Stability Analysis of a Nonlinear PID Controller

• Published in: International journal of control, automation, and systems Vol. 19; no. 10; pp. 3400 - 3408
• Main Authors: Son, Yung-Deug, Bin, Sang-Do, Jin, Gang-Gyoo
• Format: Journal Article
• Language: English
• Published: Bucheon / Seoul Institute of Control, Robotics and Systems and The Korean Institute of Electrical Engineers 01.10.2021; Springer Nature B.V; 제어·로봇·시스템학회
• Subjects: Control; Control stability; Controllers; Engineering; Feedback; Mechatronics; Nonlinear control; Nonlinearity; Optimization; Performance indices; Proportional integral derivative; Regular Papers; Robotics; Stability analysis; Tuning; 제어계측공학; nonlinear PID controller; tuning rule; Circle stability theory; FOPTD model; nonlinear gain
• ISSN: 1598-6446; 2005-4092; 2005-4092
• DOI: 10.1007/s12555-020-0599-y

In our previous work, the authors presented an effective nonlinear proportional-integral-derivative (PID) controller by incorporating a nonlinear function. The proposed controller is based on a conventional PID control architecture, wherein a nonlinear gain is coupled in series with the integral action to scale the error. Three new tuning rules for processes represented as the first-order plus time delay (FOPTD) model were obtained by solving an optimization problem formulated to minimize three performance indices. The main feature of the proposed controller is that it preserves the numbers of tuning gains even though nonlinearity is introduced and remains easy implementation in real applications. However, due to the introduction of a nonlinear element, the stability problem of the proposed controller may be raised. This paper presents one sufficient condition in the frequency domain for the absolute stability of the nonlinear PID controller, based on circle stability theory. It is proved that the nonlinear gain used is in the sector [0, 1]. The condition of the linear block F ( s ) is derived for the overall feedback system to be stable. Checking the stability and the effectiveness and robustness of the feedback system for setpoint tracking are demonstrated through a set of simulation works on three processes with uncertainty.