Stabilising PID controller for time-delay systems with guaranteed gain and phase margins

A new analytical-graphical method is proposed for computing the region of stability for proportional-integral-derivative (PID) controllers based on the Hermite–Biehler theorem. By this method, a PID controller is designed to ensure the Hurwitz stability of a time-delay system with any order of transfer function. First, the possible range of the derivative part that makes the system stable is obtained. Then, the stability region is found by applying the Hermite–Biehler theorem. It is shown that this theorem can be extended to quasi-polynomials functions in the form of $ \psi (\nu,e^\nu ) $ ψ(ν,eν) to find the stability region of systems with time delay under certain conditions. Using this, the proposed approach can guarantee specified gain and phase margins for time-delay systems; therefore, it is very beneficial and advantageous for the control of practical plants. Five different examples including two practical systems are studied throughout the paper to illustrate the applicability, performance, and efficiency of the proposed control approach.