  This paper presents a parameter formula connecting the well-known
proportional-integralderivative (PID) control and the active disturbance
rejection control (ADRC). On the one hand, this formula gives a quantitative
lower bound to the bandwidth of the extended state observer (ESO) used in ADRC,
implying that the ESO is not necessarily of high gain. On the other hand,
enlightened by the design of ADRC, a new PID tuning rule is provided, which can
guarantee both strong robustness and nice tracking performance of the
closed-loop systems under the PID control. Moreover, it is proved that the ESO
can be rewritten as a suitable linear combination of the three terms in PID,
which can give a better estimate for the system uncertainty than the single
integral term in the PID controller. Theoretical results are verified also by
simulations in the paper.
