In this paper, with the aid of the powerful Restricted Isometry Constant (RIC), a deterministic (or say non-stochastic) analysis, which includes a series of sufficient conditions (related to the RIC order) and their resultant error estimates, is established for the weighted Basis Pursuit De-Noising (BPDN) to guarantee the robust signal recovery when Partially Known Support Information (PKSI) of the signal is available. Specifically, the obtained conditions extend nontrivially the ones induced recently for the traditional constrained weighted $\ell_{1}$-minimization model to those for its unconstrained counterpart, i.e., the weighted BPDN. The obtained error estimates are also comparable to the analogous ones induced previously for the robust recovery of the signals with PKSI from some constrained models. Moreover, these results to some degree may well complement the recent investigation of the weighted BPDN which is based on the stochastic analysis.