Structural system identification using Least Mean Square (LMS) adaptive technique

Structural system identification is a critical task in resilience assessments, especially following a natural hazard. In this paper, we propose PIDynNet, a novel physicsinformed approach that produces an ordinary differential equation-constrained neural network model for structural system identification. PIDynNet improves the estimation of structural parameters of nonlinear structural systems by embedding an auxiliary physicsbased loss term into the overall loss function as well as a supervised data-driven loss term. The proposed framework has the generalization capability to predict nonlinear structural response given unseen ground excitations. Two nonlinear numerical experiments are conducted to demonstrate the advantage of PIDynNet over other identification methods in problems with or without latent variables.

Estimating the governing equation parameter values is essential for integrating experimental data with scientific theory to understand, validate, and predict the dynamics of complex systems. In this work, we propose a new method for structural system identification (SI), uncertainty quantification, and validation directly from data. Inspired by generative modeling frameworks, a neural network maps random noise to physically meaningful parameters. These parameters are then used in the known equation of motion to obtain fake accelerations, which are compared to real training data via a mean square error loss. To simultaneously validate the learned parameters, we use independent validation datasets. The generated accelerations from these datasets are evaluated by a discriminator network, which determines whether the output is real or fake, and guides the parameter-generator network. The uncertainty in each parameter is quantified by fitting a Gaussian distribution to each parameter, then computing the 95% confidence bounds. Analytical and real experiments show the parameter estimation accuracy and model validation for different nonlinear structural systems.

An identification approach for linear and nonlinear time-variant structural systems via harmonic wavelets

Development of adaptive modeling techniques for non-linear hysteretic systems

The time finite element method (TFM) is employed for parametric identification of nonlinear structural dynamic systems. An advantage of TFM is the ease with which one can calculate the sensitivity of the transient response with respect to various design parameters, a key requirement for gradient-based parameter identification schemes. The method is simple, because one obtains the sensitivities of the response to system parameters by differentiating the algebraic equations, not original differential equations. These sensitivities are used in the Levenberg-Marquardt iterative direct method to identify parameters for nonlinear single- and two-degree-of-freedom systems. The measured response was simulated by integrating the example nonlinear systems using the given values of the system parameters. The accuracy and the efficiency of the present method are compared to a previously available approach that employs a multistep method to integrate nonlinear differential equations. It is seen, for the same accuracy, that the present approach requires fewer data points.

Convergence analysis of a continuous-time nonlinear LMS-type algorithm using stochastic calculus

This paper surveys recent advances related to sparse least mean square (LMS) algorithms. Since standard LMS algorithm does not take advantage of the sparsity information about the channel being estimated, various sparse LMS algorithms that are aim at outperforming standard LMS in sparse channel estimation are discussed. Sparse LMS algorithms force the solution to be sparse by introducing a sparse penalty to the standard LMS cost function. Under the reasonable conditions on the training datas and parameters, sparse LMS algorithms are shown to be mean square stable, and their mean square error performance and convergence rate are better than standard LMS algorithm. We introduce the sparse algorithms under Gaussian noises model. The simulation results presented in this work are useful in comparing sparse LMS algorithms against each other, and in comparing sparse LMS algorithms against standard LMS algorithm.

3R12. Nonlinearity in Structural Dynamics: Detection, Identification and Modeling. - K Worden and GR Tomlinson (Univ of Sheffield, UK). Inst Phys Publ, Bristol, UK. 2001. 659 pp. ISBN 0-7503-0356-5. $130.00.Reviewed by K Yagasaki (Dept of Mech and Syst Eng, Gifu Univ, 1-1 Yanagido, Gifu, 501-1193, Japan).This book provides backgrounds in some techniques for analyzing nonlinear structural dynamical systems and is intended for engineers and scientists in the fields of structural dynamics and nonlinear systems. It is also suitable for postgraduate and senior undergraduate students in these disciplines as a textbook and for expert structural dynamicists as a survey. Many engineering examples are given. As for the mathematics of the reader, calculation of vectors and matrices, and basic knowledge of linear differential equations and Fourier analysis are almost sufficient. The most enthusiastic topics in modern nonlinear dynamics are chaos and bifurcation. There have also been remarkable advances in the field of nonlinear control. However, these subjects are not treated in this book at all. Instead, it explains useful techniques, which include the frequency response functions (FRFs) and Hilbert transformations, for detection of nonlinearity and system identification (ie, estimation of the governing equations) when the systems do not exhibit very different motions, like chaos, from ones of linear systems. The techniques are especially important from a point of view of structural engineering because really nonlinear behavior like chaos is thought to be rather uncommon in the problems of that field. The book begins by describing the relevant backgrounds including the frequency response functions (FRFs) in linear dynamics for discrete- and continuous-time systems in Chapter 1. The discussion is first given for single-degree-of-freedom (SDOF) systems and finally generalized to multi-degree-of-freedom (MDOF) systems with an outline of modal analysis. Chapter 2 gives fundamental results and classical approaches to nonlinear systems in structural dynamics. An idea of FRF distortion is also stated there. Chapter 3 discusses FRFs for nonlinear systems in detail and describes how they are used to obtain information about nonlinearity.In Chapters 4 and 5, the Hilbert transformation, which can not only detect nonlinearity but also solve system identification problems, is explained. A mathematical exposition of the method by complex analysis is also presented. In Chapters 6 and 7, several techniques of system identification for discrete- and continuous-time equations are discussed. In Chapter 8, the concept of FRFs is generalized to higher-order FRFs with the assistance of the Volterra series. The generalization presents a method of system identification for MDOF systems. Finally, three experimental examples of a built-in beam rig, automotive shock absorber, and bilinear beam rig (the last of which is motivated from constructing a system with localized damage in the benchmark of fault detection algorithms) are given, and the techniques of the earlier chapters are applied to demonstrate their effectiveness in Chapter 9. This chapter is especially interesting for the reader from a practical point of view. A substantial set of appendices is also valuable for not only the beginners, but also ordinary researchers in the field. In summary, this reviewer recommends Nonlinearity in Structural Dynamics: Detection, Identification and Modeling for students and researchers in structural dynamics who want to study techniques for detection of nonlinearity and system identification in realistic problems.

Summary Identifiability of parameters in structural system identification (SSI) is of primary importance in any SSI method. It depends on the number and the location of the measurements, which is linked with sensor configuration. In this paper, under the framework of SSI by observability method (OM), the number of necessary measurements to identify all parameters of structural system was clarified first. Then, an example was solved step by step to show the lacking constraints among unknowns in SSI by OM. In a frame example, it was found that no measurement set having as many measurements as the number of unknowns was able to identify all parameters. To further understand this phenomenon, the observability of a simply supported beam was analyzed in an exhaustive way using 252 possible measurement sets. Three quarters of these sets were not able to identify all the parameters. In order to solve this issue, for the very first time, SSI by constrained observability method (COM), which appends the nonlinear constraints to SSI by OM, was proposed. With SSI by COM applied, the observability of the structural parameters with respect to the 252 sets was greatly improved. Finally, the efficacy of SSI by COM was verified by a 13-story frame building.

Response evolutionary power spectrum determination of chain-like MDOF non-linear structural systems via harmonic wavelets

This paper introduces an innovative adaptive control approach utilizing a nonlinear filter for a specific subset of nonlinear discrete-time systems, considering the presence of both input and output noise. The system can be transformed into a nonlinear autoregressive moving average with exogenous inputs (NARMAX) model. The concept of discrete Nussbaum gain is introduced to address the theoretical constraint associated with unknown directions of feed-forward or control gains, and the extended adaptive tuning sequence is introduced to facilitate the acceleration of parameter updating. In the case of no noise, asymptotical output tracking and global stability are achieved with the adaptive control. Further, in the presence of input noise and output noise, a novel nonlinear filter is designed to generate a more accurate filtered output, which improves the control system’s ability to adapt and track accurately. Finally, examples are provided to showcase the effectiveness and precision of the method.

Identification of nonlinear structural system is an important but challenging task for structural health monitoring. Due to the complexities of structural nonlinearities, it is hard to establish proper mathematical models for some structural nonlinear behaviors. Moreover, only partial structural responses can be measured in practice; it is essential to conduct identification of nonlinear structural systems using only partial measurements of structural responses. To cope with these issues, an algorithm is proposed in this article for the identification of some model-free structural nonlinear restoring forces using only partial measurements of structural responses. First, an equivalent linear structural system is introduced for the identification of the locations of structural nonlinearities. Then, a model-free structural nonlinear restoring force is approximated by a power series polynomial. The unknown coefficients of the power series polynomials together with other structural parameters are identified by the extended Kalman filter so that the characteristics of the behaviors of the model-free of nonlinear restoring forces can be identified. Some numerical examples including the identification of two nonlinear multi-story shear frames and a planar nonlinear truss with different structural nonlinear restoring forces are used to validate the proposed algorithm.

Studies in the approximate control of flexible structures

A parameter estimation problem in a class of nonlinear systems is considered where the input-output relation of a nonlinear system is approximated by a polynomial model (e.g., a Volterra series). A least mean squares (LMS) type algorithm is utilized for the recursive estimation of the polynomial coefficients, and its resulting mean square error (MSE) convergence properties are investigated. Conditions for the algorithm stability (in the mean square sense) are established, steady-state MSE bounds are obtained, and the convergence rate is discussed. In addition, modeling accuracy versus steady-state performance is examined; it is found that an increase of the modeling accuracy may result in a deterioration of the asymptotic performance, that is, yielding a larger steady-state MSE. Linear system identification is studied as a special case.

The transversal adaptive filter using the least mean square (LMS) algorithm of Widrow and Ropf (1976) has been widely used mainly due to its relative ease of implementation. The major drawback of this time-domain LMS (TDLMS) algorithm is that as the eigenvalue spread of the input autocorrelation matrix increases, the convergence speed of the algorithm decreases. This led to the transform domain adaptive filters where the input signals are orthogonalized. Normalized frequency domain LMS algorithms (NFDLMS) are known to be faster than the time domain implementations. However, in some implementations with low signal to return noise ratio, NFDLMS algorithms can have stability problems. The stability problem can be solved by weighting the normalization gain /spl mu/. We perform computer simulations for the telephone echo channel and show that the modified /spl mu/-weighted NFDLMS algorithm is 8 times faster than the time-domain LMS (TDLMS) algorithm and more stable than the NFDLMS algorithm over a wide range of signal to noise ratios.