Optimal Tuning of Pid-deadtime Controllers for Integrating and Unstable Time-delayed Processes

A PID-deadtime controller is realized by inserting a time delay into the integral feedback circuit. Such a controller possesses a transfer function similar to a Smith predictor or an internal-model controller. Effective tuning of PID deadtime controllers remains unsolved even though they give better regulatory performance than a conventional PID control algorithm for processes with a long pure or distributed delay. In this work, a systematic method is proposed to develop optimal-ISE PID-deadtime controller tuning rules for integral and first-order unstable plus time delay plants, which cannot be controlled by a conventional PID controller along with a Smith predictor. Let the inserted time delay d be equal to the process delay θ, we derive analytical expressions for the integral-squared-error (ISE) performance index and its derivatives with respect to PID controller parameters. The optimal parameters that minimize the ISE performance index are determined from the necessary conditions of optimality, which are given in a system of nonlinear algebraic equations. By regarding the process time delay θ as the continuation parameter and tracing the one-dimensional curves defined by the equations, the optimal gain kc, integral time constant τi, and derivative time constant τd of a PID-deadtime controller are obtained as functions of time delay θ. Based on the traced curves, simple formulas are correlated to represent the optimal PID-deadtime controller settings in terms of process delay time θ. Such tuning formulas are developed for both servo and regulatory controls. As compared with optimal-ISE PID controllers, the optimal PID-deadtime controllers exhibit a significant performance improvement in the control of integral and unstable time-delay processes.