PID Control of Unstable Time-Delay Systems: Tuning and Robustness

Abstract

The paper is aimed to the design of a general class of PID like linear controllers for first order systems with time delays. Several approximations for time delay were used. All of them can be covered by a second order transfer function. Then explicit PID formulas were derived for SISO systems with a feedback as well as with a feedback-feedforward structure of the controlled system. The approach is especially suitable for integrator and unstable first order time-delay systems. The proposed control synthesis is based on general solutions of diophantine equations in the ring of proper and Hurwitz stable rational functions RPH and the Youla-Kučera parametrization of controllers is utilized. The conditions for asymptotic tracking are obtained through the divisibility conditions in the ring RPH. The proposed methodology covers stable and unstable plants with time delays. Moreover, a scalar tuning parameter which influences properties of the closed-loop system has been introduced. Robustness of proposed algorithms can be studied through the infinity (H∞) norms and through the sensitivity functions of controlled systems. A Matlab + Simulink program system for automatic design and simulation has been developed. The program system is suitable for a wide engineering utilization without a deep insight into the control theory. Some simulations and Nyquist plots of unstable time-delay systems are included.