In this article, we address the problem of estimating the state and learning of the parameters in a linear dynamic system with generalized $L_1$-regularization. Assuming a sparsity prior on the state, the joint state estimation and parameter learning problem is cast as an unconstrained optimization problem. However, when the dimensionality of state or parameters is large, memory requirements and computation of learning algorithms are generally prohibitive. Here, we develop a new augmented Lagrangian Kalman smoother method for solving this problem, where the primal variable update is reformulated as Kalman smoother. The effectiveness of the proposed method for state estimation and parameter learning is demonstrated in spectro-temporal estimation tasks using both synthetic and real data.