This approach can be used for the measured variable for any loop, not just the inner loops of cascades. But whenever this is done, the choice for the resolution can potentially have an impact on the performance of the PID controller.
Resolutions Imposed by the Manufacturer
In PLC installations, "registers" traditionally stored 16-bit integer values (or sometimes 4-digit BCD values). In older models, no options were offered. Current models usually provide the option of 32-bit registers for either integer representations or floating-point representations. However, product features often reflect past practices and are slow to change.
In one product (that will not be named), a 16-bit integer register is used to store the reset gain, which is the product of the controller gain (in %/%) and the reset time (in repeats/minute). The reset gain is represented as an integer value expressed to 0.1 (%/%)/minute. A value of zero for the reset gain disables the integral or reset mode. Otherwise, the smallest value for the reset gain is an integer value of 1, which is a reset gain of 0.1 (%/%)/minute. If the controller gain is 1.0%/% (certainly a reasonable value for some loops), the minimum reset rate is 0.1 repeats/min. This translates to a reset time of 10 minutes. An integer value of 2 (the next increment) is a reset gain of 0.2 (%/%)/minute. This translates to a reset time of 5 minutes. If the controller gain is set to 1.0%/%, it is not possible to set the reset time to 7.5 minutes. This may be acceptable for automotive applications, but not in chemical plants.
In another product (which will also not be named), the input processing software provides an option for square root extraction on the input value (to support inputs from head-type flow meters such as orifice meters). The range of the input is 0 to 4096, which is 1 part in 4096 (a reasonable resolution). However, the product simply takes the square root of the integer value. The result has a range of 0 to 64 (642 = 4096). The problem with this approach is a poor resolution for the flow, specifically, a resolution of 1 part in 64. The appropriate approach is as follows:
However, the calculation must be performed with either 32-bit integer arithmetic or floating-point.
This article has illustrated some of the ways that inadequate resolution can impair the performance of a PID controller. There are certainly others. These problems seem to appear more often in applications implemented with PLCs, but they can and do arise in DCS applications as well (the example in Figure 1 was from a DCS application).
The problem here is that those doing the implementation and/or programming are very knowledgeable regarding the systems aspects of the application, but have little or sometimes no experience with PID control. The decisions are made based on ease of programming, compatibility with the features of graphical display devices, ease of troubleshooting, etc. The impact of the decisions on the performance of PID controllers often gets no consideration whatsoever.
Cecil L. Smith, PhD, PE, has a consulting practice devoted exclusively to industrial automation, encompassing both batch and continuous processes. He also teaches continuing education courses on various aspects of process control.