### On PID Control Theory for Nonaffine Uncertain Stochastic Systems

ZHANG Jinke1,2, ZHAO Cheng1,2, GUO Lei1,2

1. 1. Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;
2. School of Mathematical Science, University of Chinese Academy of Sciences, Beijing 100049, China
• Received:2021-12-07 Online:2023-01-25 Published:2023-02-09
• Supported by:
This research was supported by the National Natural Science Foundation of China under Grant No. 12288201.

ZHANG Jinke, ZHAO Cheng, GUO Lei. On PID Control Theory for Nonaffine Uncertain Stochastic Systems[J]. Journal of Systems Science and Complexity, 2023, 36(1): 165-186.

PID (proportional-integral-derivative) control is recognized to be the most widely and successfully employed control strategy by far. However, there are limited theoretical investigations explaining the rationale why PID can work so well when dealing with nonlinear uncertain systems. This paper continues the previous researches towards establishing a theoretical foundation of PID control, by studying the regulation problem of PID control for nonaffine uncertain nonlinear stochastic systems. To be specific, a three dimensional parameter set will be constructed explicitly based on some prior knowledge on bounds of partial derivatives of both the drift and diffusion terms. It will be shown that the closed-loop control system will achieve exponential stability in the mean square sense under PID control, whenever the controller parameters are chosen from the constructed parameter set. Moreover, similar results can also be obtained for PD (PI) control in some special cases. A numerical example will be provided to illustrate the theoretical results.
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