Robustness of classical tuning correlations for proportional-integral controllers

A formal robustness stability analysis of popular proportional-integral (PI) controller tuning rules for systems approximated by a first-order-plus-time-delay model is proposed. The uncertainty in the process model is represented by multiplicative parametric perturbations in the process gain, process time constant, and process time-delay. The zero-exclusion principle is used to characterize the robustness of the uncertain system in terms of the set of all perturbations that result in stable closed-loops. The robustness results recover the standard gain and phase margin concepts as special cases. In addition, a parametric stability margin is introduced for this class of problems as a generic metric via which alternative PI controller tuning rules may be compared in terms of robustness to simultaneous variations in the all three model parameters. The results of the paper can be applied to several disturbance-rejection and tracking PI tuning rules in widespread use, and permits comparing the tuning rules in terms of their relative robustness. It is shown for example that the integral-square-error tuning rule for disturbance rejection can be destabilized by a 7% simultaneous variation in the system parameters.