  In this paper, we have performed the dynamical system analysis of $f(T)$
gravity cosmological models at both background and perturbation levels. We have
presented three models pertaining to three distinct functional forms of $f(T)$.
The first form is that of the logarithmic form of the torsion scalar $T$, the
second one is in the power law form, and the third one is the combination of
the first two forms. For all these three forms of $f(T)$, we have derived the
corresponding cosmological parameters in terms of the dynamical variables.
Subsequently, the critical points are obtained and the condition(s) of its
existence has been derived. Critical points of each model have been analysed
individually and the corresponding cosmology are derived. The stability
behaviour of these critical points are discussed from the behaviour of the
eigenvalues and the phase portraits. At least one stable node has been obtained
in each of these models. Further from the evolution plots of the cosmological
parameters, the accelerating behaviour of the cosmological models are also
verified.
