Accuracy of derivatives of control performance using a reduced structural model

The sensitivity of control system performance to structural changes is calculated for a multi-span beam with direct-rate feedback vibration control. Reduced models based on the natural modes of the structure are formed and derivatives of the damping ratios of the closed-loop eigenvalues are calculated. The convergence of the derivatives of the damping ratios with increasing number of modes is shown to be slower than the convergence of the damping ratios themselves. In particular, in some cases the convergence of finite-element approximations to the derivatives is much faster than the convergence of the modal approximations. The results indicate that the use of reduced models based on natural vibration modes may be ill-advised for calculating the sensitivity of control system performance to changes in the controlled structure.