Benchmarking Optimisation Methods for Model Selection and Parameter Estimation of Nonlinear Systems

Sina Safari,Julián Londoño Monsalve

doi:10.3390/vibration4030036

Abstract

Characterisation and quantification of nonlinearities in the engineering structures include selecting and fitting a good mathematical model to a set of experimental vibration data with significant nonlinear features. These tasks involve solving an optimisation problem where it is difficult to choose a priori the best optimisation technique. This paper presents a systematic comparison of ten optimisation methods used to select the best nonlinear model and estimate its parameters through nonlinear system identification. The model selection framework fits the structure’s equation of motions using time-domain dynamic response data and takes into account couplings due to the presence of the nonlinearities. Three benchmark problems are used to evaluate the performance of two families of optimisation methods: (i) deterministic local searches and (ii) global optimisation metaheuristics. Furthermore, hybrid local–global optimisation methods are examined. All benchmark problems include a free play nonlinearity commonly found in mechanical structures. Multiple performance criteria are considered based on computational efficiency and robustness, that is, finding the best nonlinear model. Results show that hybrid methods, that is, the multi-start strategy with local gradient-based Levenberg–Marquardt method and the particle swarm with Levenberg–Marquardt method, lead to a successful selection of nonlinear models and an accurate estimation of their parameters within acceptable computational times.

Highlights

Efficient and lightweight engineering structures have been found to behave nonlinearly; some examples of multiple applications are microelectromechanical structures represented in [1] and aerospace structures in [2,3]
Nonlinear terms are added based on the forward approach, and backward regression starts to eliminate the terms with less contribution to the system responses
This paper presents a comparative evaluation of ten optimization methods for nonlinear model selection and parameter estimation of nonlinear dynamic systems

Summary

Introduction

Efficient and lightweight engineering structures have been found to behave nonlinearly; some examples of multiple applications are microelectromechanical structures represented in [1] and aerospace structures in [2,3]. While linear system identification and modal testing techniques have been widely developed and applied [4], the lack of detailed knowledge about the structural mechanisms with nonlinear behavior has motivated the development of Nonlinear System Identification (NSI) methods. After the detection of nonlinearities [8] in the initial stages of modal testing, characterisation and quantification are crucial steps. Methods for the characterisation and quantification of nonlinearities present in the structures have been developed using the measured data whether in the time domain, frequency domain, or both; many of which have been reviewed in [9]. We are looking at which optimisation method works best when dealing with the challenge of nonlinear model selection using time domain data

Methods

Results

Conclusion