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:
The time scale is normalized by dividing by Στ, the sum of all the lags in the distributed process—it is identified by the time required for 63.2% complete response from the initiation of a step input in the open-loop. (See Ref.2 for more information on this process common to heat exchangers, multistage columns and stirred tanks.) Essentially the same values of decay ratio and overshoot represent minimum-IAE response for other processes using other controllers; the period varies with the process dynamics and the type of controller.
There are many different methods used to tune controllers, with a wide range of effectiveness. Some are intended to optimize setpoint response, but these are not recommended, as they invariably compromise load response. Once a controller has been optimized for load response, setpoint filtering can then be added where necessary. The economically important loops in a process plant—controlling composition, temperature and pressure—operate at constant setpoint all the time, but are regularly exposed to load variations.
Most tuning methods result in sluggish load response, costly in IE, peak deviation, and period. Some methods set integral time equal to the primary process time constant, whereas minimum-IAE requires it to be half that value for a PI controller and one-fourth for a PID controller. Two examples are compared in Fig. 3 against a minimum-IAE curve. Doubling the proportional band increases peak deviation by 44% as it doubles IE—heavier damping bought at a high price. Doubling the integral time has little effect on peak height as it doubles IE—it principally affects overshoot. These two responses can guide the user as to which parameter needs adjusting.