Wiener-Hammerstein Systems Research Articles

Fast identification of Wiener‐Hammerstein systems using discrete optimisation

This letter proposes a fast identification algorithm for Wiener-Hammerstein systems. The computational cost of separating the front and the back linear time invariant block dynamics is significantly improved by using discrete optimization. The discrete optimization is implemented as a genetic algorithm. Numerical results confirm the efficiency and accuracy of the proposed approach.

Parametric identification of parallel Wiener–Hammerstein systems

Block-oriented nonlinear models are popular in nonlinear modeling because of their advantages to be quite simple to understand and easy to use. To increase the flexibility of single branch block-oriented models, such as Hammerstein, Wiener, and Wiener–Hammerstein models, parallel block-oriented models can be considered. This paper presents a method to identify parallel Wiener–Hammerstein systems starting from input–output data only. In the first step, the best linear approximation is estimated for different input excitation levels. In the second step, the dynamics are decomposed over a number of parallel orthogonal branches. Next, the dynamics of each branch are partitioned into a linear time invariant subsystem at the input and a linear time invariant subsystem at the output. This is repeated for each branch of the model. The static nonlinear part of the model is also estimated during this step. The consistency of the proposed initialization procedure is proven. The method is validated on real-world measurements using a custom built parallel Wiener–Hammerstein test system.

Identification of Wiener–Hammerstein systems by a nonparametric separation of the best linear approximation

Wiener–Hammerstein models are flexible, well known and often studied. The main challenge in identifying a Wiener–Hammerstein model is to distinguish the linear time invariant (LTI) blocks at the front and the back. This paper presents a nonparametric approach to separate the front and back dynamics starting from the best linear approximation (BLA). Next, the nonparametric estimates of the LTI blocks in the model can be parametrized, taking into account a phase shift degeneration. Once the dynamics are known, the estimation of the static nonlinearity boils down to a simple linear least squares problem. The consistency of the proposed approach is discussed and the method is validated on the Wiener–Hammerstein benchmark that was presented at the IFAC SYSID conference in 2009.

Identification of parallel Wiener-Hammerstein systems with a decoupled static nonlinearity

Block-oriented models are often used to model a nonlinear system. This paper presents an identification method for parallel Wiener-Hammerstein systems, where the obtained model has a decoupled static nonlinear block. This decoupled nature makes the interpretation of the obtained model more easy. First a coupled parallel Wiener-Hammerstein model is estimated. Next, the static nonlinearity is decoupled using a tensor decomposition approach. Finally, the method is validated on real-world measurements using a custom built parallel Wiener-Hammerstein test system.

A fractional approach to identify Wiener–Hammerstein systems

Block-oriented nonlinear models are appealing due to their simplicity and parsimony. Existing methods to identify the Wiener–Hammerstein model suffer from one or several drawbacks. This paper shows that it is possible to generate initial estimates in an alternative way. A fractional model parameterization is the key to the success of this approach. Advantages are that no more than two iterative optimizations are needed and that large model orders can be handled. As illustrated through a simulation example and experimental benchmark data, it gives superior initial estimates and comparable optimized results.

Recursive identification of errors-in-variables Wiener–Hammerstein systems

This paper considers the recursive identification of errors-in-variables Wiener–Hammerstein system, which is composed of a static nonlinearity sandwiched by two linear dynamic subsystems. Both the system input and output are observed with additive noises being ARMA processes with unknown coefficients. By the stochastic approximation algorithms incorporated with the deconvolution kernel functions, the coefficients of the linear subsystems and the values of the nonlinear function are recursively estimated. All the estimates are proved to converge to the true values with probability one. A simulation example is given to verify the theoretical analysis.

New Connections Between Frequency Response Functions for a Class of Nonlinear Systems

The notion of frequency response functions has been generalized to nonlinear systems in several ways. However, a relation between different approaches has not yet been established. In this paper, frequency domain representations for nonlinear systems are uniquely connected. Specifically, by means of novel analytical results, the generalized frequency response function (GFRF) and the higher order sinusoidal input describing function (HOSIDF) for polynomial Wiener-Hammerstein systems are explicitly related. Necessary and sufficient conditions for this relation to exist and results on uniqueness and equivalence of the HOSIDF and GFRF are provided. Finally, a numerically efficient computational procedure is presented that allows to compute the GFRF from the HOSIDF and vice versa.

Classification of the Poles and Zeros of the Best Linear Approximations of Wiener-Hammerstein Systems

The parameters of a Wiener-Hammerstein model, a nonlinear block structure comprising two linear filters separated by a memoryless nonlinearity, may be identified using an iterative nonlinear least squares optimization, however avoiding suboptimal local minima in the error surface requires a good initial estimate of the parameter vector. The Best Linear Approximation (BLA) of a Wiener-Hammerstein model will contain all the poles and zeros of both linear elements, but does not provide any information regarding which poles and/or zeros should be assigned to either of the linear elements. This information is contained in the nonlinear terms in the system response. One such nonlinear term is the BLA fitted between a suitably chosen nonlinear transformation of the input, and the output residuals remaining after all linear terms have been removed. The poles and zeros present in this nonlinear transfer function are used to classify the poles and zeros in the initial linear fit as belonging to either the first or second linear element in the Wiener-Hammerstein model. The procedure is illustrated by applying it to experimental data from a Wiener-Hammerstein benchmark system.

Uniquely connecting frequency domain representations of given order polynomial Wiener–Hammerstein systems

The notion of frequency response functions has been generalized to nonlinear systems in several ways. However, a relation between different approaches has not yet been established. In this paper, frequency domain representations for nonlinear systems are uniquely connected for a class of nonlinear systems. Specifically, by means of novel analytical results, the generalized frequency response function (GFRF) and the higher order sinusoidal input describing function (HOSIDF) for polynomial Wiener–Hammerstein systems are explicitly related, assuming the linear dynamics are known. Necessary and sufficient conditions for this relation to exist and results on the uniqueness and equivalence of the HOSIDF and GFRF are provided. Finally, this yields an efficient computational procedure for computing the GFRF from the HOSIDF and vice versa.

Least-Squares Support Vector Machines for the identification of Wiener–Hammerstein systems

This paper considers the identification of Wiener–Hammerstein systems using Least-Squares Support Vector Machines based models. The power of fully black-box NARX-type models is evaluated and compared with models incorporating information about the structure of the systems. For the NARX models it is shown how to extend the kernel-based estimator to large data sets. For the structured model the emphasis is on preserving the convexity of the estimation problem through a suitable relaxation of the original problem. To develop an empirical understanding of the implications of the different model design choices, all considered models are compared on an artificial system under a number of different experimental conditions. The obtained results are then validated on the Wiener–Hammerstein benchmark data set and the final models are presented. It is illustrated that black-box models are a suitable technique for the identification of Wiener–Hammerstein systems. The incorporation of structural information results in significant improvements in modeling performance.

Analysis of Best Linear Approximation of a Wiener–Hammerstein System for Arbitrary Amplitude Distributions

For a nonlinear system with an input signal having a Gaussian probability distribution (this includes random-phase multisines), the best linear approximation (BLA) is proportional to the frequency response of the overall system. However, this is not the case for non-Gaussian input signals, for which the frequency response is biased with respect to the Gaussian BLA. In this paper, theoretical expressions for determining this bias for Wiener-Hammerstein systems are developed both for binary input signals and for white noise inputs with arbitrary probability distribution. Cubic and quintic nonlinearities are considered, but the methods can be extended to other forms of polynomial nonlinearity. Simple measures for quantifying the bias are also developed. It is shown that the bias decays rapidly to zero for a growing length of the impulse response.

MODEL PREDICTIVE CONTROL OF A LYOPHILIZATION PLANT: A SIMPLIFIED APPROACH USING WIENER AND HAMMERSTEIN SYSTEMS

Lyophilization process is widely used in pharmaceutical industries, preparing stable dried medications and important biopreparations, so they remain stable and easier to store at room temperature. Since a lyophilization cycle involves high energy demands, an improved control strategy has to be used in order to minimize the operating costs. This paper deals with the design methodology of nonlinear model predictive controllers for lyophilization plant. The controllers are based on fuzzy-neural predictive models and simplified gradient optimization algorithm. As predictive models, fuzzy-neural implementations of Hammerstein and Wiener-Hammerstein systems are used. Such structures provide fast and reliable system identification using small number of parameters which reduces the computational burden during the optimization procedure. The potential benefits of the proposed approaches are demonstrated by simulation experiments.

Tensor-based Identification of the Structure of Block-Oriented Nonlinear Systems

In this paper, we propose a new tensor-based approach to identify the structure of a block-oriented nonlinear system (Hammerstein, Wiener, and Wiener-Hammerstein systems). The proposed method makes use of one time-domain Volterra kernel of an arbitrary order higher than two, which can be viewed as a tensor. We develop a tensor analysis for carrying out the identification of the structure of block-oriented nonlinear systems. The performance of the proposed identification scheme is illustrated by means of some simulation results.

Identification of Wiener-Hammerstein Systems using LS-SVMs

This paper extends the overparametrization technique as used for Hammerstein systems employing nonlinear Least-Squares Support Vector Machines (LS-SVMs) towards the identification of Wiener-Hammerstein systems. We present some practical guidelines as well as empirical results on the performance of the method with respect to various deficiencies of the excitation signal. Finally we apply our method to the SYSID2009 Wiener-Hammerstein benchmark data set.

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.

Nonparametric initial estimates for Wiener-Hammerstein systems

This paper studies the generation of nonparametric initial estimates for Wiener-Hammerstein systems. The kernel idea is to replace the original structure by an alternative representation that is easier to identify in a two step procedure. In the first step, the best linear approximation is estimated, in the second step a structure of the form GfG–1 is identified, where G is a linear dynamic system and f a static nonlinear system. By making a nonparametric approach it is possible to reduce the required user interaction, which strongly increases the user friendliness.

Computing an Initial Estimate of a Wiener–Hammerstein System With a Random Phase Multisine Excitation

Wiener-Hammerstein systems consist of two linear dynamic systems placed around a static nonlinearity. These models are difficult to identify due to the presence of two dynamic systems. Usually, a nonlinear estimation procedure is necessary to estimate the parameters of the different parts. This nonlinear estimation procedure needs good starting values to converge quickly and/or reliably to a global minimum. This paper proposes a method to compute a first estimate based on one measurement record only.

Linear approximations of nonlinear FIR systems for separable input processes

Nonlinear systems can be approximated by linear time-invariant (LTI) models in many ways. Here, LTI models that are optimal approximations in the mean-square error sense are analyzed. A necessary and sufficient condition on the input signal for the optimal LTI approximation of an arbitrary nonlinear finite impulse response (NFIR) system to be a linear finite impulse response (FIR) model is presented. This condition says that the input should be separable of a certain order, i.e., that certain conditional expectations should be linear. For the special case of Gaussian input signals, this condition is closely related to a generalized version of Bussgang's classic theorem about static nonlinearities. It is shown that this generalized theorem can be used for structure identification and for the identification of generalized Wiener–Hammerstein systems.

Nonlinear structure identification with application to wiener-hammerstein systems

While there exists a substantial literature on the identification of Hammerstein and Wiener models, the identification of Wiener-Hammerstein models has received considerably less attention yet this is a model class of very great practical importance. This paper proposes an elegant approach to estimating Wiener-Hammerstein systems from measured data.

FIRST ESTIMATES OF WIENER-HAMMERSTEIN SYSTEMS USING A RANDOM MULTISINE EXCITATION

Wiener-Hammerstein systems consist of a linear dynamic system followed by a static nonlinearity, followed by another linear dynamic system. These models are difficult to identify due to the presence of two dynamic systems. Usually, a nonlinear estimation procedure is used to estimate the parameters of the different parts. This nonlinear estimation procedure needs good starting values to converge quickly and/or reliably to a global minimum. This paper proposes a method to compute a first estimate based on one measurement record only.