A unified approach for stabilization of arbitrary order continuous-time and discrete-time transfer functions with time delay using a PID controller

Abstract. The object in this paper is to find the stability regions of an arbitrary order discrete or continuous time transfer function with time delay. The stability bounds of the proportional integral (PI) controller are found in terms of the proportional ( pK) and integral gain ( iK). The delta operator is 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 a zero. The advantage of this method is that designers can find the stability boundaries when only the frequency response and not the parameters of the plant transfer function are known. A unified approach allows us to use 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. 1