The [tau] Decomposition Method for PID Controllers of First Order Delayed Unstable Processes

• Published in: Asian journal of control 2016-01, Vol.18 (1), p.293
• Main Authors: Cai, Tiao-Yang, Zhang, Hua-Guang, Yang, Fei-Sheng, Xie, Xiang-Peng
• Format: Article
• Language: English
• Subjects: Control systems; Decomposition
• ISSN: 1561-8625; 1934-6093
• DOI: 10.1002/asjc.1025

The stabilization of first-order delayed (FOD) unstable processes with proportional-integral-derivative (PID) controllers is considered, and all the feasible PID controllers are determined. Different from the existing results which are based on the D-partition technique and partitioning complex's real and imaginary parts, a novel procedure enlightened by the τ decomposition method is proposed to characterize the space of controller parameters. Generally speaking, the parameter space is mainly divided into four regions: delay independent stable region, delay independent unstable region, delay interval dependent stable region, and delay interval dependent unstable region. The depiction of the space partition boundaries and the determination of stable intervals become the main problems. In our work, these boundaries are related to the existence of the purely imaginary roots (PIRs) of the systems' characteristic equation, and the determination of stable intervals refers to the number and value of the PIRs. Thus, the key points are the number and calculation of PIRs. According to our discussion, analytical results on both topics are obtained in explicit form. Finally, vivid numerical simulations are given to illustrate the results.