Sparse system identification for stochastic systems with general observation sequences

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Focusing on identification, this paper develops techniques to reconstruct zero and nonzero elements of a sparse parameter vector θ of a stochastic dynamic system with general observation sequences, including stationary time series and feedback control, for which the current input may depend on the past inputs and outputs, system noises as well as exogenous dithers. First, a sparse parameter identification algorithm is introduced based on L2 norm with L1 regularization, where the adaptive weights are adopted in the optimization variables of L1 term. Second, estimates generated by the algorithm are shown to have both set and parameter convergence. That is, sets of the zero and nonzero elements in the parameter θ can be correctly identified with probability one using a finite number of observations, and further estimates of the nonzero elements converge to the true values almost surely. Third, it is shown that the results are applicable to a large number of applications, including variable selection, open-loop identification, and closed-loop control of stochastic systems. Finally, numerical examples are given to support the theoretical analysis.