Optimal Design of the Refined Ziegler-Nichols Proportional-Integral-Derivative Controller for Stable and Unstable Processes with Time Delays

In this paper, the well-known refined Ziegler-Nichols (RZN) method is analyzed and extended to the control of unstable processes with time delays. It is shown that both the RZN control structure and the inner stabilizing control structure can be equivalent to the unity feedback loop with a set-point filter or, equivalently, a two-degree-of-freedom control structure. This implies that those analysis and design techniques developed for the classical control structure can be applied to the two special control structures. The original formulas for the RZN proportional-integral-derivative (PID) controller are empirical and can only be used in restricted scope. A new procedure is developed for designing the RZN PID controller. Suboptimal formulas without the restriction are analytically derived. In the procedure, the load response can be independently optimized. Nominal performance of the closed-loop system is discussed, and sufficient and necessary conditions for internal stability and robust stability are given. Numerical examples are provided to illustrate the proposed method and compare it with the original RZN method.