The most important loops on vessels and columns typically have a near or true integrating response. A self-regulating process is classified as near integrating if the process time constant is larger than four times the process dead time. The composition, pH, and temperature response of continuous columns tend to have a time constant to dead time ratio of about 6:1 for changes in the liquid material balance.
The composition, pH, and temperature response of continuous well-mixed vessels tend to have a time constant to dead time ratio of 50:1 or larger. Level and gas pressure have a true integrating response since changes in level or pressure have a negligible effect on the discharge flow notable exceptions occurring for gravity flow for liquid level control and relatively high but non critical pressure drops for gas pressure control. The composition, pH, and temperature of batch columns and batch vessels have a true integrating response in the normal operating range.
A notable exception is the composition control of a batch reaction when there is no deficiency of any reactant concentration. Here the response is near-integrating with a very large time constant to dead time ratio making differentiation between true and near integrating inconsequential. Reactors with a potentially runaway response are treated as true integrators with the intent being that control action is sufficient to prevent the loop from seeing the acceleration from a runaway response.
The designation of having an integrating response is critical in terms of tuning and recognizing there is a window of allowable controller gains, where too low of a PID gain as well too high of a PID gain will cause excessive oscillations. For a PID gain that is too low, the oscillations tend to be much larger and 10 times slower (e.g., period is 40 dead times for low PID gain and four dead times for high PID gain). For a PID gain greater than the ultimate gain, the oscillations can grow and the loop becomes unstable. For a PID gain that is too low, the oscillations will always decay but the decay rate becomes incredibly slow as the PID gain is decreased. For a runaway reaction, too low of a PID gain is disastrous in that the process can runaway reaching a point of no return.
The Lambda tuning rules switch from a Lambda being the closed loop time constant for a setpoint change for self-regulating processes to Lambda being an arrest time for a load disturbance with the objective of stopping the ramping effect of integrating processes and potential acceleration of runaway processes.
Questions from ISA Mentor Program Participant Hector Torres
- How do you calculate Lambda for near-integrating processes? I understand we should follow the Integrating Process rules but I am not clear as of how to determine the desired arrest time. You mention that for maximum unmeasured disturbance rejection a Lambda equal to the dead time is used. Also it is stated that a Lambda equal three dead times minimizes consequences of nonlinearities, inverse response and resonance.
- Why should we consider Lambda of one or three times dead time in these rules? To be identified as a near integrating process the time constant should be four times greater than the dead time. Why should we make Lambda a factor of dead time here? I remember it was mentioned that Lambda should be set at three or four times the largest of the dead time or the time constant. Would this apply here? Am I mixing in my mind the rules for self-regulating and integrating processes?
- I understand that integrating processes can have an inverse response that is problematic. What could be an example of inverse response? What do you mean by this?
Greg McMillan’s Answers
1) In integrating tuning rules, Lambda is the arrest time, which means for a step disturbance or step change in PID output, how long does it take for the PV to halt its excursion and start its return to setpoint. If you multiply the integrating process gain (%PV/sec/%CO) by a the change in controller output (CO%) required to get a ramp rate (%PV/sec) and then Lambda, you have the peak error (maximum excursion in %PV). For level and pressure it is easier to visualize in that the maximum PV excursion (peak error) for a maximum expected change in controller output added to the setpoint must not hit an alarm or trip point. The integrated error (% sec area between the PV and SP on trend chart) is the peak error (%) multiplied by Lambda (sec). Thus, the peak error is proportional to Lambda and integrated error is proportional to Lambda squared where Lambda is the arrest time set relative to the dead time.
2) The ability of a loop to handle changes in gain and dynamics is expressed by the gain margin and phase margin, which are both a function of Lambda relative to dead time. The Lambda tuning rules reduce to tuning rules commonly used for the last six decades if you realize Lambda should always be thought of and set relative to dead time and not a time constant or an integrating process gain (as mistakenly shown in various publications). Also for large time constants you have a near-integrating process and must switch to integrating tuning rules.
In the 5th edition of the Process/Industrial Instruments and Controls Handbook (1999 edition for which I became chief editor), Bialkowski on pages 10.52 and 10.53 shows how the gain margin and phase margin are a function of Lambda varying from one to five dead times. Elsewhere he talks about the concept of near-integrating processes. I think the rule sometimes states of choosing the largest of three times the dead time or three times the time constant are not in tune with advancement in understanding of Lambda always being thought of as a value relative to dead time. While Bialkowski did not say when to switch integrating tuning rules, I estimated that the switch point of when the time constant to dead time ratio was greater than four would result in tuning rules similar to what has been practices for the last six decades. This rule plus realizing that Shinskey essentially was using a Lambda of 0.6 x dead time to give the impressive maximum disturbance rejection results he has in his articles and books. His response is oscillatory. The acknowledged practical limit for a smooth response even if you exactly know the process dynamics and they never change is a Lambda equal to the dead time.
Thinking of Lambda as being three dead times is a good rule for both self-regulating and integrating tuning rules. For self-regulating processes with a time constant to dead time ratio between one and two , there might be some advantage of using three times constants instead of three dead times for PI control but I think the advantage is minimal and is negligible compared to other issues. If you have a nonlinear valve or process, you may need to increase Lambda to be five or six dead times unless you do signal characterization, gain scheduling or adaptive control.
To summarize, for all Lambda tuning rules (self-regulating and integrating), the most aggressive tuning if the process dynamics are fixed and exactly identified (rare case), is a Lambda of 0.6 dead times. This is case is normally only used to show how well Lambda tuning can do compared to other tuning methods (showcase test for gamesmanship). Normally, to deal with unknowns and nonlinearities a Lambda of three dead times is used but may be increased for more uncertain applications and greater changes in dynamics whether due to the process or valve. For bioreactors where the disturbances are extremely slow, rise time is inconsequential for setpoint changes, and process gains can change dramatically from the pre-exponential to the exponential growth phases, Lambda may be as large as 10 dead times.
To better understand different process responses and tuning objectives, watch the three-part ISA Mentor Program webinar on PID options and solutions:
3) Inverse response is where the initial response of the PV is in the opposite direction of the final response. If a feedforward correction arrives too soon it can cause inverse response. For feedback control, inverse response originates typically occurs when the feed stream throttled that has a temperature less than the operating temperature of the equipment. The classic case of inverse response is boiler drum level. An increase in feed water flow being colder than the boiling water in down comers will cause bubbles to collapse which will cause fluid to go down from the drum into the down comers causing shrink (decrease in drum level). Eventually the increase in feed water is heated enough in the down comers to increase the drum inventory (increase drum level). For a decrease in feed water flow, the bubbles in the down comers increase in number and size pushing fluid up into the drum causing swell (increase in drum level). This shrink and swell is quite common and can be reduced by feed water preheaters. For furnace and reactor temperature an increase in air flow or reactant flow that is colder than furnace or reactor temperature will cause the equipment temperature to decrease until the firing rate and reaction rate generates enough heat to increase the equipment temperature. This is the main reason plus the time constant of the concentration response why reactor temperature should not be controlled by manipulating reactant flow. This is true for liquids and polymers. For gases in fluidized bed reactors, the reaction rate and concentration time constant are so fast, inverse response is imperceptible.