A unified approach for finding the stability boundaries of PID controllers for arbitrary order continuous or discrete time plants with time delays

Abstract

The object of this thesis was to find the stability regions for control of an arbitrary order continuous or discrete-time plan transfer function with time delay using a unified approach. The stability boundaries of the proportional-integral (PI), proportional-derivative (PD), and proportional-integral-derivative (PID) controllers were found in terms of the proportional gain Kp , integral gain Ki , and derivative gain Kd . The delta operator was used to describe the controllers because it provides not only numerical properties superior to the discrete time shift operator but also converges to the continuous time derivative operator as the sampling period approaches zero. A key advantage of this approach is that stability boundaries can be found when only the frequency response and not the parameters of the plant transfer function are known. A unified approach allows use of the same procedure for finding the discrete time and continuous time stability regions. If the plant transfer function is known, the stability regions can be found analytically. Regions where phase and gain margin specifications are met can also be found. Finally, it was shown how the number of unstable poles changes as the stability boundaries are crossed.