The term "signed error squared" (E2 < 0 if E < 0) more accurately reflects the nature of the relationship. To achieve this, we use |E| E in lieu of E2.
Figure 5 illustrates the relationship for the proportional mode. The dashed line is the error gap relationship from Figure 1. The two relationships are very similar, which suggests that if one will work, the other probably also will.
The proportional mode equation is:
M = KC |E/10| E + MR
Sensitivity = KC |E/10|
Incorporating the factor of 10 into the control equation gives the following behavior compared to the linear control equation:
E < 10%. The sensitivity is less.
E = 10%. The sensitivity is the same.
E > 10%. The sensitivity is greater.
Whereas the sensitivity of the error gap equation increases abruptly when the control error crosses the error deadband, the sensitivity of the error squared equation increases linearly with the control error.
The error squared relationship in Figure 5 provides only one adjustment, the controller gain. But for the same reasons that a high deadband and a low deadband are appropriate for the error gap equation, different sensitivities or gains are appropriate for the error squared control equation:
When PV > SP, use KC, HI as controller gain.
When PV < SP, use KC,LO as controller gain.
For level control applications, you can compute the values for the controller gains from the following values:
• normal value of the set point (40%);
• value of vessel level at the location of the high level switch (95%);
• value of vessel level at the location of the low level switch (5%); and
• typical value of controller output bias, MR (70%).
To ensure the final control element has been driven to an extreme before a trip occurs, you can compute the appropriate value of the controller gain from the error squared control equation:
KC = [(M – MR)/E] |10/E|
To determine KC,HI:
PV = 95% (high level trip point)
E = PV – SP = 55%
M = 100%
KC,HI= [(100% - 70%)/55%] |10%/55%| = 0.099 %%
To determine KC,LO:
PV = 5% (low level trip point)
E = PV – SP = -35%
M = 0%
KC,LO = [(0% - 70%)/-35%] |10%/-35%| = 0.57 %%
Figure 6 illustrates the performance of the error squared control equation for KC,HI= 0.1 %/% and KC,LO = 0.5 %/%. The low sensitivity above the set point makes the controller more tolerant of errors when PV > SP. Consequently, the vessel level is above its set point more than it is below it. Because of the higher sensitivity below the set point, rather rapid decreases occur in the controller output and the discharge flow when the vessel level drops below its set point. Reducing KC,LO would slow these changes — but would not ensure that the control valve is closed should the vessel level drop to the location of the low-level switch.
CECIL L. SMITH is president of Cecil L. Smith, Inc., Houston, Tex. E-mail him at email@example.com. This article is based on concepts from his book "Practical Process Control," published by John Wiley & Sons.