International Journal of Control, Vol.92, No.12, 2939-2951, 2019
Robustness evaluation and robust design for proportional-integral-plus control
Proportional-integral-plus (PIP) control provides a logical extension to conventional two- or three-term (proportional-integral-derivative) industrial control, with additional dynamic feedback and input compensators introduced when the process has second order or higher dynamics, or time delays. Although PIP control has been applied in a range of engineering applications, evaluation of closed-loop robustness has generally relied on empirical methods. In the present article, expressions for the norm of two commonly used PIP control implementations, the feedback and forward path forms, are used, for the first time, to quantify closed-loop robustness. It is shown that the forward path form is not robust for unstable plants. Additional expressions for the norm that encompass frequency weightings of generalised disturbance inputs are also determined. Novel analytical expressions to minimise the norm are derived for the simplest plant, while simulation results based on numerical optimisation are provided for higher order examples. We show that, for certain plants, there are (non-unique) sets of PIP control gains that minimise the norm. The norm is introduced in these cases to determine the controller that balances performance with robustness. Finally, the norm is used as a design parameter for a practical example, namely control of airflow in a 2 m by 1 m by 1 m forced ventilation chamber. The performance of the new PIP controller is compared to previously developed PIP controllers based on pole placement and linear quadratic design.