Most of the control literature focuses on minimizing the integrated absolute error (IAE) for a step disturbance, often in a linear system. In the process industry, there are many other objectives and complications that require special attention. Here we quickly review when and how to achieve minimum IAE and see how a rule of 5 comes into play for applications with different challenges.
Step disturbances are rare if the automation system is well designed and complete. The fastest disturbances are liquid pressure and flow. Often a pressure change shows up as a flow disturbance. If flow controllers and throttling valves are installed rather than on-off valves discretely opened and closed on all the streams, the flow loops correct for pressure upsets and provide a closed loop time constant for changes in flow. Pressure controllers can be tuned for tight control (maximum transfer of variability from controlled variable to manipulated variable) to reduce the propagation of pressure upsets. Level controllers can be tuned for maximum absorption of variability (minimum transfer of variability) to reduce the propagation of flow upsets as detailed at the end of the white paper. (Access the white paper here.)
Tight control (minimum IAE and/or peak error) is important for pressure and many loops for product formation (e.g. reaction) and purification (e.g. distillation). If the system is linear and the dynamics are well known (Big “Ifs”), the PID gain could approach being about half the PID gain from Ziegler-Nichols reaction curve method turning. This PID gain setting corresponds to a closed loop time constant for self-regulating and an arrest time for near-integrating and true integrating processes that is about equal to the dead time. For runaway reactors, the PID gain may be further maximized for worst case dynamics by an arrest time equal to half of the dead time. Lambda is the closed loop time constant in self-regulating tuning rules for processes with a time constant to dead time ratio less than 4 and Lambda is the arrest time in integrating process tuning rules for all other processes (near-integrating, true integrating, and runaway). In all cases, Lambda rather than a Lambda factor is used and Lambda is set relative to the dead time. This makes sense in that dead time is the primary fundamental limitation to loop performance and the ultimate period is a factor of the dead time (e.g., approximately 4 dead times for balanced and 2 dead times for dead time dominant self-regulating processes).
For many challenging situations, the rule of 5 is useful. Table D-1 Lambda Tuning Solutions for Difficult Situations and Different Objectives shows that to minimize the effect of resonance, inverse response, and nonlinearities, Lambda should be greater than 5 dead times. To prevent the violation of the cascade rule, the upper (primary) loop Lambda should be at least 5 times larger than the lower (secondary) loop Lambda. To minimize interaction, the slower loop Lambda should be at least five times larger than the faster loop Lambda. In these last two cases, it is in general preferable to make the lower and faster loop Lambda smaller rather than making the upper and slower loop Lambda larger. Note that in these situations, what may seem as a small disturbance can be amplified and propagated when these rules are violated. While the use of external reset feedback can help minimize consequences inherently for cascade control and by the use of directional move suppression for interactions, paying attention to the rule of five helps provide better control.
If pressure disturbances are an issue, equal setpoint filters rather than the equal lambdas shown in Table D-1 can be used for the coordination of flow loops to maintaining a blend or reactant feed stoichiometric ratio. This allows each loop to be tuned for the fastest response to pressure upsets.
For a perceptive, concise and direction approach to choosing tuning settings for a large variety of applications see the 2015 version of Good Tuning: A Pocket Guide – A Pocket Guide - Fourth Edition.
For more information on how to minimize the sources and adverse consequences of disturbances, see the October 2013 Control Talk Blogs “Disturbance Dynamics Recommendations” and “Effect of Disturbance Dynamics Perspective”.