Shapiro's Conjecture and the placement problem

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2.iv. Shapiro's Conjecture and the placement problem

An important variant of the pole placement problem is that when the matrices A, B, and C are real, and when f(s) has all (mp) roots real, which (if any) real feedback laws satisfy

f(s) = det(sI_n - A - BFC).

Using the coprime factorization N(s)D(s)^-1 of the transfer function, as in Section 2.iii, and letting s₁,s₂,...,s_n be the (real) roots of the polynomial f, the real pole placement problem is equivalent to the following question: Which real m by p-matrices F satisfy

for each i=1, 2, ..., n?

The conjecture of Shapiro and Shapiro is then seen to be a special case of the real pole placement problem. Let n=mp and note that the matrix K(s) of their conjecture:

has the same form as the top submatrix in the previous equation.