Optimal PI and PID control of first-order plus delay processes and evaluation of the original and improved SIMC rules

Abstract

The first-order plus delay process model with parameters k (gain), τ (time constant) and θ (delay) is the most used representation of process dynamics. This paper has three objectives. First, we derive optimal PI- and PID-settings for this process. Optimality is here defined as the minimum Integrated Absolute Error (IAE) to disturbances for a given robustness level. The robustness level, which is here defined as the sensitivity peak (Ms), may be regarded as a tuning parameter. Second, we compare the optimal IAE-performance with the simple SIMC-rules, where the SIMC tuning parameter τc is adjusted to get a given robustness. The “original” SIMC-rules give a PI-controller for a first-order with delay process, and we find that this SIMC PI-controller is close to the optimal PI-controller for most values of the process parameters (k, τ, θ). The only exception is for delay-dominant processes where the SIMC-rule gives a pure integrating controller. The third objective of this paper is to propose and study a very simple modification to the original SIMC-rule, which is to add a derivative time τd = θ/3 (for the serial PID-form). This gives performance close to the IAE-optimal PID also for delay-dominant processes. We call this the “improved” SIMC-rule, but we put “improved” in quotes, because this controller requires more input usage, so in practice the original SIMC-rule, which gives a PI-controller, may be preferred.