On consistency of stochastic gradient algorithms for ARMAX models with disturbances

In this paper, we give a new method to prove in detail the consistency of the residual based and innovation based stochastic gradient algorithms for identifying CARMA/ARMAX models with disturbances under weaker conditions on statistical properties of the noise, e.g., the mean value is non-zero, and the variance is time-varying, and/or high-order moments are possibly nonexistent. The analysis indicates that the parameter estimation error is consistently bounded, and consistently converges to zero under persistent excitation conditions.