Efficient Stability and Robustness Analysis of Uncertain Nonlinear Systems using a New Response-based Method

Abstract

A stable dynamic system implies safety, reliability, and satisfactory performance. However, the determination of stability is very difficult when the system is nonlinear and when the ever present uncertainties in the components must be considered. Herein a response-based approach that uses both system and time information obtained through singular value decomposition is presented to determine the stability space of nonlinear, uncertain dynamic systems: any approximating linearization of the nonlinearities has been obviated. The approach extends previous work for linear systems that invoked only the variability of the left singular vectors to predict stability. In the new approach, the variability of the right singular vectors is augmented to that of the left singular vectors and it is shown that a simulation time span, as short as two or three periods, is sufficient to predict stability over the entire life-time dynamics rendering the method very efficient. The stability space is a subset of the design space and its robustness is proportional to the tolerances assigned to the random design variables. Errors due to sampling size, time increments, and number of singular vectors used are controllable. The method can be implemented with readily available software. A study of a practical engineering system with different tolerances and different time spans shows the efficacy of the proposed approach.