The Case Against Lambda Tuning

On these pages, in May 2010 (Maximizing Control-Loop Performance), I described two measures of economic performance for regulatory control loops based on the direction of a load disturbance:

1. If the controlled variable is driven from setpoint in the "safe" direction—away from specification limits—the economic penalty varies directly with its integrated error in returning to setpoint.
2. If the controlled variable is driven in the "unsafe" direction—toward the specification limit—the economic penalty varies with peak deviation, as that determines how far the setpoint must be positioned from that limit.

Fortunately, the integrated error (IE) is easy to calculate for any PID controller, ideal or not, as derived in that reference:

where ∆m is the percent change in manipulated variable m required to return the controlled variable to setpoint following a disturbance; P is the percent proportional band; and I is the integral time of the controller. This formula applies with or without derivative. In the presence of a sampling element of scan time ∆t, or a filter of time constant τ_f, or a dead-time compensator τ_dc, the integral time in Equation 1 is augmented thereby.

Figure 1 plots the response of an integrating process, such as liquid level with dead time, following a step-load change introduced at the controller output. The deviation grows during the process dead time just as it would in the open loop. From that point onward, the proportional action of the controller reduces its growth rate as determined by the width of the band, ideally reaching a peak before the next dead time elapses. But the wider the proportional band, the longer this takes, and the larger is the resulting peak.

Optimum Tuning

Regardless of the direction of the upset, the obvious strategy is to minimize the proportional band (maximize proportional gain), but this works only to the point where oscillation develops. Minimizing IE alone would result in continuous cycling. Damping must be provided to insure stability and allow for some variation in process parameters—a property known as "robustness." So an optimum proportional band must be found that provides the best combination of minimal peak deviation and IE, along with damping and robustness. These properties are combined in the red response curve in Figure 1, which happens to represent the minimum integrated absolute error (IAE) in response to a load step for a PI controller on this process, an integrator with dead time τ_d

Minimum-IAE response is a mathematical fiction that just happens to combine these desirable properties. IAE is not usually calculated on-line, as it grows with noise level and must be divided by the size of the disturbance to indicate performance. Its primary use is in simulations as in Figure 1. Then, once having accepted this as a desirable response, its pattern can be emulated while tuning on-line. The optimum response curve has a symmetrical first peak, a setpoint overshoot about one-tenth of the first peak, and a decay ratio of the damped oscillation of about 0.2. The robustness of the tuning is indicated by the approach to continuous oscillation when the proportional band is reduced by two-thirds—a change in loop gain equivalent to a 50% increase in process gain.

The tuning rules for attaining minimum-IAE step-load response on an integrating process with dead time are very simply:

where τ_d and τ₁ are the dead time and integrating time of the process respectively. The proportional setting determines the symmetry of the first peak and the damping as shown; the integral setting determines the degree of overshoot or undershoot. The optimum integral time was used for all of the curves in Figure 1.

The focus here is on load response being the primary objective of a regulatory control loop. This is especially true when controlling liquid level, pressure, temperature and composition in a continuous process plant or power plant. These loops operate at a single setpoint for months at a time. Their setpoint response is therefore irrelevant. (For those loops that must respond to setpoint changes, a solution is recommended later that can be effectively applied without compromising load response.)

Model-Based Control

Over the last two decades it has become fashionable to include a process model or its inverse into the controller, resulting in such a configuration as internal-model control (IMC) and model-predictive control (MPC). Academics especially have promoted these technologies as generally superior to PID control in providing more robust and predictable response to setpoint changes and other disturbances. But this practuce results from overlooking the dynamic functions in the path of load disturbances (see Control, May 2011, "Meditating on Disturbance Dynamics." When model-based controller parameters are simply matched to process parameters, load response curves decay exponentially along the time constant in the load path. Aggressive tuning can overcome this limitation, but this practice conflicts with the intent of the model-matching procedure.