Iterative Design of Centralized PID Controllers Based on Equivalent Loop Transfer Functions and Linear Programming

This paper deals with the design problem of multivariable Proportional-Integral-Derivative (PID) controllers for square and stable multivariable processes when a linear margin at the Nyquist plot is considered as robustness specification for each closed loop. A tuning method is developed based on the new concept of equivalent loop transfer function, which is proposed for centralized control and allows an independent design for each loop considering the interactions with the other loops through an iterative procedure. For the k-th loop, the PID parameters of the k-th column of the control matrix are calculated in each iteration by a linear programming optimization that maximizes the integral gains while fulfilling the robustness specification and achieving static decoupling. The method uses a frequency response array as representation of the process, which allows its applicability to systems with multiple time delays without requiring model reductions or approximations. The effectiveness of the method is illustrated by means of two simulation examples with dimensions 2×2 and 3×3. Comparisons with other centralized control methodologies show that the proposed approach achieves similar or greater performance and a remarkable better disturbance rejection response.

NOTE: The Figure 9 of this publication is partially wrong because it does not show the second loop signals and the legend. A corrected version of this figure can be found, together with the code and data files of this manuscript, in https://doi.org/10.5281/zenodo.14767872.