Delay-difference Approximation of PD-Controllers. Insights into Improperly-posed Closed-loop Systems

Delay‐difference approximations of PD‐controllers. Improperly‐posed systems in multiple delays case

“…To analyze the effects of small delays on the discretization of the derivative action, we will use the so‐called Newton's Diagram Method 30 to find an appropriate change of variable, such an approach follows similar steps to those presented in References 31, 23 and 25. In fact, the change of variable aims to simplify the asymptotic behavior analysis with respect to the delay parameter.…”

“…Such an observation stems from the fact that, for $$$ \ell \in \left\{1,2\right\} $$$, any potential crossings take place exclusively at the origin. This distinctive characteristic has been extensively explored in previous works such as References 23 and 25 to characterize regions of properly/improperly‐posedness. Bearing in mind the previous observations, in the following we wish to express the crossing direction of the solution $$$ {s}^{\ast }=\boldsymbol{i}{\omega}^{\ast } $$$ as a function of $$$ \alpha $$$, seen as a parameter .…”

“…Additionally, the singular behavior of the unstable characteristic root was observed via a numerical example but without any further discussion of the observed phenomenon. Next, following a similar idea, an extension of the analysis to a second‐order approximation has been proposed in Reference 25. In such a work, it is important to emphasize an interesting property known as the locking real unstable roots mechanism .…”