Variable knot-based spline approximation recursive Bayesian algorithm for the identification of Wiener systems with process noise

In this paper, we study the identification problem for the Wiener systems with internal noises. If the internal noise is iid Gaussian, then the Wiener system can be transformed to the one without internal noise by the Weierstrass transform. The linear subsystems for both the original and transformed systems are the same. Recursive algorithms are proposed to identify the transformed Wiener system. The strong consistency of the estimates is proved under reasonable conditions. Nonparametric estimate for the nonlinear block of the original Wiener system is given trough the inverse Weierstrass transform of the nonlinear function of the transformed Wiener system, and it is proved that the estimate is consistent if the nonlinearity of the original Wiener system is a polynomial. A numerical example is presented showing that the simulation result is consistent with theoretical analysis.

To identify the OEAR model with non-uniformly sampled input data, a recursive Bayesian identification algorithm with covariance resetting is proposed in this paper. Comparing with the conventional recursive least squares algorithm based on auxiliary model, the presented algorithm considers the variance of the colored noise and can estimate the parameter with high accuracy. Furthermore, the algorithm integrates the prior probability density function of the parameters and the prior probability density function of the process data together, and achieves better performance than the maximum likelihood algorithm. To improve the convergence rate, a new covariance resetting method is also integrated in the algorithm. A simulation example demonstrates the performance of the proposed algorithm.

SummaryThe paper considers a block‐oriented nonlinear Hammerstein system that consists of a linear block with real‐valued time‐varying unknown coefficients and a pure time delay, preceded by the static nonlinearity. The joint tracking of linear block coefficients and the time delay while processing observation samples is under consideration. The difficulty of the tracking process consists in the initial nonconvexity of the objective function to be minimized, in respect of the time delay. To obtain current estimates of the true coefficients and the time delay it is necessary to overcome the multiextremality of the objective function. A global optimization problem could be solved by changing a form of the nonconvex objective function into a convex one. The optimization problem has been subdivided into two problems that are analyzed in this article, as well. The first one consists of removal of the objective function multiextremality in respect of the time delay, while hunting for the global extremum. Therefore, the well‐known approach based on the method of corrective operators, which is used to transform the multiextremal criterion into a unimodal function for the nonstationary Wiener system, is developed here for the Hammerstein one. The other problem consists of the applicability of unified adaptive algorithms, used for Wiener systems in the case of Hammerstein systems. The recursive parametric identification technique, based on the adaptive algorithm procedures, is developed here for the nonstationary block‐oriented Hammerstein systems, too. The applicability of algorithms is supported by various simulation tests solved by computer for distinct inputs and for two static nonlinearities.

The recursive algorithms are given for identifying the single‐input single‐output Wiener system which consists of a moving average type linear subsystem followed by a static nonparametric nonlinearity. The input is defined to be a sequence of mutually independent Gaussian random variables. The estimates for coefficients of the linear subsystem as well as for f(v) at any v are proved to converge to the true values with probability one. A numerical example is given, justifying the theoretical analysis. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

Identification of a Wiener System via Semidefinite Programming

This paper concerns the identification of MISO (multiple inputs single output) Wiener systems. For each input-output path, the linear dynamics are modeled by a set of orthonormal basis functions (OBFs). The static nonlinearity is modeled through a multivariate polynomial. The parameters of the model are the coefficients of this polynomial. In this paper, an identification procedure for SISO (single input single output) Wiener systems is extended towards MISO Wiener systems. The poles of the OBFs are estimated using an extension of the best linear approximation (BLA) towards MIMO (multiple input multiple output) systems. As the number of parameters can increase significantly compared to the SISO case, a parameter reduction step, first developed for the SISO case, is extended in this paper to the MISO case. In each set of OBFs, one OBF is replaced by the BLA of the input-output path corresponding to that set. It is shown that in this way the number of relevantly contributing terms in the multivariate polynomial is significantly reduced. Simulation results show a major reduction of the number of parameters, with only a minor increase in the rms error on the simulated output.

A two-experiment approach to Wiener system identification

State Space Approach for Two-Dimensional Data Smoothing

This paper considers identification of Wiener systems for which the internal variables and output are corrupted by noises. When the internal noise is a sequence of independent and identically distributed (iid) Gaussian random variables, by the Weierstrass transformation (WT) the system under consideration turns to be a Wiener system without internal noise. The nonlinear part of the latter is nothing else than the WT of the nonlinear function of the original system, while the linear subsystem is the same for both systems before and after WT. Under reasonable conditions, the recursive identification algorithms are proposed for the transformed Wiener system, and strong consistency for the estimates is established. By using the inverse WT the nonparametric estimates for the nonlinearity of the original system are derived, and they are strongly consistent if the nonlinearity in the original system is a polynomial. Similar results also hold in the case where the internal noise is non-Gaussian. Simulation results are fully consistent with the theoretical analysis.

The impact of static output nonlinearities on the control strategies that humans use in command-following tasks

The paper concerns the identification of the multiple-input multiple-output (MIMO) Wiener system consisting of a linear subsystem followed by a static nonlinearity. Based on Fourier transform, the coefficient matrices of the linear subsystem is recursively estimated by stochastic approximation based principal component analysis (SABPCA) method with random inputs with possibly non-Gaussian distribution. The main contribution of the paper also consists in establishing convergence with probability one of the proposed algorithm to the true values. A numerical example is given to justify the theoretical analysis.

To identify the Box---Jenkins systems with non-uniformly sampled input data, a recursive Bayesian algorithm with covariance resetting was proposed in this paper. Considering the prior probability density functions of parameters and the observed input---output data, the parameters were estimated by maximizing the posterior probability distribution function. During the estimation, the variance of the noise was taken as a weighting factor, and the proposed algorithm was formulated as a weighted least squares. As a result, the accuracy of the estimates increased. Meanwhile, a modified covariance resetting strategy was integrated into the algorithm to improve the convergence rate, and the convergence of the algorithm was also analyzed. A simulation example was applied to validate the proposed algorithm.

This paper addresses the problem of recursive identification of Wiener nonlinear systems whose linear subsystems are observable state-space models. The maximum likelihood principle and the recursive identification technique are employed to develop a recursive maximum likelihood identification algorithm which estimates the unknown parameters and the system states interactively. In comparison with the developed recursive maximum likelihood algorithm, a recursive generalized least squares algorithm is also proposed for identification of such Wiener systems. The performance of the developed algorithms is validated by two illustrative examples.

The class of nonlinear systems studied in this paper is assumed to be modelled by parallel block-cascades. Such models are composed of parallel branches where each branch has a linear block in cascade with a zero-memory nonlinear block followed by another linear block. These types of models are extensively used to represent nonlinear dynamic systems and are known in the literature as Wiener-Hammerstein models. Using a zero-mean stationary white gaussian sequence as an input to such models, a structure identification criterion is developed, utilizing the bispectrum estimate of the output sequence only. The application of this criterion is shown by several simulation examples. Also, impulse response estimation of an example of such a model is considered to show the effectiveness of the proposed identification technique.

Parameter estimation for Non-uniformly Sampled Wiener Systems with Dead-zone Nonlinearities