Response Sensitivity Approach Research Articles

NEURAL NETWORK METHOD FOR PARAMETER ESTIMATION OF FRACTIONAL DISCRETE-TIME UNIFIED SYSTEMS

Data-driven learning of the fractional discrete-time unified system is studied in this paper. A neural network method is suggested in the parameter estimation of fractional discrete-time chaotic systems. An optimization problem is obtained and the famous Adam algorithm is employed to train the neural network’s weights and parameters. The parameter estimation result is compared with that of the stepwise response sensitivity approach (SRSA). This paper provides a high accuracy method for parameter inverse problems. The method also can be applied to data-driven learning of other fractional chaotic systems.

High-precision semi-analytical solution for the quasi-periodic nanobeam system based on the weight time-domain minimum residual method

Nanobeams have been widely used in various micro-electromechanical systems because of their small size and high sensitivity. At the nanoscale, quasi-periodic (QP) vibration often occurs in nanobeams due to the surface effects such as surface elasticity, residual surface tension and geometric nonlinearity. According to the QP motion properties of the nanobeam system, a weight time-domain minimum residual method (WTMRM) is presented in this paper by introducing the weight vector into the original time-domain minimum residual method (TMRM), and the high-precision semi-analytical QP (SAQP) solution is obtained by the WTMRM. This method mainly includes four aspects. Firstly, according to the properties of QP motion of the nanobeam system, a new SAQP solution constructed by the dominant frequency and frequency interval is presented. Then, the problem of solving the SAQP solution is converted into a nonlinear minimum optimization issue by deducing the presumptive solution and substituting the state variables back to the original control equations, and this optimization issue is resolved by the enhanced response sensitivity approach (ERSA) iteratively. Next, the weight vector is introduced to deal with the contradiction of the residual between the control equation and the initial condition constraints. Finally, the numerical results of the Runge–Kutta (RK) method are compared with those of the proposed method. The comparison results show that the precision of the SAQP solution achieved by the WTMRM is much higher than the existing results.

Parameter Identification Method for Nonsmooth Aeroelastic System

In this paper, a parameter identification method merged with the precise integration method and the predictor–corrector strategy is proposed to identify the parameters of a nonsmooth aeroelastic system quickly and accurately. Considering the aeroelastic system subject to unsteady aerodynamic force, the parameters that need to be identified include the structural and nonsmooth parameters. The work of this paper mainly includes three parts. First, based on the precise integration method and combined with the predictor–corrector strategy, a new technology that can accurately locate the switching point is proposed to obtain the response of the aeroelastic system with high precision. Then, the problem for identification of parameters of the nonsmooth aeroelastic system is modeled as a nonlinear least-squares problem between the measured data and the calculated response. A new strategy that can obtain the nonsmooth parameters sensitivity matrix accurately is proposed, and then the parameters are identified through the enhanced response sensitivity approach. Finally, the nonsmooth aeroelastic system with freeplay and hysteresis nonlinearity are taken as the numerical examples to demonstrate the feasibility and robustness of the present method.

Parameters identification of Iwan bolted joint models based on enhanced hysteretic force response sensitivity approach

An enhanced hysteretic force response sensitivity approach is proposed to identify the Iwan model of a bolted joint under the quasi-static cyclic loading. The parameters identification is formulated as a least-squares problem that finds the parameters to minimize the residuals between the measured and predicted hysteretic force. To solve this problem, the sensitivity analysis of hysteretic force response is derived in the discrete form. In doing so, the Heaviside function and smoothing strategy are utilized to overcome the discontinuity of Iwan model and make the response sensitivity accessible. Then, the least-squares problem is linearized by the hysteretic force response sensitivity and solved subsequently by the Tikhonov regularization, in which trust region constraint is employed to enhance the convergence. Numerical simulations and experiment study show that the proposed methodology is accurate and efficient for identifying the Iwan bolted joint models.

Time-domain minimum residual method combined with energy balance for nonlinear conservative systems

This paper presents a new semi-analytical method that combines the time-domain minimum residual method with energy balance for nonlinear conservative systems. Due to work done only by the conservative force, the total energy of the nonlinear conservative system will remain constant. The response frequency is also the function of the initial conditions and it is difficult to achieve the semi-analytical solutions accurately. The proposed method aims to quickly obtain the higher-order approximate semi-analytical periodic or quasi-periodic solutions of both weakly and strongly nonlinear conservative systems. To this end, the approximate analytical solution is expressed as a group of trigonometric series with undetermined coefficients firstly. Next, by deriving the assumed trigonometric series and substituting the state variables back into the original nonlinear conservative equation, the goal of solving the semi-analytical solution is transformed into an optimization problem that minimizes the residual objective function in a period. Finally, the energy constraints are introduced into the optimization process, and the enhanced response sensitivity approach is called to resolve the nonlinear objective function iteratively. A conservative Duffing system and a nonlinear symmetrically conservative two-mass system are taken as two numerical examples to illustrate the performance of the proposed method.

Residual-tuned analytical approximation for the limit cycle of aeroelastic systems with hysteresis nonlinearity

In this paper, the limit cycle of the two-dimensional aeroelastic system with hysteresis nonlinearity was obtained by the time-domain minimum residual method. Compared with the existing semi-analytical methods, the proposed approach does not need to extend the hysteresis nonlinearity into smooth functions, so the approximate solutions obtained by this approach have higher accuracy. The method is mainly three-fold. Firstly, the approximate analytical solution of the aeroelastic system is expanded as a trigonometric series with N terms and the unknown coefficients. Then, substituting the approximate solutions into the original aeroelastic system, and the problem of solving the analytical solution is transformed into a nonlinear least-square optimization problem, which is to find a set of trigonometric coefficients to minimize the residual vector over a period. Finally, the unknown coefficients are iteratively solved by the enhanced response sensitivity approach. Moreover, the Tikhonov regularization and the trust-region constraint are called to tackle ill-position and enhance the convergences. Numerical examples show that the approximate analytical solutions obtained by the proposed approach are in good agreement with numerical results, and their accuracy is much higher than that of the IHB method.

A new semi-analytical approach for quasi-periodic vibrations of nonlinear systems

This paper presents a new semi-analytical approach, namely the enhanced time-domain minimum residual method to solve the semi-analytical quasi-periodic solution for nonlinear system. This approach does not require multiple numerical integration and can be applied to strongly nonlinear systems. The approach is mainly three-fold. Firstly, the semi-analytical solution of the nonlinear quasi-periodic system is expanded into a set of trigonometric series with unknown coefficients, i.e., x(t)≈∑k=1N[bkcos(ωkt)+cksin(ωkt)]. Then, the problem of solving quasi-periodic solution can be expressed as: determining the coefficients of the trigonometric series so that the residual objective function R=Mx¨+Cx˙+Kx+N(x¨,x˙,x,t)−F(t) is minimum over a period, i.e., mina∈A∫0TR(a,t)TR(a,t)dt. Finally, the nonlinear minimum optimization problem is solved iteratively through the enhanced response sensitivity approach. Moreover, the “adaptive zero-setting curve” is introduced to accelerate the convergence. Two numerical examples, a van der Pol-Duffing system and a nonlinear energy sink system are adopted to verify the feasibility of the proposed approach.

Parameter identification of nonlinear systems with time-delay from time-domain data

The present paper develops an enhanced response sensitivity approach for parameter identification of nonlinear time-delay systems from the time-domain data. Accurate and rapid identification of the time-delay system’s parameters is the key to achieve effective control and synchronization, especially for the delay parameters. The work of this paper is mainly threefold. Firstly, taking a general nonlinear time-delay system as the investigated object, the characteristics of the nonlinear time-delay system are introduced, and the response sensitivity analysis with respect to the system parameters is derived. Then, the problem of parameter identification is modeled as a nonlinear least-squares optimization function. To this end, a novel iteration-based approach is developed to solve such problem, and the ill-posed situation is tackled by Tikhonov regularization. Besides, the trust-region constraint is implemented to enhance the convergence of the algorithm. Finally, the feasibility of the proposed approach is verified by two numerical examples and a real electronic circuit experiment. The results proved that the proposed approach can identify parameters of nonlinear time-delay systems accurately, robustly and effectively.

A new semi-analytical technique for nonlinear systems based on response sensitivity analysis

In this paper, a new semi-analytical method, namely the time-domain minimum residual method, is proposed for the nonlinear problems. Unlike the existing approximate analytical method, this method does not depend on the small parameter and can converge to the exact analytical solutions quickly. The method is mainly threefold. Firstly, the approximate analytical solution of the nonlinear system $${\varvec{F\left( \ddot{x},{\dot{x}},x\right) }}={\varvec{0}}$$ is expanded as the appropriate basis function and a set of unknown parameters, i.e., $${\varvec{x(t)}}\approx \sum _{i=0}^{N}{\varvec{a_i\chi _i(t)}}$$ . Then, the problem of solving analytical solutions is transformed into finding a set of parameters so that the residual $${\varvec{R}}={\varvec{F}}\left( \sum _{i=0}^{N}a_i\ddot{\chi }_i,\sum _{i=0}^{N}a_i{\dot{\chi }}_i,\sum _{i=0}^{N}a_i\chi _i\right) $$ is minimum over a period, i.e., $$\underset{{\varvec{a}}\in {\mathscr {A}}}{\min }\int _{0}^T {\varvec{R}}({\varvec{a}},t)^{T} {\varvec{R}}({\varvec{a}},t) \mathrm {d} t$$ . The nonlinear equation $${\varvec{F\left( \ddot{x},{\dot{x}},x\right) }}={\varvec{0}}$$ is regarded as the objective function to optimize, and the process of solving the analytic solution is transformed into a nonlinear optimization process. Finally, the optimization process is iteratively solved by the enhanced response sensitivity approach. Four numerical examples are employed to verify the feasibility and effectiveness of the proposed method.

Parameter Identification of Nonlinear Aeroelastic System with Time-Delayed Feedback Control

In this paper, an enhanced response sensitivity approach is proposed for parameter identification of nonlinear time-delay aeroelastic systems. Two common aeroelastic models are investigated, namely, the aeroelastic system under the unsteady Theodorsen aerodynamic forces and the airfoil-store system under the steady incompressible flow. The work of this paper is mainly threefold. First, the motion equations of two aeroelastic models including time-delay signals are derived, and the response sensitivity analysis with respect to the system parameters is conducted. Then, parameter identification is formulated as a nonlinear least-squares problem, and the enhanced response sensitivity approach is developed to get the solution. Finally, numerical examples are studied to verify the effectiveness of the proposed method. The results reveal that the proposed approach can give accurate, robust, and efficient identification of the parameters for time-delay aeroelastic systems.

Physical‐based parametrization and local damage identification for frame‐type structures using response sensitivity approach in time domain

Physical‐based parametrization and local damage identification for frame‐type structures using response sensitivity approach in time domain

Parameter identification of nonlinear fractional-order systems by enhanced response sensitivity approach

The fractional-order derivative is a powerful and promising concept to describe many physical phenomena due to its heredity/memory feature. This paper aims to establish a general methodology for parameter identification of nonlinear fractional-order systems based on the time domain response data and the sensitivity analysis. The development of the enhanced response sensitivity approach is mainly threefold. Firstly, a computational scheme based on the Adams-type discretization and the Newmark- $$\beta $$ method is presented to get the numerical solution of the nonlinear fractional-order systems. Thereafter, a hybrid strategy is developed to proceed the sensitivity analysis where the sensitivity to the fractional-order parameters is obtained through finite different calculation, while the sensitivity to other parameters is analyzed via direct differentiation. Secondly, the trust-region constraint is incorporated into the response sensitivity approach, and as a result, a weak convergence is reached. Thirdly, the optimal choice of the weight matrix within the framework of the response sensitivity approach is derived by minimizing the identification error, and eventually, the reciprocal of the measurement error covariance is found to be the optimal weight matrix. Numerical examples are conducted to testify the feasibility and efficiency of the present approach for parameter identification of nonlinear fractional-order systems and to verify the improvement in the identification accuracy brought up by the optimal weight matrix.

Identification of an Airfoil-Store System with Cubic Nonlinearity via Enhanced Response Sensitivity Approach

This paper presents an investigation on parameters identification of an airfoil-store system with cubic nonlinearity via enhanced response sensitivity approach. The nonlinear aeroelastic behavior of the nonlinear airfoil system depends on the airfoil’s parameters, and thus the structural parameters of the airfoil should be identified accurately before aeroelastic analysis. The identification problem of the airfoil-store system is modeled as a nonlinear least-squares optimization problem. The enhanced response sensitivity approach has been recognized for parameter identification with time domain response data and is to be followed up in the present work. To do so, the autonomous vibration response of the nonlinear motion equation is obtained by the numerical integration, and then the response sensitivity with respect to the unknown parameters is calculated likewise. Numerical tests are performed to show the effectiveness, the convergence, the sensitivity to initial parameter values, and measurement noises of the present approach in the airfoil-store system.

Parameter identification of fractional order system using enhanced response sensitivity approach

This paper presents an investigation on parameters identification of fractional order system based on the enhanced response sensitivity approach. Compared with the integer order differential operator, fractional order operator can describe the characteristics of some phenomena more accurately such as hereditary and memory properties. In order to achieve the solution of the fractional order system, a new computational scheme is presented in this paper based on Adams-type scheme and the Newmark -β method. Such an identification problem is classed as an inverse problem and can be modeled as an nonlinear optimization problem. In this paper, the enhanced response sensitivity approach is derived and used to identify the parameters of the fractional order system. To this end, the time domain response of the fractional order system is obtained by the proposed computational scheme, then the response sensitivity with respect to the parameter is achieved as well. Numerical examples illustrated the computing and identification procedure and shown the accuracy and the convergence of identification scheme.

Detection of spatially sparse damage using impulse response sensitivity and LASSO regularization

ABSTRACTThis paper presents an impulse response sensitivity approach enhanced with a least absolute shrinkage and selection operator regularization in order to detect spatially sparse (localized) damage. The analytical expression for impulse response sensitivity was derived using Vetter calculus. The proposed algorithm exploits the fact that when damage is sparse, an -norm regularization is more suitable than the common least squares (-norm) minimization. The proposed methodology is successfully applied in the context of a simulated 21 degree of freedom non-uniform shear beam with noise-contaminated measurements, limited modal parameters and limited sensor locations. Single input–single output and single input-two output cases are investigated.

Identification of nonlinear hysteretic parameters by enhanced response sensitivity approach

Hysteresis is a ubiquitous phenomenon describing the special nonlinear memory-based relation between the input and the output in many physical systems. Identifying the hysteretic parameters is the first step towards practical application of hysteretic models. In this paper, a general framework for parameter identification of nonlinear hysteretic models is developed based on the enhanced response sensitivity approach. To do so, three typical hysteretic models—Bouc–Wen model, bilinear model with kinematic hardening and bilinear model with equal yielding force are analyzed at first and the general way to model a structure with such hysteretic components is established thereafter. Then, the enhanced response sensitivity approach is presented for inverse parameter identification where the key lies in the sensitivity analysis and the trust-region constraint. Particularly, smoothing procedure is introduced to overcome the non-differentiability of bilinear hysteretic functions for sensitivity analysis of bilinear models. Numerical examples are studied to testify the feasibility and performance of the proposed approach.

Identification of Bouc-Wen hysteretic parameters based on enhanced response sensitivity approach

This paper aims to identify parameters of Bouc-Wen hysteretic model using time-domain measured data. It follows a general inverse identification procedure, that is, identifying model parameters is treated as an optimization problem with the nonlinear least squares objective function. Then, the enhanced response sensitivity approach, which has been shown convergent and proper for such kind of problems, is adopted to solve the optimization problem. Numerical tests are undertaken to verify the proposed identification approach.

Incremental response sensitivity approach for parameter identification of chaotic and hyperchaotic systems

This paper develops an incremental response sensitivity approach for parameter identification of chaotic and hyperchaotic systems. In doing so, there hinge on two points. The first point is the establishment of the incremental objective function; this origins from the idea that long-term behavior of (hyper-)chaotic systems is often sensitive to the change of system parameters, but the short-term behavior is not. The advantage over the general objective function resides in that the incremental objective function is often well-defined, that is, having only one local minimal in the parametric space. The second is the utilization of the response sensitivity approach to get the minimal or identified parameters of the incremental objective function. In order to guarantee the performance, the trust-region constraint is additionally enforced to enhance the response sensitivity approach and it is practically dealt with by the Tikhonov regularization. Numerical tests are performed to verify the proposed approach.

An enhanced response sensitivity approach for structural damage identification: convergence and performance

SummaryThis paper develops an enhanced response sensitivity approach for structural damage identification. The whole work is mainly two‐fold. Firstly, the general response sensitivity approach has been shown to perform well for small damage, but may not work for moderate or large damage. Therefore, to overcome this drawback, the trust‐region restriction is additionally enforced to enhance the general response sensitivity approach. In doing so, the Tikhonov regularization is invoked which is simple for practical manipulation and is shown equivalent to trust‐region considerations. Secondly, concrete convergence analysis is presented to guarantee the performance of the enhanced response sensitivity approach. Numerical examples are studied to verify the proposed approach. Copyright © 2016 John Wiley & Sons, Ltd.

Exploiting Spatial Sparsity in Vibration-based Damage Detection

One of the main limitations traditionally encountered in vibration-based structural health monitoring (SHM) is detecting, localizing and quantifying localized damage using global response measurements. This paper presents an impulse response sensitivity approach enhanced with a LASSO regularization in order to detect spatially sparse (localized) damage. The analytical expression for impulse response sensitivity was derived using Vetter calculus. The proposed algorithm exploits the fact that when damage is sparse, an l1-norm regularization is more suitable than the common least squares (l2-norm) minimization. The proposed methodology is successfully applied in the context of a simulated 21 degree of freedom non-uniform shear beam with noise-contaminated measurements, limited modal parameters, and single input-single output and single input-two output systems.