Impact of Delay-Difference Approximation on PID Controllers: Stability and Performance Insights

This paper aims to explore the advantages and disadvantages associated with implementing derivative action in a PID controller using a delay-difference approximation scheme in continuous time. Despite the simplicity of the approximation, the characteristic function of the closed-loop system is a quasipolynomial including delay-dependent coefficients, which results in a challenging task. The proposed study primarily focuses on the way the controller gains and the delay affect the key characteristics of the dynamics such as asymptotic stability, noise rejection, and closed-loop performance for linear-time-invariant (LTI) minimum-phase second-order systems. Additionally, the paper provides some results concerning the delay margin of the delay-difference approximation of the closed-loop system. Finally, the proposed results are illustrated via numerical examples.