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.
In the steady state, there is no difference between the load and its lagged value. This is considered operationally desirable, in that the output of the controller and the manipulated flow have the same steady-state value, as in a conventional cascade system.
However, it has a profound defect: After a load change has passed, the feedforward compensation gradually returns to zero, requiring the PID controller to move its output to the new steady-state value dictated by the load. In essence, feedforward correction is only dynamic—there will be a Δm required by the controller, which will have to integrate an error to get there. The IE is, therefore, no different with or without impulse feedforward. The impulse acts to flatten the disturbance, with the result that the same IE is spread out over a longer time. Peak deviation alone is reduced—but there is a better way.
Fig. 5. Three possible methods of applying feedforward control.
In a linear feedforward system, the load variable passes through a dynamic compensator (if required) and then is added to the controller output with a gain adjustment as shown. Again, the gain is set so that the effect of the load change is canceled by the resulting change in setpoint of the manipulated flow. This is the preferred arrangement when the process is linear and its gains are constant.
The best example of its application is that of boiler drum-level control. The level responds equally to the load—steam flow—and the manipulated flow of feedwater. The feedforward gain is thereby constant at 1.0. Linear feedforward control of boiler drum level has been applied as early as 1929, in the "three-element" feedwater system.
In the control of composition or temperature, however, the gain of the controlled variable to load flow is not constant. It varies with both the feed condition and the product composition or temperature setpoint. Feedforward gain can be set to match the process gain initially, but it will be incorrect when feed conditions or product specifications change. For example, the ratio of high-octane blending ingredient to the flow of the main stock varies with the octane rating of that stock and the setpoint of the octane controller.