The alternative is to adjust the setpoint of the composition controller so that it exceeds the specification limit by the expected variations in quality under all but the worst cases. This is also costly in that it results in quality giveaway all the time. Yet this is the normal condition of most product-quality controllers in most plants, and offers the largest economic potential for tighter control in processing plants universally. To minimize giveaway in the face of unsafe-side variations, the amplitude of those variations must be minimized.
The curves in Fig. 1 are typical of a loop responding to a step change in load—the flow or composition of one of the ingredients entering the process—and the subsequent reaction of a well-tuned PID controller to that upset. A step was selected as it is the most difficult disturbance for a control loop, containing all the frequencies from zero to infinity—a step has the same power spectrum as white noise. At the zero-frequency end, it requires a permanent change in controller output, necessitating integration. At the other end, its rise time is zero, requiring vigorous proportional and derivative action. Finally, it is commonly encountered when equipment trips or is suddenly switched, and is the easiest test to administer. In their monumental work on controller tuning in 1941, Ziegler and Nichols used a pneumatic bias regulator in the controller output to simulate a load change at the process input. We can do it by transferring the controller to Manual while in a steady state, stepping its output and immediately transferring to Auto.
The well-damped step-load response curve has a first peak much higher than all the rest. If that can be cut down, the setpoint can be moved closer to the specification limit. Since it is a single peak, it is easily attenuated by any downstream capacity, allowing the margin between setpoint and specification limit to be biased somewhat less than the expected peak deviation. The amplitude ratio Ar of output to input for a cycle passing through downstream capacity of time constant Τ1 is
where Τo is the period of the wave. Where the time constant of the downstream capacity is long compared to the period, the operating margin can therefore be small. For example, a 1-minute cycle passing through a 10-minute capacity is attenuated by a factor of 62.8. With very large capacity, load upsets are equally likely in both directions, and IE over time can approach zero.
Peak deviation and period are then the key factors affecting the operating cost for unsafe-side variations. The first step in minimizing peak variation and period following a load change is to be sure that the controller is paired with the manipulated variable having the most influence over it, and less over other associated variables. (This concept was presented in Ref.1 with regard to distillation and will not be repeated here.) Having made that pairing, performance is then determined by controller selection and tuning.
As useful as IE is in evaluating loop performance, it is incomplete. IE can be lowered by successive reductions in proportional band and integral time according to Eq. (3), but that also reduces the damping of the loop, leading to instability. A more complete performance criterion is Integrated Absolute Error (IAE), which has a minimum value that can be reached by proper tuning. The sign of the deviation e is simply removed before integration. In this way, the IAE of an oscillating loop will increase without end—IAE penalizes both load response and instability, reconciling them. For the curve of Fig. 2, IAE is only 9% higher than IE, so IE is being effectively minimized by minimizing IAE.
(Other related criteria are Integrated Square Error (ISE), where the deviation is squared before integrating, also eliminating the sign, and the Integral of Time and Absolute Error (ITAE) that penalizes errors increasing with their duration. The square function has no economic significance, but is simply mathematically convenient, and the time function of ITAE is not applicable to continuous processes.)
There can be difficulties in calculating IAE in the field; for example, any noise will cause it to increase without limit. However, it does not need to be monitored to achieve minimum-IAE tuning, because its closed-loop load-step response curves have characteristics readily identified in simulations that can be duplicated in the field. After having simulated many of these loops, the characteristics shown in Fig. 2 are found to be common to most: They are low damping ratio δ, small overshoot Ω, and short period Τo. Fig. 2 was obtained by stepping the load to a distributed lag under PID control with these results: