Enforcing a System model to be Negative Imaginary via Perturbation of Hamiltonian Matrices

Flexible structure dynamics with collocated force actuators and position sensors lead to negative imaginary (NI) systems. However, in some cases, the models obtained for these systems may not satisfy the NI property. This paper provides a new method for enforcing such models to be NI. The results are based on a study of the spectral properties of related Hamiltonian matrices. A test for the negativity of the imaginary part of a corresponding transfer function matrix is first performed by checking for the existence of imaginary eigenvalues of the associated Hamiltonian matrix. In the presence of imaginary eigenvalues, the system is not NI. In such cases, a first-order perturbation is presented for the precise characterization of frequency bands where violations of the NI property occur. This characterization is then used for the design of an iterative perturbation scheme for state matrices aimed at displacing the imaginary eigenvalues of the Hamiltonian matrix away from the imaginary axis.