Abstract
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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2 \times 2$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$3 \times 3$ </tex-math></inline-formula> . Comparisons with other centralized control methodologies show that the proposed approach achieves similar or greater performance and a remarkable better disturbance rejection response.
Highlights
Despite most industrial processes are multiple inputs and multiple outputs (MIMO) systems, most of the thousands of works about design methods of Proportional-IntegralDerivative (PID) control, which has been the preferred one in industrial process control for decades [1], have been developed only for single input single output (SISO) loops [2]
Since the proposed methodology is based on a frequency response representation, frequency domain specifications on the Nyquist diagram are preferable as design requirements, and the robust linear margin described in [32] is selected in this work
The methodology is based on the developed concept of equivalent loop transfer function (ELTF) for centralized control, which allows to design separately the columns of the controller matrix considering the interactions with the other loops through an
Summary
Despite most industrial processes are multiple inputs and multiple outputs (MIMO) systems, most of the thousands of works about design methods of Proportional-IntegralDerivative (PID) control, which has been the preferred one in industrial process control for decades [1], have been developed only for single input single output (SISO) loops [2]. For nondiagonally dominant systems with great couplings decentralized controllers have a worse performance as they cannot reduce enough the interaction level and important transitory responses can arise in other loops when some reference changes To face such processes, a centralized control scheme, which uses a full matrix controller, is recommended. One problem of most of these multivariable methodologies is the increasing complexity of the elements for high dimensional systems, where irrational or complicated transfer functions can arise They usually need model approximations or controller reductions that affect the robustness of the control system or can lead to conservative control designs. Based on the ELTF concept, a new iterative methodology for designing centralized PID controllers for stable and square MIMO systems is proposed and formulated for each loop as a linear programming optimization that maximizes integral gains of PID controllers subject to constraints on robustness linear margin and static decoupling.
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