Approximated gramians and balanced realisation of flexible structures with light damping

A method of approximating the controllability and observability gramians, and the balancing transformation for lightly damped flexible structures is presented. The approximation uses the special structure of the system i.e. the positive definiteness of the inertia, damping and stiffness matrices, and the fact that the damping is small. The damped system is treated as a perturbation from the undamped case and the gramians as well as the balancing transformation are given as a power series in the perturbation scaling factor, with closed form expressions for the coefficient matrices. The results can be obtained at any level of accuracy by-appropriate truncation of the series.