kp-stable regions in the space of time delay and PI controller coefficients

The destabilizing effects of time delays on linear systems make the control design for such systems a challenging task. This study aims at developing a new graphical method to define the stable regions in the space of uncertain delay and controller coefficients. The features of the Nyquist plot are used to design the proportional-integral controllers which guaranty the closed-loop stability for the first order and second order plants containing an uncertain time delay. The main aim is to determine the regions in the space of the uncertain delay and the controller coefficients for which the stable gain intervals exist. This is achieved by computing the values of the parameters at which the Nyquist plot gets tangent to the real axis. As a result, there is no need to sweep the gain or delay parameter to sketch the stability boundaries. Rekasius substitution is employed to convert the characteristic quasi-polynomial to a polynomial and then the polynomial results are used to derive the stability boundaries.