Novel Proportional–Integral–Derivative Control Framework on Continuous-Time Positive Systems Using Linear Programming

Qingbo Li (), Xiaoyue Zhou, Fengyu Lin, Yahao Yang and Junfeng Zhang
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Qingbo Li: School of Mathematics and Statistics, Hainan Normal University, Haikou 571158, China
Xiaoyue Zhou: School of Automation, Hangzhou Dianzi University, Hangzhou 310018, China
Fengyu Lin: School of Automation, Hangzhou Dianzi University, Hangzhou 310018, China
Yahao Yang: School of Information and Communication Engineering, Hainan University, Haikou 570228, China
Junfeng Zhang: School of Information and Communication Engineering, Hainan University, Haikou 570228, China

Mathematics, 2024, vol. 12, issue 4, 1-15

Abstract: This paper considers the proportional–integral–derivative (PID) control for continuous-time positive systems. A three-stage strategy is introduced to design the PID controller. In the first stage, the proportional and integral components of the PID control are designed. A matrix decomposition approach is used to describe the gain matrices of the proportional and integral components. The positivity and stability of the closed-loop systems without the derivative component of PID control are achieved by the properties of a Metzler and Hurwitz matrix. In the second stage, a non-negative inverse matrix is constructed to maintain the Metzler and Hurwitz properties of the closed-loop system matrix in the first stage. To deal with the inverse of the derivative component of PID control, a matrix decomposition approach is further utilized to design a non-negative inverse matrix. Then, the derivative component is obtained by virtue of the designed inverse matrix. All the presented conditions can be solved by virtue of a linear programming approach. Furthermore, the three-stage PID design is developed for a state observer-based PID controller. Finally, a simulation example is provided to verify the effectiveness and validity of the proposed design.

Keywords: PID control; continuous-time positive systems; linear programming (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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