Graphical abstract

The paper investigates the possibilities of using the singularly perturbation method in a multivariable proportional-integral-derivative (MPID) controller design. The MPID methods of Davison, Penttinen-Koivo and Maciejowski are implemented and the effective of each method is tested on wastewater treatment plant (WWTP). Basically, this work involves modeling and control. In the modeling part, the original full order system of the WWTP was decomposed to a singularly perturbed system. Approximated slow and fast models of the system were realized based on eigenvalue of the identified system. The estimated models are then used for controller design. Mostly, the conventional MPID considered static inverse matrix, but this singularly perturbed MPID considers dynamic matrix inverse. The stability of the singularly perturbed system is established by using Bode analysis, whereby the bode plot of the model system is compared to the original system. The simulation results showed that the singularly perturbed method can be applied into MPID. The three methods of MPID have been compared and the Maciejowski shows the best closed loop performance