable, considerably reducing overshoot.