Analytical Design of PI Controller for First Order Transfer Function Plus Time Delay: Stability Triangle Approach

In this study, proportional-integral (PI) controller design with a geometric approach for first order time-delayed systems is presented. This method can be expressed as an improved version of the weighted geometric center method. The method is based on calculating the center of gravity of a stability triangle selected inside the stability boundary locus (SBL). Stability boundary is determined along the
$(k_{p},k_{i})$
axes with the SBL. A stability triangle is formed with the two boundary points of the SBL intersecting the
$k_{p}$
axis for
$k_{i} =0$
and the point corresponding to the weighted geometric center value of the angular frequency. The center of gravity of the determined stability triangle gives the optimum PI control gains. Also, an analytical solution method for the PI controller design is presented. The proposed method (stability triangle) is examined on numerical examples. The time response performances of the controller calculated with the proposed method and the controllers calculated using the weighted geometric center method were compared. In addition, the comparisons of the PI controller determined by the proposed method and the different controllers selected in the neighborhood of this point are included. Comparisons with the studies in the literature including the PI controller design of the time-delayed first-order open-loop unstable transfer function are presented. As a result, it has been seen that the proposed method determines the parameters that provide optimum system performance in the tested region.