## Pseudo-Derivative Feedback Control

In 1977 Phelan  published a book titled "Automatic Control Systems" which emphasizes a simple yet effective control structure. This structure provides all the control aspects of PID control, but without system zeros that are normally introduced by a PID compensator. Phelan named this structure "Pseudo-derivative feedback (PDF) control from the fact that the rate of the measured parameter is fed back without having to calculate a derivative.

The differences between PID and PDF control are best illustrated by comparing the application of each structure on a first order plant having a pole at "a" rad/sec. This diagram shows the application of PI control, and by clicking the block, it's step response can be observed.

The closed loop transfer function for this system is The equivalent PDF control structure feed's back a proportional component of the measurement rather than the error, following the integrator.

The closed loop transfer function for this system is It's immediately apparent that the characteristic equation for the PDF and PI structures are identical, the difference being that the PI compensated system contains a zero introduced by the compensator at Ki/Kp. Click the diagram for the PDF control, and note the overshoot of the PI compensated system does not occur in the PDF compensated system.

If overshoot response to commanded input cannot be tolerated in the system, a pre-filter can be used on the system input as shown in this block diagram The overall transfer function for this system is identical to the transfer function for the PDF compensated system shown above. Note that the gains selected for the PI and PDF examples are the same, and they set the damping at about 0.9 for each system. The PDF compensated system has absolutely no overshoot, but the PI system having identical damping, has overshoot caused by the zero at Ki/Kp. The zero introduced by the PI control does however provide more lead the control, and thus a faster rise time, but at the expense of overshoot.

Conclusion: The PDF structure internalizes a pre-filter one would apply to cancel the zeros introduced in the PI (or PID) equivalent system. The PDF structure can therefore be used in lieu of a PI (or PID) compensator where one may also be using a pre-filter to eliminate overshoot.

 Automatic Control Systems Richard M. Phelan, Cornell University Press, Ithaca, New York, 1977