Recursive identification for ARMAX systems

For the multivariate ARMAX systems A(z)y
k
= B(z)u
k
+ C(z)w
k
the recursive algorithms are proposed for estimating coefficients of A(z),B(z), and C(z) and the covariance matrix R
w
of w
k
, assuming that the upper bounds for the orders of A(z), B(z), and C(z) are available and the control u
k
is allowed to choose for the identification purpose. The algorithms are easily computable and are proved to converge to the true values under reasonable conditions. The restrictive SPR condition is not used.