An efficient inverse solution for MIMO dynamic systems by a Moving Least Squares meta-model

Abstract

In the inverse problem, the component parameters and excitations (e.g. typical inputs), are to be determined given corresponding system query responses (e.g. usual outputs). Parameter and excitation identification are two aspects of the inverse problem. For example, component parameter identification is crucial in designing mass-spring-damper systems (e.g., optimizing stiffness and damping for driver comfort in smart truck seats). Similarly, excitation identification is vital for improving engineering structural designs by determining large loads and displacements. Present solutions to the inverse problem include both model-based methods and data-based methods. The model-based methods invoke optimization using the so-called forward model - typically differential equations. The data-based approaches, or model-free methods, typically apply feature extraction methods such as, Support Vector Machines (SVM) and artificial neural networks (ANN). These methods are computationally intensive and provide inconsistent results. Lastly, a Least-Squares approach has greatly improved the computing efficiency but failed to fully address prediction accuracy. For the contribution of this paper, a novel solution to the inverse problem for dynamic systems, that addresses the present deficiencies, is developed through a Moving Least Squares (MLS) methodology. The impact of the work is far-reaching. The approach greatly improves prediction accuracies of the requisite component parameters and excitations. The improved accuracy is consistent for all query responses. The computation is straightforward and competitively fast. Only a hyper-linear surface fit is required, thus keeping the data matrices at manageable sizes. A larger design space is permitted and only the standard number of training sets is needed. The steps needed to solve the inverse problem herein are straightforward. First, the continuous time signals, including the excitations and the responses from the mechanistic model, are sampled and stored in vectors: the lengths may be quite large depending on the simulation period and the sample interval. The meta-model training provides the usual output matrix from the input data. Then, these matrices are interchanged to provide a new causal relationship. Now, the inputs can be found for given query outputs. To enable the nature of MLS, the pertinent closest responses to the query response are found by using a simple but effective distance metric created from the integral of squared error. The MLS weights are found from matrix inversion using Singular-Value Decomposition (SVD). Finally, the meta-model predicts the component parameters and excitations from multiple query responses. The efficacy of the work herein is illustrated by several examples. First, a simple electric network details the methodology needed to apply the MLS mathematics and compares the new approach with the main alternative methods. Next, a nonlinear, MIMO (multiple input and multiple output), dynamic, mechanical system is invoked to demonstrate simultaneous parameter design and excitation allocation. The new inverse MLS meta-model is compared to the original least squares model and the mechanistic model. The results show significant error mitigation with negligible extra computation time.