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

Abstract

The object of this paper is to find the stability regions of an arbitrary order transfer function with time delay using a unified approach for continuous and discrete-time systems. The stability boundaries of the proportional-integral (PI) controller, proportional-derivative (PD) controller and proportional-integral-derivative (PID) controller are found in terms of the proportional gain ( K <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</inf> ), integral gain ( K <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> ) and derivative gain ( K <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</inf> ). 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 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 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.