The performance criteria primarily used in control theory text books is integrated absolute error (IAE) for a step disturbance. The IAE performance criteria can translate to the total amount of material not at setpoint if the changes in feed flow are included.
The peak error is important when exceeding a limit can trigger a Safety Instrumented System (SIS), relief device, an environmental violation, or a hazardous reaction. The peak error is the maximum error for a disturbance. Again a step disturbance is used for comparison of tuning methods and algorithms.
The rise time is the time required for the process variable (PV) to reach a new setpoint (e.g. PV within 0.5% of setpoint). Overshoot is the error of the first peak after the PV reaches setpoint. Undershoot is the largest consequential absolute error amplitude of opposite sign. The settling time is the time for the error to remain within a specified limit.
The minimization of IAE, peak error, and rise time is often at the expense of overshoot, undershoot, and settling time. Consequently, unit operations where PV overshoot is of primary concern (e.g. chemical reactor temperature), the PID tuning is oriented towards a cautious approach to a new setpoint.
Setpoint response is the primary focus for batch processes. For continuous operations, the IAE and peak error for disturbances were traditionally more of a concern since setpoint changes were relatively infrequent. The need for automated startup and flexible manufacturing has created more setpoint changes due to production rate and grade changes. Setpoint response is becoming increasingly important for all processes. Options are discussed here to eliminate overshoot without detuning and increasing the integrated and peak errors from unmeasured disturbances.
One option is to use a "Two Degrees of Freedom" PID structure that allows the user to set the setpoint weight factors beta and gamma for the proportional and derivative modes, respectively. If these factors are both zero, the proportional and derivative action is on PV (PD on PV, I on error). If these factors are both one, the proportional and derivative action is on error (PID on error). What seems to work best in practice are both factors set about equal to 0.5 for minimal overshoot without too much of an increase in rise time.
Another option is to use a setpoint lead-lag where the lag is set equal to the PID reset time and the lead is set to be ¼ of the lag. A lead of zero is equivalent to setpoint weight factors of zero (PD action on PV, I on error). In this case a setpoint filter could have used instead of a lead-lag. A filter or the corresponding structure is the right choice if overshoot must be eliminated even at the expense of a significant increase in rise time (e.g. bioreactor temperature).
Slide 2 of Options-to-Improve-Setpoint-Response.ppt shows how a PID tuned for maximum disturbance rejection performs for 4 loops for a process time constant of 50 seconds and total loop dead time of 10 seconds.
Loop 1: no options to reduce overshoot (beta = 1.0 and gamma = 0.0)
Loop 2: setpoint lag = reset time and lead = ¼ lag
Loop 3: same as loop 2 except setpoint lead = 1/5 lag
Loop 4: beta and gamma both equal to 0.5 with no setpoint lead-lag
Slide 3 shows the rise time and overshoot identified by a Metrics block for the 4 loops.
Loop 1: rise time = 22 seconds, overshoot = 3.06%
Loop 2: rise time = 44 seconds, overshoot = 0.71%
Loop 3: rise time = 46 seconds, overshoot = 0.67%
Loop 4: rise time = 41 seconds, overshoot = 0.76%
Loops 2, 3, and 4 achieve about the same results but loop 4 shows an initial kick and an additional peak in the PID output from derivative action on error. The kick and subsequent peak in the PID output may be psychologically or physically disruptive depending upon operators and applications.
If a set point filter or a beta and gamma were both equal to zero (PD on PV and I on error), overshoot would be eliminated but the rise time would have increased to about 100 sec for this case.
If you have a DeltaV Simulate Pro system, the above options and many more can be run from an operator interface provided to develop a better understanding and process control improvements by a download of a Virtual Plant. The configuration has the composite template blocks for simulating metrics, control valve dynamics, and wireless devices.
If rise time and overshoot must both be at the absolute minimum possible, an output tracking strategy can be used to first position the PID output at the output limit, next position the output to the final resting value when the PV one dead time into the future is projected to reach setpoint, then hold the final resting value for one dead time, and finally release the output back to PID control. This strategy described in the Control May 2006 article "Full Throttle Batch and Startup Response" works best for processes with a slow gradual response (near-integrating and true-integrating processes).