Consistent Approximations of Linear Stochastic Models

This paper considers the problem of approximating a stochastic process $\{ y(t)\} $ with state space X. The desired process $\{ y_1 (t)\} $ has state space $X_1 $, of dimension as small as possible, such that, in mean square norm, \[ \left\| {y(t) - y_1 (t)} \right\| \leqq \varepsilon \] for a given $\varepsilon \geqq 0$. The solution given here has the inclusion property, i.e., $X_1 \subset X$ and is consistent, that is, it reduces to the problem of finding a minimal realization of $y(t)$ when $\varepsilon $ is set equal to zero.