The ABCs of Controller Tuning
All of the major tuning methods end up with the same expressions for PID gain, reset time, and rate time when the tuning objective is maximum disturbance rejection. Differences come down to tweaking of the a, b, and c coefficients. Furthermore the use of external-reset feedback (dynamic reset limit), and options such as a setpoint filter, setpoint rate limit, and an enhanced PID for wireless eliminate the need to retune for different objectives.
The ISA Automation Week 2012 paper “Effective Use of PID Features” discusses how diverse process objectives such as minimizing interaction and overshoot and maximizing coordination, equipment protection, production rate, and process efficiency can be met by the use of PID features without having to retune the PID. Furthermore, these features enable expressions for tuning to be simplified.
The major discrepancies between the methods predominantly used in the chemical industry up to the 1990s, the newer Lambda tuning method developed in the pulp and paper industry, and the Internal Model Control (IMC) tuning method popular in universities have been resolved. The same gain expression can be achieved by setting the closed loop time constant (Lambda) equal to the loop deadtime. The same expression for reset time can be attained by treating the loop as a near-integrator and setting the closed loop arrest time (Lambda) equal to the loop deadtime. Finally, the same expression for integral time can be obtained by using the Lambda or IMC tuning rules for integrating processes and realizing that the secondary process time constant in processes where derivative is used is about half of the deadtime (e.g. mixing and thermal lags are about half of the total deadtime).
The controller gain expression depends upon whether the process is self-regulating or integrating. The two expressions can be equated by realizing the integrating process gain for a near-integrator is the self-regulating process open loop gain divided by the open loop time constant.
For deadtime dominant processes the b coefficient is reduced towards 0.5 and the c coefficient is reduced towards 0.0 as the total loop deadtime becomes increasingly larger than the open loop time constant. Expressions have been developed to compute b as a function of the degree of deadtime dominance. People tend to simply set c to zero because the improvement is so marginal from derivative action.
In the next blog we will discuss more of the tuning aspects for deadtime dominant processes. In the meantime have fun with the a, b, and c coefficients for tuning an ISA standard form PID. Just remember to use the dynamic reset limit to prevent the PID output from changing faster than a secondary loop or final control element can respond.
Kc = a * [to / ( Ko *θo )] (self–regulating)
Kc = a * [1 / ( Ki *θo )] (integrating)
Ti = b *θo
Td = c *θo
a = controller gain coefficient (e.g. 0.4)
b = controller reset time coefficient (e.g. 3)
c = controller rate time coefficient (e.g. 0.5)
Kc = controller gain (dimensionless)
Ki = integrating process gain (%/sec/%)
Ko = self-regulating process open loop gain (%/%)
Ti = integral reset time (sec)
Td = derivative rate time (sec)
to = self-regulating process open loop time constant (sec)
θo = total loop time deadtime (sec)