Asymptotic unbounded root loci--Formulas and computation

We study the asymptotic behavior of the dosed loop eigenvalues (root loci) of a strictly proper linear time-invariant control system as loop gain goes to
\infty
. The formulas are stated in terms of the eigenvalues of nested restricted linear maps of the form
A(\mod S_{2})|_{S_{1}}
where S
1
and S
2
are subspaces of complementary dimension. Additional geometrical insight into the formulas is obtained by mechanizing the formulas using orthogonal projections. Our method and formulas are useful in other asymptotic calculations as well, e.g., hierarchical multiple-time scales aggregation of Markov chains with some infrequent transitions.