## Pseudo-Derivative Feedback Control

In 1977 Phelan [1] 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.

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

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