The manipulated-variable overshoot required for best load response follows Equation 3, as for the self-regulating process, but with the time constant being negative, the overshoot exceeds 100%, or 165% in this example.
Figure 3 compares the load responses for the same two processes where no manipulated-variable (MV) overshoot is applied—typical of model-based control. The self-regulating process recovers in a long exponential curve, reflecting the time lag in the load path. In this example, its value is only two dead times, but it's more commonly around seven dead times, resulting in poor load regulation indeed.
But the absence of MV overshoot with the unstable process is catastrophic—temperature runaway! A familiar analog of an unstable process is the inverted pendulum—any disturbance will cause it to accelerate away from the vertical position. Attempting to balance an inverted pendulum on the hand will only be successful if the hand moves farther in correcting a deviation of the tip of the pendulum than the size of that deviation: Over 100% MV overshoot is essential in stabilizing an unstable process. But effective control is possible, as any trained seal can demonstrate.
This limitation is not inconsequential. Writing on the control of unstable petroleum hydrocracking reactors in "Update Hydrocracking Reactor Controls for Improved Reliability," Hydrocarbon Processing, October 2012, noted expert Allan Kern reports: "MPC [model-predictive control] is often considered a comprehensive solution for the type of control concerns raised here. However, none of the critical excursion control, depressure prevention or auto-quench functions are of the type provided by MPC."
Grading Real Controllers
The unstable process is especially demanding of regulators. Just as liquid-level loops can develop cycles when the controller's proportional band is too wide as well as too narrow, the unstable process can as well (See Process Control Systems, p. 320). If the dead time is sufficiently short relative to the lag time, an unstable process like a stirred-tank reactor can be regulated under proportional + integral (PI) control, but not the example considered here. No combinations of settings will be successful; derivative action (D) is essential.
Two candidates are now proposed for the unstable process described in Figures 2 and 3. The first to be considered adds dead-time compensation to the reset-feedback loop of a series-connected PID (interacting) controller, creating a PIDτ controller. Figure 4 describes its step-load response when tuned to minimize integrated absolute error (IAE). Its output cannot reproduce the step in load, but does overshoot it by 256% (compared to the best at 165%). The resulting peak deviation is only 1.14eb, reached 0.3 dead times later than the best. The IE of the response curve is about 18% higher than the best.
Along with its high performance comes limited robustness—the loop can destabilize if the process gain or dead time change in either direction. Yet the controller was successfully applied to a steam superheater in a 500-MW power boiler by gain-scheduling all four tuning parameters as a function of measured steam flow (See my "PID-Dead Time Control of Distributed Processes," Control Eng. Practice, 9(2001), 1177-1183).
Figure 4 also reveals a ripple in the controller output, which appears to decay—evidence of a "hot" controller. Secondary lags common to real processes will probably filter those out.
The more familiar alternative would be the ISA standard (non-interacting) PID controller (Figure 5). Its MV overshoot is 190%, causing deviation to peak at 1.25eb, reached 0.56 dead times later than the best. The IE of the response curve is also about 56% higher than IEb. While its performance is lower than the dead-time compensated PID, its robustness is higher by a factor of nearly three.
There are reasons why the PID controller continues to dominate the process-control field. It is not archaic or obsolete or simply a mathematical construct soon to be replaced by model-based control. Adding dead-time compensation improves its performance in the same way as it contributes to model-based control—and diminishes robustness for the same reasons, too. The success of high-level control loops for product quality and economic efficiency depends on a foundation of well-performing regulatory loops.