  Focusing on identification, this paper develops techniques to reconstruct
zero and nonzero elements of a sparse parameter vector of a stochastic dynamic
system under 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 can be correctly identified with probability one
using a finite number of observations, and 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.
