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However, this type of test is only accurate for processes where the load dynamics are essentially identical to those in the controller-output path. Figure 5 shows what can happen when those dynamics are not the same. A process was simulated from 20 interacting time lags, representative of distributed processes such as heat exchangers, stirred tanks, and distillation columns. (Here, Στ is the 63.2%, open-loop response time of a multi-lag processes, also the residence time of a stirred tank.) With the total lag Στ equal in the two paths, a PID controller was tuned to produce a minimum integrated absolute error (IAE) in response to a step change in load—this is the central curve (red), where Στq/Στm = 1. A minimum-IAE curve has a symmetrical peak (similar to a Gaussian distribution curve), followed by minimal overshoot and good damping.
When the step was repeated with the lags Στq in the load path reduced by half, the resulting peak was twice as high as expected, but also showed much more overshoot, increasing its IAE above the minimum, but only by 6%. Then, with the load time constants doubled, the response peak is half as high as the original, but now undershoots—the IAE for this curve is 53% above the minimum. To minimize IAE in response to a step change in load, the controller must be retuned, with the correction based on the ratio of the time constants in the load path to those in the loop.
The multiple-lag process simulated to generate the response curves of Fig. 5 can be approximated as dead time plus lag, and the results apply equally well. In changing the multiple time constants in the load path, the effective dead time and effective lag are both changed in the same proportion, as the initial part of the response curves—up to the first peak—reveal. However, in duplicating the tests on a lag-plus-dead-time process, it was observed that variations in the dead time alone in the load path have no effect on the response curve, other than to shift its location in time.
Optimum load response requires that the integral and derivative times of the controller be set as a function of the dead time in the loop, and the proportional band as a function of the ratio of the dead time to the dominant time constant in the loop. These rules, originally developed by Ziegler and Nichols, have been refined by later writers (See Process Control Systems, May issue 2011 pp 117-120). Now, however, it is observed that the optimum PID settings also depend on the time constant (but not the dead time) in the disturbance path. (The term “disturbance path” is used here as applying more generally to loop response rather than simply to load response, because the principal distinction between load and setpoint response happens to be the time constant in the path of the disturbing variable.)
First, the PI or PID controller should be tuned for optimum load response with equal dynamics in the load and in the loop. This can be done using identification of the dynamics in the path of the controller output, either in the open or closed loop, and verifying the minimum-IAE response by simulating a load step using the controller output. Having done this, the controller then requires retuning if the time constant(s) in the load path differ from those in the path of the controller output. Whether there is a difference can be determined by stepping the load itself and observing the resulting closed-loop response, comparing it with one of the curves in Fig. 5. If the load cannot be stepped easily or without some cost, an estimate of the difference may be required based on a process model.
For the simulated multiple-lag process described above, load dynamics were changed relative to loop dynamics, and the controller was retuned for minimum-IAE response. It was discovered that for PI controllers, only the integral term needed readjustment for load dynamics. Proportional action was unaffected. The center curve in Fig. 6 shows the correction needed for integral time as a function of the ratio of the total lag in the load path to that in the loop, Στq/Στm. For example, if the load lag is only half as great as that in the path of the controller output, the integral time of the controller needs to be increased by a factor of about 1.5; conversely, if the load lag is twice as great, integral time should be reduced by about the same factor (note the logarithmic scales). This is the extent of variation in this ratio likely to be encountered in fluid processes.
PID controllers need no correction to the derivative-time setting as a function of load lag. The integral-time correction is similar to that for PI controllers where Στq/Στm ≥ 0.5, but more correction is required where the ratio falls below this value, as indicated by the upper curve in Fig. 6.
There are basically two types of PID controllers, however—interacting (or series or cascade) and noninteracting (or parallel or ideal). Pneumatic controllers, such as those used by Ziegler and Nichols to develop their tuning rules, are of the interacting type, and many manufacturers still offer this type of PID controller, even implemented digitally. However, the noninteracting type is becoming more common, and some manufacturers even offer both. The noninteracting PID controller requires a slight correction to the proportional band as a function of load lag, as shown in the flattest of the three curves. For example, if Στq/Στm = 0.1, the proportional band of the non-interacting PID controller would have to be increased by a factor of about 1.3; but in the normal range of Στq/Στm between 0.5 and 2, little readjustment is required.
The setpoint normally enters the controller without any dynamics in its path. As a result, a controller that has been optimized for load changes will typically produce a sizeable overshoot following a change in setpoint. Based on the integral correction factor for load lag in Fig. 6, more than doubling of the integral time might seem necessary to produce minimum IAE following a change in setpoint. Simulation of a PI loop, however, indicates that minimizing IAE can be accomplished by only a doubling of the integral time. Unfortunately, the controller is then no longer optimized for load disturbances, which should be its primary objective.
Accommodating both objectives—load and setpoint response—requires a setpoint filter or its equivalent. Some controllers offer the option of integral action alone on the setpoint. This is the equivalent of applying a first-order filter to it, using a time constant equal to the integral time. While suppressing overshoot, it extends the time to reach the new setpoint considerably, actually increasing IAE substantially above the unfiltered response. A better approach is to apply a lead-lag filter. With the lag matched to the integral time and the lead set to about half that value, the setpoint response will approach the IAE produced by doubling the integral time. In essence, the controller is first tuned for optimum load response, and then the lead-lag ratio (which is also the dynamic gain) of the setpoint filter is tuned to optimize setpoint response. In effect, the lead-lag filter reduces the proportional gain of the controller to changes in setpoint by a factor equal to the lead/lag ratio.
Figure 7 summarizes the setpoint response for a loop simulated as dead time plus lag. The highest overshoot is produced by a PI controller optimized for load response, and having no setpoint filter. Doubling the integral time minimizes the IAE while reducing the overshoot. Unfortunately, this also doubles the integrated error in response to the load. When lag filtering is applied to the setpoint (equivalent to integral action alone on the setpoint), undershoot results, along with the largest IAE of all the curves. Lead-lag filtering is shown to minimize setpoint overshoot and minimize IAE without compromising load response.
F. G. Shinskey is a process control consultant and a member of the Process Automation Hall of Fame.