Construction and parameterization of all static and dynamic H 2-optimal state feedback solutions, optimal fixed modes and fixed decoupling zeros

The authors consider an H₂ optimization problem via state feedback. The problems dealt with are of general singular type which have a left invertible transfer matrix function from the control input to the controlled output. This class subsumes the regular H₂ optimization problems. The authors construct and parameterize all the static and dynamic H₂ optimal state feedback solutions. Moreover, all the eigenvalues of an optimal closed-loop system are characterized. All optimal closed-loop systems share a set of eigenvalues called the optimal fixed modes. Every H₂ optimal controller must assign among the closed-loop eigenvalues the set of optimal fixed modes. This set of optimal fixed modes includes a set of optimal fixed decoupling zeros which shows the minimum absolutely necessary number and locations 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 depending upon whether the static or the dynamic controllers are used