Delay-Difference Approximations of PD-Controllers. Improperly-posed Systems in Multiple Delays Case

In this paper, we study the behavior of characteristic roots of a class of strictly proper LTI SISO systems that are subject to a PD controller for which the derivative action has been implemented by means of a delay-difference approximation scheme that uses multiple delays. To such an end, the improperly-posed case is discussed in depth, i.e. the situations where small delays induce instability in the closed-loop system. The effect of adding more delays on the improperly-posed case is analyzed and an algorithm for determining the set of parameters that produces such a situation for a given number of delays is derived. Some illustrative examples complete the presentation