Adaptive prediction under control (Q1287300)

Consider the signal-plus-noise model \(z_n=s_n+w_n\) where the signal \(s_n\) is a sum of two components \(y_n+v_n\); \(y_n\) is determined by the control \(u_n\) and is a solution to the equation \[ f_n+a_1f_{n-1}+\dots+a_pf_{n-p} = b_1u_{n-1-d}+\dots+b_ru_{n-r-d} ; \] and either \(v_n\) itself or some of its linear transformations is stationary. The problem under study is the prediction of \(s_{n+\nu}\), \(\nu>0\), given the observations \(z_k\) and controls \(u_k\) up to time \(n\). Recursive adaptive algorithms are developed and their asymptotic properties are studied. The algorithms work without full knowledge of the covariance structure of \(w_n\) and \(v_n\).