New adaptive controllers are coming soon to your control room to individually schedule the tuning as a function of any variable. So given all the choices, what process variable generally works best taking into account what we have recently learned about mixing and process dynamics?
The short answer is controller output for continuous processes and level for batch processes. Of course this is just a best guess and doesn't replace the need to test any variable or calculation used to set controller tuning.
If you are curious as to how I arrived at the above conclusions, read on.
Let's consider first the flow loop. Nothing complicated here, we just need to remember that the controller gain is inversely proportional to the product of the valve gain, process gain, and measurement gain for a control loop. The valve gain is generally nonlinear since it is the slope of the installed characteristic of the valve. For flow, the process gain is one (how lucky can one be). Just like for other loops, the measurement gain is simply 100% divided by the scale span of the PID. So the only nonlinearity in a flow loop (barring a missing square root extractor for a head meter) is the valve. A good choice for the controller gain would be to schedule it as a function of controller output (position on the installed characteristic of the valve). The reset time is set equal to the largest time constant in the loop. For liquid flow, the process time constant is only 50 to 100 milliseconds, which is generally smaller than the effective time constant associated with the valve, measurement or DCS. Thus, the reset time depends upon on the slowest part of the automation system. If a signal filter in the DCS becomes the largest time constant in the loop, the reset time is approximately the filter time setting. For aggressively tuned flow loops or big valves, it is a good idea to enable the Dynamic Reset Limit and use the read back of actual valve position as the external reset to prevent the PID reset action from outrunning the speed of the valve.
Let's further consider that we put this flow loop to good use as a secondary controller for cascade control where the primary loop is level, temperature, or concentration. A secondary flow loop removes the control valve nonlinearity from the primary loop and makes the primary loop ready, willing, and able to use flow feedforward (e.g. a flow ratio corrected by the output of the primary loop).
Finally, let's focus on volumes with different types of mixing. The two major types are inline (e.g. pipeline) volumes that have only some radial mixing from bafflis or pipe fittings and vessel (tank) volumes that have axial mixing as the result of an agitator, eductor, and/or sparger. The inline systems have a uniform composition and temperature in a cross section but not along the length of the pipeline. The process dead time is much larger than the process time constant. These inline volumes provide little to no smoothing with respect to time and are called "plug flow." Well mixed vessel volumes have a uniform composition and temperature throughout the vessel volume. The process dead time is much smaller than the process time constant. These vessel volumes provide maximum smoothing with respect to time and are called "back mixed." For batch operations, these "back mixed" volumes have an integrating response. The following figures show the self-regulating response for "plug flow" and "back mixed" volumes for continuous processes and the integrating response for batch operations.
Mixing Effect on Open Loop Responses
For plug flow volumes, the residence time (volume/flow) becomes a process dead time making the dead time inversely proportional to flow. The process gain is also inversely proportional to flow. As a result, the primary controller gain for composition control is proportional to flow and flow squared, if the Lambda is set equal to a factor of the integral time and dead time, respectively. Since most applications set Lambda equal to multiple of the integral time, controller output would be a good choice again for gain scheduling. Examples of plug flow systems are pipelines, static mixers, desuperheaters, sheet lines, web lines, extruders, and sheet lines. The process dead time is larger than the process time constant in these primary composition loops. Like the secondary flow loop, the integral time depends upon the valve, measurement, or filter time lag.
One word of caution, these primary loops may not be much faster than the flow loop, so the primary loop may have to be tuned to be slower than expected to avoid violating the cascade rule (primary loop should be at least 4 times slower than the secondary loop). Using gain scheduling in the flow loop helps makes make the flow loop faster, which reduces the need to make the primary loop slower.
For back mixed volumes, the residence time (volume/flow) almost entirely becomes a process time constant for composition control. If the primary loop's integral time is set to be a factor of the time constant, it is then inversely proportional to flow. This assumes the injection delay associated with the dip tube or pipeline feed is small (not a good assumption for small additive or reagent flows). The process gain is also inversely proportional to flow. The process dead time is the turn over time and is relatively fixed for a constant agitator speed. Good gosh, controller output is again a good choice for scheduling tuning settings. Examples of back mixed volumes are agitated reactors and fermentors (except mammalian cell). Most agitated blend tanks, crystallizers, and evaporators behave more like a stirred reactor than a pipeline. The dynamics can be approximated by splitting the total volume into a small plug flow volume combined with a large back mixed volume.
For pH, I would use signal characterization to translate the controlled variable from pH to reagent demand based on the titration curve. This makes it just a reagent concentration loop whose process gain like other composition loops is inversely proportional to flow, which means I can again schedule the controller gain as a function of controller output.
Hmm, I wonder what the default variable will be for scheduling controller tuning for these self-regulating loops. Could it be controller output?
Composition loops of large back mixed volumes and batches have a "near" and true integrating response, respectively. The process gain is inversely proportional to liquid volume. For liquid temperature, the change in heat transfer surface area covered by liquid may cancel this effect out. For gas pressure, the process gain increases as the liquid level decreases. So for integrating loops, the variable for scheduling tuning is often level.
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