Nonlinear System Identification Method Research Articles

Nonparametric Volterra Series Estimate of the Cascaded Water Tanks Using Multidimensional Regularization

This paper presents an efficient nonparametric time domain nonlinear system identification method applied to the measurement benchmark data of the cascaded water tanks. In this work a method to estimate efficiently finite Volterra kernels without the need of long records is presented. This work is a novel extension of the regularization methods that have been developed for impulse response estimates of linear time invariant systems. Due to the limited number of available data samples, the highest considered Volterra order is limited. In the paper the results for different scenarios varying from a simple Finite Impulse Response (FIR) model to a 3rd degree Volterra series are compared and studied. In each case, the transients are removed by a special regularization method based on the novel ideas of transient removal for Linear Time-Varying (LTV) systems. Using the proposed methodologies, the nonparametric Volterra models provide a very good data-fit, and their performance is comparable with the white-box (physical) models.

Nonlinear system identification based on ANFIS-Hammerstein model using Gravitational search algorithm

The identification of real-world plants and processes, which are nonlinear in nature, represents a challenging problem. Currently, the Hammerstein model is one of the most popular nonlinear models. A Hammerstein model involves the combination of a nonlinear element and a linear dynamic system. On the other hand, the Adaptive-network-based fuzzy inference system (ANFIS) represents a powerful adaptive nonlinear network whose architecture can be divided into a nonlinear block and a linear system. In this paper, a nonlinear system identification method based on the Hammerstein model is introduced. In the proposed scheme, the system is modeled through the adaptation of an ANFIS scheme, taking advantage of the similarity between it and the Hammerstein model. To identify the parameters of the modeled system, the proposed approach uses a recent nature-inspired method called the Gravitational Search Algorithm (GSA). Compared to most existing optimization algorithms, GSA delivers a better performance in complex multimodal problems, avoiding critical flaws such as a premature convergence to sub-optimal solutions. To show the effectiveness of the proposed scheme, its modeling accuracy has been compared with other popular evolutionary computing algorithms through numerical simulations on different complex models.

A new analytical method for characterizing nonlinear visual processes with stimuli of arbitrary distribution: Theory and applications

Characterization of the functional relationship between sensory inputs and neuronal or observers' perceptual responses is one of the fundamental goals of systems neuroscience and psychophysics. Conventional methods, such as reverse correlation and spike-triggered data analyses are limited in their ability to resolve complex and inherently nonlinear neuronal/perceptual processes because these methods require input stimuli to be Gaussian with a zero mean. Recent studies have shown that analyses based on a generalized linear model (GLM) do not require such specific input characteristics and have advantages over conventional methods. GLM, however, relies on iterative optimization algorithms and its calculation costs become very expensive when estimating the nonlinear parameters of a large-scale system using large volumes of data. In this paper, we introduce a new analytical method for identifying a nonlinear system without relying on iterative calculations and yet also not requiring any specific stimulus distribution. We demonstrate the results of numerical simulations, showing that our noniterative method is as accurate as GLM in estimating nonlinear parameters in many cases and outperforms conventional, spike-triggered data analyses. As an example of the application of our method to actual psychophysical data, we investigated how different spatiotemporal frequency channels interact in assessments of motion direction. The nonlinear interaction estimated by our method was consistent with findings from previous vision studies and supports the validity of our method for nonlinear system identification.

Nonlinear Identification of Electronic Brake Pedal Behavior Using Hybrid GMDH and Genetic Algorithm in Brake-By-Wire System

In this paper, we represent a nonlinear identification of electronic brake pedal behavior in the brake-by-wire (BBW) system based on hybrid group method of data handling (GMDH) and genetic algorithm (GA). A GMDH is a kind of multi-layer network with a structure that is determined through training and which can express nonlinear dynamics as a mathematical model. The GA is used in the GMDH, enabling each neuron to search for its optimal set of connections with the preceding layer. The results obtained with this hybrid approach were compared with different nonlinear system identification methods. The experimental results showed that the hybrid approach performs better than the other methods in terms of root mean square error (RMSE) and correlation coefficients. The hybrid GMDH/GA approach was effective for modeling and predicting the brake pedal system under random braking conditions.

System identification of nonlinear system subjected to white random excitation using Fokker-Planck equation

This paper proposes an identification method of symmetry nonlinear system using Fokker-Planck equation. The proposed identification method will be derived from the least square estimation based on probability density function of Fokker-Planck equation. In order to estimate the probability density function, response of target nonlinear system is supposed to be subject to a constrained Gaussian Mixture Model, which has a constraint about symmetrical distribution. The estimation algorithm will be derived by using a method of maximum logarithmic likelihood, i.e. ExpectationMaximization algorithm. In a numerical simulation based on a nonlinear one degree-of-freedom system subjected to white random excitation, the estimation result of probability density function shows good agreement with the ground truth. In addition, relationship between restoring force and displacement shows good agreement with the ground truth.

Milling gate vibrations analysis via Hilbert-Huang transform

The study aimed at identification of the milling gate modal parameters: natural frequencies and mode shapes, in order to establish the working range free from resonant areas. The evaluation was performed under exploitation loads, generated in the course of the milling process. System responses, in the form of the acceleration time histories, were analysed with the application of the Hilbert-Huang method data analysis. Selection of this method was motivated by the fact that it does not require stationarity of the analysed and provides information about nonlinearities in vibration empirical modes. In contrast to classical nonlinear system identification methods, the Hilbert-Huang method does not require measurements of excitations acting on the system of interest nor linear system dynamic behaviour in a broad frequency range around any operating point.

A novel APSO-aided weighted LSSVM method for nonlinear hammerstein system identification

Identification of Hammerstein nonlinear models has received much attention due to its ability to describe a wide variety of nonlinear systems. A novel identification approach based on intelligent optimal weighted least squares SVM (WLSSVM) is proposed for Hammerstein system, where a new adaptive particle swarm optimization algorithm (APSO) using the evolutionary state estimation technique and mutation operator is applied. The proposed method not only has fast convergence to the global optimal solution but also has good identification results. The comparison researches are carried out among the proposed method, WLSSVM, LSSVM and RIV methods in detail. The research results show the effectiveness of proposed APSO-WLSSVM method.

Bias compensation principle based recursive least squares identification method for Hammerstein nonlinear systems

This paper presents a bias compensation principle based recursive least squares (BCP-RLS) identification method for Hammerstein nonlinear autoregressive moving average with exogenous variable (ARMAX) systems. By introducing a non-singular matrix and an auxiliary vector uncorrelated with the noise term, we firstly establish the BCP-RLS unified framework. Next the convergence and consistency properties of the achieved BCP-RLS method are rigorously analyzed without the martingale difference sequence assumption or the strictly positive real condition. Furthermore, some discussions on the flexibility of the BCP-RLS method and its comparisons with some other existing methods are presented. Finally, some representative simulation examples are conducted to verify the obtained results.

Recursive identification of discrete-time nonlinear cascade systems with time-varying output hysteresis

A recursive identification method for discrete-time nonlinear cascade systems with a linear dynamic system followed by a time-varying output hysteresis is presented. The mathematical model is based on the application of a special form of Coleman–Hodgdon hysteresis model and a compound operator decomposition technique and is linear-in-parameters. The recursive least-squares algorithm with internal variable estimation is used for the time-varying parameter identification. Simulation studies show the feasibility of proposed approach to estimate the model parameters and track their changes.

Particle filtering and marginalization for parameter identification in structural systems

In structural health monitoring, one wishes to use available measurements from a structure to assess structural condition, localize damage if present, and quantify remaining life. Nonlinear system identification methods are considered that use a parametric, nonlinear, physics-based model of the system, cast in the state-space framework. Various nonlinear filters and parameter learning algorithms can then be used to recover the parameters and quantify uncertainty. This paper focuses on the particle filter (PF), which shows the advantage of not assuming Gaussianity of the posterior densities. However, the PF is known to behave poorly in high dimensional spaces, especially when static parameters are added to the state vector. To improve the efficiency of the PF, the concept of Rao–Blackwellisation is applied, that is, we use conditional linearities present in the equations to marginalize out some of the states/parameters and infer their conditional posterior pdf using the Kalman filtering equations. This method has been studied extensively in the particle filtering literature, and we start our discussion by improving upon and applying two well-known algorithms on a benchmark structural system. Then, noticing that in structural systems, high nonlinearities are often localized while the remaining equations are bilinear in the states and parameters, a novel algorithm is proposed, which combines this marginalization approach with a second-order extended Kalman filter. This new approach enables us to marginalize out all the states/parameters, which do not contribute to any high nonlinearity in the equations and, thus, improve identification of the unknown parameters. Copyright © 2016 John Wiley & Sons, Ltd.

Output-Only Identification of Vibratory Machine Suspension Parameters under Exploitational Conditions

The paper concerns model based identification of vibratory machine body suspension system on the basis of dynamic responses measured under exploitational conditions. The research was carried out by means of the restoring force, boundary perturbation and direct parameter estimation techniques which, on the contrary to classical nonlinear system identification methods, requires neither excitation measurements nor linear behaviour of the considered system around an operating point. At the first stage of the research, parameters of the machine body suspension system were identified. Results accuracy was verified by determining percentage relative error of mass estimation with respect to the value calculated based on the machine geometrical and material properties. In the next step, the suspension system was modified by introduction of a nonlinear damping system. Obtained results proved that the assumed identification method is convenient for vibratory machine suspension condition monitoring and determining forces transferred on machine foundations.

Identification of Nonlinearities in Joints of a Wing Structure

Nonlinear structural identification is essential in engineering. As new materials are being used and structures become slender and lighter, nonlinear behaviour of structures becomes more important. There have been many studies into the development and application of system identification methods for structural nonlinearity based on changes in natural frequencies, mode shapes and damping ratios. A great challenge is to identify nonlinearity in large structural systems. Much work has been undertaken in the development of nonlinear system identification methods (e.g. Hilbert Transform, NARMAX, and Proper Orthogonal Decomposition), however, it is arguable that most of these methods are cumbersome when applied to realistic large structures that contain mostly linear modes with some local nonlinearity (e.g. aircraft engine pylon attachment to a wing). In this paper, a multi-shaker force appropriation method is developed to determine the underlying linear and nonlinear structural properties through the use of the measurement and generation of restoring force surfaces. One undamped mode is excited in each multi-shaker test. Essentially, this technique is a derivative of the restoring surface method and involves a non-linear curve fitting performed in modal space. A reduced finite element model is established and its effectiveness in revealing the nonlinear characteristics of the system is discussed. The method is demonstrated through both numerical simulations and experiments on a simple jointed laboratory structure with seeded faults, which represents an engine pylon structure that consists of a rectangular wing with two stores suspended underneath.

Simultaneous Identification of Structural Parameter and External Excitation with an Improved Unscented Kalman Filter

The structural parameters identification without the prior known time history of external force is still a challenging problem for nonlinear structures. In this study, a new nonlinear system identification method is proposed to evaluate the structural parameters without the prior information of excitation time history. Excitation decomposition algorithm is utilized to the system identification. The structural excitation time history is decomposed by orthogonal approximation. With this approximation, the inverse problem of the external excitation identification is transformed into the inverse problem of parameter identification. An improved Unscented Kalman filter (UKF) is proposed to identify the structural parameters and coefficients of the orthogonal decomposition only based on the acceleration measurements. The computation of the square roots of matrix may be ill-posed. The singular value decomposition (SVD) method instead of Cholesky decomposition is applied to calculate the square roots of the covariance matrix in the updating step of UKF identification to improve the robustness. The calculation of the covariance matrix is also modified to ensure its symmetric property so that error propagation can be mitigated. For the complicate excitation identification, the proposed method can be improved for practical application. The structural response would be divided into several windows. In each window the proposed method is performed iteratively for accuracy identification result. A single degree of freedom structure and a three-story hysteretic nonlinear shear frame are investigated numerically with the proposed identification method. Results from the simulation studies demonstrate that the structural parameters and excitation can be accurately identified. The proposed improved UKF can also simultaneously identify the structural parameters and dynamic excitation fairly accurately with measurement noise.

Nonlinear system identification using least squares support vector machine tuned by an adaptive particle swarm optimization

In this paper, we present a method for nonlinear system identification. The proposed method adopts least squares support vector machine (LSSVM) to approximate a nonlinear autoregressive model with eXogeneous (NARX). First, the orders of NARX model are determined from input–output data via Lipschitz quotient criterion. Then, an LSSVM model is used to approximate the NARX model. To obtain an efficient LSSVM model, a novel particle swarm optimization with adaptive inertia weight is proposed to tune the hyper-parameters of LSSVM. Two experimental results are given to illustrate the effectiveness of the proposed method.

Nonlinearity of mechanical damping and stiffness of a spring-suspended sectional model system for wind tunnel tests

The wind tunnel test of spring-suspended sectional models (SSSM) is an important means in the research of wind engineering, which is very frequently employed to check the performances of flutter and vortex-induced resonance of bridges as well as to identify the various aerodynamic and aeroelastic parameters of bridge components, such as aerodynamic derivatives of self-excited forces. However, in practice, the mechanical damping ratios and natural frequencies of SSSM system are prevailingly supposed to be constant in the whole procedure of a test. This assumption often leads to notable errors of the test results or dispersion of the identified aerodynamic parameters because the mechanical damping ratios and natural frequencies of SSSM system are proved to vary in fact to some extent with the change of oscillating amplitude. On that account, the mechanical nonlinearity of SSSM system is investigated and discussed in this paper by taking a flat-closed box section as a research background. The conventional linear model is firstly proved to fail to predict precisely the long-duration free decay responses of the SSSM system. The formulae of equivalent linearization approximation (ELA) are then derived by using a multiple-scale method to model the mechanical nonlinearities in the first-order approximate sense, and a time-domain system identification method is proposed on this basis to identify equivalent amplitude-dependent (EAD) damping ratio and frequency. The proposed ELA and nonlinear system identification methods are then found to be precise enough to model the mechanical nonlinearities of SSSM system. The characteristics of EAD damping ratio and frequency of both the bending and torsional modes are then discussed in detail. It is then found that the major energy dissipation of SSSM vibrations at both the bending and torsional modes generally comes from the combined effect of viscous damping and quadratic damping. However, for the vibration at the bending mode with small to moderate amplitudes, the coulomb friction damping becomes the major source of energy dissipation. Furthermore, different types of additional dampers added on the SSSM system will introduce extra mechanical nonlinearities in manners very different from each other. The liquid viscous damper with oil tank form employed in this study is found to be an ideal linear damper, whereas the steel wire-rope-circle damper of friction type brings in significant mechanical nonlinearities.

Bayesian Methods for Nonlinear System Identification of Civil Structures

This paper presents a new framework for the identification of mechanics-based nonlinear finite element (FE) models of civil structures using Bayesian methods. In this approach, recursive Bayesian estimation methods are utilized to update an advanced nonlinear FE model of the structure using the input-output dynamic data recorded during an earthquake event. Capable of capturing the complex damage mechanisms and failure modes of the structural system, the updated nonlinear FE model can be used to evaluate the state of health of the structure after a damage-inducing event. To update the unknown time-invariant parameters of the FE model, three alternative stochastic filtering methods are used: the extended Kalman filter (EKF), the unscented Kalman filter (UKF), and the iterated extended Kalman filter (IEKF). For those estimation methods that require the computation of structural FE response sensitivities with respect to the unknown modeling parameters (EKF and IEKF), the accurate and computationally efficient direct differentiation method (DDM) is used. A three-dimensional five-story two-by-one bay reinforced concrete (RC) frame is used to illustrate the performance of the framework and compare the performance of the different filters in terms of convergence, accuracy, and robustness. Excellent estimation results are obtained with the UKF, EKF, and IEKF. Because of the analytical linearization used in the EKF and IEKF, abrupt and large jumps in the estimates of the modeling parameters are observed when using these filters. The UKF slightly outperforms the EKF and IEKF.

An identification method for nonlinear systems with colored measurement noise

An identification method for nonlinear systems with colored measurement noise

Identification of Suspension Bridge Parameters under Exploitational Conditions

1. Department of Robotics and Mechatronics, Faculty of Mechanical Engineering and Robotics, AGH University of Science and Technology, Mickiewicz Alley 30, 30-059, Krakow, Poland 2. Department of Process Control, Faculty of Mechanical Engineering and Robotics, AGH University of Science and Technology, Mickiewicz Alley 30, 30-059, Krakow, Poland Received: ** **, 2014 / Accepted: ** **, 2014 / Published: ** **, 2014. Abstract: The paper concerns research into dynamic properties of the steel suspension bridge across Opolska street in Krakow, Poland. Parameter identification was carried out with the application of the nonlinear system identification method on the basis of system responses to exploitational excitation resulting from pedestrian traffic. In order to verify obtained results, on the basis of the geometrical and material properties of the considered system, the finite elements model was created. Created FEM model was updated through the comparison with the model determined by the use of experimental modal analysis method and then applied to analytical evaluation of the considered suspension bridge natural frequencies. Key words: steel bridge, nonlinear system identification, exploitational excitation.

Recurrent Spiking Neural Networks the Third Generation in Identification of Systems

paper the modified identification method for nonlinear systems is proposed based on Recurrent Spiking Neural Networks (RSNN). Spike Response Model (SRM) has been employed in the modification method. The learning of the parameters of RSNN is based on modified backpropagation algorithm which is known as SpikeProp. In the identification of a variety of types of nonlinear systems, a coding equation is applied to convert real numbers into spike times. The RSNN structure is tested for the identification of the nonlinear systems. The simulation results show that the proposed modification method provides a good performance in terms of execution time and minimizing error in the training phase. . KeywordsNeural Networks, Identification, nonlinear systems, SpikeProp

Identifying parameters of multi-degree-of-freedom nonlinear structural dynamic systems using linear time periodic approximations

The authors recently presented a new nonlinear system identification method, here dubbed the NL-LTP method, in which the system of interest is forced harmonically so that it responds in a stable periodic orbit, and then it is perturbed slightly and its response is recorded as it returns to the orbit. Under mild assumptions the response about the periodic orbit can be approximated using a linear time periodic system model, which can be identified from the measurements using techniques that are akin to linear modal analysis. While the prior work focused on simulated measurements from single degree-of-freedom systems, this work presents several tools that are needed in order to use this approach on multi-degree-of-freedom systems and focuses on applying the method to experimental hardware. The proposed system identification methodology is unique in that it identifies both the order of the nonlinear system and a mathematical model for the nonlinear restoring forces without assuming the mathematical form for the nonlinearities a priori. Towards these ends, this work explains how to extract the underlying nonlinear system model, or nonlinear restoring force versus displacement relationships, from the time periodic model that governs deviations of the system from its periodic orbit, and presents various metrics that can be used to determine which terms in the model are meaningful. These new tools are used to apply the identification method to a continuous, multi-degree-of-freedom structure with a discrete geometric nonlinearity, using both simulated and experimental measurements. The experimental hardware consists of a cantilever beam with a nonlinear spring attached to its tip, which is driven in a periodic limit cycle by an electromagnetic shaker.