Nonparametric identification of Wiener systems by orthogonal series

A Wiener system, i.e., a system comprising a linear dynamic and a nonlinear memoryless subsystems connected in a cascade, is identified. Both the input signal and disturbance are random, white, and Gaussian. The unknown nonlinear characteristic is strictly monotonous and differentiable and, therefore, the problem of its recovering from input-output observations of the whole system is nonparametric. It is shown that the inverse of the characteristic is a regression function and a class of orthogonal series nonparametric estimates recovering the regression is proposed and analyzed. The estimates apply the trigonometric, Legendre, and Hermite orthogonal functions. Pointwise consistency of all the algorithms is shown. Under some additional smoothness restrictions, the rates of their convergence are examined and compared. An algorithm to identify the impulse response of the linear subsystem is proposed     

Continuous-time Hammerstein system identification

A continuous-time Hammerstein system, i.e., a system consisting of a nonlinear memoryless subsystem followed by a linear dynamic one, is identified. The system is driven and disturbed by white random signals. The a priori information about both subsystems is nonparametric, which means that functional forms of both the nonlinear characteristic and the impulse response of the dynamic subsystem are unknown. An algorithm to estimate the nonlinearity is presented and its pointwise convergence to the true characteristic is shown. The impulse response of the dynamic part is recovered with a correlation method. The algorithms are computationally independent. Results of a simulation example are given     

Combined parametric-nonparametric identification of Hammerstein systems

A novel, parametric-nonparametric, methodology for Hammerstein system identification is proposed. Assuming random input and correlated output noise, the parameters of a nonlinear static characteristic and finite impulse-response system dynamics are estimated separately, each in two stages. First, the inner signal is recovered by a nonparametric regression function estimation method (Stage 1) and then system parameters are solved independently by the least squares (Stage 2). Convergence properties of the scheme are established and rates of convergence are given.     

Parameter identification in a class of nonlinear systems

Two approaches are proposed for on-line identification of parameters in a class of nonlinear discrete-time systems. The system is modeled by state equations in which state and input variables enter nonlinearly in general polynomial form, while unknown parameters and random disturbances enter linearly. All states and inputs must be observed with measurement errors represented by white Gaussian noise having known covariance. System disturbances are also white and Gaussian with finite, but unknown, covariance. One method of parameter estimation is based upon a least squares approach, the second is a related stochastic approximation algorithm (SAA). Under fairly mild conditions the estimate derived from the least squares algorithm (LSA) is shown to converge in probability to the correct parameter; the SAA yields an estimate which converges in mean square and with probability 1. Examples illustrate convergence of the LSA which even in recursive form requires inversion of a matrix at each step. The SAA requires no matrix inversions, but experience with the algorithm indicates that convergence is slow relative to that of the LSA.     

The Laplace Image of Random Processes and Nonparametric Identification of Continuous Systems

The Laplace image of stationary random normal processes is studied. The covariance function of the Laplace image of white noise is converted by a linear shaping filter into the covariance function of the Laplace image of the filter output process. The relationship between the covariance function of the Laplace image of a random process and the autocovariance function and spectral density is determined. The covariance function of the Laplace image of measurement errors of a transition process in a stationary linear system is applied in optimal nonparametric identification of the transfer function.     

Identification of non-linear systems by recursive kernel regression estimates

Identification of non-linear, dynamical systems described by the Hammerstein model are discussed. Such a system consists of a multi-input single-output nonlinear, memoryless subsystem followed by a dynamic, linear subsystem. Outputs of both subsystems are corrupted by random noise. The parameters of the linear subsystem are identified by a correlation technique. The main contribution lies in estimating the non-linear, memoryless subsystem. The identification algorithm is based on the recursive kernel regression estimate. No restrictions are imposed on the functional form of the non-linearity as well on its continuity. We prove global convergence of the algorithm regardless of the distribution of the random input and for outputs with bounded moment of order greater than 2. The rate of convergence is obtained for the Lipschitz non-linearities and all input distributions.     

The spectral density of the nonlinear damping model: single DOFcase

An analytical method of computing the frequency response of single degree of freedom (DOF) oscillators with nonlinear damping is described. The author proposes an energy-type nonlinear damping model and the corresponding stationary probability density with white noise input can be obtained explicitly. A theorem is presented which gives an interesting result, in terms of the Krylov-Bogoliubov approximation, concerning the modeling and identification of nonlinear internal damping in flexible structures. This analysis also serves as a contribution to random vibration theory by providing a method of computing the first- and second-order statistics (steady-state probability density, correlation function, and spectral density) of nonlinearity damped oscillators with white noise input     

Identification of non-linear characteristics of a class of block-oriented non-linear systems via Daubechies wavelet-based models

The paper deals with the identification of non-linear characteristics of a class of block-oriented dynamical systems. The systems are driven by random stationary white processes (i.i.d. random input sequences) and disturbed by a zero-mean stationary, white or coloured, random noise. The prior knowledge about non-linear characteristics is non-parametric excluding implementation of standard parametric identification methods. To recover non-linearities, a class of Daubechies wavelet-based models using only input-output measurement data is introduced and their accuracy is investigated in the global MISE error sense. It is shown that the proposed models converge with a growing collection of data to the true non-linear characteristics (or their versions), provided that the complexity of the models is appropriately fitted to the number of measurements. Suitable rules for optimum model size selection, maximizing the convergence speed, are given and the asymptotic rate of convergence of the MISE error for optimum models is established. It is shown that in some circumstances the rate is the best possible that can be achieved in non-parametric inference. We also show that the convergence conditions and the asymptotic rate of convergence are insensitive to the correlation of the noise and are the same for known and unknown input probability density function (assumed to exist). The theory is illustrated by simulation examples.     

Stochastic functional fourier series, Volterra series, and nonlinear systems analysis

A functional Fourier series is developed with emphasis on applications to the nonlinear systems analysis. In analogy to Fourier coefficients, Fourier kernels are introduced and can be determined through a cross correlation between the output and the orthogonal basis function of the stochastic input. This applies for the class of strict-sense stationary white inputs, except for a singularity problem incurred with inputs distributed at quantized levels. The input may be correlated if it is zero-mean Gaussian. The Wiener expansion is treated as an example corresponding to the white Gaussian input and this modifies the Lee-Schetzen algorithm for Wiener kernel estimation conceptually and computationally. The Poisson-distributed white input is dealt with as another example. Possible links between the Fourier and Volterra series expansions are investigated. A mutual relationship between the Wiener and Volterra kernels is presented for a subclass of analytic nonlinear systems. Connections to the Cameron-Martin expansion are examined as well The analysis suggests precautions in the interpretation of Wiener kernel data from white-noise identification experiments.     

有色噪声干扰下Hammerstein非线性系统两阶段辨识

针对实际工业过程中普遍存在有色噪声,提出了有色噪声干扰下Hammerstein非线性系统两阶段辨识方法。采用设计的组合式信号实现Hammerstein系统各模块参数辨识分离,简化了辨识过程。在第一阶段,基于可分离信号的输入输出数据,利用相关分析算法估计线性模块参数,减少了有色噪声对辨识的干扰。在第二阶段,基于随机信号的...     

On the existence of stationary points for the Steiglitz-McBridealgorithm

Most convergence results for adaptive identification algorithms have been developed in sufficient order settings, involving an unknown system with known degree. Reduced-order settings, in which the degree of the unknown system is underestimated, are more common, but more difficult to analyze. Deducing stationary points in these cases typically involves solving nonlinear equations, hence the sparseness of results for reduced-order cases. If we allow ourselves the tractable case in which the input to an identification experiment is white noise, we shall show that the Steiglitz-McBride method (1965) indeed admits a stationary point in reduced-order settings for which the resulting model is stable. Our interest in this study stems from a previous result, showing an attractive a priori bound on the mismodeling error at any such stationary point     

Towards identification of Wiener systems with the least amount of a priori information: IIR cases

In this paper, we investigate what constitutes the least amount of a priori information on the nonlinearity so that the linear part is identifiable in the non-Gaussian input case. Under the white noise input, three types of a priori information are considered: quadrant information, point information, and monotonic information. In all three cases, identifiability has been established, and the corresponding nonparametric identification algorithms are developed along with their convergence proofs.     

Adaptive deconvolution and identification of nonminimum phase FIRsystems based on cumulants

A novel adaptive deconvolution and system identification scheme is introduced for a linear, non-minimum-phase finite-impulse-response (FIR) system driven by non-Gaussian white noise. The adaptive scheme is based on approximating the FIR system by noncausal autoregressive (AR) models and using higher order cumulants of the system output. As such, it is a blind equalization (deconvolution) scheme. The set of updated AR parameters is obtained by using a gradient-type algorithm and by using higher order cumulants instead of time samples of the output signal. It is demonstrated by means of extensive simulations that the adaptive scheme works well for both stationary and nonstationary cases. As expected, it outperforms the autocorrelation-based gradient method for nonminimum-phase system identification and deconvolution. Performance comparisons to existing methods are given, using as figures of merit the probability of errors in the restored input sequence, computational complexity, and convergence rate     

An integro-differential formula on the Wiener kernels and its application to sandwich system identification

It is well known that a nonlinear system with a white Gaussian noise input can be characterized in terms of kernels using the celebrated Wiener theory. In a practical use of the method, however, one may encounter difficulty in obtaining higher order kernels except for the first few because of, for instance, the excessive computational requirement. In this paper, we give an integro-differential formula on the kernels and as its application, an algorithm to identify a cascade system of a linear, a memoryless nonlinear, and linear subsystems, which we call a sandwich system as a whole. According to the formula, kernels up to the second order for different power levels of the input noise are required to identify the subsystems. Impulse response functions of the two linear subsystems can be obtained under appropriate normalization conditions, while the nonlinear subsystem is estimated in the form of a truncated Hermite polynomial expansion. As illustrated examples, two such systems are identified using the algorithm.     

Identification and prediction of a class of non-Gaussian time-series via transformation

Whereas optimal prediction of Gaussian sequences requires the employment of a linear filter with consistently identifiable parameters and with Gaussian white noise input, the optimal predictor of non-Gaussian sequences is n nonlinear filter, having an independent noise input. Since the latter cannot be identified directly without prior knowledge of the non-linearity, the optimal linear predictor is usually identified where a non-Gaussian white noise input is considered and which is fully optimal only when that input turns out to be independent in all moments. However, if the non-Gaussian sequence is the outcome of a Gaussian sequence passed through a zero memory non-linearity or through non-linear measurement elements, a transformation of the non-Gaussian sequence into a Gaussian one is possible, such that optimal non-linear prediction may be approximated to any required degree, as is shown by the analysis of the present work. Furthermore, the parameters of that predictor may be consistently identified in the absence of any parameter information.     

Identification of discrete Hammerstein systems by the Fourier series regression estimate

The identification of a single-input, single-output (SISO) discrete Hammerstein system is studied. Such a system consists of a non-linear memoryless subsystem followed by a dynamic, linear subsystem. The parameters of the dynamic, linear subsystem are identified by a correlation method and the Newton-Gauss method. The main results concern the identification of the non-linear, memoryless subsystem. No conditions are imposed on the functional form of the non-linear subsystem, recovering the non-linear using the Fourier series regression estimate. The density-free pointwise convergence Of the estimate is proved, that is.algorithm converges for all input densities The rate of pointwise convergence is obtained for smooth input densities and for non-linearities of Lipschitz type.Globle convergence and its rate are also studied for a large class of non-linearities and input densities     

Nonparametric algorithm for input signals identification in static distributed-parameter systems

In this correspondence, a nonparametric algorithm for identification of input signals in linear, static distributed-parameter systems is proposed and investigated. Integral mean-square convergence of the algorithm is proved for an infinite number of point measurements of the system state. The algorithm is a generalized version of the one recently proposed by Rutkowski [10] for nonparametric function fitting, and in a common area, the presented results are complementary.     

Robust identification of MISO neuro‐fractional‐order Hammerstein systems

This paper introduces a multiple‐input–single‐output (MISO) neuro‐fractional‐order Hammerstein (NFH) model with a Lyapunov‐based identification method, which is robust in the presence of outliers. The proposed model is composed of a multiple‐input–multiple‐output radial basis function neural network in series with a MISO linear fractional‐order system. The state‐space matrices of the NFH are identified in the time domain via the Lyapunov stability theory using input‐output data acquired from the system. In this regard, the need for the system state variables is eliminated by introducing the auxiliary input‐output filtered signals into the identification laws. Moreover, since practical measurement data may contain outliers, which degrade performance of the identification methods (eg, least‐square–based methods), a Gaussian Lyapunov function is proposed, which is rather insensitive to outliers compared with commonly used quadratic Lyapunov function. In addition, stability and convergence analysis of the presented method is provided. Comparative example verifies superior performance of the proposed method as compared with the algorithm based on the quadratic Lyapunov function and a recently developed input‐output regression‐based robust identification algorithm.     

Hammerstein system identification by non-parametric instrumental variables

A mixed, parametric–non-parametric routine for Hammerstein system identification is presented. Parameters of a non-linear characteristic and of ARMA linear dynamical part of Hammerstein system are estimated by least squares and instrumental variables assuming poor a priori knowledge about the random input and random noise. Both subsystems are identified separately, thanks to the fact that the unmeasurable interaction inputs and suitable instrumental variables are estimated in a preliminary step by the use of a non-parametric regression function estimation method. A wide class of non-linear characteristics including functions which are not linear in the parameters is admitted. It is shown that the resulting estimates of system parameters are consistent for both white and coloured noise. The problem of generating optimal instruments is discussed and proper non-parametric method of computing the best instrumental variables is proposed. The analytical findings are validated using numerical simulation results.     

Correlation analysis based MIMO neuro-fuzzy Hammerstein model with noises

A novel identification algorithm for neuro-fuzzy based MIMO Hammerstein system with noises by using the correlation analysis method is presented in this paper. A special test signal that contains independent separable signals and uniformly random multi-step signal is adopted to identify the MIMO Hammerstein system, resulting in the identification problem of the linear model separated from that of nonlinear part. As a result, it can circumvent the problem of initialization and convergence of the model parameters encountered by the existing iterative algorithms used for identification of MIMO Hammerstein model. Moreover, least square method based parameter identification algorithms of dynamic linear part and static nonlinear part are proposed to avoid the influence of noise. Examples are used to illustrate the effectiveness of the proposed method.