Nonlinear Parameter Identification Research Articles

Nonparametric identification of multi-degree-of-freedom nonlinear systems from partially measured responses under uncertain dynamic excitations

With the rapid advent of new materials and novel structures, it becomes difficult, if not impossible, to accurately model and simulate the nonlinear response of complex systems under uncertain dynamic excitations based on close-formed nonlinear functions and parametric identification. In this study, a two-step structural nonlinearity localization and identification approach for multi-degree-of-freedom (MDOF) nonlinear systems under uncertain dynamic excitations is developed by integrating an extended Kalman filter with unknown inputs into the equivalent linearized systems. In the first step, unmeasured responses and excitations are estimated as well as unknown structural parameters and nonlinearity locations are identified by fusing acceleration with displacement time histories at the observed degrees of freedom (DOFs). In the second step, the nonlinear restoring force of the detected nonlinear structural members is identified nonparametrically using three polynomial models, including a power series polynomial model (PSPM), a double Chebyshev polynomial model (DCPM), and a Legendre polynomial model (LPM). Linear multi-story shear frames controlled by nonlinear magnetorheological (MR) dampers are modelled computationally to demonstrate the generality of the proposed methodology. The multi-source uncertainties considered in these representative examples include the location and the type of nonlinearities represented by a Bingham model and a modified Dahl model of the dampers, the location of response measurements, the location and intensity of dynamic excitations, the level of measurement noise, and the initial assignment of structural parameters. The acceleration, velocity, and displacement time histories of structures can be evaluated accurately with a maximum error of 2.62% even with the presence of 8% measurement noise, while the external excitations can be estimated within an error of 1.77%. The location of nonlinear elements can be detected correctly. The structural parameters, the NRFs provided by MR dampers and the corresponding energy dissipation can be identified with a maximum error of 2.08%, 1.19% and 0.39%, respectively, even 8% measurement noise and very rough initial assignment of structure parameters (−70%) are considered. Moreover, the numerical results change little (<0.20%) even the initial assignment of structural parameters varies from 50% to 30% of their original values, no matter which nonparametric model is employed. Results indicate that the presented algorithm can effectively identify unmeasured dynamic responses, structural parameters, unknown excitations, nonlinear locations, and NRF of nonlinear elements in a nonparametric way.

AN ADAPTIVE DIFFERENTIAL EVOLUTION ALGORITHM WITH A BOUND ADJUSTMENT STRATEGY FOR SOLVING NONLINEAR PARAMETER IDENTIFICATION PROBLEMS

Real-world parameter identification problems require determining the bounds that cover the unknown solutions. This paper presents an adaptive differential evolution algorithm with a bound adjustment strategy (ADEBAS) for solving nonlinear parameter identification problems. The adjustment strategy detects the parameter-bound violations of mutant vectors during the evolution process and gradually extends the bounds. The algorithm adaptively uses two mutation strategies and two ranges of crossover rate to balance the population diversity and convergence speed. Experimental results show that ADEBAS can solve 24 nonlinear regression tasks from the National Institute of Standards and Technology benchmark with accurate estimation and reliability. It also outperforms the compared methods on real-world parameter identification problems.

Non-linear modeling and parameter identification of a bolted joint using substructure synthesis theory

Engineering applications require accurate modeling and parameter identification to anticipate bolted joint behavior under different loading circumstances. Bolted joints are extensively used in civil and mechanical structures. In this research study, cubic stiffness non-linearity is examined and a comprehensive model, which can exhibit the non-linearity of joint behavior, is developed. This is achieved by modeling the bolted joint which is connected at one end of a cantilever beam. Mathematical model is established using concept of sub structure synthesis theory by considering non-linear translational stiffness and non-linear rotational stiffness as a boundary condition. Non-dimensional natural frequencies are estimated by assuming stiffness parameters and joint parameter estimation for translational spring having nonlinear stiffness is calculated using second order curve fitting technique. Furthermore, estimated and exact value of cubic stiffness coefficient are compared. The experimental results shows that the percentage error of estimation for all the modes are less than 8%. The experimentally determined natural frequencies are further contrasted with an analytical results to validate a precision of generated model. Present work contributes in improving the bolted joint analysis by providing a solid non-linear modeling and parameter identification methodology. This methodology enhances design practices and ensures the durability of engineering structures and mechanical systems by improving bolted joint comprehension.

Synchronous identification of nonlinear structural parameters and unknown external excitation based on improved UKF

The traditional Unscented Kalman Filter (UKF) approach for nonlinear structural identification usually requires knowledge of external excitation. However, in real application, obtaining the input excitation of nonlinear structures can be challenging due to measurement error or difficulty in acquiring the accurate information. Consequently, the real-time application of the UKF method will be constrained. To address this issue, this paper develops an improved UKF-based synchronous identification method of nonlinear structural parameters and unknown external excitation. The proposed method starts with a preliminary estimate of the unknown excitation using the current predicted values of structural dynamic responses and parameters. Subsequently, the external excitation is further identified based on the updated state vector. In addition, to reduce the effect of measurement noise on the nonlinear structural identification and ensure the accuracy of the identified results, a Kalman Filter (KF) process is embedded in the UKF to optimize measurement noise covariance matrix in real time. The practicability and accuracy of the improved UKF method are confirmed by conducting numerical simulations on a single degree-of-freedom (S-DOF) and a six degree-of-freedom (six-DOF) nonlinear structural systems due to unknown seismic excitation. Moreover, a shake table test of a five-storey steel frame subjected to seismic excitation is employed to attest the feasibility of the proposed method. Both numerical and experimental results demonstrate that the improved UKF method can effectively achieve synchronous identification for nonlinear structural parameters and unknown excitation.

Frequency splitting of hemispherical resonators trimmed with focused ion beams

This paper studies a high-precision etching process based on focused ion beam static trimming of the defective mass of fused silica hemispherical resonators to eliminate frequency splitting. This trimming method effectively overcomes the shortcomings of inefficiency and poor robustness, and the accuracy is seriously dependent on the stability of the trimming equipment in the traditional approach based on the harmonic form of trimming. The vibration model of an imperfect hemispherical resonator is firstly established by equating structural defects into point mass defective units, revealing that the frequency splitting is a function of the size of the equivalent defect mass and that the angular position of the stiffness axis is determined by the spatial distribution of the defect mass, which further elucidates that the frequency splitting of the resonator is the equivalent form of the defect mass on its low-frequency axis. To validate the accuracy of the present method, a simulation model of the imperfect resonator is established, and the results show that the theoretical model is in good agreement with the finite element method. Then, a high-precision nonlinear parameter identification model containing frequency splitting and low-frequency axis azimuth parameters is established based on the standing wave oscillation effect and spectral analysis, and the frequency splitting and low-frequency axis azimuth are identified by analyzing the vibration function of the hemispherical resonator. Finally, based on the parameter identification results and the ion beam removal function analysis, an ion beam trimming process is designed to eliminate frequency splitting by trimming the defective mass on the low-frequency axis. The results show that the frequency splitting after mass trimming is better than 0.001 Hz, significantly improving the hemispherical resonator's performance.

Time-varying nonlinear parameters identification of high-speed train suspension system based on WMA

The parameters identification of suspension components provides a foundation for health assessment of the high-speed train. Considering the time-varying nonlinearity of suspension parameters caused by changing temperature, this work identifies the time-varying nonlinear parameters based on the wavelet multiresolution analysis (WMA) combined with the proposed improved Akaike information criterion (AIC). Firstly, the parameters variation law against time-temperature are fitted based on test data and the vibration responses of a two-degree-of-freedom suspension model are obtained. Secondly, the time-varying nonlinear parameters are expanded into scale coefficients using WMA. Thus, the identification of time-varying nonlinear parameters is transformed into the identification of time-invariant wavelet scale coefficients. Then the improved AIC is utilized to select the optimal basis functions and decomposition scales in noisy environments. The identified parameters vectors are used by the improved AIC as evaluation index of identification model to reduce the effect of noise. Finally, the time-varying nonlinear parameters are reconstructed based on the optimal wavelet scale coefficients. The results show that the time-varying nonlinear stiffness and damping coefficients can be accurately identified based on the WMA combined with the proposed improved AIC.

The Unscented Kalman Filter for Nonlinear Parameter Identification of Adaptive Cruise Control Systems

This paper develops and investigates a dual unscented Kalman filter (DUKF) for the joint nonlinear state and parameter identification of commercial adaptive cruise control (ACC) systems. Although the core functionality of stock ACC systems, including their proprietary control logic and parameters, is not publicly available, this work considers a car-following scenario with a human-driven vehicle (leader) and an ACC engaged ego vehicle (follower) that employs a constant time-headway policy (CTHP). The objective of the DUKF is to determine the CTHP parameters of the ACC by using real-time observations of space-gap and relative velocity from the vehicle's onboard sensors. Real-time parameter identification of stock ACC systems is essential for assessing their string stability, large-scale deployment on motorways, and impact on traffic flow and throughput. In this regard, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\mathfrak {L}}_{2}$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\mathfrak {L}}_\infty$</tex-math></inline-formula> string stability conditions are considered. The observability rank condition for nonlinear systems is adopted to evaluate the ability of the proposed estimation scheme to estimate stock ACC system parameters using empirical data. The proposed filter is evaluated using empirical data collected from the onboard sensors of two 2019 SUV vehicles, namely Hyundai Nexo and SsangYong Rexton, equipped with stock ACC systems; and is compared with batch and recursive least-squares optimization. The set of ACC model parameters obtained from the proposed filter revealed that the commercially implemented ACC system of the considered vehicle (Hyundai Nexo) is neither <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\mathfrak {L}}_{2}$</tex-math></inline-formula> nor <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\mathfrak {L}}_\infty$</tex-math></inline-formula> string stable.

Parametric identification of flat stochastic systems for effective connectivity characterization

Some of the mechanisms that generate neuronal signals are known at the cellular level and rest on a balance of excitatory and inhibitory interactions within and between populations of neurons. Neural mass models assume that a neuronal population can be approximated using very few state variables, generally limited to mean membrane currents, potentials, and firing rates. This article deals with nonlinear parametric identification problems in neurophysiologically based models simulating brain effective connectivity. We propose a novel approach which utilizes optimal control theory for partially flat stochastic differential systems. The optimization-based approach to effective connectivity characterization has been tested through simulation experiments and compared with the extended and unscented Kalman filters. A variety of case studies have been successfully used for connectivity parameter identification: constant functions, step functions, periodic functions and random functions.

Swarm Intelligence Based Nonlinear Friction and Dynamic Parameters Identification for a 6-DOF Robotic Manipulator

Swarm Intelligence Based Nonlinear Friction and Dynamic Parameters Identification for a 6-DOF Robotic Manipulator

Rational Continuous Neural Network Identifier for Singular Perturbed Systems With Uncertain Dynamical Models.

This study aims at designing a robust nonparametric identifier for a class of singular perturbed systems (SPSs) with uncertain mathematical models. The identifier structure uses a novel identifier based on a differential neural network (DNN) with rational form, which can take into account the multirate nature of SPS. The identifier uses a mixed learning law including a rational formulation of neural networks which is useful to solve the identification of the fast dynamics in the SPS dynamics. The rational form of the design is proposed in such a way that no-singularities (denominator part of the rational form never touches the origin) are allowed in the identifier dynamics. A proposed control Lyapunov function and a nonlinear parameter identification methodology yield to design the learning laws for the class of novel rational DNN which appears as the main contribution of this study. A complementary matrix inequality-based optimization method allows to get the smallest attainable convergence invariant region. A detailed implementation methodology is also given in the study with the aim of clarifying how the proposed identifier can be used in diverse SPSs. A numerical example considering the dynamics of the enzymatic-substrate-inhibitor system with uncertain dynamics is showing how to apply the DNN identifier using the multirate nature of the proposed DNN identifier for SPSs. The proposed identifier is compared to a classical identifier which is not taking into account the multirate nature of SPS. The benefits of using the rational form for the identifier are highlighted in the numerical performance comparison based on the mean square error (MSE). This example justifies the ability of the suggested identifier to reconstruct both the fast and slow dynamics of the SPS.

Design and Development of Structural Parameter Identification Software for Nonlinear Vibration Isolators

This paper developed an object-oriented software to facilitate parameter calibration of the quasi-zero stiffness (QZS) vibration isolator for engineering applications. Considering its advantages of convenient programming and high noise resistance, we employ the harmonic balance method to construct the identification core algorithm. The developed software consists of data pre-processing, parameter identification, and post-processing modules. The data pre-processing module imports and classifies all the required data. Then, the parameter identification module conducts preliminary nonlinearity analysis so that nonlinear structural parameter identification can be successfully carried out. After parameter identification, the data post-processing module displays and stores the identified results. The performance of the software is tested by a numerical example of a two-degree-of-freedom QZS vibration isolator, proving the software can give accurate identification results within a few user-friendly graphical interfaces. Based on this software, we hope that the application and promotion of nonlinear system parameter identification become more convenient.

A High-Order Sliding-Mode Adaptive Observer for Uncertain Nonlinear Systems

A high-order sliding-mode adaptive observer is proposed to solve the problem of an adaptive estimation, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i.e.,</i> the simultaneous estimation of the state and parameters, for a class of uncertain nonlinear systems in the presence of external disturbances, which do not need to satisfy a relative degree condition equal to one. This approach is based on a high-order sliding-mode observer and a nonlinear parameter identification algorithm. The practical, global, and uniform asymptotic stability of the adaptive estimation error, despite the external disturbances, is guaranteed through the small-gain theorem. The convergence proofs are developed based on the Lyapunov and input-to-state stability theories. Some simulation results illustrate the performance of the proposed high-order sliding-mode adaptive observer.

Identification of a monitoring nonlinear oil damper using particle filtering approach

Oil dampers have been used in recent years for passive structural control and shock mitigation in dynamic structural systems. However, machining technical and economic problems limit its application for new and existing buildings due to the necessary precision machining of incorporating shaft bearings and pressure sealings. A kind of nonlinear oil damper that is quite different from traditional oil damper is investigated and applied to a steel building. Vibration monitoring system was instrumented on the dampers to explore their actual performance and effectivity under strong earthquakes. Based on monitoring response of the nonlinear dampers under various excitations, Bayesian model selection is employed to analyze the most probable model class which can capture main dynamic characteristics of the nonlinear oil dampers and can also be used for predicting future response as well as reliability. Then, a particle filtering approach is proposed to identify the nonlinear model of the damper and quantify the model uncertainty. The developed particle filter is capable of re-parameterizing joint posterior distribution of states and parameters of the nonlinear oil damper without augmented state estimation, which combined with Markov chain Monte Carlo algorithm so as to be able to sample high-dimensional posterior distribution. The identified models and posterior distributions of parameters show that the developed particle filter approach can be appropriately used for nonlinear parameter identification without stuck to special particles. Furthermore, the dynamic properties of the nonlinear oil damper with respect to various excitations involving different spectral characteristics are discussed.

Structural system identification and damage detection using adaptive hybrid Jaya and differential evolution algorithm with mutation pool strategy

Engineering applications of swarm intelligence optimization algorithms have been widely employed to identify structural systems and damages owing to their merits of simplicity, flexibility, robustness etc., while they often suffer from the defects of slow efficiency, premature convergence or even trapping into local optima in solving the inverse problem of nonlinear optimization-based parameter identification with partial noise-contaminated measurements. To deal with this issue, an adaptive hybrid Jaya and differential evolution (AHJDE) algorithm is proposed based on Jaya algorithm and differential evolution (DE) by effectively combining the advantages of both algorithms. In the proposed AHJDE, four improvements including adaptive mutation strategy, dynamic mutation and crossover operators, sampling-based resizing search space and linear resizing population size are integrated. The effectiveness of the proposed method is verified using a numerical example of 20-DOF linear system by comparing its performance with particle swarm optimization, modified artificial bee colony algorithm, clustering tree seeds algorithm, improved butterfly optimization algorithm etc. considering known mass case and unknown mass case. In addition, numerical studies on a nonlinear single degree-of-freedom system with classical Bouc-Wen hysteretic model and improved Bouc-Wen model are implemented to investigate the applicability in the field of nonlinear system identification. Finally, a series of experimental tests on a five-story steel frame structure are conducted in the laboratory to further validate the performance of the proposed approach in damage identification. Identification results demonstrate the proposed AHJDE can accurately and effectively identify the unknown system parameters and damages with limited sensors and noise-polluted responses.

Nonlinear dynamical modeling of neural activity using volterra series with GA-enhanced particle swarm optimization algorithm.

In order to improve the modeling performance of Volterra sequence for nonlinear neural activity, in this paper, a new optimization algorithm is proposed to identify Volterra sequence parameters. Algorithm combines the advantages of particle swarm optimization (PSO) and genetic algorithm (GA) improve the performance of the identification of nonlinear model parameters from rapidity and accuracy. In the modeling experiments of neural signal data generated by the neural computing model and clinical neural data set in this paper, the proposed algorithm shows its excellent potential in nonlinear neural activity modeling. Compared with PSO and GA, the algorithm can achieve less identification error, and better balance the convergence speed and identification error. Further, we explore the influence of algorithm parameters on identification efficiency, which provides possible guiding significance for parameter setting in practical application of the algorithm.

Simultaneous identification of structural damage and nonlinear hysteresis parameters by an evolutionary algorithm-based artificial neural network

An evolutionary algorithm-based artificial neural network (ANN) to identify structural damage and nonlinear hysteresis parameters simultaneously is presented. To avoid the ‘dimensional disaster’, the principal component analysis technique is applied to eliminate the redundant dimensionality of the acceleration data. The acquired principal components, which covers over 95% of the variability in data, are then employed as the input to the ANN, while the system parameters to be identified are defined as the output. ANN is an effective tool to tackle complex problems in numerous fields. However, if using the gradient-descend algorithm to train the ANN model, with vanishing or exploding gradients, ANN may suffer the local minimal during the training process. To address this drawback, a new evolutionary algorithm, termed the K-means Jaya, is employed to train the ANN model to obtain optimal weights and biases by minimizing the discrepancies between real outputs and desired ones. The optimal weights and biases are then used to configure the ANN. The proposed method is applied to single and multiple degree-of-freedom nonlinear systems subjected to different external loadings. Results demonstrate the structural damage and nonlinear hysteresis parameters can be accurately identified.

Parameter Identification of DOC Model Based on Variable Forgetting Factor Least Squares

In diesel engine after-treatment control technology, the accurate real-time control of Diesel Oxidation Catalyst (DOC) outlet temperature is an important topic. To find a high-precision parameter identification algorithm for the DOC system, this paper establishes zero-dimensional (0D) and one-dimensional (1D) mathematical models of DOC, introduces Variable Forgetting Factor Least Squares(VFFRLS) and Nonlinear Least Squares parameter identification for comparison and analysis. The results show that the 0D determination coefficient R-square of Nonlinear Least Squares parameter identification results is around 0.9, the root mean square error (RSME) mean is 23.682, the R-square of 1D is mostly less than 0.9, and the mean value of RSME is 32.649; The R-square of VFFRLS algorithm is 1, and the RSME is below 0.02. Therefore, the VFFRLS algorithm is more suitable for the parameter identification of the DOC temperature model.

Practical tracking control of piezoelectric actuators with time-delay estimation and nonsingular terminal sliding mode

A novel type of nonlinear robust control strategy is proposed in view of uncertain nonlinear factors, such as hysteresis, creep, and high-frequency vibration, of piezoelectric actuators (PEAs). This strategy can be used for the precise trajectory tracking of PEAs. The Bouc–Wen dynamic model is reasonably simplified to facilitate engineering application. The hysteresis term is summarized as an unknown term to avoid its nonlinear parameter identification. The controller robustness is achieved due to the nonsingular terminal sliding mode control, and the online estimation of unknown disturbances is realized because of the delay estimation technology; thus, no prior knowledge of the unknown boundary of the system is required. The precision robust differentiator is used to estimate the speed and acceleration signals in real time on the basis of the obtained displacement signals. The closed-loop stability of the system is proved by the Lyapunov criterion. Experimental results show that the proposed control strategy performs better than the traditional time-delay estimation control in terms of control accuracy and energy conservation. Therefore, the proposed control strategy can play an important role in the micro/nanofield driven by PEAs.

Identification of Key Nonlinear Parameters and Research on Dynamic Characteristics of Air Spring Dynamic Stiffness Model

Identification of Key Nonlinear Parameters and Research on Dynamic Characteristics of Air Spring Dynamic Stiffness Model

A Nonlinear Model and Parameter Identification Method for Rubber Isolators under Shock Excitation in Underwater Vehicles

Rubber isolators are usually used to protect high-precision equipment of autonomous underwater vehicles (AUVs), avoiding damage from overlarge dynamic excitation. Considering the nonlinear properties of the rubber material, the nonlinear behavior of rubber isolators under shock exaltation is hard to be predict accurately without the available modal and accurate parameters. In view of this, the present study proposes a nonlinear model and parameter identification method of rubber isolators to present their transient responses under shock excitation. First, a nonlinear model of rubber isolators is introduced for simulating their amplitude and frequency-dependent deformation under shock excitation. A corresponding dynamic equation of the isolation system is proposed and analytically solved by the Newmark method and the Newton-arithmetic mean method. Secondly, a multilayer feed-forward neural network (MFFNN) is constructed with the current model to search the parameters, in which the differences between the estimated and tested responses are minimized. The sine-sweep and drop test are planned with MFFNN to build the parameter identification process of rubber isolators. Then, a T-shaped isolator composed of high-damping silicon rubber is selected as a sample, and its parameters were determined by the current identification process. The transient responses of the isolation system are reconstructed by the current mode with the identified parameter, which show good agreement with measured responses. The accuracy of the proposed model and parameter identification method is proved. Finally, the errors between the reconstructed responses and tested responses are analyzed, and the main mode of energy attenuation in the rubber isolator is discussed in order to provide an inside view of the current model.