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 H-infinity 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 H-infinity norm that encompass frequency weightings of generalised disturbance inputs are also determined. Novel analytical expressions to minimise the H-infinity 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 H-infinity norm. The H-2 norm is introduced in these cases to determine the controller that balances performance with robustness. Finally, the H-infinity 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 H-infinity controller is compared to previously developed PIP controllers based on pole placement and linear quadratic design