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Given the huge installed base of PID controllers, it was logical to expect that some model-based proponents would develop tuning methods that could make a PID appear to emulate a model-based controller in a closed loop. This approach seems to have been encouraged by the false assumption that load disturbances enter the loop directly at the controlled variable with no dynamics in their path, as described by Chia and Lefkowitz in "Robust PID Tuning Using IMC Technology," InTech, Oct., Nov. 1992.
In practice, load variables are very much like manipulated variables, being flow rates or properties of flowing streams, like temperature and composition, and often are driven by the output of another controller. Their dynamics are similar and, in fact, usually identical to those of the manipulated variable, consisting of dead time and dominant lag. For example, in a liquid-level loop, inflow could be manipulated to balance against the outflow as load, or vice-versa.
In recent years, this practice of emulating model-based response using PID controllers has been simplified with the introduction of "lambda" (λ) tuning, wherein λ represents the closed-loop time constant of the loop’s response to setpoint changes. Proponents suggest values for λ ranging from one to three times the dominant process lag. With λ = 1, setpoint response would be essentially the same as stepping the controller output in manual; higher values will only make it slower. When controlling flow, whose response time may be 5 seconds or thereabouts, this approach is reasonable.
But there are lambda-tuning rules for PI liquid-level controllers which make no sense at all. Liquid-level controllers are in practice never required to respond to setpoint changes, and interestingly, require no integral action to do so if required. A level controller’s output must match the load in the steady state, therefore, its output will be the same after a setpoint change as before. Then ∆m in Equation 1 is zero and, therefore, IE is also zero, resulting in an equal-area overshoot. A proportional controller can drive level to a new setpoint without overshoot, but with integral action, setpoint overshoot is unavoidable. Consequently, lambda tuning of a PI controller on an integrating process cannot achieve its intended result of exponential setpoint response.
Lambda-tuning rules for an integrating process such as liquid level are given by Blevins, McMillan, Wojsznis and Brown in Advanced Control Unleashed, ISA (2003), p. 220:
where closed-loop time constant λ can be as high as 3t1. These rules give a clue to the weakness of lambda tuning: the controller settings vary directly with λ, which is chosen arbitrarily, usually as a multiple of the dominant process time constant τ1. By contrast, the minimum-IAE rules vary controller settings with dead time, which is the limiting factor in the controllability of a process.
Compare the integrated error for the two methods by solving Equation 1 for each set of rules. For lambda tuning, IE/∆m = λ2/τ1, whereas for minimum-IAE tuning, IE/∆m = 4.2τd2/τ1. To achieve equivalent performance, λ would have to be set to 2td, which for lag-dominant processes is always a fraction of τ1. If λ is, in fact, set there, then both tuning rules arrive at essentially the same results. So the real problem with lambda tuning is that λ is linked arbitrarily to the process time constant, rather than fixed at twice its dead time.
Applying the value of one time constant to lambda results in a proportional band of twice the optimum and an integral time also twice the optimum. Load responses are simulated in Figure 2 for an integrating process whose time constant is five dead times.
With the proportional band and integral time both doubled under lambda tuning, IE is now four times the optimum, and peak deviation is higher by a factor of 1.58 and takes more than twice as long to be reached. Since IE increases with λ2, investigating higher values is pointless.
Lambda-tuning rules for self-regulating processes are given by Blevins, et. al., as:
where Kp is the steady-state process gain. IE/∆m then comes out to be Kpλ. Again, observe the arbitrary setting of the proportional band as a function of λ, which could be anywhere from 1τ1 to 3τ1. By comparison, the minimum-IAE settings for a lag-dominant process are similar to those given in Equation 2, except that the coefficient for the integral time reduces from 4 to 3.4 to 2.9 as the τd/τ1 ratio increases from 0 to 0.1 to 0.2. This covers the bulk of fluid processes such as heat exchangers, columns and stirred tanks.
Consider a self-regulating process where τd/τ1 = 0.2. Minimum-IAE tuning gives Popt = 105Kptd/τ1 and Iopt = 2.9τd, with a resulting IE/∆m = 3.05Kptd2. To approach this result with lambda tuning, λ would have to be reduced to 0.12τ1, or 0.6τd. However, Equation 4 does not allow an integral time less than τ1, so minimum-IAE results cannot be approached. Figure 3 compares the load response for the case where λ= τ1