Determination of all stabilizing analog and digital PID controllers

In this paper, a unified approach is introduced for finding the stability boundary and the number of unstable poles in the integral derivative (ID) plane for continuous-time or discrete-time PID controllers. The ID plane is particularly important because in this plane it is easier than in the PI or PD planes to determine the entire stability region. These problems can be solved by finding all achievable PID controllers that stabilize the closed-loop polynomial of a single-input single-output (SISO) linear time invariant (LTI) system. This method is used to predict the number of unstable poles of the closed-loop system in any region of the parameter space of a PID controller. 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 case as the sampling period approaches zero. A key advantage of this approach is that the stability boundary 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 continuous-time or discrete-time stability region and the number of unstable poles of the closed-loop system.