Nonlinear System Identification Techniques Research Articles

Predicting physico-mechanical properties of LLDPE and PDMS rubber blends using ANFIS-based model

Blends of linear low density polyethylene (LLDPE) and poly dimethyl siloxane (PDMS) rubber have been identified as a promising material for cable insulation. The influence of processing parameters on the physico-mechanical properties of these blends has been investigated and the same has been optimised for preparing these blends. In practice, it is not possible to carry out several experiments to identify the relationship between the intermediate processing parameters and physical properties. To address this issue, the relationship between the processing parameters and the mechanical properties of the optimised LLDPE and PDMS rubber blends have been mapped using a non-linear system identification technique namely, adaptive-network-based fuzzy inference system (ANFIS) without carrying further experiments, but with the available experimental data. In this model, the effect of the number of fuzzy rules on the model performance has been investigated to predict the properties and to correlate the above relationship. This paper reports the development of ANFIS-based model for predicting the LLDPE/PDMS rubber blends in reducing the experimental trials and its success over the ANN model.

Identification of Low Order Models for Large Scale Systems

Abstract Abstract In this paper we propose a novel procedure for obtaining a low order non-linear model of a large scale multi-phase, non-linear, reactive fluid flow systems. Our approach is based on the combinations of the methods of Proper Orthogonal Decomposition (POD), and non-linear System Identification (SID) techniques. The problem of non-linear model reduction is formulated as parameter estimation problem. In the first step POD is used to separate the spatial and temporal patterns and in the second step a model structure and it's parameters of linear and of non-linear polynomial type are identified to approximate the temporal patterns obtained by the POD in the first step. The proposed model structure treats POD modal coefficients as states rather than outputs of the identified model. The state space matrices which happens to be the parameters of a black-box to be identified, comes linearly in parameter estimation process. For the same reason, Ordinary Least Square (OLS) method is used to estimate the model parameters. The simplicity and reliability of the proposed method gives computationally very efficient linear and non-linear low order models for extremely large scale processes. The method is of generic nature. The efficiency of proposed approach is illustrated on a very large scale benchmark problem depicting Industrial Glass Manufacturing Process (IGMP). The results show good performance of the proposed method.

Structure Computation of Quiet Spike Flight-Test Data During Envelope Expansion

System identification or mathematical modeling is used in the aerospace community for development of simulation models for robust control law design. These models are often described as linear time-invariant processes. Nevertheless, it is well known that the underlying process is often nonlinear. The reason for using a linear approach has been due to the lack of a proper set of tools for the identification of nonlinear systems. Over the past several decades, the controls and biomedical communities have made great advances in developing tools for the identification of nonlinear systems. These approaches are robust and readily applicable to aerospace systems. In this paper, we show the application of one such nonlinear system identification technique, structure detection, for the analysis of F-15B Quiet Spike(TradeMark) aeroservoelastic flight-test data. Structure detection is concerned with the selection of a subset of candidate terms that best describe the observed output. This is a necessary procedure to compute an efficient system description that may afford greater insight into the functionality of the system or a simpler controller design. Structure computation as a tool for black-box modeling may be of critical importance for the development of robust parsimonious models for the flight-test community. Moreover, this approach may lead to efficient strategies for rapid envelope expansion, which may save significant development time and costs. The objectives of this study are to demonstrate via analysis of F-15B Quiet Spike aeroservoelastic flight-test data for several flight conditions that 1) linear models are inefficient for modeling aeroservoelastic data, 2) nonlinear identification provides a parsimonious model description while providing a high percent fit for cross-validated data, and 3) the model structure and parameters vary as the flight condition is altered.

A review of nonlinear dynamics applications to structural health monitoring

The process of implementing a damage detection strategy for aerospace, civil and mechanical engineering infrastructure is referred to as structural health monitoring (SHM). In many cases damage causes a structure that initially behaves in a predominantly linear manner to exhibit nonlinear response when subject to its operating environment. The formation of cracks that subsequently open and close under operating loads is an example of such damage. The damage detection process can be significantly enhanced if one takes advantage of these nonlinear effects when extracting damage-sensitive features from measured data. This paper will provide a review of examples from nonlinear dynamical systems theory and from nonlinear system identification techniques that are used for the feature-extraction portion of the damage detection process. This paper is not intended as a comprehensive review of all damage detection methods rooted in nonlinear dynamics, but rather to provide a number of illustrations of complimentary approaches where damage-sensitive data features are based on nonlinear system response. These features, in turn, can either be used as a direct diagnosis of damage or as input to statistical damage classifier. Copyright © 2007 John Wiley & Sons, Ltd.

Experimental Identification of Nonlinear Continuous Vibratory Systems Using Nonlinear Principal Component Analysis

This paper presents a new experimental identification technique for nonlinear continuous vibratory systems. The governing equations of motion of a nonlinear continuous system are described by a set of nonlinear partial differential equations and boundary conditions. Determining both of them simultaneously is a quite difficult task. Thus, one has to discretize the governing equations of motion, and reduce the order of the equations as much as possible. In analysis of nonlinear vibratory systems, it is known that one can reduce the order of the system by using the nonlinear normal modes preserving the effect of the nonlinearity accurately. The nonlinear normal modes are description of motion as nonlinear functions of the coordinates for analysis. In identification, if one can express the data as nonlinear functions of the coordinates for identification, it is expected that accurate mathematical model with minimum degrees of freedom can be determined. Based on this idea, this paper proposes an identification technique which uses nonlinear principal component analysis by a neural network. Applicability of the proposed technique is confirmed by numerical simulation.

Learning to Transform Time Series with a Few Examples

We describe a semi-supervised regression algorithm that learns to transform one time series into another time series given examples of the transformation. This algorithm is applied to tracking, where a time series of observations from sensors is transformed to a time series describing the pose of a target. Instead of defining and implementing such transformations for each tracking task separately, our algorithm learns a memoryless transformation of time series from a few example input-output mappings. The algorithm searches for a smooth function that fits the training examples and, when applied to the input time series, produces a time series that evolves according to assumed dynamics. The learning procedure is fast and lends itself to a closed-form solution. It is closely related to nonlinear system identification and manifold learning techniques. We demonstrate our algorithm on the tasks of tracking RFID tags from signal strength measurements, recovering the pose of rigid objects, deformable bodies, and articulated bodies from video sequences. For these tasks, this algorithm requires significantly fewer examples compared to fully-supervised regression algorithms or semi-supervised learning algorithms that do not take the dynamics of the output time series into account.

Electric Load Forecasting

This article illustrates the application of a nonlinear system identification technique to the problem of STLF. Five NARX models are estimated using fixed-size LS-SVM, and two of the models are later modified into AR-NARX structures following the exploration of the residuals. The forecasting performance, assessed for different load series, is satisfactory. The MSE levels on the test data are below 3% in most cases. The models estimated with fixed-size LS-SVM give better results than a linear model estimated with the same variables and also better than a standard LS-SVM in dual space estimated using only the last 1000 data points. Furthermore, the good performance of the fixed-size LS-SVM is obtained based on a subset of M = 1000 initial support vectors, representing a small fraction of the available sample. Further research on a more dedicated definition of the initial input variables (for example, incorporation of external variables to reflect industrial activity, use of explicit seasonal information) might lead to further improvements and the extension toward other types of load series.

Monthly streamflow prediction in the Volta Basin of West Africa: A SISO NARMAX polynomial modelling

Single-input-single-output (SISO) non-linear system identification techniques were employed to model monthly catchment runoff at selected gauging sites in the Volta Basin of West Africa. NARMAX (Non-linear Autoregressive Moving Average with eXogenous Input) polynomial models were fitted to basin monthly rainfall and gauging station runoff data for each of the selected sites and used to predict monthly runoff at the sites. An error reduction ratio (ERR) algorithm was used to order regressors for various combinations of input, output and noise lags (various model structures) and the significant regressors for each model selected by applying an Akaike Information Criterion (AIC) to independent rainfall-runoff validation series. Model parameters were estimated from the Matlab REGRESS function (an orthogonal least squares method). In each case, the sub-model without noise terms was fitted first followed by a fitting of the noise model. The coefficient of determination ( R-squared), the Nash-Sutcliffe Efficiency criterion (NSE) and the F statistic for the estimation (training) series were used to evaluate the significance of fit of each model to this series while model selection from the range of models fitted for each gauging site was done by examining the NSEs and the AICs of the validation series. Monthly runoff predictions from the selected models were very good, and the polynomial models appeared to have captured a good part of the rainfall-runoff non-linearity. The results indicate that the NARMAX modelling framework is suitable for monthly river runoff prediction in the Volta Basin. The several good models made available by the NARMAX modelling framework could be useful in the selection of model structures that also provide insights into the physical behaviour of the catchment rainfall-runoff system.

Nonlinear Model for Sub- and Superharmonic Motions of a MDOF Moored Structure, Part 2—Sensitivity Analysis and Comparison

The nonlinear R-MI/SO system identification procedure and the parameters of the MDOF system identified in Part 1 are examined in detail in this paper. A parametric study is conducted and the results are presented on the sensitivity of the system parameters for two key nonlinear responses—subharmonic and superharmonic resonances. The parameters are compared to determine the appropriateness of using a single set of system parameters for both response regions. A detailed comparison of the MDOF and the corresponding SDOF system results is performed. The knowledge gained from the SDOF and MDOF studies on the applicability of the R-MISO technique for the system identification of MDOF submerged moored structures is discussed. The results show that the MDOF extension of the R-MI/SO nonlinear system identification technique works well; the resulting system parameters are relatively constant and can be applied to the both the sub- and superharmonic regions.

Identification of the non-linear behaviour of liquefied and non-liquefied soils during the 1995 Kobe earthquake

A method based on the non-linear system identification technique is suggested for estimating the contents of the non-linear components, which are the result of quadratic, cubic and higher-order non-linearities, in the ground response during the strong motion. The method is applied to data from the near-fault zone of the 1995 Kobe earthquake, and the contents of linear and non-linear components in the ground response, changing with time during the strong motion, are estimated for Port Island (PI), SGK and TKS sites, located at 2, 6 and 15 km from the fault plane, respectively. At PI, the non-linear part of the response increased with developing liquefaction, it was as high as ˜ 40-60 per cent of the intensity of the response. At SGK and TKS sites, the non-linear components of the response did not exceed ˜40 and ˜13 per cent of the intensity of the response, respectively. Odd-order non-linear components predominated in the soil response, whereas even-order non-linear components increased and became comparable with odd-order ones in liquefied soils and in cases of high intensity of the strong motion, when the loading parts of the stress-strain relations of the upper layers gained noticeable even components. As a whole, the contents of odd-order and even-order components in the soil response are determined by the shapes of the stress-strain relations in the upper most non-linear soil layers. At the three sites, changes in spectra of earthquake signals in subsurface soil layers were substantial as a result of the high non-linearity of the soil behaviour and spectra of signals on the surface tend to take the form of E(f) ˜f-k. The limiting spectral shape was achieved during the Kobe earthquake at PI and SGK sites. The proposed methods for processing vertical array records allow understanding of seismic wave transformations in subsurface soils and are useful for predicting the soil behaviour during future earthquakes.

Frequency Domain Identification of Tip-sample van der Waals Interactions in Resonant Atomic Force Microcantilevers

Hamaker constants are characteristic material properties that determine the magnitude of the nonlinear van der Waals force between atoms, molecules and nanoscale aggregates of atoms. This paper explores the novel possibility of using Harmonic Balance based nonlinear system identification methods to extract from the nonlinear vibration spectrum of resonant atomic force silicon microcantilevers, the Hamaker constants between a few atoms at the tip of the microcantilever and graphite, gold and silicon carbide samples. First, the nonlinear dynamics of a diving board microcantilever coupled to the samples through van der Waals force potentials are investigated through a discretized model of the system. Next, the feasibility of using Harmonic Balance based nonlinear system identification techniques are demonstrated using simulations of the discretized model. Finally the method is implemented on an AFM system. The results indicate that the proposed method provides a novel alternative way to measure Hamaker constants and the measured results are within the range of known experimental data.

Estimation of Nonlinear Time-dependent Soil Behavior in Strong Ground Motion Based on Vertical Array Data

To improve our understanding of nonlinear elastic properties of soils, a method is proposed of estimation of stress-strain relations of soils in situ in strong ground motion based on vertical array data. Strong motion records provided by seismic vertical arrays allow estimation of nonlinear stress-strain relations in soil layers at different depths, from the surface down to the location of the deepest device. As an example, records obtained during the main shock of the 1995 Kobe earthquake at Port-Island, SGK, and TKS sites were used to estimate the stress-strain relations in the soil profiles. For different layers, different types of nonlinear stress-strain relations were selected, according to the profiling data. To account for temporal changes in the soil behavior, consecutive parts of records were examined, and for successive time intervals, the relations were found showing the best-fit approximation to the observed data. At Port Island and SGK sites, where the strongest accelerations were recorded, the obtained stress-strain relations showed systematic changes in the upper layers (0–14 m), such as, a progressive reduction of the slopes of the stress-strain curves due to liquefaction at Port Island and reduction and recovery of the slopes at SGK and TKS sites. At the three sites, the stress-strain relations remained stable in layers below 11–14 m. Thus, the proposed approach gives us a representation of the soil behavior in layers at different depths in strong ground motion; it allows calculation of the propagation of arbitrary seismic signals in the studied profiles and estimation of nonlinear components in the ground response by the nonlinear system identification technique. The method can also be applied to evaluate the ground response at sites where profiling data are available and an imposed motion can be estimated.

Wavelet‐based nonparametric identification technique for nonlinear dynamic systems

AbstractNew parametric and non‐parametric identification techniques, based on wavelet expansion, for dynamic systems is shown. The identification results of the parameters of models obtained by the least‐squares method using the Haar wavelet packet analysis of randomly disturbed displacement signal is presented. New differentiation procedure of the measured data exploiting the properties of the wavelet packet analysis of a signal is applied. The non‐parametric identification results of models are known in the form of the restoring force surfaces.

Empirical Modeling of a Hydraulic Actuator in a Vehicle Control System

The pressure information of a hydraulic actuator plays a key role in modern vehicle control and fault diagnosis. The difficulty in measuring pressure directly naturally motivates such an indirect approach as an observer whose accuracy depends heavily upon the availability of a high fidelity model of a hydraulic actuator. Notwithstanding its success in understanding the dynamics of the first principle model of a hydraulic actuator, it is not suited to controller/observer design due to its complexity. This paper presents an alternative to the first principle modeling methodology: an empirical approach to hydraulic actuator modeling. Linear and nonlinear system identification techniques are applied to obtain low-order models of a hydraulic actuator. Experimental results show how the empirical models reproduce the key features of a hydraulic actuator with sufficient accuracy.

System identification of the Vincent Thomas suspension bridge using earthquake records

AbstractThe Vincent Thomas Bridge in the Los Angeles metropolitan area, is a critical artery for commercial traffic flow in and out of the Los Angeles Harbor, and is at risk in the seismically active Southern California region, particularly because it straddles the Palos Verdes fault zone. A combination of linear and non‐linear system identification techniques is employed to obtain a complete reduced‐order, multi‐input–multi‐output (MIMO) dynamic model of the Vincent Thomas Bridge based on the dynamic response of the structure to the 1987 Whittier and 1994 Northridge earthquakes.Starting with the available acceleration measurements (which consists of 15 accelerometers on the bridge structure and 10 accelerometers at various locations on its base), an efficient least‐squares‐based time‐domain identification procedure is applied to the data set to develop a reduced‐order, equivalent linear, multi‐degree‐of‐freedom model. Although not the main focus of this study, the linear system identification method is also combined with a non‐parametric identification technique, to generate a reduced‐order non‐linear mathematical model suitable for use in subsequent studies to predict, with good fidelity, the total response of the bridge under arbitrary dynamic environments.Results of this study yield measurements of the equivalent linear modal properties (frequencies, mode shapes and non‐proportional damping) as well as quantitative measures of the extent and nature of non‐linear interaction forces arising from strong ground shaking. It is shown that, for the particular subset of observations used in the identification procedure, the apparent non‐linearities in the system restoring forces are quite significant, and they contribute substantially to the improved fidelity of the model. Also shown is the potential of the identification technique under discussion to detect slight changes in the structure's influence coefficients, which may be indicators of damage and degradation in the structure being monitored. Difficulties associated with accurately estimating damping for lightly damped long‐span structures from their earthquake response are discussed. The technical issues raised in this paper indicate the need for added spatial resolution in sensor instrumentation to obtain identified mathematical models of structural systems with the broadest range of validity. Copyright © 2003 John Wiley & Sons, Ltd.

Characterizing cochlear mechano-electric transduction with a nonlinear system identification technique: the influence of the middle ear.

Previously a third-order polynomial equation characterizing mechano-electric transduction was obtained from a nonlinear system identification procedure applied to an ear canal acoustic signal and cochlear microphonic (CM/AC). In this paper, we examine the influence of the linearity and frequency response of the intervening middle ear on the nonlinearity, frequency response, and coherence of the third-order polynomial model of mechano-electric transduction (MET). Ear canal sound pressure (AC), cochlear microphonics (CM), and stapes velocity (SV) were simultaneously recorded from Mongolian gerbils. Linear and nonlinear transfer and coherence functions relating stapes velocity to the acoustic signal (SV/AC), CM to the acoustic signal (CM/AC), and CM to the stapes velocity (CM/SV) were computed. The results showed that SV/AC was linear while CM/AC and CM/SV were not, indicating that the nonlinearity of CM/AC was not due to nonlinearity of the middle ear. The frequency response of the linear term of CM/AC was similar to that of ST/AC but differed from that of CM/SV while the cubic term of CM/AC was similar to that of CM/SV. This indicates that the frequency dependence of CM/AC was due to both the middle ear and frequency dependence of the inner ear. Finally the fit of the polynomial model of MET without the middle ear (CM/SV) did not improve from the fit including the middle ear (CM/AC). A cochlear model of the CM indicated that the lack of improvement was due to the limitations of a third-order polynomial equation characterizing the hair cell transducer function.

EXPERIMENTAL VALIDATION OF THE CONSTANT LEVEL METHOD FOR IDENTIFICATION OF NON-LINEAR MULTI-DEGREE-OF-FREEDOM SYSTEMS

System identification for non-linear dynamical systems could find use in many applications such as condition monitoring, finite element model validation and determination of stability. The effectiveness of existing non-linear system identification techniques is limited by various factors such as the complexity of the system under investigation and the type of non-linearities present. In this work, the constant level identification approach, which can identify multi-degree-of-freedom systems featuring any type of non-linear function, including discontinuous functions, is validated experimentally. The method is shown to identify accurately an experimental dynamical system featuring two types of stiffness non-linearity. The full equations of motion are also extracted accurately, even in the presence of a discontinuous non-linearity.

Nonlinearity in the response of soils in the 1995 Kobe earthquake in vertical components of records

Strong motion records provided by seismic vertical arrays allow estimation of stress–strain relations in soils at depths from the surface to the location of the deepest device. As an example, time-dependent nonlinear soil behavior was estimated in vertical components of records obtained in the epicentral area of the 1995 Kobe earthquake. Degradation of the rigidity of soils in the strong motion was observed. The constructed nonlinear models of the soil behavior were used for estimating the nonlinear parts in the ground response by the nonlinear system identification technique. Nonlinear parts in the ground response were found to be as high as ∼50% at 2 km from the fault and ∼10% at 6–15 km from the fault plane. Odd types of nonlinearity, such as cubic, the fifth, seventh, etc. order, were found to be typical for soils, whereas, nonlinearities of even types are usually weak, but increase in liquefied soils.

Types of elastic nonlinearity of sedimentary soils

Vertical array records of the 1995 Kobe earthquake showed clearly nonlinearity of soil behavior. Although there have been few observational validations, earlier scientific publications on soil nonlinearity appeared since 1970's. In theoretical studies, nonlinearity has been introduced in constitutive relations as the quadratic correction to Hooke's law, but the question of the types of soil nonlinearity was not investigated. Here we show, on the basis of data processing and nonlinear system identification technique, that sediments represent physical systems possessing mostly odd types of nonlinearity. Even‐order nonlinearities become comparable with odd‐order ones only in special cases, like liquefied soils. Though the even and odd nonlinearities have much in common, the frequencies of combination harmonics, shapes of seismic solitary waves, and other nonlinear phenomena are different in these two cases. On the whole, the nonlinearity in soils leads to changes in spectra of propagating seismic signals: the spectra tending to take the form of E(f)∼f–n.

Simulation of Aircraft Pilot Flight Controls Using Nonlinear System Identification

This paper is concerned with modelling of the front end of an aircraft pilot flight control system's behav iour using a nonlinear system identification technique known as parallel cascade identification. Using this technique, we are able to model a critical part of a pi lot flight control system with sufficient accuracy to meet the objective test requirements of the U.S. Fed eral Aviation Administration for certifying full flight simulators. Traditional approaches to modelling such aircraft systems involve extensive analytical studies of the design of the system, lengthy and detailed empiri cal testing and recording of data from the physical system, and then considerable analysis to fit paramet ric models to the data. The approach presented in this paper virtually eliminates the need for analysis of the system in question, significantly reduces the number of signals that need be recorded from the real aircraft flight control system, and provides an extremely fast method of identifying the mathematical model based on these data. Overall, the time and costs associated with building an effective model are greatly reduced.