Riccati equation computation af supremal (F,G)-Invariant and reachability subspaces

We consider two Riccati equations, and associate with each Riccati equation three recursions for generating subspace sequences. It is shown that, with appropriate choice of parameters and initial conditions, several of these recursions generate suhspace sequences which are important in system theory. Particular examples are the supremal unobservable suhspace, the supremal teachability subspace, and the smallest reachable subspace obtained by output injection. The recursions may be implemented by solving the associated Riccati equations, and thus we are able to derive methods for computing the above-mentioned subspaces. In addition, we derive a feedback which makes a system maximally unobservable, and also present nonrecursive expressions for the supremal unobservable subspace and for the subspace of
k
-predictable directions.