Identification of both coefficients and orders for ARMAX system

Let the observed sequence {y_k} be generated by the multivariate ARMAX system A(z)y_k = B(z)u_k-1 + C(z)w_k, where {w_k} is the system noise with unknown covariance matrix R_w > 0, and {u_k} is a sequence of mutually independent and identically distributed (iid) random vectors. Based on {y_k} and {u_k}, identification algorithms are proposed to simultaneously estimate the orders (p,q,r), the covariance matrix R_w, and the coefficients of A(z), B(z), and C(z). Under reasonable conditions the estimates are proved to converge to the true values with probability one. The advantage of the proposed algorithms is that the estimates can be easily updated for online identification.