On the notions of normality, locality, and operational stability in ADRC

Treating plant dynamics as an ideal integrator chain disturbed by the total disturbance is the hallmark of active disturbance rejection control (ADRC). To interpret its effectiveness and success, to explain why so many vastly different dynamic systems can be treated in this manner, and to answer why a detailed, accurate, and global mathematical model is unnecessary, is the target of this paper. Driven by a motivating example, the notions of normality and locality are introduced. Normality shows that, in ADRC, the plant is normalized to an integrator chain, which is called local nominal model and locally describes the plant's frequency response in the neighborhood of the expected gain crossover frequency. Locality interprets why ADRC can design the controller only with the local information of the plant. With normality and locality, ADRC can be effective and robust, and obtain operational stability discussed by T. S. Tsien. Then viewing proportional-integral-derivative (PID) control as a low-frequency approximation of second-order linear ADRC, the above results are extended to PID control. A controller design framework is proposed to obtain the controller in three steps: (1) choose an integrator chain as the local nominal model of the plant; (2) select a controller family corresponding to the local nominal model; and (3) tune the controller to guarantee the gain crossover frequency specification. The second-order linear ADRC and the PID control are two special cases of the framework.