An Interpretation of Rosenbrock's Theorem Via Local Rings
نویسندگان
چکیده
the invariant factors of its state-space matrix A + BF . This result can be seen as the solution of an inverse problem; that of finding a non-singular polynomial matrix with prescribed in‐ variant factors and left Wiener–Hopf factorization indices at infinity. To see this we recall that the invariant factors form a complete system of invariants for the finite equivalence of polynomial matrices (this equivalence relation will be revisited in Section ▭) and it will be seen in Section ▭ that any polynomial matrix is left Wiener–Hopf equivalent at infinity to a diagonal matrix Diag(s 1, ..., s km), where the non-negative integers k1, ..., km (that can be as‐ sumed in non-increasing order) form a complete system of invariants for the left Wiener– Hopf equivalence at infinity. Consider now the transfer function matrix G(s) = (sI (A + BF ))-1B of (▭). This is a rational matrix that can be written as an irreducible matrix fraction description G(s) = N (s)P(s)-1, where N (s) and P(s) are right coprime polyno‐
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