It is well known that Fourier series of discontinuous functions converge slowly and that the derivatives of the series may not converge at all. Since modal expansion of structural response is a generalization of the Fourier series, slow convergence of modal expansion can be expected when the applied loads exhibit discontinuities in time or space. Thus, in a structure controlled by point actuators, slow convergence of derivatives of structural response with respect to system parameters can be expected. To demonstrate this, the sensitivity of the closed-loop response to structural changes is calculated for a multi-span beam with three control systems of increasing complexity that utilize point actuators. 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. As expected, the convergence of the derivatives of the damping ratios with increasing number of modes is slower than the convergence of the damping ratios themselves. The convergence is improved when distributed actuators replace the point actuators. When the control system is designed based on a reduced model, the damping ratios also converge slowly. In transient response problems, it is known that complementing the vibration modes with a mode representing static response to the loads can greatly improve convergence. Indeed, for the examples studied, when Ritz vectors corresponding to static responses due to unit loads at the actuators are added to the basis vectors, the convergence of the reduced-model derivatives is greatly enhanced. Also, when the control system is designed using a reduced model containing both vibration modes and Ritz vectors, its prediction of the full-model response is greatly improved.

Ph. D.