# Model-Free Identification of Nonlinear Restoring Force with Modified Observation Equation
## 1 Introduction
### Literature Survey  
Since the restoring force can act as an indicator of the extent of the nonlinearity and be used to quantitatively evaluate the absorbed energy during vibration, the research on nonlinear restoring force (NRF) identification has been actively conducted over years and the comprehensive literature reviews can be found [3; 4]. Broadly speaking, the existing NRF identification methods can be divided into two groups, i.e., parametric identification and nonparametric identification. In parametric identification methods, a model structure is specified, whereas the primary quantities obtained from the nonparametric identification methods do not directly specify equations of motion [5]. Once the structural model is fixed, the nonlinear identification problem is reduced to parameter estimation as only the coefficients of the model terms remain unspecified. The subject of parametric nonlinear identification is extremely broad, and an extensive literature exists (e.g., [6; 7; 8; 9; 10]). It is not intended to provide a comprehensive overview of the past and current approaches for the parametric identification of nonlinearity in this paper. Instead, some relevant research regarding extended Kalman filter (EKF)-based nonlinear parameter identification methods is introduced herein. For example, [PERSON] et al. [11] proposed an EKF-based adaptive tracking technique for the identification of time-vary parameters of linear and nonlinear hysteretic structures. The performance of EKF and Unscented Kalman filter (UKF) for the nonlinear identification was discussed in details by [PERSON] and [PERSON] [12]. Based on the sequential application of extended Kalman estimator and least squares estimation (LSE), [PERSON] et al. [13] proposed an algorithm for the identification of nonlinear structural parameters. On the basis of the orthogonal decomposition of excitation, [PERSON] et al. [14] proposed an EKF-based method for the condition assessment of linear and nonlinear hysteretic structures. By using EKF to estimate time-invariant parameters of the nonlinear material constitutive models, a novel nonlinear finite element model updating framework was proposed by [PERSON] et al. [15]. A two-stage NRF identification method was proposed by [PERSON] et al. [16] by using EKF for the determination of nonlinear locations in the first stage and the estimation of NRFs in the second stage. This method was extended by [PERSON] et al. [17] for identifying strong nonlinearity parameters. With the combination of EKF and dynamic statistical process control, [PERSON] et al. [18] presented a novel real-time damage detection method for identifying the parameters of both linear and nonlinear structures under different damage scenarios. [PERSON] et al. [19] proposed an adaptive three-stage EKF method to solve state and fault estimation in nonlinear discrete-time system. A dual adaptive filtering approach was proposed by [PERSON] [20] for nonlinear finite element model updating with the consideration of modeling uncertainty. The performance of different model updating techniques using EKF, UKF and iterated EKF were further compared by [PERSON] et al. [21] in terms of convergence, accuracy, robustness and computational demand. The methods mentioned above are all capable of identifying the NRF of a preset nonlinear model. However, accessing prior knowledge of nonlinear characteristics may prove difficult in many circumstances owing to the highly individualistic nature of real nonlinearities. Moreover, it is sometimes difficult to define a specific physically-motivated model to precisely describe the encountered nonlinear phenomena such as the initiation and development of crack. The lack of high-fidelity, physics-based and robust models that could accurately reflect the multidimensional behavior of these nonlinear systems would hinder the application of the computational tools or methods in identifying the complex nonlinearities in the design stage. Thus, nonparametric nonlinear identification solely using input and output data is necessary.