Optimal tuning of PID controllers: And the verification of the SIMC rules

Abstract

Optimal tuning of PID controllers and the verification of the SIMC rules

The simple PI/PID controller is the most used controller in the process industry. However, finding good settings for this simple controller is not trivial. One simple way of finding settings for a PID controller is the popular SIMC tuning rule. This thesis deals with the following main topics, •Finding optimal parameters for fixed-order controllers. •Validating that the SIMC tuning rules are close-to optimal for first-order with delay processes. •Validating that the SIMC tuning rules are close-to optimal for double integrating with delay processes. •Showing that a well tuned PID controller is a better choice than a Smith predictor (SP).

A central part of this thesis is defining and quantifying an optimal PID controller. We optimize performance (using IAE for input and output disturbances as the cost function) subject to constraints on robustness (using the sensitivity peaks Ms and Mt). We provide analytical gradients for the cost function and the constraints, which give increased accuracy and better convergence properties than numerical gradients.

Using this, we derive optimal PI- and PID-settings for first-order plus delay processes. The optimal PID IAE-performance is compared with the IAE-performance for controllers that have been tuned using the SIMC-rules, where the SIMC tuning parameter is adjusted such that the two controllers have the same robustness. The original SIMC-rules give a PI-controller for a first-order plus delay process, and we find that this controller is close to the optimal PI-controller.

The only exception is for delay-dominant processes where the SIMC-rule gives a pure integrating controller. We propose and study a very simple modification to the original SIMC-rule, which is to add a derivative time = delay/3. This gives performance close to the IAE-optimal PID-controller 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.

We also investigate optimal PID control of a double integrating with delay process and compare with the SIMC rules. What makes the double integrating process special is that derivative action is actually necessary for stabilization. Surprisingly, the SIMC PID controller is almost identical to the optimal PID controller. We also propose a generalized SIMC rule which includes second-order processes with large time constants.

For processes with large time delays, the common belief is that PI and PID controllers give poor performance, and that a SP or a similar dead-time compensator can significantly improve performance. In the last chapter, we argue that this is a myth. For a given robustness level, we find that the performance improvement with the SP is small even for a pure time delay process. For other first-order with delay processes a PID controller is generally better for a given robustness level. In addition, the Smith predictor is much more sensitive to time delay errors than PI and PID controllers.