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.
A common operating concern is robustness, that is, to remain well away from stability limits. With minimum-IAE tuning of the PID controller, the gain Kp of this process can increase by 80% before instability is reached, and Στ can increase by 75%—very acceptable margins both. If the process parameters are more variable than that, compensation should be applied, either through valve characterization or adapting the controller settings as a function of flow, needed when controlling heat exchangers2.
Another objection to tight tuning is measurement noise, which the proportional and derivative gains can pass along to the valve, causing excessive wear. Rather than detuning the controller, filtering may be applied, with the caution to use as little as necessary, for it augments the IE function:
where Τf is the filter time constant, and Δt is the sampling interval of the controller. In addition, the filter time adds to the dead time in the loop, requiring a further increase in all three controller settings. Sampling has a similar effect.
Fig. 4 gives the step-load curves for three controllers, all tuned for minimum-IAE response. Derivative action is very effective against the multiple lags of distributed processes, as demonstrated in the substantial improvement of PID over PI control. Derivative is under-utilized in industry for a variety of reasons, but should be used whenever possible in high-value composition and temperature loops, for its obvious advantages in reducing peak deviation, IE and period.
Further reductions in those same curve features can be achieved by adding dead-time compensation τ to the PID controller, a capability that has been available for about twenty years, but rarely used. The controller dead time essentially delays integration by an amount similar to the process dead time, allowing a much shorter integral time. The IE formula for this controller is the same as Eq. (6), with controller dead time taking the place of Δt. Optimum settings for all three controllers on a distributed process are given in Table 1, along with the resulting IE, peak deviation e1, and period of the loop Τo, all normalized by division by process gain Kp, time constant Στ, and disturbance size expressed in terms of equivalent change in controller output Δm.
Table 1. Optimum controller settings and resulting response characteristics
The PIDτ controller has the highest performance available in a commercial product, but is rarely used owing to its robustness limitations. It is difficult to tune, because a mismatch between its dead time and that of the process in either direction can cause instability. Beginning from a condition of optimum tuning, limits of stability are reached when Στ increases by 37% or decreases by 8.5%, and when process gain increases by 25%. It has been successfully applied to controlling steam temperature from a superheater by programming all four settings as functions of steam flow3.
Feedforward Control Structure
If feedback alone cannot satisfy the economic objective, feedforward will be able to, but only if properly engineered. Figure 5 illustrates three methods of applying feedforward to control composition or temperature, two of which have serious limitations. In the top illustration, the process load—the feed rate to the process, for example—is added as an impulse function to the output of the PID controller. The impulse function is the difference between the measured value of the load and its lagged value; this is then multiplied by an adjustable gain K, and added to the PID output to set the manipulated flow. The gain adjustment is used to calibrate the system, so that a sudden change in load will produce the correct change in the manipulated flow that will cancel its effect on the controlled variable.