8.1 Derivative compensation

Thus far, our primary tool for affecting closed-loop transient response performance has been proportional control, thereby restricting us to the transient responses available on the preexisting root locus. Recall from sections 7.5 and 7.6 that our goal for integral compensation is to improve the steady-state error while changing the preexisting root locus as little as possible. To improve the transient response (e.g., settling time) we must alter the shape of the root locus.

Suppose that we want, for transient response requirements, a dominant closed-loop pole \(p\) to be in a location where the root locus does not intersect. We pursue the following method: introduce a cascade ideal derivative, or proportional-derivative (PD) compensator acting on the error signal, with the form \[\begin{align} K (s-Z_D), \end{align}\] where \(K\in\mathbb{R}\) is a gain. The zero location \(Z_D\) is chosen such that the root locus intersects \(p\) …