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

This paper focuses on the study of the behavior of critical roots when a dynamical system is stabilized by a PD-controller, for which the derivative action has been approximated by using two commensurate delays. The use of such an approximation leads to a characteristic quasipolynomial whose coefficients depend explicitly on the delay parameter. The aim of the paper is to address the way the delay parameter may affect the location of the roots of the corresponding characteristic function, and in particular the cases when "small" delays induce instability in the closed-loop systems. Such an analysis is performed by expressing the critical solution of the system as a delay-dependent power series. Illustrative examples complete the presentation