Formulas for Computing Closed-Loop Time Responses of Flexible Systems Using Truncated Models

In this paper we derive formulas for computing the closed-loop time response of an uncertain flexible mechanical system with a truncated model. It is assumed that the dynamics of the flexible mechanical system is described by an infinite series of lightly damped vibration modes. The modal parameters, damping ratio, and natural frequency, are assumed to lie within known ranges. Given a truncated model and a robust control solution, our formulaes define upper and lower bounds for the time response of single-input/output systems about the closed-loop response of the truncated model.