  Using the observation data of SNeIa, CMB and BAO, we establish two concrete
$f(T)$ models with nonminimal torsion-matter coupling extension. We study in
detail the cosmological implication of our models and find they are successful
in describing the observation of the Universe, its large scale structure and
evolution. In other words, these models do not change the successful aspects of
$\Lambda$CDM scenario under the error band of fitting values as describing the
evolution history of the Universe including radiation-dominated era,
matter-dominated era and the present accelerating expansion. Meanwhile, the
significant advantage of these models is that they could avoid the cosmological
constant problem of $\Lambda$CDM. A joint analysis is performed by using the
data of CMB+BAO+JLA, which leads to $\Omega_{m0}=0.255\pm 0.010,
\Omega_{b0}h^2=0.0221\pm 0.0003$ and $H_0=68.54\pm 1.27$ for model I and
$\Omega_{m0}=0.306\pm 0.010, \Omega_{b0}h^2=0.0225\pm 0.0003$ and $H_0=60.97\pm
0.44$ for model II at 1$\sigma$ confidence level. The evolution of the
decelaration parameter $q(a)$ and the effective equation of state $w_{DE}(a)$
are displayed. Furthermore, The resulted age of the Universe from our models is
consistent with the ages of the oldest globular clusters. As for the fate of
the Universe, model I results in a de Sitter accelerating phase while model II
appears a power-law one, even though $w_{DE0}< -1$ makes model I look like a
phantom at present time.
