On multiple-delay approximations of multiple-derivative controllers

We study approximation of multiple-derivative output feedback for linear time-invariant (LTI) plants using multiple-delay approximations. We obtain a condition on the plant and feedback that yields an equivalence between the closed-loop spectra for the approximate feedbacks and the desired multiple-derivative feedbacks. On the other hand, we use a scalar example to illustrate that multiple-delay approximations of sufficiently high derivatives may in some cases yield closed-loop spectra that differ greatly from the dynamics upon derivative feedback (for instance, containing many and very large right half plane poles), while in other cases replicating the derivative feedback perfectly. Finally, through understanding this dichotomy, we present a condition for stabilizing a SISO relative degree-1 plant when a delay implementation of the first-order derivative is used in the output feedback control law.