The sparse signal recovery in the standard compressed sensing (CS) problem requires that the sensing matrix be known a priori. Such an ideal assumption may not be met in practical applications where various errors and fluctuations exist in the sensing instruments. This paper considers the problem of compressed sensing subject to a structured perturbation in the sensing matrix. Under mild conditions, it is shown that a sparse signal can be recovered by $\ell_1$ minimization and the recovery error is at most proportional to the measurement noise level, which is similar to the standard CS result. In the special noise free case, the recovery is exact provided that the signal is sufficiently sparse with respect to the perturbation level. The formulated structured sensing matrix perturbation is applicable to the direction of arrival estimation problem, so has practical relevance. Algorithms are proposed to implement the $\ell_1$ minimization problem and numerical simulations are carried out to verify the result obtained.