  Control systems are inevitably affected by external disturbances, and a major
objective of the control design is to attenuate or eliminate their adverse
effects on the system performance. This paper presents a disturbance rejection
approach with two main improvements over existing results: 1) it relaxes the
requirement of calculating or measuring the state derivatives, which are not
available for measurement, and their calculation is corrupted by noise, and 2)
it achieves finite-time disturbance rejection and control. To this end, the
disturbance is first modeled by an unknown dynamics, and an adaptive
disturbance observer is proposed to estimate it. A filtered regressor form is
leveraged to model the nonlinear system and the unknown disturbance. It is
shown that using this filtered regressor form, the disturbance is estimated
using only measured state of the regressor. That is, contrary to the existing
results on disturbance rejection, the presented approach does not require the
state derivative measurements. To improve the convergence speed of the
disturbance estimation, an adaptive law, equipped with experience replay, is
presented. The disturbance observer is then augmented with an adaptive integral
terminal sliding mode control to assure the finite-time convergence of tracking
error to zero. A verifiable rank condition on the history of the past
experience used by the experience-replay technique provides a sufficient
condition for convergence. Compared to the existing results, neither the
knowledge of the disturbance dynamics nor the state derivatives are required,
and finite-time stability is guaranteed. A simulation example illustrates the
effectiveness of the proposed approach.
