Multivariable PID Using Singularly Perturbed System

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.

Singularly Perturbation Method Applied To Multivariable PID Controller Design

“…Dynamic terms of , , and are expressed in (39), where , , and are Maciejowski tuning parameters. Due to a complex gain obtained from the calculation of ( ) −1 , a real approximation of ( ) −1 is necessary which can be done by solving the following optimization problem:…”

“…Therefore, an enhancement is presented based on the dynamic inverse matrix and singularly perturbation method to the designs of MPID controller. Essentially, this enhancement has been considered in the previous work reported in [39] and it shows that the enhancement is able to control dynamic system where the output is able to track the set point change and produced less proses interaction. Nevertheless, it only considered that three controller designs instead of four and the selection of parameter tuning are done without optimization technique.…”

“…The aim of this case study is to control the concentration of substrate and dissolved oxygen at the desired value by manipulating the manipulate variables of dilution rate and air flow rate, respectively. The state space form of the nonlinear wastewater treatment plant is linearized from [39] as follows:…”