New results on the eigenstructure assignment in the computation of reachability output nulling subspaces

This paper provides a system theoretic investigation of the building blocks of the supremal output-nulling, reachability and stabilizability subspaces that are obtained from the eigenstructure assignment approach based on projecting the bases of the null-spaces of the Rosenbrock matrix pencil on the state space. In particular, we show that the use of a certain set of eigenvalues to build such bases guarantees that their span is the largest output-nulling subspace of the system such that the closed-loop mapping restricted to it has exactly these eigenvalues, and that all the output-nulling subspaces constructed in this way share the same reachability subspace. This paper also casts some light into the fact that, even if these building blocks depend on the values of the indeterminate used to compute the kernels of the Rosenbrock matrix, the dimension of their sums for different values is independent of the specific values.