Synthesis and design of PID controllers

controllers for discrete-time systems and time-delayed systems. By using bilinear transformation and orthogonal transformation, earlier research results obtained in the continuous-time case are extended to discrete-time situation. The complete set of stabilizing PID controllers for the discrete-time systems is thus obtained. Moreover, this set remains to be a union of convex sets when one particular parameter is fixed. Thus a method to design robust and non-fragile digital PID controllers is proposed by following a similar design procedure for the continuous-time systems. In order to find the stabilizing controller set for systems with time-delays, the relationship between the Nyquist Criterion and Pontryagin?s theory is investigated. The conditions under which one can correctly apply the Nyquist Criterion to time-delayed systems are derived. Then, the complete set of stabilizing PID controllers for arbitrary order LTI systems with time-delay up to a given value is obtained. Furthermore, the stability issue of a system with fixed-delay is also studied and a formula which provides complete knowledge of the distribution of the closed-loop poles is presented. Based on this formula, stabilizing P and PI controller sets for the system with fixed-delay can be computed.