The integrating process gain (Ki or R) is very small or, equivalently, the process time constant (Tp) is very large relative to the dead time (Td) for temperature, composition, and gas pressure control of well-mixed volumes. For temperature loops on columns, evaporators, crystallizers and reactors, the process time constant is typically 10 times larger than the dead time. Users can choose very high controller gain or tight control. However, this assumes rapid movement of the controller output doesn’t upset the operator or other loops.
The integrated absolute error (IAE) for a disturbance is proportional to the integral time divided by the product of the process gain and controller gain for a non-oscillatory response. If the controller is tuned for maximum load rejection, the IAE is proportional to the dead time squared divided by the process time constant. A large time constant in the process slows the excursion rate of the actual process variable disturbances, and allows the controller to catch up with the disturbance.
The relationships for controller tuning and performance work no matter where the largest time constant is located. If it’s in the measurement, the measured process variable looks good, but the actual has gone off to parts unknown. If it’s in a sensor, transmitter or signal filter, it gives the illusion of good control, but is merely an attenuated version of the real world. A large time constant in the measurement converts part of the next smaller time constant in the loop to dead time. Dead time slows down the ability of the controller to see and compensate for disturbances (see Control Talk,” Aug. ’06).
Some confusion comes from the varied nomenclature and lack of detail for the three parameters used for a first-order-plus-dead-time model of a self-regulating process response. The process gain for controller tuning is a dimensionless static or steady-state gain that’s actually the product of the final-element gain, process gain and measurement gain (100%/span) for the controller in manual. It’s the final change in the process variable in percent divided by the change in manual controller output in percent, assuming no disturbances. I prefer to call it an open loop gain (Ko).
This Month's Puzzler
If a controller is tuned with a Lambda factor of 2, and the total loop dead time is one-tenth of the largest time constant in the loop, what wireless communication interval would start to cause a degradation in load response? Send an e-mail with your answer, questions, or comments to The Puzzler.
The process time constant is the largest time constant in the loop for a change in manual controller output. I call it the open-loop time constant (To) to distinguish it from the closed-loop time constant, which is the time constant for a set point change with the controller in automatic. A time constant is the time to reach 63% of the final change after the loop dead time.
The process dead time is the total dead time in the loop, or the time to the start of a distinguishable response in the process variable after the change in controller output. Sometimes the total loop dead time is called the “delay,” and the open loop time constant is called the “lag.”
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