Computation of All Stabilizing PID Controllers for HighOrder Systems with Time Delay in a Graphical Approach

In the present paper, the problem of computation of all stabilizing high-order time delay systems using well known and efficient proportionalintegral-derivative (PID) controller is investigated in a graphical approach. An efficient approach to this important problem is presented. Based on this approach, all PID controllers that will ensure stability are determined in a ( , , ) p i d k k k plane and then the stability boundary in a ( , ) i d k k plane for a constant value of p k is determined and analytically described. It is shown that the stabilizing ( , ) i d k k plane consists of triangular regions. The generalized HermiteBiehler theorem which is applicable to quasipolynomials and finite root boundaries which will be described in detail in continuation are studied to establish results for this design and also determining region of stability of designed PID controllers. Bode diagram criterion is used to show the stabilizing PID gain and phase margins. Step response is also used to show the correctness and advantage of the approach in two examples which are given to illustrate the method.