  In this paper, a novel adaptive smooth disturbance observer-based fast
finite-time adaptive backstepping control scheme is presented for the attitude
tracking of the 3-DOF helicopter system subject to compound disturbances.
First, an adaptive smooth disturbance observer (ASDO) is proposed to estimate
the composite disturbance, which owns the characteristics of smooth output,
fast finite-time convergence, and adaptability to the disturbance of unknown
derivative boundary. Then, a finite-time backstepping control protocol is
construct to drive the elevation and pitch angles to track reference
trajectories. To tackle the "explosion of complexity" and "singularity"
problems in the conventional backstepping design framework, a fast finite-time
command filter (FFTCF) is utilized to estimate the virtual control signal and
its derivative. Moreover, a fractional power-based auxiliary dynamic system is
introduced to compensate the error caused by the FFTCF estimation. Furthermore,
an improved fractional power-based adaptive law with the $\sigma $-modification
term is designed to attenuate the observer approximation error, such that the
tracking performance is further enhanced. In terms of the fast finite-time
stability theory, the signals of the closed-loop system are all fast
finite-time bounded while the attitude tracking errors can fast converge to a
sufficiently small region of the origin in finite time. Finally, a contrastive
numerical simulation is carried out to validate the effectiveness and
superiority of the designed control scheme.
