We consider the sensitivity of semidefinite programs (SDPs) under perturbations. It is well known that the optimal value changes continuously under perturbations on the right hand side in the case where the Slater condition holds in the primal problems. In this manuscript, we observe by investigating a concrete SDP that the optimal value can be discontinuous if the dual problem is not strictly feasible and one perturbs the SDP with coefficient matrices. We show that the optimal value of such an SDP changes continuously if the perturbations preserve the rank of the space spanned by submatrices of the coefficient matrices and do not change the minimal face which is obtained by facial reduction algorithm. In addition, we determine the kinds of perturbations that make minimal faces invariant. Our results allow us to classify change of the minimal face of an SDP obtained from a control problem under linear perturbations which preserve matrix structures that appear in the associated dynamical systems.