As it is well known, proportional-integral (PI) and proportional-integral-derivative (PID) controllers have been widely used in most practical industrial processes. It is stated that, more than 95% of the control loops in process control are of PID type, and most loops are actually PI control. In particular, 98% of control loops in the pulp and paper industries are controlled by PI controllers. Actually, the PI control technique has been used from the time when windmills and steam engines were the dominated technology. 

　　In the past years, how to design PI/PID controller for desired performance despite various kinds of uncertainties, including unknown nonlinear dynamics and external disturbance, has been well studied. Lots of effective tuning methods for PI/PID controller were developed, with most of them being established for linear systems. Optimizing the gain of PID controller is an important topic drawing lots of attentions. Automatic tuning of PID controllers is another effective approach. Also, it is significant to analyze the set of PID gains to ensure the robustness requirements of closed-loop systems. In addition, the improvement of the PID control by introducing fractional order has been investigated. Besides, a novel method of designing nonlinear PID control to achieve desired performance is presented in. We also remark the pioneer work on the necessary and sufficient conditions for PID control to stabilize the systemswhere the nonlinear uncertain dynamics has a linear growth rate. It is shown that the integral part of PI control is able to force the tracking error to approach zero despite constant disturbance. There are other and newer results that automatically and optimally generate the multivariable gains of PID-type controllers. In addition, Saab provides necessary and sufficient conditions for convergence. However, few results are provided on ensuring PI control systems’ output to have satisfactory transient performance, which is greatly important in practice, especially under unknown nonlinear dynamics. To ensure desired satisfactory transient performance of systems’ output and tackle larger unknown nonlinear dynamics, this article considers the uncertainty estimator integrated PI control for a class of multi-input–multi-output (MIMO) nonlinear uncertain systems. 

　　Actually, many effective uncertainty estimators for control object have been proposed in the last decades, including the extended state observer (ESO), the disturbance observer, the nonlinear disturbance observer, the extended high-gain observer, the uncertainty and disturbance estimator, and many others. The conditions for these uncertainty estimators to stabilize the nonlinear uncertain systems have been well studied, especially for the ESO-based control or active disturbance rejection control (ADRC). However, most of existing results only demonstrate qualitative condition or tuning laws (e.g., designing large enough gain of observer/estimator) for stabilization. The quantitative relationship between the parameters of observer/estimator and the size of uncertainties to be tackled for control object, which is very concerned by engineers, has not been shown. Since these uncertainty observers/estimators are mainly for estimating the uncertainties to be compensated in systems, it is natural to combine them and the conventional tracking error feedback laws, such as the popular PI methods. Nevertheless, how to quantitatively tune the observer/estimator integrated into the typical PI loop for better performance has not yet been studied. 

　　This article focuses on integrating uncertainty estimator into PI controller for better robustness and transient performance against uncertain nonlinear coupling dynamics and time-varying disturbances. First, the descriptions for the sizes of three kinds of uncertainties in a class of multi-input–multi-output nonlinear systems are discussed. Then, the tuning laws of the typical uncertainty estimator, i.e., extended state observer (ESO), are quantitatively presented to ensure the stability of closed-loop systems. More importantly, it is shown that the desired transient performance of tracking error can be ensured by tuning the bandwidth of ESO. In addition, it is proven that much stronger disturbance rejection at low frequency can be achieved by integrating the uncertainty estimator module. The simulation results for calibration-free robotic eye-hand coordination system show the effectiveness of the proposed method. 

　　Publication: 

　　-    IEEE Transactions on Automatic Control, 66, 7, 3409-3416 (2021). 

　　Authors: 

　　-    Wenchao Xue (Institute of Systems Sciences, AMSS, Chinese Academy of Sciences) 

　　-    Sen Chen (Shaanxi Normal University) 

　　-    Cheng Zhao (Shandong University) 

　　-    Yi Huang (Institute of Systems Sciences, AMSS, Chinese Academy of Sciences) 

　　-    Jianbo Su (Shanghai Jiao Tong University)