FIGURE 5: INSERTING DEADTIME
Adding deadtime compensation to the integral lag can greatly improve performance.
For example, this controller’s performance is shown in its simulated step-load response curve in Figure 6, compared to the curves of PI and PID controllers tuned to minimize integrated absolute error (IAE). Its integrated error is only 40% that of the PID and 14% that of the PI controller, while its proportional band is less than half of the PID and a third of the PI controller. What’s even more remarkable is that the process being controlled in this simulation is a distributed lag, containing no deadtime at all! Its dynamic response is typical of heat exchangers, stirred tanks, and distillation columns, consisting of multiple lags distributed throughout their mass.
FIGURE 6: MINIMIZING IAE
The PID controller outperforms others, even on a distributed process.
However, this performance enhancement comes with a price—low robustness. Applied to a heat exchanger, the optimally tuned PID controller will reach its stability limit when process flow either decreases by 21%, or increases by 14%. By contrast, optimally tuned PI and PID controllers will only reach their stability limit when flow decreases by 36% and 32%, respectively. If applied to a variable-parameter process such as this one, then its tuning needs to be scheduled as a function of flow.
Perhaps the most valuable role of external reset is in cascade control. The two controllers have demonstrated difficulty in transferring from full-manual to full-automatic operation. And, whenever the secondary controller is placed in manual or reaches an output limit, the primary controller winds up. This is solved with the configuration shown in Figure 7.
FIGURE 7: CONTROLLING BATCHES
Cascade control is improved by reset feedback of the secondary variable.
The secondary controlled variable is sent to the primary controller as external-reset feedback. If the secondary loop is then open for any reason, the primary controller stops integrating because its positive-feedback loop is open, and so it can be left in automatic all the time. Because represents the current process condition, it keeps the primary controller current and ready to resume integration whenever permitted by a closed secondary loop.
The secondary controller must have integral action, however, so that primary output and feedback will be equal in the steady state. Any offset in the secondary loop will produce offset in the primary, as Equation 1 attests. Some cascade systems include a feed-forward calculation inserted between the controllers, such as in a multiplier (Equation 4), where:
and q(t) is the process measured load.
Then, the external-reset feedback path must include a back-calculation of feedback signal (f1) using a divider, by substituting for m1 and for (Equation 5):
This is necessary to insure that primary output and feedback are equal in the steady state. If the selected output from the override system in Figure 4 is a set point to a secondary controller, the secondary controlled variable should be sent to all controllers as reset feedback, instead of the selected output signal.
Observe that the entire secondary loop, including the secondary part of the process, lies within the integral term of the primary controller. This turns out to be a distinct advantage. When a deadtime block was inserted in the integral feedback path, the performance of the controller improved markedly. The secondary loop now inserted in that path may include some deadtime and other lags, which also improve the performance of the primary controller. Not only does this configuration allow the integral time of the primary controller to be reduced, but also allows its proportional band to be reduced.
Adding deadtime to a controller was found to reduce its robustness, which is the ability of the control loop to remain stable while process parameters vary. However, including the secondary loop in the primary integral path actually improves robustness because it includes some potentially variable process parameters. The thermal time constant of a vessel is (Equation 6):
where M is the mass of the process, C is its heat capacity, U the overall heat-transfer coefficient, and A the heat-transfer area.
Several years ago, tests were carried out on a polymerization reactor, where batch temperature was controlled by manipulating jacket outlet temperature in cascade with the controllers connected as in Figure 7. The value of could be varied over a range of 4:1 by changing batch size M and area A. It was expected that a different set of tuning constants would be required for each combination, but the same controller settings produced acceptable results over the entire range of thermal time constants. A larger time constant meant slower heating and cooling, and slower integration as well. With this configuration, retuning wasn’t required with changes in recipe or heat-transfer coefficient, both variations common to batch reactions.