Recursive estimation based on an extended observation vector (Q790782)

The author considers the construction of a suboptimal filtering algorithm of assigned order N with a memory of \(\tau\) steps for a partially observable stochastic sequence \(z_ t=(x_ t,y_ t)\) described by the system of difference equations \[ x_ t=a_ 1(t,\epsilon_ t)+b_ 1(t,\epsilon_ t)x_{t-1}+u_ 1(t-1)+d_ 1(t,\epsilon_ t)y_{t-1}, \] \[ y_ t=a_ 2(t,\epsilon_ t)+b_ 2(t,\epsilon_ t)x_{t- 1}+u_ 2(t-1)+d_ 2(t,\epsilon_ t)y_{t-1}, \] where the controls \(u_ 1(t-1)\) and \(u_ 2(t-1)\) are expressed linearly in terms of linear estimates of the unobservable component \(x_ t\) of \(z_ t\) on the basis of realizations of the observable component \(y_ t\) of \(z_ t\) and \(\epsilon_ t\) is a sequence of independent random vectors with independent components whose distribution is assumed to be known. The proposed algorithm gives estimates whose accuracy is higher than that of linear estimates, this being due to the use of polynomial functions of \(y_ t\).