A convergence theorem for a class of stochastic gradient type algorithms with application to robust system identification

The recursive algorithms of stochastic gradient type for estimating the parameters of linear discrete-time systems in the presence of disturbance uncertainty has been considered in the paper. Problems related to the construction of min-max optimal recursive algorithms are demonstrated. In addition, the robustness of the proposed algorithms has been addressed. Since the min-max optimal solution cannot be achieved in practice, a simple procedure for constructing a practically applicable robustified recursive algorithm based on a suitable nonlinear transformation of the prediction error and convenient approximations is suggested. The convergence of the robustified recursive algorithm is established theoretically using the martingale theory.