Construction and parameterization of all static and dynamic H2 -optimal state feedback solutions, optimal fixed modes and fixed decoupling zeros

A problem of H₂ optimization via state feedback is considered. The problems dealt with are of the general singular type, with a left invertible transfer matrix function from the control input to the controlled output. All the static and dynamic H ₂ optimal state feedback solutions are constructed and parameterized, and all the eigenvalues of an optimal closed-loop system are characterized. All optimal closed-loop systems share a set of eigenvalues which are termed the optimal fixed modes, which must be assigned among the closed-loop eigenvalues. This set includes a set of optimal fixed decoupling zeros which shows the minimum absolutely necessary number and location of pole-zero cancellations present in any H₂ optimal design. It is shown that both the sets of optimal fixed modes and optimal fixed decoupling zeros do not vary