Over the past 60 years, remarkable progresses in modern control theory have been made, e.g., numerous advanced control techniques including optimal control, robust control, adaptive control, nonlinear control, intelligent control, etc., have been introduced and investigated. However, the classical PID (proportional-integral-derivative) controller (or its variations), which has nearly 100 years of history, is still the most widely and successfully used one in engineering systems by far, which exhibits its lasting vitality.  In fact, a recent survey shows that the PID controller has much higher impact rating than the advanced control technologies.  However, it has also been reported that most of the practical PID loops are poorly tuned, and there is strong evidence that PID controllers remain poorly understood. Therefore, it is of vital importance to investigate the rationale of the ubiquitous controller PID. 

　　In fact, the PID controller has been investigated extensively in the literature by numerous control scientists and engineers, but most existing works are focused on linear systems, albeit almost all practical systems are nonlinear with uncertainties. Therefore, to justify the remarkable effectiveness of the PID controllers for real world systems, one has to face with nonlinear uncertain dynamical systems and to understand the rationale and capability of PID controller. 

　　Recently, the authors have investigated the capability of the classical PID control for second-order nonlinear uncertain systems and provided some analytic design methods for the choices of PID parameters, where the system is assumed to be in the form of cascade integrators. In this article, they have considered the natural extension of the classical PID control for high-order affine-nonlinear uncertain systems. Specifically, the authors have shown that the extended PID controller can semi-globally stabilize the nonlinear uncertain systems, and at the same time the regulation error converges to zero exponentially fast, as long as the control parameters are chosen from an open unbounded parameter set. These theoretical results do not need such special system structures as normal or triangular forms used extensively in the literature for global or semi-global stabilization, thanks to the inherent robustness of the extend PID controller. 

　　Publication: 

　　-    IEEE Transactions on Automatic Control, 66, 8, 3840-3847 (2021) 

　　Authors: 

　　-    Cheng Zhao (Shandong University) 

　　-    Lei Guo (Institute of Systems Science, AMSS, Chinese Academy of Sciences)