Formal Verification of Fractional-Order PID Control Systems Using Higher-Order Logic

• Published in: Fractal and fractional Vol. 6; no. 9; p. 485
• Main Authors: Zhao, Chunna, Jiang, Murong, Huang, Yaqun
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.09.2022
• Subjects: Accuracy; Calculus; Control systems design; Control theory; Controllers; Design techniques; Differential calculus; Feedback control; fractional-order calculus; fractional-order PID control; higher-order logic; Logic; Mathematical models; Proportional integral derivative; Robots; Robust control; Semantics; Simulation; theorem proving; Theorems; Verification
• ISSN: 2504-3110; 2504-3110
• DOI: 10.3390/fractalfract6090485

Fractional-order PID control is a landmark in the development of fractional-order control theory. It can improve the control precision and accuracy of systems and achieve more robust control results. As a theorem-proving formal verification method, it can be applied to an arbitrary system represented by a mathematical model. It is the ideal verification method because it is not subject to limits on state numbers. This paper presents the higher-order logic (HOL) formal verification and modeling of fractional-order PID controller systems. Firstly, a fractional-order PID controller was designed. The accuracy of fractional-order PID control can be supported by simulation, comparing integral-order PID controls. Secondly, the superior property of fractional-order PID control is validated via higher-order logic theorem proofs. An important basic property, the relationship between fractional-order differential calculus and integral-order differential calculus, was analyzed via a higher-order logic theorem proof. Then, the relations between the fractional-order PID controller and integral-order PID controller were verified based on the fractional-order Grünwald–Letnikov definition for higher-order logic theorem proofs. Formalization models of the fractional-order PID controller and the fractional-order closed-loop control system were established. Finally, the stability of the fractional-order control systems was verified based on established formal models and theorems. The results show that the fractional-order PID controllers can be conducive to the control performance of control systems, and the higher-order logic formal verification method can ensure the reliability and security of fractional-order control systems.