Polynomial approach to pole shifting to infinity in singular systems by feedbacks. (Q1847686)

Necessary and sufficient conditions are established for the existence of a state feedback gain matrix \(K\) such that the closed-loop characteristic polynomial is of zero degree, \(\det[Es-A+BK] =\alpha\neq 0\) \((\alpha\) is independent of \(s)\). Necessary and sufficient conditions are also found for the existence of a solution \(X=I_n\), \(Y=K\) to the polynomial matrix equation \([Es-A]X+ BY=U(s)\) for a unimodular matrix \(U(s)\) \((\det U(s)= \alpha)\). Algorithms for the computation of \(K\) are proposed and illustrated by several numerical examples. It is shown that the complete controllability of the system is sufficient but not necessary and the strong controllability of the system is not a necessary condition for the existence of a solution to the first problem.