Optimal gyroscopic stabilization of vibrational system: algebraic approach

The paper deals with LTI vibrational systems with positive definite stiffness matrix $K$ and symmetric damping matrix $D$. Gyroscopic stabilization means the existence of gyroscopic forces with a skew-symmetric matrix $G$, such that a closed loop system with damping matrix $D+G$ is asymptotically stable. The feature of characteristic polynomial in the case predetermines such stabilization as a low order control design. Assuming the necessary condition of gyroscopic stabilization is fulfilled, we pose the problem of achieving relative stability maximum using a stabilizer $G$. The stability maximum value is determined by a matrix $D$ trace, but its reachability depends on the coincidence of all pole real parts with the corresponding minimal value, i.e. equality of characteristic and root polynomials. We illustrate a root polynomial technique application to optimal gyroscopic stabilizer design by examples of dimension 3–5.