Identification Of Nonstationary Systems Research Articles

An efficient non-negative least mean squares algorithm based on q-gradient for system identification

The system identification under non-negative constraints problem is a common and important one in the real-life problems. A non-negative least mean squares algorithm was proposed to address such problems. However, it suffers from slow and unbalanced convergence. Motivated by the introduction of non-Newtonian gradient in least mean squares algorithm can accelerate the convergence. In this paper, an efficient non-negative least mean squares algorithm based on q-gradient is developed. At first, the q-gradient of the mean square error cost function is derived from the definition of Jackson's derivative. Then, the modified algorithm is developed by replacing the conventional gradient with the q-gradient on a fixed-point iteration scheme using Karush-Kuhn-Tucker condition. Simulations for stationary and non-stationary system identification are conducted to illustrate the effectiveness of the developed algorithm for constrain the system parameters to be non-negative and accelerate the convergence speed. Besides, it can balance the convergence rate for widely distributed parameters, especially for the convergence of small parameters. For practical application, the accuracy of system identification when the order of the adaptive filter is larger than the order of the system is also verified by the simulation.

Transient Analysis of Signed LMS Algorithms With Cyclostationary Colored Gaussian Inputs

It is of significant importance to investigate the transient behaviors of signed LMS algorithms under the cyclostationary colored Gaussian inputs for the design of corresponding filters in practical applications. In this brief, we derive the analytical models of a family of signed LMS algorithms with the cyclostationary colored Gaussian inputs in the mean and mean-square sense for the nonstationary system identification. Simulation results illustrate the excellent consistency between the simulated results and the theoretical findings.

A new look at the statistical identification of nonstationary systems

The paper presents a new, two-stage approach to identification of linear time-varying stochastic systems, based on the concepts of preestimation and postfiltering. The proposed preestimated parameter trajectories are unbiased but have large variability. Hence, to obtain reliable estimates of system parameters, the preestimated trajectories must be further filtered (postfiltered). It is shown how one can design and optimize such postfilters using the basis function framework. The proposed solution to adaptive tuning of postfilter settings is based on parallel estimation and cross-validatory analysis. When compared with the classical solutions to the problem of parameter tracking, the new approach offers, without compromising good tracking performance, significant computational savings, higher numerical robustness and greater flexibility.

Selective ensemble of multiple local model learning for nonlinear and nonstationary systems

This paper proposes a selective ensemble of multiple local model learning for modeling and identification of nonlinear and nonstationary systems, in which the set of local linear models are self adapted to capture the newly emerging process characteristics and the prediction of the process output is also self adapted based on an optimally selected ensemble of subset linear local models. Specifically, our selective ensemble of multiple local model learning approach performs the model adaptation at two levels. At the level of local model adaptation, a newly emerging process state in the incoming data is automatically identified and a new local linear model is fitted to this newly emerged process state. At the level of online prediction, a subset of candidate local linear models are optimally selected and the prediction of the process output is computed as an optimal linear combiner of the selected subset local linear models. Two case studies involving chaotic time series prediction and modeling of a real-world industrial microwave heating process are used to demonstrate the effectiveness of our proposed approach, in comparison with other existing methods for modeling and identification of nonlinear and time-varying systems.

Generalized Savitzky–Golay filters for identification of nonstationary systems

The problem of identification of nonstationary systems using noncausal estimation schemes is considered and a new class of identification algorithms, combining the basis functions approach with local estimation technique, is described. Unlike the classical basis function estimation schemes, the proposed local basis function estimators are not used to obtain interval approximations of the parameter trajectory, but provide a sequence of point estimates corresponding to consecutive instants of time. Based on the results of theoretical analysis conducted for nonstationary finite impulse response systems the paper proposes two mechanisms for adaptive selection of the number of basis functions and the size of the local analysis window.

Time-Varying System Identification Using an Ultra-Orthogonal Forward Regression and Multiwavelet Basis Functions With Applications to EEG.

A new parametric approach is proposed for nonlinear and nonstationary system identification based on a time-varying nonlinear autoregressive with exogenous input (TV-NARX) model. The TV coefficients of the TV-NARX model are expanded using multiwavelet basis functions, and the model is thus transformed into a time-invariant regression problem. An ultra-orthogonal forward regression (UOFR) algorithm aided by mutual information (MI) is designed to identify a parsimonious model structure and estimate the associated model parameters. The UOFR-MI algorithm, which uses not only the observed data themselves but also weak derivatives of the signals, is more powerful in model structure detection. The proposed approach combining the advantages of both the basis function expansion method and the UOFR-MI algorithm is proved to be capable of tracking the change of TV parameters effectively in both numerical simulations and the real EEG data.

Identification of non-stationary dynamical systems using multivariate ARMA models

This paper is concerned by the modal identification of time-varying mechanical systems. Based on previous works about autoregressive moving average models in vector form (ARMAV) for the modal identification of linear time invariant systems, and time-varying autoregressive moving average models (TV-ARMA) for the identification of nonstationary systems, a time-varying ARMAV (TV-ARMAV) model is presented for the multivariate identification of time-varying systems. It results in the identification of not only the time-varying poles of the system but also of its respective time-varying mode shapes. The method is applied on a time-varying structure composed of a beam on which a mass is moving.

Potential of Acoustic Emission in Unsupervised Monitoring of Gas-Fuelled Engines

Abstract: The aim of this paper is to scrutinise closely information that contains in the acoustic emission of a gas-fuelled engine and to define the potential of knock condition detection. The recently introduced technique of nonstationary system identification is investigated. At first, it utilises the wavelet transform to reveal time-frequency energy density of data. Then the modified version of singular value decomposition is applied to extract dominant frequency components from the data buried in background noise. The efficacy of knock feature extraction is investigated using three sources of data: in-cylinder pressure, engine structure vibration and engine noise sensed by a microphone.

Non-stationary frequency domain system identification using time–frequency representations

System properties of buildings and bridges may vary with time due to temperature changes, aging or extreme loadings. To identify these time-varying system properties, this study proposes a new output-only non-stationary system identification (SI) framework based on instantaneous or marginal spectra derived from the time–frequency representation, e.g., short time Fourier or wavelet transform. Spectra derived from these time–frequency representations are very popular in tracking time-varying frequencies; however, they have seldom been used to identify the time-varying damping ratio because a short window needed to capture the time-varying information amplifies the bandwidth significantly, which may lead to considerably overestimating the damping ratio. To overcome this shortcoming, this study modifies the theoretical frequency response function (FRF) to explicitly account for the windowing effect, and therefore enables SI directly using instantaneous or marginal spectra derived from the wavelet or short time Fourier transform. The response spectrum estimated using the short time window and the modified FRF are both influenced by the same time window, thus the instantaneous or time-localized marginal spectrum of response can be fitted to the modified FRF to identify frequency and damping ratio at each time instant. This spectral-based SI framework can reliably identify damping in time-varying systems under non-stationary excitations. The efficacy of the proposed framework is demonstrated by both numerical and full-scale examples, and also compared to the time-domain SI method, stochastic subspace identification (SSI), since the time-domain SI approaches and their extensions are popular in identifying time-varying systems utilizing recursive algorithms or moving windows.

On-line parameter and delay estimation of continuous-time dynamic systems

Abstract The problem of on-line identification of non-stationary delay systems is considered. The dynamics of supervised industrial processes are usually modeled by ordinary differential equations. Discrete-time mechanizations of continuous-time process models are implemented with the use of dedicated finite-horizon integrating filters. Least-squares and instrumental variable procedures mechanized in recursive forms are applied for simultaneous identification of input delay and spectral parameters of the system models. The performance of the proposed estimation algorithms is verified in an illustrative numerical simulation study.

System Identification through Nonstationary Response: Wavelet and Transformed Singular Value Decomposition—Based Approach

AbstractThe full-scale data measured from tall buildings and long-span bridges are usually nonstationary due to transient earthquake ground motion, sudden changes in wind speed and/or direction, or time-varying motion of vehicles. The dynamic properties (i.e., frequency, damping, etc.) of these structures during nonstationary events closely relate to their operational safety, so their accurate identification is critical. Traditional system identification methods may fail to provide reliable results (especially for damping estimation) in these situations due to a lack of long segments of stationary data. In addition, these methods are not able to track the time-varying system properties caused by large amplitude response due to strong earthquakes or winds. This highlights the necessity of developing system identification methods suitable for nonstationary/transient data. In response to this need, this paper proposes a new nonstationary system identification scheme which first utilizes the wavelet transform...

Unsupervised identification of damage and load characteristics in time-varying systems

Parameters identification of nonstationary systems is a very challenging topic that has only recently received critical attention from the research community. Aim of the paper is the structural health monitoring of bridge-like structures excited by a massive moving load, whose characteristics, such as the mass and speed, are unknown, in the presence of a localized damage along the structure. A novel method for the simultaneous identification of both the load characteristics and damage parameters from vibration measurements is proposed: the data processing relies on the ensemble empirical mode decomposition and the normalized Hilbert transform. Neither a priori information about the response of the undamaged structure nor the free decay of the damaged system is required, only a single-point measurement is needed. The empirical instantaneous frequency is firstly employed to estimate the load characteristics; secondly, the effect of the moving mass is filtered from the instantaneous frequency, and then, the damage position is identified. The performance of the technique are studied varying the load characteristics, damage locations, and crack depths. The effect of ambient noise is also taken into account. Numerical experiments show the identification is rather accurate, results are not very sensitive to the crack location and depth, while they are sensibly affected by the speed of the moving load.

Identification and frequency domain analysis of non-stationary and nonlinear systems using time-varying NARMAX models

This paper introduces a new approach for nonlinear and non-stationary (time-varying) system identification based on time-varying nonlinear autoregressive moving average with exogenous variable (TV-NARMAX) models. The challenging model structure selection and parameter tracking problems are solved by combining a multiwavelet basis function expansion of the time-varying parameters with an orthogonal least squares algorithm. Numerical examples demonstrate that the proposed approach can track rapid time-varying effects in nonlinear systems more accurately than the standard recursive algorithms. Based on the identified time domain model, a new frequency domain analysis approach is introduced based on a time-varying generalised frequency response function (TV-GFRF) concept, which enables the analysis of nonlinear, non-stationary systems in the frequency domain. Features in the TV-GFRFs which depend on the TV-NARMAX model structure and time-varying parameters are investigated. It is shown that the high-dimensional frequency features can be visualised in a low-dimensional time–frequency space.

Damage assessment of pre-stressed structures: A SVD-based approach to deal with time-varying loading

Vibration-based methods are well-established and effective tools to assess the health state of civil, mechanical and aerospace engineering structures. However, their reliability is still affected by the variability of the features commonly used for damage detection. Environmental effects and changes in operational conditions are the main sources of variability in the structural response. As a consequence, the modal identification used to extract damage sensitive features has to face constricting requirements in terms of signals stationarity and performance accuracy. Moreover, with reference to the damage assessment, large variations of monitored features mask subtle effects due to damage, which remain undetected. This study is conceived to address both these issues by focusing, in particular, on the non-stationarity of the loading conditions of tensioned structures, such as cables and pre-stressed beams. The capability of spectral methods to deal with the modal identification of non-stationary systems is enhanced by a curve-fitting procedure based on nonlinear least squares optimization. Wavelet analysis is applied for comparison and validation of the FFT-based technique. Identified natural frequencies are then used for the damage detection, exploiting the capacity of singular values decomposition to discriminate between damage-related events and the intrinsic non-stationary nature of the structural response. A reduced-order realization of the features set is performed to amplify changes not belonging to measurement variability but deriving from exogenous events, such as damage. The proposed methodology is validated by experimental analyses carried out on beams subjected to time-varying loading conditions in order to simulate the health monitoring of quasi and non-stationary systems.

Online sequential extreme learning machine in nonstationary environments

System identification in nonstationary environments represents a challenging problem to solve and lots of efforts have been put by the scientific community in the last decades to provide adequate solutions on purpose. Most of them are targeted to work under the system linearity assumption, but also some have been proposed to deal with the nonlinear case study. In particular the authors have recently advanced a neural architecture, namely time-varying neural networks (TV-NN), which has shown remarkable identification properties in the presence of nonlinear and nonstationary conditions. TV-NN training is an issue due to the high number of free parameters and the extreme learning machine (ELM) approach has been successfully used on purpose. ELM is a fast learning algorithm that has recently caught much attention within the neural networks (NNs) research community. Many variants of ELM have been appeared in recent literature, specially for the stationary case study. The reference one for TV-NN training is named ELM-TV and is of batch-learning type. In this contribution an online sequential version of ELM-TV is developed, in response to the need of dealing with applications where sequential arrival or large number of training data occurs. This algorithm generalizes the corresponding counterpart working under stationary conditions. Its performances have been evaluated in some nonstationary and nonlinear system identification tasks and related results show that the advanced technique produces comparable generalization performances to ELM-TV, ensuring at the same time all benefits of an online sequential approach.

Locally-adaptive Kalman smoothing approach to identification of nonstationary stochastic systems

The problem of noncausal identification of nonstationary, linear stochastic systems, i.e., identification based on prerecorded input/output data, is considered. It is shown how several competing Kalman-type parameter smoothers, differing in smoothness constraints and memory settings, can be combined together to yield a better and more reliable smoothing algorithm. The resulting parallel estimation scheme automatically adjusts its smoothing bandwidth to the unknown, and possibly time-varying, rate of nonstationarity of the identified system. It also allows one to account for parameter jumps and for the distribution of measurement noise.

Locally Adaptive Cooperative Kalman Smoothing and Its Application to Identification of Nonstationary Stochastic Systems

One of the central problems of the stochastic approximation theory is the proper adjustment of the smoothing algorithm to the unknown, and possibly time-varying, rate and mode of variation of the estimated signals/parameters. In this paper we propose a novel locally adaptive parallel estimation scheme which can be used to solve the problem of fixed-interval Kalman smoothing in the presence of model uncertainty. The proposed solution is based on the idea of cooperative smoothing-the Bayesian extension of the leave-one-out cross-validation approach to model selection. Within this approach the smoothed estimates are evaluated as a convex combination of the estimates provided by several competing smoothers. We derive computationally attractive algorithms allowing for cooperative Kalman smoothing and show how the proposed approach can be applied to identification of nonstationary stochastic systems.

On noncausal weighted least squares identification of nonstationary stochastic systems

In this paper, we consider the problem of noncausal identification of nonstationary, linear stochastic systems, i.e., identification based on prerecorded input/output data. We show how several competing weighted (windowed) least squares parameter smoothers, differing in memory settings, can be combined together to yield a better and more reliable smoothing algorithm. The resulting parallel estimation scheme automatically adjusts its smoothing bandwidth to the unknown, and possibly time-varying, rate of nonstationarity of the identified system. We optimize the window shape for a certain class of parameter variations and we derive computationally attractive recursive smoothing algorithms for such an optimized case.

On noncausal identification of nonstationary stochastic systems

AbstractIn this paper we consider the problem of noncausal identification of nonstationary, linear stochastic systems, i.e., identification based on prerecorded input/output data. We show how several competing weighted least squares parameter smoothers, differing in memory settings, can be combined together to yield a better and more reliable smoothing algorithm. The resulting parallel estimation scheme automatically adjusts its smoothing bandwidth to the unknown, and possibly time-varying, rate of nonstationarity of the identified system. It also allows one to account for the distribution of measurement noise, and in particular – to cope with heavy-tailed disturbances, such as Laplacian noise, or light-tailed disturbances, such as uniform noise.

Optimal and suboptimal smoothing algorithms for identification of time-varying systems with randomly drifting parameters

Noncausal estimation algorithms, which involve smoothing, can be used for off-line identification of nonstationary systems. Since smoothing is based on both past and future data, it offers increased accuracy compared to causal (tracking) estimation schemes, incorporating past data only. It is shown that efficient smoothing variants of the popular exponentially weighted least squares and Kalman filter-based parameter trackers can be obtained by means of backward-time filtering of the estimates yielded by both algorithms. When system parameters drift according to the random walk model and the adaptation gain is sufficiently small, the properly tuned two-stage Kalman filtering/smoothing algorithm, derived in the paper, achieves the Cramér–Rao type lower smoothing bound, i.e. it is the optimal noncausal estimation scheme. Under the same circumstances performance of the modified exponentially weighted least-squares algorithm is often only slightly inferior to that of the Kalman filter-based smoother.