Stability regions in the parameter space for a unified PID controller

In this paper a unified approach is presented for finding the stability boundary and the number of unstable poles for an arbitrary order transfer function with time delay in continuous-time or discrete-time systems. These problems can be solved by finding all achievable proportional integral derivative (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 system. If the plant transfer function is known, the stability regions can be found analytically.