Online sparse Bayesian identification of nonlinear time-varying systems

Sparse regression provides a compact and interpretable route for nonlinear system modeling by selecting a small number of active terms from a candidate dictionary. Most sparse regressors, however, are constructed offline and then used as static predictors. In online operation, changing load, material properties, ambient conditions, or equipment states may alter both the coefficient values and the effective active support within the dictionary. Moreover, a direct recursive update over a rich dictionary may spread the adaptation over many weakly relevant terms, causing an initially sparse model to become increasingly dense. The key problem is therefore to maintain a sparse regressor online, so that it can absorb streaming data while keeping a compact but revisable active structure. This paper develops a Bayesian recursive sparse learning (BRSL) method for online sparse identification over candidate dictionary terms. The coefficient distribution is updated through a Bayesian posterior recursion, where sliding-window likelihood-ratio information recursion incorporates new samples, removes expired samples, and discounts historical information in a unified update. To preserve sparsity during recursion, posterior-guided shrinkage is introduced to suppress weakly supported dictionary terms and revise the active structure according to posterior evidence. The posterior update is performed in a candidate subspace with an adaptive information floor to keep the recursive solve well posed, and a bounded-error relation is given to clarify the influence of shrinkage, residual information, coefficient drift, and information conditioning. The proposed method is evaluated on sparse coefficient tracking and a power-plant-oriented multi-input multi-output (MIMO) nonlinear time-varying identification benchmark.