Frequency domain identification of Wiener models

This paper proposes a frequency domain algorithm for Wiener model identifications based on exploring the fundamental frequency and harmonics generated by the unknown nonlinearity. The convergence of the algorithm is established in the presence of white noise. No a priori knowledge of the structure of the nonlinearity is required and the linear part can be nonparametric.     

Towards identification of Wiener systems with the least amount of a priori information on the nonlinearity

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

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.     

Computational complexity analysis of set membership identification of Hammerstein and Wiener systems

This paper analyzes the computational complexity of set membership identification of Hammerstein and Wiener systems. Its main results show that, even in cases where a portion of the plant is known, the problems are generically NP-hard both in the number of experimental data points and in the number of inputs (Wiener) or outputs (Hammerstein) of the nonlinearity. These results provide new insight into the reasons underlying the high computational complexity of several recently proposed algorithms and point out the need for developing computationally tractable relaxations.     

Combined frequency-prediction error identification approach for Wiener systems with backlash and backlash-inverse operators

Wiener systems identification is studied in the presence of possibly infinite-order linear dynamics and memory nonlinear operators of backlash and backlash-inverse types. The latter is laterally bordered with polynomial lines of arbitrary-shape. It turns out that the borders are allowed to be noninvertible and crossing making possible to account, within a unified theoretical framework, for memory and memoryless nonlinearities. Moreover, the prior knowledge of the nonlinearity type, being backlash or backlash-inverse or memoryless, is not required. Using sine excitations, and getting benefit from model plurality, the initial complex identification problem is made equivalent to two tractable (though still nonlinear) prediction-error problems. These are coped with using linear and nonlinear least squares estimators which all are shown to be consistent.     

Support vector method for identification of Wiener models

Support vector regression is applied to identify nonlinear systems represented by Wiener models, consisting of a linear dynamic system in series with a static nonlinear block. The linear block is expanded in terms of basis functions, such as Laguerre or Kautz filters, and the static nonlinear block is determined using support vector machine regression.     

Fast approximate identification of nonlinear systems

In this paper, a method is presented to extend the classical identification methods for linear systems towards nonlinear modelling of linear systems that suffer from nonlinear distortions. A well chosen, general nonlinear model structure is proposed that is identified in a two-step procedure. First, a best linear approximation is identified using the classical linear identification methods. In the second step, the nonlinear extensions are identified with a linear least-squares method. The proposed model not only includes Wiener and Hammerstein systems, it is also suitable to model nonlinear feedback systems. The stability of the nonlinear model can be easily verified. The method is illustrated on experimental data.     

Study of the LTI relations between the outputs of two coupled Wiener systems and its application to the generation of initial estimates for Wiener-Hammerstein systems

This paper consists of two parts. In the first, more theoretic part, two Wiener systems driven by the same Gaussian noise excitation are considered. For each of these systems, the best linear approximation (BLA) of the output (in mean square sense) is calculated, and the residuals, defined as the difference between the actual output and the linearly simulated output is considered for both outputs. The paper is focused on the study of the linear relations that exist between these residuals. Explicit expressions are given as a function of the dynamic blocks of both systems, generalizing earlier results obtained by Brillinger [Brillinger, D. R. (1977). The identification of a particular nonlinear time series system. Biometrika, 64(3), 509-515] and Billings and Fakhouri [Billings, S. A., & Fakhouri, S. Y. (1982). Identification of systems containing linear dynamic and static nonlinear elements. Automatica, 18(1), 15-26]. Compared to these earlier results, a much wider class of static nonlinear blocks is allowed, and the efficiency of the estimate of the linear approximation between the residuals is considerably improved. In the second, more practical, part of the paper, this new theoretical result is used to generate initial estimates for the transfer function of the dynamic blocks of a Wiener-Hammerstein system. This method is illustrated on experimental data.     

A blind approach to the Hammerstein-Wiener model identification

In this paper, we propose a blind approach to the sampled Hammerstein-Wiener model identification. By using the blind approach, it is shown that all internal variables can be recovered solely based on the output measurements. Then, identification of linear and nonlinear parts can be carried out. No a priori structural knowledge about the input nonlinearity is assumed and no white noise assumption is imposed on the input.     

Maximum likelihood identification of Wiener models

The Wiener model is a block oriented model, having a linear dynamic system followed by a static nonlinearity. The dominating approach to estimate the components of this model has been to minimize the error between the simulated and the measured outputs. We show that this will, in general, lead to biased estimates if there are other disturbances present than measurement noise. The implications of Bussgang’s theorem in this context are also discussed. For the case with general disturbances, we derive the Maximum Likelihood method and show how it can be efficiently implemented. Comparisons between this new algorithm and the traditional approach, confirm that the new method is unbiased and also has superior accuracy.     

Mixed parametric/non-parametric identification of systems with discontinuous nonlinearities

The subject of this paper is identification of discrete time nonlinear dynamical systems when the system dynamics are defined by a discontinuous nonlinear function with the location of the discontinuity unknown. By representing the nonlinear function using both a parametric term to capture the continuous part and a non-parametric term to capture the discontinuous part, we present an identification algorithm along with conditions for recovery of the true nonlinearity.     

Wiener models of direction-dependent dynamic systems

Analysis of systems with direction-dependent dynamics is currently limited to cases in which the dynamics in the two directions of the output are first order; results for such systems have been published for both pseudo-random maximum-length binary (MLB) and inverse-repeat maximum-length binary (IR-MLB) inputs. These relatively limited analytical results make it useful to examine alternative ways of modelling such systems and in this paper, Wiener models are considered for this purpose. Methods for optimising the Wiener model parameters by matching the system and model cross-correlation functions, outputs, and discrete Fourier transforms of the outputs are considered, and the results are compared. These methods are also applied to a first-order direction-dependent system with a maximum-length ternary (MLT) input, for which no analytical results are currently available, and to a second-order system with an IR-MLB input.     

Representation and identification of non-parametric nonlinear systems of short term memory and low degree of interaction

In this paper, we propose a new representation which is particularly useful for a class of non-parametric nonlinear systems that have short term memory and low degree of interaction. Advantages and disadvantages of this representation are discussed and compared to existing methods both theoretically and numerically. Furthermore, results regarding structural estimation based on the analysis of variance and on full scale identification are also provided.     

On linear models for nonlinear systems

Best linear time-invariant (LTI) approximations are analysed for several interesting classes of discrete nonlinear time-invariant systems. These include nonlinear finite impulse response systems and a class of nonsmooth systems called bi-gain systems. The Fréchet derivative of a smooth nonlinear system is studied as a potential good LTI model candidate. The Fréchet derivative is determined for nonlinear finite memory systems and for a class of Wiener systems. Most of the concrete results are derived in an ?_∞ signal setting. Applications to linear controller design, to identification of linear models and to estimation of the size of the unmodelled dynamics are discussed.     

Parametric identification of structured nonlinear systems

In this paper, identification of structured nonlinear systems is considered. Using linear fractional transformations (LFT), the a priori information regarding the structural interconnection is systematically exploited. A parametric approach to the identification problem is investigated, where it is assumed that the linear part of the interconnection is given and the input to the nonlinear part is measurable. An algorithm for the identification of the nonlinear part is proposed. The uniqueness properties of the estimate provided by the algorithm are examined. It is shown that the estimate converges asymptotically to its true value under a certain persistence of excitation condition. Two simulated examples and a real-data example are presented to show the effectiveness of the proposed algorithm.     

Strong consistence of recursive identification for Wiener systems

The paper concerns identification of the Wiener system consisting of a linear subsystem followed by a static nonlinearity f(·) with no invertibility and structure assumption. Recursive estimates are given for coefficients of the linear subsystem and for the value f(v) at any fixed v. The main contribution of the paper consists in establishing convergence with probability one of the proposed algorithms to the true values. This probably is the first strong consistency result for this kind of Wiener systems. A numerical example is given, which justifies the theoretical analysis.     

Variance results for identification of cascade systems

The objective of this contribution is to analyze statistical properties of estimated models of cascade systems. Models of such systems are important in for example cascade control applications. The aim is to present and analyze some fundamental limitations in the quality of an identified model of a cascade system under the condition that the true subsystems have certain common dynamics. The model quality is analyzed by studying the asymptotic (large data) covariance matrix of the Prediction Error Method parameter estimate. The analysis will focus on cascade systems with three subsystems. The main result is that if the true transfer functions of the first and second subsystem are identical, the output signal information from the second and third subsystems will not affect the asymptotic variance of the estimated model of the first subsystem. This result implies that for a cascade system with two subsystems, where the dynamics of the first subsystem is a factor of the dynamics of the second one, the output signal information from the second subsystem will not improve the asymptotic quality of the estimate of the first subsystem. The results are illustrated by some simple FIR examples.