Least-squares identification for ARMAX models without the positive real condition

Recursive identification problems of linear stochastic feedback control systems described by ARMAX models are studied without imposing the strictly positive real condition on the noise model. An increasing lag least squares is used in the first step to estimate the noise process, while the parameter estimate is formed in the second step by an extended least squares. In the algorithm, there is an increase in computational cost in comparison to traditional algorithms. However the results of this note do not need any a priori information or conditions on the noise model except that of stability, and they are applicable to the identification of general feedback control systems