A Robust Semi-Analytical Method for Calculating the Response Sensitivity of a Time Delay System

It is often necessary to establish the sensitivity of an engineering system’s response to variations in the process/control parameters. Applications of the calculated sensitivity include gradient-based optimization and uncertainty quantification, which generally require an efficient and robust sensitivity calculation method. In this paper, the sensitivity of the milling process, which can be modeled by a set of time delay differential equations, to variations in the input parameters is calculated. The semi-analytical derivative of the maximum eigenvalue provides the necessary information for determining the sensitivity of the process stability to input variables. Comparison with the central finite difference derivative of the stability boundary shows that the semi-analytical approach is more efficient and robust with respect to step size and numerical accuracy of the response. An investigation of the source of inaccuracy of the finite difference approximation found that it is caused by discontinuities associated with the iterative process of root finding using the bisection method.