An efficient sparse identification algorithm for stochastic systems with general observation sequences

This paper studies the sparse identification problem of unknown sparse parameter vectors in stochastic dynamic systems. Firstly, a novel sparse identification algorithm is proposed, which can generate sparse estimates based on least squares estimation by adaptively adjusting the threshold. Secondly, under a possibly weakest non-persistent excited condition, we prove that the proposed algorithm can correctly identify the zero and nonzero elements of the sparse parameter vector using a finite number of observations, and further estimates of the nonzero elements almost surely converge to the true values. Compared with the related works, e.g., LASSO, our method only requires the weakest assumptions and does not require solving additional optimization problems. Besides, our theoretical results do not require any statistical assumptions on the regression signals, including independence or stationarity, which makes our results promising for application to stochastic feedback systems. Thirdly, the number of finite observations that guarantee the convergence of the zero-element set of unknown sparse parameters of the Hammerstein system is derived for the first time. Finally, numerical simulations are provided, demonstrating the effectiveness of the proposed method. Since there is no additional optimization problem, i.e., no additional numerical error, the proposed algorithm performs much better than other related algorithms.