Designing Proportional Delayed Integral Control for Fast Regulation in Second-Order Systems: A Geometric Approach

This paper presents a geometric approach to designing a modified version of the classical PI control called Proportional Delayed Integral (PδI) to obtain the fastest regulation in a type of second-order linear system. Specifically, we first show that by adding an appropriate time delay in the integral component of the PI control, the system achieves a faster response to what can be obtained by using a classical PI control. Then, we outline the conditions under which our delay-based control stabilizes a specific class of second-order systems through an analysis of the spectral abscissa and a graphical approach called D-decomposition. Furthermore, we apply our results to determine the parameters of the fastest regulator.