Calculation of frequency response envelope for dynamic systems with uncertain parameters

Frequency response finds a wide range of applications in many engineering sectors. When variations and uncertainties exist during the operation and lifetime of an engineering system, the calculation of frequency response for an uncertain dynamic system is required in order to assess the worst cases in terms of various criteria, for example gain and phase margins in control engineering, or peak magnitude at different modes in the finite element analysis of structures. This paper describes an analytical approach toward the identification of critical interior lines that possibly contribute to the boundary of the frequency response. It is allowed that uncertain parameters perturb transfer function coefficients in a nonlinear form. Conditions for critical interior lines contributing to the boundary of the frequency response are presented. An invariant property of these critical lines under open-loop and closed-loop configurations augmented by control systems or compensators is established, which greatly simplifies the analysis, design, and verification process when using frequency domain techniques. A procedure for computing frequency response and identifying the worst cases is then developed based on the combination of symbolic and numerical computation.