Los ecos de la historia: Esclavitud y cimarronaje

In chapter 4, we stated that if the derivative mode is used in the presence of measurement noise, then the derivative action will amplify the noise and produce an excessive noise compo-nent on the controller output. For this reason, we recommended that the derivative mode not be used on control loops that have excessive noise.

If a moderate amount of noise is present, however, it is still possible to use derivative, provided that a filter is used to attenuate the noise. The actual physical construction of a filter can take many different forms, such as a restriction and a bellows in a pneumatic controller, an R-C net-work (or an R-C netnet-work with an operational amplifier) in an electronic analog controller, or a few lines of code in a digital control algorithm. Mathematically, the filter is often represented as a first-order lag, although more complex forms of filters are available in some commercial controllers.

Structurally, the filter can be incorporated into the controller in several ways, as shown in Fig-ure 5-7. The filter can either be placed on the incoming measFig-urement signal to the controller, as shown in Figure 5-7a, or merely in series with the derivative unit, as shown in Figure 5-7b.

When the controller uses the interactive form, the filter can be placed on the error itself, as shown in Figure 5-7c. This controller can be represented in Laplace notation by Equation 5-16.

Many texts state that this form more nearly represents traditional analog controllers than does Equation 5-6:

Figure 5-7. Internal Filter Configurations

ment can be represented by Figure 5-7d and Equation 5-17:

(5-17)

If the filter is formulated as a first-order lag and the filter time constant is short relative to the process dynamics, then the filter placement will not have an appreciable effect on the behavior of the control loop. On the other hand, if there is heavy filtering (i.e., the filter time constant is the same order of magnitude as the dead time of the process), and if the filter is on the overall measurement signal, then the process’s apparent dead time will be increased by approximately one-half of the filter time constant. This will significantly affect the controller tuning.

The filter time constant is often made proportional to the user-adjustable derivative time. For example, the filter time constant is given by T_D/α. The parameter α can have a value of from 6 to 20; it often has a built-in value of 10. Thus, the filter time constant is automatically set at 0.1 times the derivative time. This limits the amplitude of the derivative spike, when the step change in set point is made, to 10 times the proportional response. This is often called the “derivative gain.” Some manufacturers make the derivative gain accessible so users can adjust it.

™ NONLINEARIZATION

A PID algorithm in any of the forms described in this chapter is termed a “linear” controller.

For a given amount of error, it will always respond in the same manner. There are circum-stances where one might want the algorithm to perform in a different fashion. If a PID is used as a liquid-level controller, it may be desirable for the controller to have very conservative action when the error is small, to avoid excessive fluctuations of the flow rate. If, however, the error is large, more aggressive controller action may be desired to prevent the vessel from draining or overflowing. To achieve this, the error signal can be modified by a characterizer function. The modified (pseudo) error signal, designated ê, is then used by the PID modes, as shown in Figure 5-8.

One popular nonlinearization function is called “error-squared” or “absquare” (for “absolute value of the square of the error”). Neither of these terms is a precise mathematical description of the function; that is best provided by Equation 5-18 and by the graphical relation shown by curve A in Figure 5-9.

(5-18)

The effect of this nonlinearization function is both to lower the controller gain when the error is near zero and to gradually increase the controller gain as the deviation from set point increases. In fact, when the error is zero, the effective controller gain is zero. Possible varia-tions of Figure 5-8 would be to place the non-linear characterization on only the proportional mode, or on only the integral mode, rather than on the common error signal. A slightly less Figure 5-8. Insertion of a Characterizer in a PID Algorithm

Figure 5-9. Error versus Pseudo Error Relationship 3 P

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absquare error, as shown by curve B in Figure 5-9 and described by the following equation:

(5-19)

where θ is an adjustable parameter that can take any value from 0 to 1. With this arrangement, if θ > 0 there is a finite controller gain, even at zero error.

Page 168 contains further discussion related to the use of error-squared algorithms for level control.

In addition to its use in liquid-level control, the “error-squared” or “absquare” form of nonlin-earization has also been used in pH control. Suppose you are neutralizing a waste stream by adding a reagent. The titration curve is very nonlinear and is essentially the opposite nonlin-earity provided by Figure 5-9. Therefore, using this nonlinear control feature will tend to can-cel out the nonlinearity of the process.

Another form of nonlinearization is to characterize the error with straight-line segments. For a custom application, this may be nonsymmetrical. Commercial systems would have symmetri-cal characterization, as shown in Figure 5-10, curve A. If the central segment has zero slope as shown in Figure 5-10, curve B, this is called a “gap action” or “dead zone” algorithm. An application for this algorithm would be for positioning a final actuator that has a reversible electric motorized valve. If the process variable is near the set point (deviation < ± b), the con-troller acts as if there were zero error, hence it doesn’t move the valve. The valve motor is only activated if the deviation exceeds the break-point limits. The objective of this form of control is to prevent “chatter” of the valve motor, albeit at the cost of control of somewhat reduced quality.