Robustness and parametrization of the proportional plus double-integral compensator

The proportional plus double-integral (PI²) controller was proposed by Belanger and Luyben (Ind. Eng. Chem. Res. 1997, 36, 5339-5347) as a low- frequency compensator. The objective of the additional double-integral compensation is to reject the effects of ramplike disturbances. The aim of this work is to provide a parametrization of the PI² controller gains in terms of a nominal closed-loop and disturbance estimation time constants. A reduced-order observer is constructed to estimate ramplike disturbances, which is subsequently coupled with a feedback loop to counteract the effects of the disturbance. It is shown that such control configuration is equivalent to a PI² compensator. Some robustness issues of the PI² compensator with respect to unmodeled dynamics and nonlinearities are discussed and illustrated with two numerical examples.