Silverman algorithm and the structure of discrete-time stochastic systems (Q1611904)

The authors consider a linear discrete-time stochastic model \(x(t+1)=Ax(t)+Bw(t), y(t)= Cx(t)+Dw(t)\), driven by a normalized \(p\)-dimensional white-noise process \(w\). This model represents a certain \(m\)-dimensional wide-sense stationary process \(y\), which may be a measured signal. Only minimal stochastic realizations where the dimension of the state vector \(x\) is as small as possible are considered. The process \(y\) is supposed to be full-rank, i.e. its density matrix is invertible almost everywhere. If the matrix \(D\) is singular, the problem of estimating the state \(x\) based on the (past) observations of \(y\) is known as ``cheap'' (or singular) filtering. It has been observed that, related to the singularity of \(D\), there is a possible reduction in the order of an associated Riccati equation. This reduction has been investigated in a series of papers by L. Silverman and co-authors. Here, a dual (filtering) version of the Silverman structure algorithm is discussed, and a new necessary and sufficient condition of regularity of a stochastic process (that means the existence of a right-invertible matrix \(D\) for all minimal realizations of \(y\)) is derived. The case of processes admitting models with \(D=0\) is also discussed. A reduced-order algorithm for filtering non-regular processes is the main result of the paper. It is proved that if \(y\) is a non-regular process, the optimal filter may be derived by solving a reduced-order Riccati equation. The structure of this equation and the amount of reduction are analyzed and related to the parameters \(A,B,C,D\) of the given model and to the spectral density of the process \(y\). Examples of implementation of the proposed algorithm are presented.