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In this paper, we propose a robust reinforcement learning method for a class of linear discrete-time systems to handle model mismatches that may be induced by sim-to-real gap. Under the formulation of risk-sensitive linear quadratic Gaussian control, a dual-loop policy optimization algorithm is proposed to iteratively approximate the robust and optimal controller. The convergence and robustness of the dual-loop policy optimization algorithm are rigorously analyzed. It is shown that the dual-loop policy optimization algorithm uniformly converges to the optimal solution. In addition, by invoking the concept of small-disturbance input-to-state stability, it is guaranteed that the dual-loop policy optimization algorithm still converges to a neighborhood of the optimal solution when the algorithm is subject to a sufficiently small disturbance at each step. When the system matrices are unknown, a learning-based off-policy policy optimization algorithm is proposed for the same class of linear systems with additive Gaussian noise. The numerical simulation is implemented to demonstrate the efficacy of the proposed algorithm.