  This paper theoretically investigates the closed-loop performance of active
disturbance rejection control (ADRC) on a third-order linear plant with
relative degree 3, subject to a class of exogenous disturbances. While PID
control cannot be guaranteed to be capable of stabilizing such plants, ADRC
offers a model-free alternative. However, many existing works on ADRC consider
the observer gains to be taken arbitrarily large, in order to guarantee desired
performance, such as works which consider parameterizing ADRC by bandwidth.
This work finds that, for constant exogenous disturbances, arbitrary eigenvalue
assignment is possible for the closed-loop system under linear ADRC, thus
guaranteeing the existence of an ADRC controller for desired performance
without taking any gains arbitrarily large. We also find that stabilization is
possible when the exogenous disturbance is stable, and show how ADRC can
recover the performance of model-based observers. We demonstrate aspects of the
resulting closed-loop systems under ADRC in simulations.
