Identification of Nonlinear Block-Oriented Systems starting from Linear Approximations: A Survey
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paper_title: Linear approximations of nonlinear FIR systems for separable input processes

paper_content:

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.
---
paper_title: System Identification of Nonlinear State-Space Models

paper_content:

This paper is concerned with the parameter estimation of a general class of nonlinear dynamic systems in state-space form. More specifically, a Maximum Likelihood (ML) framework is employed and an Expectation Maximisation (EM) algorithm is derived to compute these ML estimates. The Expectation (E) step involves solving a nonlinear state estimation problem, where the smoothed estimates of the states are required. This problem lends itself perfectly to the particle smoother, which provides arbitrarily good estimates. The maximisation (M) step is solved using standard techniques from numerical optimisation theory. Simulation examples demonstrate the efficacy of our proposed solution.
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paper_title: System Identification: A Frequency Domain Approach

paper_content:

System identification is a general term used to describe mathematical tools and algorithms that build dynamical models from measured data. Used for prediction, control, physical interpretation, and the designing of any electrical systems, they are vital in the fields of electrical, mechanical, civil, and chemical engineering. Focusing mainly on frequency domain techniques, System Identification: A Frequency Domain Approach, Second Edition also studies in detail the similarities and differences with the classical time domain approach. It high??lights many of the important steps in the identification process, points out the possible pitfalls to the reader, and illustrates the powerful tools that are available.
---
paper_title: Linear Models of Nonlinear Systems

paper_content:

Linear time-invariant approximations of nonlinear systems are used in many applications and can be obtained in several ways. For example, using system identification and the prediction-error method, it is always possible to estimate a linear model without considering the fact that the input and output measurements in many cases come from a nonlinear system. One of the main objectives of this thesis is to explain some properties of such approximate models. More specifically, linear time-invariant models that are optimal approximations in the sense that they minimize a mean-square error criterion are considered. Linear models, both with and without a noise description, are studied. Some interesting, but in applications usually undesirable, properties of such optimal models are pointed out. It is shown that the optimal linear model can be very sensitive to small nonlinearities. Hence, the linear approximation of an almost linear system can be useless for some applications, such as robust control design. Furthermore, it is shown that standard validation methods, designed for identification of linear systems, cannot always be used to validate an optimal linear approximation of a nonlinear system. In order to improve the models, conditions on the input signal that imply various useful properties of the linear approximations are given. It is shown, for instance, that minimum phase filtered white noise in many senses is a good choice of input signal. Furthermore, the class of separable signals is studied in detail. This class contains Gaussian signals and it turns out that these signals are especially useful for obtaining approximations of generalized Wiener-Hammerstein systems. It is also shown that some random multisine signals are separable. In addition, some theoretical results about almost linear systems are presented. In standard methods for robust control design, the size of the model error is assumed to be known for all input signals. However, in many situations, this is not a realistic assumption when a nonlinear system is approximated with a linear model. In this thesis, it is described how robust control design of some nonlinear systems can be performed based on a discrete-time linear model and a model error model valid only for bounded inputs. It is sometimes undesirable that small nonlinearities in a system influence the linear approximation of it. In some cases, this influence can be reduced if a small nonlinearity is included in the model. In this thesis, an identification method with this option is presented for nonlinear autoregressive systems with external inputs. Using this method, models with a parametric linear part and a nonparametric Lipschitz continuous nonlinear part can be estimated by solving a convex optimization problem.
---
paper_title: Nonlinear system identification using neural state space models, applicable to robust control design

paper_content:

Prediction error learning algorithms for neural state space models are developed, both for the deterministic case and the stochastic case with measurement and process noise. For the stochastic case, a predictor with direct parametrization of the Kalman gain by a neural net architecture is proposed. Expressions for the gradients are derived by applying Narendra's sensitivity model approach. Finally a linear fractional transformation representation is given for neural state space models, which makes it possible to use these models, obtained from input/output measurements on a plant, in a standard robust performance control scheme.
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paper_title: Structure Discrimination in Block-Oriented Models Using Linear Approximations: A Theoretic Framework

paper_content:

In this paper we show that it is possible to retrieve structural information about complex block-oriented nonlinear systems, starting from linear approximations of the nonlinear system around different setpoints. The key idea is to monitor the movements of the poles and zeros of the linearized models and to reduce the number of candidate models on the basis of these observations. Besides the well known open loop single branch Wiener-, Hammerstein-, and Wiener-Hammerstein systems, we also cover a number of more general structures like parallel (multi branch) Wiener-Hammerstein models, and closed loop block oriented models, including linear fractional representation (LFR) models.
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paper_title: Input-output parametric models for non-linear systems Part II: stochastic non-linear systems

paper_content:

Recursive input-output models for non-linear multivariate discrete-time systems are derived, and sufficient conditions for their existence are defined. The paper is divided into two parts. The first part introduces and defines concepts such as Nerode realization, multistructural forms and results from differential geometry which are then used to derive a recursive input-output model for multivariable deterministic non-linear systems. The second part introduces several examples, compares the derived model with other representations and extends the results to create prediction error or innovation input-output models for non-linear stochastic systems. These latter models are the generalization of the multivariable ARM AX models for linear systems and are referred to as NARMAX or Non-linear AutoRegressive Moving Average models with exogenous inputs.
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paper_title: Structure identification of nonlinear dynamic systems - A survey on input/output approaches

paper_content:

Abstract In the past several methods have been elaborated for the identification of nonlinear dynamic systems. Most of the methods assume that the structure of the system is given a priori . Therefore they are in reality parameter estimation algorithms and structure identification is thus usually performed by repeated parameter estimation. However in nonlinear system theory several methods are known to determine the structure of a system. In this paper structure identification of block-oriented (especially cascade) models, of semi-linear dynamic models with signal-dependent parameters and of nonlinear dynamic models being linear in the parameters will be considered. Different structure selection methods are summarized based on step and impulse tests, frequency response measurements, correlation analysis, repeated reproducible tests and normal operating data.
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paper_title: Three free data sets for development and benchmarking in nonlinear system identification

paper_content:

System identification is a fundamentally experimental field of science in that it deals with modeling of system dynamics using measured data. Despite this fact many algorithms and theoretical results are only tested with simulations at the time of publication. One reason for this may be a lack of easily available live data. This paper therefore presents three sets of data, suitable for development, testing and benchmarking of system identification algorithms for nonlinear systems. The data sets are collected from laboratory processes that can be described by block - oriented dynamic models, and by more general nonlinear difference and differential equation models. All data sets are available for free download.
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paper_title: Nonlinear Structure Analysis Using the Best Linear Approximation

paper_content:

In this paper, we propose a method to distinguish between some nonlinear system structures using the bestlinear approximation (BLA) of the system in order to select an appropriate model structure. The main ideaof the method is to apply a Gaussian input signal and to vary the root mean square (rms) value and thepower spectrum of this signal. Depending on the resulting changes of the amplitude and phasecharacteristics of the BLA, an appropriate model structure for the Device Under Test can be selected. Atheoretical analysis of the method is presented for some block-oriented structures.
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paper_title: Mixed parametric-nonparametric identification of Hammerstein and Wiener systems - a survey1

paper_content:

Abstract The paper surveys the ideas of cooperation between parametric and nonparametric (kernel-based) algorithms of nonlinear block-oriented system identification. Various strategies are proposed, dependently on the system structure, number of data and the prior knowledge. The estimates are consistent and their rates of convergence are presented. The aim of the paper is to show some recent results in the field in a systematic, ordered way.
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paper_title: Structure Discrimination in Block-Oriented Models Using Linear Approximations: A Theoretic Framework

paper_content:

In this paper we show that it is possible to retrieve structural information about complex block-oriented nonlinear systems, starting from linear approximations of the nonlinear system around different setpoints. The key idea is to monitor the movements of the poles and zeros of the linearized models and to reduce the number of candidate models on the basis of these observations. Besides the well known open loop single branch Wiener-, Hammerstein-, and Wiener-Hammerstein systems, we also cover a number of more general structures like parallel (multi branch) Wiener-Hammerstein models, and closed loop block oriented models, including linear fractional representation (LFR) models.
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paper_title: Generalised Hammerstein-Wiener System Estimation and a Benchmark Application ✩

paper_content:

Abstract This paper examines the use of a so-called “generalised Hammerstein–Wiener” model structure that is formed as the concatenation of an arbitrary number of Hammerstein systems. The latter are taken here to be memoryless non-linearities followed by linear time invariant dynamics. Hammerstein, Wiener, Hammerstein–Wiener and Wiener–Hammerstein models are all special cases of this structure. The parameter estimation of this model is investigated using a standard prediction error criterion coupled with a robust gradient based search algorithm. This approach is profiled using a Wiener–Hammerstein Benchmark example, which illustrates it to be effective and, via Monte-Carlo simulation, relatively robust against capture in local minima.
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paper_title: Initial estimates for block structured nonlinear systems with feedback

paper_content:

This paper studies the identification of a block structured nonlinear Wiener-Hammerstein system that is captured in the feedforward or the feedback path of a feed- back loop. Nonparametric initial estimates are generated for the three dynamic blocks, modelled by their frequency response function, and the static nonlinear system. The method can be applied to input/output data resulting from random or periodic excitations.
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paper_title: Combined Parametric-Nonparametric Identification of Block-Oriented Systems

paper_content:

Hammerstein system.- Wiener system.- Wiener-Hammerstein (sandwich) system.- Large-scale interconnected systems.- Structure detection and model order selection.- Time-varying systems.- Simulation studies.- Summary.
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paper_title: Identification of Wiener-Hammerstein systems by a nonparametric separation of the best linear approximation

paper_content:

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.
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paper_title: Combining the best linear approximation and dimension reduction to identify the linear blocks of parallel Wiener systems

paper_content:

Abstract A Wiener model is a fairly simple, well known, and often used nonlinear block-oriented black-box model. A possible generalization of the class of Wiener models lies in the parallel Wiener model class. This paper presents a method to estimate the linear time-invariant blocks of such parallel Wiener models from input/output data only. The proposed estimation method combines the knowledge obtained by estimating the best linear approximation of a nonlinear system with the MAVE dimension reduction method to estimate the linear time-invariant blocks present in the model. The estimation of the static nonlinearity boils down to a standard static nonlinearity estimation problem starting from input-output data once the linear blocks are known.
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paper_title: Identification of FIR Wiener systems with unknown, non-invertible, polynomial non-linearities

paper_content:

Wiener systems consist of a linear dynamic system whose output is measured through a static non-linearity. In this paper we study the identification of single-input single-output Wiener systems with finite impulse response dynamics and polynomial output non-linearities. Using multi-index notation, we solve a least squares problem to estimate products of the coefficients of the non-linearity and the impulse response of the linear system. We then consider four methods for extracting the coefficients of the non-linearity and impulse response: direct algebraic solution, singular value decomposition, multi-dimensional singular value decomposition and prediction error optimization.
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paper_title: An iterative method for the identification of nonlinear systems using a Hammerstein model

paper_content:

An iterative method is proposed for the identification of nonlinear systems from samples of inputs and outputs in the presence of noise. The model used for the identification consists of a no-memory gain (of an assumed polynomial form) followed by a linear discrete system. The parameters of the pulse transfer function of the linear system and the coefficients of the polynomial non-linearity are alternately adjusted to minimize a mean square error criterion. Digital computer simulations are included to demonstrate the feasibility of the technique.
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paper_title: Frequency identification of nonparametric Wiener systems containing backlash nonlinearities

paper_content:

This paper addresses the problem of identifying Wiener systems constituted of nonparametric linear dynamics and backlash nonlinearities. The linear subsystem is arbitrary but stable. The backlash nonlinearity borders are almost arbitrary-shape. In particular, these are allowed to be nonsmooth, noninvertible and crossing. A frequency identification method is developed to estimate the system frequency response function (at a given set of frequencies) and the backlash nonlinearity borders (within a given working domain). The identification method involves sine excitations and consistent estimators designed using analytic geometry tools, e.g. fictive limit cycles; informative backlash limit cycles; spread- and orientation-compatibility.
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paper_title: Bayesian semiparametric Wiener system identification

paper_content:

We present a novel method for Wiener system identication. The method relies on a semiparametric, i.e. a mixed parametric/nonparametric, model of a Wiener system. We use a state-space model for the linear dynamical system and a nonparametric Gaussian process model for the static nonlinearity. We avoid making strong assumptions, such as monotonicity, on the nonlinear mapping. Stochastic disturbances, entering both as measurement noise and as process noise, are handled in a systematic manner. The nonparametric nature of the Gaussian process allows us to handle a wide range of nonlinearities without making problem-specic parameterizations. We also consider sparsity-promoting priors, based on generalized hyperbolic distributions, to automatically infer the order of the underlying dynamical system. We derive an inference algorithm based on an ecient particle Markov chain Monte Carlo method, referred to as particle Gibbs with ancestor sampling. The method is proled on two challenging identication problems with good results. Blind Wiener system identication is handled as a special case.
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paper_title: Maximum Likelihood Identification of Wiener Models

paper_content:

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.
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paper_title: Identification of wiener systems with process noise is a nonlinear errors-in-variables problem

paper_content:

This paper considers the identification of stochastic Wiener dynamic systems, that is linear dynamic systems with process noise, where the measurable output signal is a nonlinear function of the output from the linear system corrupted with additive measurement noise. It is shown how stochastic Wiener system identification can be viewed as a particular non-linear model errors-in-variables problem, for which there exists a large literature. We compare the maximum likelihood method with prediction error minimization methods based on the conditional mean predictor for Wiener systems. Related methods have previously been studied in the framework of identification of non-linear error-in-variables models. We extend these results by taking the input signal to the Wiener system into consideration. For example, the input will affect the variance of the prediction errors. Hence, a prediction error method with a variance weighting is derived to obtain more reliable parameter estimates. An advantage with the prediction error method is that for certain special cases we can avoid numerical integration. We also discuss how the unscented transform can be used to obtain an approximate predictor for the prediction error method. The numerical evaluation of these methods is performed on a simple first order FIR system with a cubic nonlinearity, for which some illustrative analytic properties are derived.
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paper_title: System Identification: A Frequency Domain Approach

paper_content:

System identification is a general term used to describe mathematical tools and algorithms that build dynamical models from measured data. Used for prediction, control, physical interpretation, and the designing of any electrical systems, they are vital in the fields of electrical, mechanical, civil, and chemical engineering. Focusing mainly on frequency domain techniques, System Identification: A Frequency Domain Approach, Second Edition also studies in detail the similarities and differences with the classical time domain approach. It high??lights many of the important steps in the identification process, points out the possible pitfalls to the reader, and illustrates the powerful tools that are available.
---
paper_title: Robustness Issues of the Equivalent Linear Representation of a Nonlinear System

paper_content:

In many engineering applications, linear models are preferred, even if it is known that the system is nonlinear. A large class of nonlinear systems, excited with a 'Gaussian' random excitation, can be represented as a linear system GBLA plus a nonlinear noise source YS. The nonlinear noise source represents that part of the output that is not captured by the linear approximation. It was shown before that the GBLA is an invariant for a wide class of excitations with a user specified power spectrum. In this paper it is shown that also the power spectrum SYS of the nonlinear noise source YS is a robust characteristic. This shows that the alternative 'linear representation' of a nonlinear system is a robust one, making its use in the daily engineering practice very attractive.
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paper_title: Linear approximations of nonlinear FIR systems for separable input processes

paper_content:

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.
---
paper_title: Theory and application of the separable class of random processes

paper_content:

"May 26, 1958." Issued also as a thesis, M.I.T. Dept. of Electrical Engineering, May 19, 1958.
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paper_title: Identification of Block-oriented Systems Using the Invariance Property

paper_content:

Identification of systems that can be written as interconnected linear time-invariant (LTI) dynamical subsystems and static nonlinearities has been an active research area for several decades. These systems are often referred to as block-oriented systems since their structures can be characterised using linear dynamical and static nonlinear blocks. In particular, block-oriented systems where the blocks are connected in series have received special attention. For example, Wiener and Hammerstein systems are common examples of series connected block-oriented systems.
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paper_title: Elliptically symmetric distributions

paper_content:

Elliptically symmetric distributions are second-order distributions with probability densities whose contours of equal height are ellipses. This class includes the Gaussian and sine-wave distributions and others which can be generated from certain first-order distributions. Members of this class have several desirable features for the description of the second-order statistics of the transformation of a random signal by an instantaneous nonlinear device. In particular, they are separable in Nuttall's sense, so that the output of the device may be described in terms of equivalent gain and distortion. These distributions can also simplify the evaluation of the output autovariance because of their similarity to the Gaussian distribution. For a certain class of functions, elliptically symmetric distributions yield averages which are simply proportional to those obtained with a Gaussian distribution. Furthermore, these distributions satisfy a relation analogous to Price's theorem for Gaussian distributions. Finally, a certain subclass of these distributions can be expanded in the series representation studied by Barrett and Lampard.
---
paper_title: Linear approximations of nonlinear FIR systems for separable input processes

paper_content:

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.
---
paper_title: System Identification: A Frequency Domain Approach

paper_content:

System identification is a general term used to describe mathematical tools and algorithms that build dynamical models from measured data. Used for prediction, control, physical interpretation, and the designing of any electrical systems, they are vital in the fields of electrical, mechanical, civil, and chemical engineering. Focusing mainly on frequency domain techniques, System Identification: A Frequency Domain Approach, Second Edition also studies in detail the similarities and differences with the classical time domain approach. It high??lights many of the important steps in the identification process, points out the possible pitfalls to the reader, and illustrates the powerful tools that are available.
---
paper_title: Linear Models of Nonlinear Systems

paper_content:

Linear time-invariant approximations of nonlinear systems are used in many applications and can be obtained in several ways. For example, using system identification and the prediction-error method, it is always possible to estimate a linear model without considering the fact that the input and output measurements in many cases come from a nonlinear system. One of the main objectives of this thesis is to explain some properties of such approximate models. More specifically, linear time-invariant models that are optimal approximations in the sense that they minimize a mean-square error criterion are considered. Linear models, both with and without a noise description, are studied. Some interesting, but in applications usually undesirable, properties of such optimal models are pointed out. It is shown that the optimal linear model can be very sensitive to small nonlinearities. Hence, the linear approximation of an almost linear system can be useless for some applications, such as robust control design. Furthermore, it is shown that standard validation methods, designed for identification of linear systems, cannot always be used to validate an optimal linear approximation of a nonlinear system. In order to improve the models, conditions on the input signal that imply various useful properties of the linear approximations are given. It is shown, for instance, that minimum phase filtered white noise in many senses is a good choice of input signal. Furthermore, the class of separable signals is studied in detail. This class contains Gaussian signals and it turns out that these signals are especially useful for obtaining approximations of generalized Wiener-Hammerstein systems. It is also shown that some random multisine signals are separable. In addition, some theoretical results about almost linear systems are presented. In standard methods for robust control design, the size of the model error is assumed to be known for all input signals. However, in many situations, this is not a realistic assumption when a nonlinear system is approximated with a linear model. In this thesis, it is described how robust control design of some nonlinear systems can be performed based on a discrete-time linear model and a model error model valid only for bounded inputs. It is sometimes undesirable that small nonlinearities in a system influence the linear approximation of it. In some cases, this influence can be reduced if a small nonlinearity is included in the model. In this thesis, an identification method with this option is presented for nonlinear autoregressive systems with external inputs. Using this method, models with a parametric linear part and a nonparametric Lipschitz continuous nonlinear part can be estimated by solving a convex optimization problem.
---
paper_title: Estimating Linear Time-invariant Models of Nonlinear Time-varying Systems

paper_content:

The standard machinery for system identification of linear time-invariant (LTI) models delivers a nominal model and a confidence (uncertainty) region around it, based on (second order moment) residual analysis and covariance estimation. In most cases this gives an uncertainty region that tends to zero as more and more data become available, even if the true system is nonlinear and/or time-varying. In this paper, the reasons for this are displayed, and a characterization of the limit LTI model is given under quite general conditions. Various ways are discussed, and tested, to obtain a more realistic limiting model, with uncertainty. These should reflect the distance to the true possibly nonlinear, time-varying system, and also form a reliable basis for robust LTI control design.
---
paper_title: Linear System Identification in a Nonlinear Setting - Nonparametric analysis of the nonlinear distortions and their impact on the best linear approximation

paper_content:

This article addresses the following problems: 1) First, a nonlinearity analysis is made looking for the presence of nonlinearities in an early phase of the identification process. The level and the nature of the nonlinearities should be retrieved without a significant increase in the amount of measured data. 2) Next it is studied if it is safe to use a linear system identification approach, even if the presence of nonlinear distortions is detected. The properties of the linear system identification approach under these conditions are studied, and the reliability of the uncertainty bounds is checked. 3) Eventually, tools are provided to check how much can be gained if a nonlinear model were identified instead of a linear model. Addressing these three questions forms the outline of this article. The possibilities and pitfalls of using a linear identification framework in the presence of nonlinear distortions will be discussed and illustrated on lab-scale and industrial examples. In this article, the focus is on nonparametric and parametric black box identification methods, however the results might also be useful for physical modeling methods. Knowing the actual nonlinear distortion level can help to choose the required level of detail that is needed in the physical model. This will strongly influence the modeling effort. Also, in this case, significant time can be saved if it is known from experiments that the system behaves almost linearly. The converse is also true. If the experiments show that some (sub-)systems are highly nonlinear, it helps to focus the physical modeling effort on these critical elements.
---
paper_title: Structure Discrimination in Block-Oriented Models Using Linear Approximations: A Theoretic Framework

paper_content:

In this paper we show that it is possible to retrieve structural information about complex block-oriented nonlinear systems, starting from linear approximations of the nonlinear system around different setpoints. The key idea is to monitor the movements of the poles and zeros of the linearized models and to reduce the number of candidate models on the basis of these observations. Besides the well known open loop single branch Wiener-, Hammerstein-, and Wiener-Hammerstein systems, we also cover a number of more general structures like parallel (multi branch) Wiener-Hammerstein models, and closed loop block oriented models, including linear fractional representation (LFR) models.
---
paper_title: Linear approximations of nonlinear FIR systems for separable input processes

paper_content:

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.
---
paper_title: System Identification: A Frequency Domain Approach

paper_content:

System identification is a general term used to describe mathematical tools and algorithms that build dynamical models from measured data. Used for prediction, control, physical interpretation, and the designing of any electrical systems, they are vital in the fields of electrical, mechanical, civil, and chemical engineering. Focusing mainly on frequency domain techniques, System Identification: A Frequency Domain Approach, Second Edition also studies in detail the similarities and differences with the classical time domain approach. It high??lights many of the important steps in the identification process, points out the possible pitfalls to the reader, and illustrates the powerful tools that are available.
---
paper_title: Linear Models of Nonlinear Systems

paper_content:

Linear time-invariant approximations of nonlinear systems are used in many applications and can be obtained in several ways. For example, using system identification and the prediction-error method, it is always possible to estimate a linear model without considering the fact that the input and output measurements in many cases come from a nonlinear system. One of the main objectives of this thesis is to explain some properties of such approximate models. More specifically, linear time-invariant models that are optimal approximations in the sense that they minimize a mean-square error criterion are considered. Linear models, both with and without a noise description, are studied. Some interesting, but in applications usually undesirable, properties of such optimal models are pointed out. It is shown that the optimal linear model can be very sensitive to small nonlinearities. Hence, the linear approximation of an almost linear system can be useless for some applications, such as robust control design. Furthermore, it is shown that standard validation methods, designed for identification of linear systems, cannot always be used to validate an optimal linear approximation of a nonlinear system. In order to improve the models, conditions on the input signal that imply various useful properties of the linear approximations are given. It is shown, for instance, that minimum phase filtered white noise in many senses is a good choice of input signal. Furthermore, the class of separable signals is studied in detail. This class contains Gaussian signals and it turns out that these signals are especially useful for obtaining approximations of generalized Wiener-Hammerstein systems. It is also shown that some random multisine signals are separable. In addition, some theoretical results about almost linear systems are presented. In standard methods for robust control design, the size of the model error is assumed to be known for all input signals. However, in many situations, this is not a realistic assumption when a nonlinear system is approximated with a linear model. In this thesis, it is described how robust control design of some nonlinear systems can be performed based on a discrete-time linear model and a model error model valid only for bounded inputs. It is sometimes undesirable that small nonlinearities in a system influence the linear approximation of it. In some cases, this influence can be reduced if a small nonlinearity is included in the model. In this thesis, an identification method with this option is presented for nonlinear autoregressive systems with external inputs. Using this method, models with a parametric linear part and a nonparametric Lipschitz continuous nonlinear part can be estimated by solving a convex optimization problem.
---
paper_title: Estimating Linear Time-invariant Models of Nonlinear Time-varying Systems

paper_content:

The standard machinery for system identification of linear time-invariant (LTI) models delivers a nominal model and a confidence (uncertainty) region around it, based on (second order moment) residual analysis and covariance estimation. In most cases this gives an uncertainty region that tends to zero as more and more data become available, even if the true system is nonlinear and/or time-varying. In this paper, the reasons for this are displayed, and a characterization of the limit LTI model is given under quite general conditions. Various ways are discussed, and tested, to obtain a more realistic limiting model, with uncertainty. These should reflect the distance to the true possibly nonlinear, time-varying system, and also form a reliable basis for robust LTI control design.
---
paper_title: Combining the best linear approximation and dimension reduction to identify the linear blocks of parallel Wiener systems

paper_content:

Abstract A Wiener model is a fairly simple, well known, and often used nonlinear block-oriented black-box model. A possible generalization of the class of Wiener models lies in the parallel Wiener model class. This paper presents a method to estimate the linear time-invariant blocks of such parallel Wiener models from input/output data only. The proposed estimation method combines the knowledge obtained by estimating the best linear approximation of a nonlinear system with the MAVE dimension reduction method to estimate the linear time-invariant blocks present in the model. The estimation of the static nonlinearity boils down to a standard static nonlinearity estimation problem starting from input-output data once the linear blocks are known.
---
paper_title: Identification of Linear Systems with Nonlinear Distortions

paper_content:

Abstract In this paper the impact of nonlinear distortions on the linear system identification framework is studied. In the first part the nonlinear system is replaced by a linear model plus a nonlinear noise source. The properties of this representation are studied. Next a method to detect, qualify and quantify the nonlinear distortions is presented. In the second part, the (non)-parametric identification of the best linear approximation is studied. In the last part, the linear modelling approach is extended towards nonlinear modelling. A fast approximate nonlinear modelling framework is set up that is a natural extension of the linear framework, and bridges the gap between the linear and the nonlinear identification approaches.
---
paper_title: Structure Discrimination in Block-Oriented Models Using Linear Approximations: A Theoretic Framework

paper_content:

In this paper we show that it is possible to retrieve structural information about complex block-oriented nonlinear systems, starting from linear approximations of the nonlinear system around different setpoints. The key idea is to monitor the movements of the poles and zeros of the linearized models and to reduce the number of candidate models on the basis of these observations. Besides the well known open loop single branch Wiener-, Hammerstein-, and Wiener-Hammerstein systems, we also cover a number of more general structures like parallel (multi branch) Wiener-Hammerstein models, and closed loop block oriented models, including linear fractional representation (LFR) models.
---
paper_title: On linear models for nonlinear systems

paper_content:

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 Frechet derivative of a smooth nonlinear system is studied as a potential good LTI model candidate. The Frechet 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.
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paper_title: Structure Discrimination in Block-Oriented Models Using Linear Approximations: A Theoretic Framework

paper_content:

In this paper we show that it is possible to retrieve structural information about complex block-oriented nonlinear systems, starting from linear approximations of the nonlinear system around different setpoints. The key idea is to monitor the movements of the poles and zeros of the linearized models and to reduce the number of candidate models on the basis of these observations. Besides the well known open loop single branch Wiener-, Hammerstein-, and Wiener-Hammerstein systems, we also cover a number of more general structures like parallel (multi branch) Wiener-Hammerstein models, and closed loop block oriented models, including linear fractional representation (LFR) models.
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paper_title: System Identification: A Frequency Domain Approach

paper_content:

System identification is a general term used to describe mathematical tools and algorithms that build dynamical models from measured data. Used for prediction, control, physical interpretation, and the designing of any electrical systems, they are vital in the fields of electrical, mechanical, civil, and chemical engineering. Focusing mainly on frequency domain techniques, System Identification: A Frequency Domain Approach, Second Edition also studies in detail the similarities and differences with the classical time domain approach. It high??lights many of the important steps in the identification process, points out the possible pitfalls to the reader, and illustrates the powerful tools that are available.
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paper_title: Practical Methods of Optimization

paper_content:

Preface Table of Notation Part 1: Unconstrained Optimization Introduction Structure of Methods Newton-like Methods Conjugate Direction Methods Restricted Step Methods Sums of Squares and Nonlinear Equations Part 2: Constrained Optimization Introduction Linear Programming The Theory of Constrained Optimization Quadratic Programming General Linearly Constrained Optimization Nonlinear Programming Other Optimization Problems Non-Smooth Optimization References Subject Index.
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paper_title: Convergence of the iterative algorithm for a general Hammerstein system identification

paper_content:

The convergence of the iterative identification algorithm for a general Hammerstein system has been an open problem for a long time. In this paper, it is shown that the convergence can be achieved by incorporating a regularization procedure on the nonlinearity in addition to a normalization step on the parameters.
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paper_title: Initial estimates for the dynamics of a Hammerstein system

paper_content:

This paper studies the generation of initial estimates for the dynamic part of a Hammerstein model. It will be shown that ARMAX or Box-Jenkins models result in better initial estimates than ARX or output-error (OE) models even in the absence of disturbing noise. This will be proven by noticing that a static nonlinear system can be replaced by a static gain plus a nonlinear noise source that acts in a completely similar way to disturbing noise for the study of the second-order properties of the estimators in the prediction error framework.
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paper_title: Blind Maximum Likelihood Identification of Hammerstein Systems

paper_content:

Abstract This paper is about the identification of discrete-time Hammerstein systems from output measurements only (blind identification). Assuming that the unobserved input is white Gaussian noise, that the static nonlinearity is invertible, and that the output is observed without errors, a Gaussian maximum likelihood estimator is constructed. Its asymptotic properties are analyzed and the Cramer-Rao lower bound is calculated. In practice, the latter can be computed accurately without using the strong law of large numbers. A two-step procedure is described that allows to find high quality initial estimates to start up the iterative Gauss-Newton based optimization scheme. The paper includes the illustration of the method on a simulation example.
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paper_title: Combined Parametric-Nonparametric Identification of Block-Oriented Systems

paper_content:

Hammerstein system.- Wiener system.- Wiener-Hammerstein (sandwich) system.- Large-scale interconnected systems.- Structure detection and model order selection.- Time-varying systems.- Simulation studies.- Summary.
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paper_title: A blind approach to the Hammerstein-Wiener model identification

paper_content:

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.
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paper_title: Parameter identification of Hammerstein systems containing backlash operators with arbitrary-shape parametric borders

paper_content:

A two-stage parameter identification method is developed for Hammerstein systems containing backlash nonlinearities bordered by parametric arbitrary-shape lines. In the first stage, a persistently exciting input is designed so that the linear subsystem can be made decoupled from the nonlinear element. Therefore, linear subsystem identification is coped with using a least squares estimator enjoying consistency, due to input persistent excitation. Then, the backlash parameters are estimated using appropriate periodic exciting signals and consistent parameter estimators.
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paper_title: Mixed parametric-nonparametric identification of Hammerstein and Wiener systems - a survey1

paper_content:

Abstract The paper surveys the ideas of cooperation between parametric and nonparametric (kernel-based) algorithms of nonlinear block-oriented system identification. Various strategies are proposed, dependently on the system structure, number of data and the prior knowledge. The estimates are consistent and their rates of convergence are presented. The aim of the paper is to show some recent results in the field in a systematic, ordered way.
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paper_title: An optimal two stage identification algorithm for Hammerstein-Wiener nonlinear systems

paper_content:

An optimal two stage identification algorithm is presented for Hammerstein-Wiener systems where two static nonlinear elements surround a linear block. The proposed algorithm consists of two steps: The first one is the recursive least squares and the second one is the singular value decomposition of two matrices whose dimensions are fixed and do not increase as the number of the data point increases. Moreover, the algorithm is shown to be convergent in the absence of noise and convergent with probability one in the presence of white noise.
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paper_title: An iterative method for the identification of nonlinear systems using a Hammerstein model

paper_content:

An iterative method is proposed for the identification of nonlinear systems from samples of inputs and outputs in the presence of noise. The model used for the identification consists of a no-memory gain (of an assumed polynomial form) followed by a linear discrete system. The parameters of the pulse transfer function of the linear system and the coefficients of the polynomial non-linearity are alternately adjusted to minimize a mean square error criterion. Digital computer simulations are included to demonstrate the feasibility of the technique.
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paper_title: A Hammerstein-based model for rate-dependent hysteresis in piezoelectric actuator

paper_content:

Most smart materials used in engineering applications have rate-dependent hysteresis nonlinearity. In this paper, a Hammerstein-based model is proposed to describe the dynamic characteristics of rate-dependent hysteresis in piezoelectric actuator. A Bouc-Wen model is used to approximate the static nonlinear characteristic while a linear dynamic model is constructed to capture the rate-dependent property of the hysteresis. Firstly, Bouc-Wen model parameters are optimized with particle swarm optimization (PSO) algorithm to model the static hysteresis nonlinearity. Based on this constructed static hysteresis nonlinear model, a recursive least squares (RLS) algorithm is utilized to identify the dynamic linear model parameters of Hammerstein model according to the input-output data with rich frequency information. Finally, the experimental results of applying the proposed method to the modeling of rate-dependent hysteresis in a piezoelectric actuator are presented with a 100Hz sinusoidal scanning signal. The model generation capability is verified in the given frequency range from 1Hz to100Hz when the excitation voltage are 40V, 80V, 120V, respectively.
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paper_title: Computational complexity analysis of set membership identification of Hammerstein and Wiener systems

paper_content:

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.
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paper_title: Convergence of the iterative Hammerstein system identification algorithm

paper_content:

The convergence of the iterative identification algorithm for the Hammerstein system has been an open problem for a long time. In this paper, a detailed study is carried out and various convergence properties of the iterative algorithm are derived. It is shown that the iterative algorithm with normalization is convergent in general. Moreover, it is shown that convergence takes place in one step (two least squares iterations) for finite-impulse response Hammerstein models with i.i.d. inputs.
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paper_title: Nuclear norm regularization for overparametrized Hammerstein systems

paper_content:

In this paper we study the overparametrization scheme for Hammerstein systems [1] in the presence of regularization. The quality of the convex approximation is analysed, that is obtained by relaxing the implicit rank one constraint. To obtain an improved convex relaxation we propose the use of nuclear norms [2], instead of using ridge regression. On several simple examples we illustrate that this yields a solution close to the best possible convex approximation. Furthermore the experiments suggest that ridge regression in combination with a projection step yield a generalization performance close to the one obtained by nuclear norms.
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paper_title: A noniterative method for identification using Hammerstein model

paper_content:

A noniterative method developed by Hsia for the particular case where the transfer function in the Hammerstein model has no zeros is extended to the general case where the transfer function may have zeros. Numerical examples show that the computation time by this method is considerably less than by the iteratire procedure proposed by Narendra and Gallman while the accuracy of the estimates is comparable.
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paper_title: Linear approximation and identification of MIMO Wiener-Hammerstein systems

paper_content:

This paper considers the linear approximation and identification of multi-input multi-output (MIMO) Wiener-Hammerstein systems, or LNL systems. Evaluating the input-output cross-covariance matrix of the MIMO LNL system for Gaussian inputs, we show that the best linear approximation of the MIMO LNL system in the mean square sense can be obtained by the orthogonal projection (ORT) subspace identification method. For each allocation of the poles of the best linear approximation between the two linear subsystems, the unknown parameters in the numerators of the linear subsystems and the coefficients of a basis function expansion of the nonlinearity are estimated by applying the separable least-squares. The best LNL system is the one that gives the minimum mean square output error. A numerical example is included to show the feasibility of the present approach.
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paper_title: A convex relaxation of a dimension reduction problem using the nuclear norm

paper_content:

The estimation of nonlinear models can be a challenging problem, in particular when the number of available data points is small or when the dimension of the regressor space is high. To meet these challenges, several dimension reduction methods have been proposed in the literature, where a majority of the methods are based on the framework of inverse regression. This allows for the use of standard tools when analyzing the statistical properties of an approach and often enables computationally efficient implementations. The main limitation of the inverse regression approach to dimension reduction is the dependence on a design criterion which restricts the possible distributions of the regressors. This limitation can be avoided by using a forward approach, which will be the topic of this paper. One drawback with the forward approach to dimension reduction is the need to solve nonconvex optimization problems. In this paper, a reformulation of a well established dimension reduction method is presented, which reveals the structure of the optimization problem, and a convex relaxation is derived.
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paper_title: Initial estimates for block structured nonlinear systems with feedback

paper_content:

This paper studies the identification of a block structured nonlinear Wiener-Hammerstein system that is captured in the feedforward or the feedback path of a feed- back loop. Nonparametric initial estimates are generated for the three dynamic blocks, modelled by their frequency response function, and the static nonlinear system. The method can be applied to input/output data resulting from random or periodic excitations.
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paper_title: Identification Of Nonlinear Physiological Systems

paper_content:

Preface. 1. Introduction. 1.1 Signals. 1.2 Systems and Models. 1.3 System Modeling. 1.4 System Identification. 1.5 How Common are Nonlinear Systems? 2. Background. 2.1 Vectors and Matrices. 2.2 Gaussian Random Variables. 2.3 Correlation Functions. 2.4 Mean-Square Parameter Estimation. 2.5 Polynomials. 2.6 Notes and References. 2.7 Problems. 2.8 Computer Exercises. 3. Models of Linear Systems. 3.1 Linear Systems. 3.2 Nonparametric Models. 3.3 Parametric Models. 3.4 State-Space Models. 3.5 Notes and References. 3.6 Theoretical Problems. 3.7 Computer Exercises. 4. Models of Nonlinear Systems. 4.1 The Volterra Series. 4.2 The Wiener Series. 4.3 Simple Block Structures. 4.4 Parallel Cascades. 4.5 The Wiener-Bose Model. 4.6 Notes and References. 4.7 Theoretical Problems. 4.8 Computer Exercises. 5. Identification of Linear Systems. 5.1 Introduction. 5.2 Nonparametric Time-Domain Models. 5.3 Frequency Response Estimation. 5.4 Parametric Methods. 5.5 Notes and References. 5.6 Computer Exercises. 6. Correlation-Based Methods. 6.1 Methods for Functional Expansions. 6.2 Block Structured Models. 6.3 Problems. 6.4 Computer Exercises. 7. Explicit Least-Squares Methods. 7.1 Introduction. 7.2 The Orthogonal Algorithms. 7.3 Expansion Bases. 7.4 Principal Dynamic Modes. 7.5 Problems. 7.6 Computer Exercises. 8. Iterative Least-Squares Methods. 8.1 Optimization Methods. 8.2 Parallel Cascade Methods. 8.3 Application: Visual Processing in the Light Adapted Fly Retina. 8.4 Problems 8.5 Computer Exercises. References. Index. IEEE Press Series in Biomedical Engineering.
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paper_title: Identification of Wiener-Hammerstein systems by a nonparametric separation of the best linear approximation

paper_content:

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.
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paper_title: A fractional approach to identify Wiener-Hammerstein systems

paper_content:

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.
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paper_title: A Time-Domain Fractional Approach for Wiener-Hammerstein Systems Identification

paper_content:

This paper describes a new approach to initialize Wiener-Hammerstein models for iterative prediction error minimization. The key idea is to parameterize the division of poles and zeros of the best linear approximation (BLA), between the two linear subsystems. Taylor expansion is used to handle the parameterization in the time-domain. Identifiability aspects are investigated on a low order example. Results regarding uniqueness of the estimation are proved. The initial estimate generated with this approach allows to avoid problems with local minima and the identification problem is solved performing only two iterative optimizations.
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paper_title: Initial estimates of the linear subsystems of Wiener-Hammerstein models

paper_content:

The iterative optimizations often used to identify Wiener-Hammerstein models, pairs of linear filters separated by memoryless nonlinearities, require good initial estimates of the linear elements in order to avoid them getting caught in local minima. Previous work has shown that initial estimates of the two linear elements can be formed by splitting the poles and zeros of the best linear approximation of the Wiener-Hammerstein system between the two linear elements, an approach which can generate a large number of initializations. This paper develops a scanning technique that can efficiently evaluate each of the proposed initializations using estimates of some carefully constructed nonlinear characteristics of the system, estimates which can be formed using linear system identification techniques after some data pre-processing. This approach results in a much smaller number, often only one, of potential starting points for the optimization. The proposed algorithm is demonstrated using a Monte Carlo simulation using data from the SYSID 2009 Wiener-Hammerstein Benchmark system.
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paper_title: Identification of a class of nonlinear systems using correlation analysis

paper_content:

An algorithm is presented for the identification of nonlinear systems which can be described by a model consisting of a linear system in cascade with a nonlinear element followed by another linear system. Cross-correlation techniques are employed to decouple the identification of the linear dynamics from the characterisation of the nonlinear element when the input is a white Gaussian signal. Parameterisation of both the linear and nonlinear component subsystems is discussed and the results of a simulation study are included to illustrate the validity of the algorithm.
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paper_title: First estimates of Wiener and Hammerstein systems using multisine excitation

paper_content:

This paper presents fast methods to obtain a first estimate of Wiener or Hammerstein systems. For this purpose, a special excitation signal is introduced: the random phase multisine. Its properties are briefly stated and used in both methods. The methods are tested with simulations and compared with similar methods available in the literature. The paper finishes with an application of the presented methods to real-world measurements.
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paper_title: A subspace-based identification of Wiener–Hammerstein benchmark model

paper_content:

Abstract This paper develops a subspace-based method of identifying the Wiener–Hammerstein system, where a nonlinearity is sandwiched by two linear subsystems. First, a state space model of the best linear approximation of it is identified by using a subspace identification method and the poles of the best linear model are allocated between two linear subsystems by a state transformation. Unknown system parameters and coefficients of a basis function expansion of the nonlinearity are estimated by using the separable least-squares for all possible allocations of poles, so that there is a possibility that many iterative minimization problems should be solved. Finally, the best Wiener–Hammerstein system that yields the minimum mean square error is selected. Numerical results for a benchmark model show the applicability of the proposed method.
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paper_title: Identification of a Block-Structured Nonlinear Feedback System, Applied to a Microwave Crystal Detector

paper_content:

This paper studies the identification of a block-structured nonlinear Wiener-Hammerstein system that is captured in the feedforward or the feedback path of a closed-loop system. First, nonparametric initial estimates are generated for the three dynamic blocks, which are modeled by their frequency response function, and the static nonlinear system, which is described using an input-output table. Next, a parametric model is identified. Random or periodic excitations can be used in the experiments. The method is illustrated on the identification of an Agilent HP420C crystal detector.
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paper_title: Modelling and Identification with Rational Orthogonal Basis Functions

paper_content:

Abstract Decomposing dynamical systems in terms of orthogonal expansions enables the modelling/approximation of a system with a finite length expansion. By flexibly tuning the basis functions to underlying system characteristics, the rate of convergence of these expansions can be drastically increased, leading to highly accurate models (small bias) being represented by few parameters (small variance). Additionally algorithmic and numerical aspects are favourable. A recently developed general theory for basis construction will be presented, that is a generalization of the classical Laguerre theory. The basis functions are applied in problems of identification, approximation, realization, uncertainty modelling, and adaptive filtering, particularly exploiting the property that basis function models are linearly parametrized. Besides powerful algorithms, they also provide useful analysis tools for understanding the underlying identification/approximation algorithms.
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paper_title: Iterative Update of the Pole Locations in a Wiener-Schetzen Model

paper_content:

Abstract A large variety of nonlinear systems can be described by a Wiener-Schetzen model. In this model, the linear dynamics are formulated in terms of orthonormal basis functions (OBFs), while the nonlinearity is modeled by a multivariate polynomial. The coefficients of the polynomial are the parameters of the model. The use of OBFs allows to incorporate prior knowledge about the system dynamics into the model, via user-specified pole locations. A mismatch between the poles used to construct the OBFs and the true poles of the underlying system can be handled by increasing the number of OBFs. Nevertheless, this results in a large number of parameters to be estimated, and eventually in a larger uncertainty of the estimated model. We propose an iterative method to update the pole locations of the model without making repeated experiments. An initial estimate of the pole locations is obtained from the best linear approximation (BLA) of the system. This paper analyzes the causes for the mismatch between the estimated and the true pole locations. Each of the error sources is tackled, resulting in the iterative method. The method is illustrated on a simulation example.
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paper_title: Fast Identification of Wiener-Hammerstein Systems using Discrete Optimization

paper_content:

A fast identification algorithm for Wiener-Hammerstein systems is proposed. The computational cost of separating the front and the back linear time-invariant (LTI) block dynamics is significantly improved by using discrete optimisation. The discrete optimisation is implemented as a genetic algorithm. Numerical results confirm the efficiency and accuracy of the proposed approach.
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paper_title: Fast identification of systems with nonlinear feedback

paper_content:

Abstract In this paper a method is proposed to identify LTI systems combined with a static nonlinearity in the feedback loop. By rearranging this structure and imposing one delay tab for the linear system, the identification is reduced to a linear problem which allows a fast estimation of the feedback parameters. The results are applied on a physical system.
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paper_title: Identification of nonlinear systems using the Wiener model

paper_content:

An algorithm for the identification of nonlinear systems which can be described by a Wiener model consisting of a linear system followed by a single-valued nonlinearity is presented. Crossconolation techniques are employed to decouple the identification of the linear dynamics from the characterisation of the nonlinear element.
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paper_title: Combined frequency-prediction error identification approach for Wiener systems with backlash and backlash-inverse operators

paper_content:

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.
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paper_title: Bayesian semiparametric Wiener system identification

paper_content:

We present a novel method for Wiener system identication. The method relies on a semiparametric, i.e. a mixed parametric/nonparametric, model of a Wiener system. We use a state-space model for the linear dynamical system and a nonparametric Gaussian process model for the static nonlinearity. We avoid making strong assumptions, such as monotonicity, on the nonlinear mapping. Stochastic disturbances, entering both as measurement noise and as process noise, are handled in a systematic manner. The nonparametric nature of the Gaussian process allows us to handle a wide range of nonlinearities without making problem-specic parameterizations. We also consider sparsity-promoting priors, based on generalized hyperbolic distributions, to automatically infer the order of the underlying dynamical system. We derive an inference algorithm based on an ecient particle Markov chain Monte Carlo method, referred to as particle Gibbs with ancestor sampling. The method is proled on two challenging identication problems with good results. Blind Wiener system identication is handled as a special case.
---
paper_title: Identification Of Nonlinear Physiological Systems

paper_content:

Preface. 1. Introduction. 1.1 Signals. 1.2 Systems and Models. 1.3 System Modeling. 1.4 System Identification. 1.5 How Common are Nonlinear Systems? 2. Background. 2.1 Vectors and Matrices. 2.2 Gaussian Random Variables. 2.3 Correlation Functions. 2.4 Mean-Square Parameter Estimation. 2.5 Polynomials. 2.6 Notes and References. 2.7 Problems. 2.8 Computer Exercises. 3. Models of Linear Systems. 3.1 Linear Systems. 3.2 Nonparametric Models. 3.3 Parametric Models. 3.4 State-Space Models. 3.5 Notes and References. 3.6 Theoretical Problems. 3.7 Computer Exercises. 4. Models of Nonlinear Systems. 4.1 The Volterra Series. 4.2 The Wiener Series. 4.3 Simple Block Structures. 4.4 Parallel Cascades. 4.5 The Wiener-Bose Model. 4.6 Notes and References. 4.7 Theoretical Problems. 4.8 Computer Exercises. 5. Identification of Linear Systems. 5.1 Introduction. 5.2 Nonparametric Time-Domain Models. 5.3 Frequency Response Estimation. 5.4 Parametric Methods. 5.5 Notes and References. 5.6 Computer Exercises. 6. Correlation-Based Methods. 6.1 Methods for Functional Expansions. 6.2 Block Structured Models. 6.3 Problems. 6.4 Computer Exercises. 7. Explicit Least-Squares Methods. 7.1 Introduction. 7.2 The Orthogonal Algorithms. 7.3 Expansion Bases. 7.4 Principal Dynamic Modes. 7.5 Problems. 7.6 Computer Exercises. 8. Iterative Least-Squares Methods. 8.1 Optimization Methods. 8.2 Parallel Cascade Methods. 8.3 Application: Visual Processing in the Light Adapted Fly Retina. 8.4 Problems 8.5 Computer Exercises. References. Index. IEEE Press Series in Biomedical Engineering.
---
paper_title: Combined Parametric-Nonparametric Identification of Block-Oriented Systems

paper_content:

Hammerstein system.- Wiener system.- Wiener-Hammerstein (sandwich) system.- Large-scale interconnected systems.- Structure detection and model order selection.- Time-varying systems.- Simulation studies.- Summary.
---
paper_title: Maximum Likelihood Identification of Wiener Models

paper_content:

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.
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paper_title: Instrumental variables approach to identification of a class of MIMO Wiener systems

paper_content:

A new approach to identification of multi-input multi-output (MIMO) Wiener systems using the instrumental variables method is presented. It is assumed that static nonlinear elements are invertible and their inverse characteristics can be expressed or approximated by polynomials of known orders. It is also assumed that the linear part of the Wiener system can be represented by a matrix polynomial form. Based on these assumptions, the Wiener system is transformed introducing a new parameterization and its parameters are estimated using a linear-in-parameters model. To solve the problem of non-consistency of least squares parameter estimates, an instrumental variables method is employed. A numerical example is included to show the effectiveness and the practical feasibility of the presented approach.
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paper_title: Identification of FIR Wiener systems with unknown, non-invertible, polynomial non-linearities

paper_content:

Wiener systems consist of a linear dynamic system whose output is measured through a static non-linearity. In this paper we study the identification of single-input single-output Wiener systems with finite impulse response dynamics and polynomial output non-linearities. Using multi-index notation, we solve a least squares problem to estimate products of the coefficients of the non-linearity and the impulse response of the linear system. We then consider four methods for extracting the coefficients of the non-linearity and impulse response: direct algebraic solution, singular value decomposition, multi-dimensional singular value decomposition and prediction error optimization.
---
paper_title: Mixed parametric-nonparametric identification of Hammerstein and Wiener systems - a survey1

paper_content:

Abstract The paper surveys the ideas of cooperation between parametric and nonparametric (kernel-based) algorithms of nonlinear block-oriented system identification. Various strategies are proposed, dependently on the system structure, number of data and the prior knowledge. The estimates are consistent and their rates of convergence are presented. The aim of the paper is to show some recent results in the field in a systematic, ordered way.
---
paper_title: Identification of wiener systems with process noise is a nonlinear errors-in-variables problem

paper_content:

This paper considers the identification of stochastic Wiener dynamic systems, that is linear dynamic systems with process noise, where the measurable output signal is a nonlinear function of the output from the linear system corrupted with additive measurement noise. It is shown how stochastic Wiener system identification can be viewed as a particular non-linear model errors-in-variables problem, for which there exists a large literature. We compare the maximum likelihood method with prediction error minimization methods based on the conditional mean predictor for Wiener systems. Related methods have previously been studied in the framework of identification of non-linear error-in-variables models. We extend these results by taking the input signal to the Wiener system into consideration. For example, the input will affect the variance of the prediction errors. Hence, a prediction error method with a variance weighting is derived to obtain more reliable parameter estimates. An advantage with the prediction error method is that for certain special cases we can avoid numerical integration. We also discuss how the unscented transform can be used to obtain an approximate predictor for the prediction error method. The numerical evaluation of these methods is performed on a simple first order FIR system with a cubic nonlinearity, for which some illustrative analytic properties are derived.
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paper_title: MINLIP for the Identification of Monotone Wiener Systems

paper_content:

This paper studies the MINLIP estimator for the identification of Wiener systems consisting of a sequence of a linear FIR dynamical model, and a monotonically increasing (or decreasing) static function. Given $T$ observations, this algorithm boils down to solving a convex quadratic program with $O(T)$ variables and inequality constraints, implementing an inference technique which is based entirely on model complexity control. The resulting estimates of the linear submodel are found to be almost consistent when no noise is present in the data, under a condition of smoothness of the true nonlinearity and local Persistency of Excitation (local PE) of the data. This result is novel as it does not rely on classical tools as a 'linearization' using a Taylor decomposition, nor exploits stochastic properties of the data. It is indicated how to extend the method to cope with noisy data, and empirical evidence contrasts performance of the estimator against other recently proposed techniques.
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paper_title: First estimates of Wiener and Hammerstein systems using multisine excitation

paper_content:

This paper presents fast methods to obtain a first estimate of Wiener or Hammerstein systems. For this purpose, a special excitation signal is introduced: the random phase multisine. Its properties are briefly stated and used in both methods. The methods are tested with simulations and compared with similar methods available in the literature. The paper finishes with an application of the presented methods to real-world measurements.
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paper_title: Identification of Stochastic Wiener Systems using Indirect Inference

paper_content:

We study identification of stochastic Wiener dynamic systems using so-called indirect inference. The main idea is to first fit an auxiliary model to the observed data and then in a second step, often by simulation, fit a more structured model to the estimated auxiliary model. This two-step procedure can be used when the direct maximum-likelihood estimate is difficult or intractable to compute. One such example is the identification of stochastic Wiener systems, i.e.,~linear dynamic systems with process noise where the output is measured using a non-linear sensor with additive measurement noise. It is in principle possible to evaluate the log-likelihood cost function using numerical integration, but the corresponding optimization problem can be quite intricate. This motivates studying consistent, but sub-optimal, identification methods for stochastic Wiener systems. We will consider indirect inference using the best linear approximation as an auxiliary model. We show that the key to obtain a reliable estimate is to use uncertainty weighting when fitting the stochastic Wiener model to the auxiliary model estimate. The main technical contribution of this paper is the corresponding asymptotic variance analysis. A numerical evaluation is presented based on a first-order finite impulse response system with a cubic non-linearity, for which certain illustrative analytic properties are derived.
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paper_title: Recursive prediction error identification using the nonlinear wiener model

paper_content:

The nonlinear Wiener model, consisting of a linear dynamic block in cascade with a static nonlinearity, is considered. A recursive prediction error identification algorithm, based on the Wiener model, is derived. The linear dynamic block is modelled as a SISO transfer function operator, and the static nonlinearity is approximated with a piecewise linear function. A theoretical analysis of the method is carried out, and conditions for local convergence to the true parameter vector are given. In particular, the analysis shows that the input signal should be such that there is signal energy in the whole range of the piecewise linear approximation. A numerical example illustrates the performance of the algorithm further. Practical guidelines on how to apply the algorithm are also included in the paper.
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paper_title: Computational complexity analysis of set membership identification of Hammerstein and Wiener systems

paper_content:

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.
---
paper_title: Blind Maximum-Likelihood Identification of Wiener Systems

paper_content:

This paper is about the identification of discrete-time Wiener systems from output measurements only (blind identification). Assuming that the unobserved input is white Gaussian noise, that the static nonlinearity is invertible, and that the output is observed without errors, a Gaussian maximum-likelihood estimator is constructed. Its asymptotic properties are analyzed and the Cramer-Rao lower bound is calculated. A two-step procedure for generating high-quality initial estimates is presented as well. The paper includes the illustration of the method on a simulation example.
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paper_title: Nonlinear Problems in Random Theory

paper_content:

A series of lectures on the role of nonlinear processes in physics, mathematics, electrical engineering, physiology, and communication theory.From the preface:"For some time I have been interested in a group of phenomena depending upon random processes. One the one hand, I have recorded the random shot effect as a suitable input for testing nonlinear circuits. On the other hand, for some of the work that Professor W. A. Rosenblith and I have been doing concerning the nature of the electroencephalogram, and in particular of the alpha rhythm, it has occurred to me to use the model of a system of random nonlinear oscillators excited by a random input...At the beginning we had contemplated a series of only four or five lectures. My ideas developed pari passu with the course, and by the end of the term we found ourselves with a set of fifteen lectures. The last few of these were devoted to the application of my ideas to problems in the statistical mechanics of gases. This work is both new and tentative, and I found that I had to supplement my course by the writing over of these with the help of Professer Y. W. Lee. "
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paper_title: Frequency Identification of Hammerstein-Wiener Systems with Piecewise Affine Input Nonlinearity

paper_content:

Abstract The problem of identifying Hammerstein-Wiener systems is addressed in the presence of linear subsystem of structure totally unknown and piecewise affine (or hard) input nonlinearity. A frequency identification approach is developed that determines the system frequency response (at a number of frequencies). A three-stage frequency identification method is developed to get estimates of the linear subsystem phase and modulus as well as estimates of the input and output nonlinearities. Finally, all suggested estimators are shown to be consistent.
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paper_title: Refined instrumental variable method for Hammerstein-Wiener continuous-time model identification

paper_content:

This study presents the first attempt of direct continuous-time model identification using instrumental variable method for Hammerstein–Wiener systems from sampled data. Under the assumption of monotonic function for the Wiener part, the whole non-linear model is first estimated as an augmented multiple-input single-output linear model, from which the model parameters are then extracted by singular value decomposition. A refined instrumental variable method is proposed to consistently identify this non-linear system acting in a coloured noisy environment. Monte Carlo simulation analysis is presented to illustrate the effectiveness of the proposed method.
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paper_title: Behaviour modelling of wideband RF transmitters using Hammerstein–Wiener models

paper_content:

The authors concern the application and use of a class of block-oriented systems, known as Hammerstein-Wiener, to model the dynamic non-linear behaviour of radio frequency (RF) transmitters. This class of system models is found to better mimic strong dynamic non-linear systems, as compared to simpler individual Hammerstein and Wiener models. The identification, which may look complicated at first glance, is skillfully tackled in an iterative two-stage identification approach by applying a class of global pattern search techniques along with the Narendra-Gallman method. The number of parameters processed at each iteration is therefore reduced, which considerably facilitates the progress and convergence of the identification algorithm. The model is identified and validated for two classes of power amplifiers. The validation results are obtained for various orders and are compared to the results of memory polynomial models. The Hammerstein-Wiener model achieves comparable results with a much lower level of complexity, in terms of the number of parameters associated with the model.
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paper_title: A blind approach to the Hammerstein-Wiener model identification

paper_content:

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.
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paper_title: Estimation of the parameters of a bilinear model with applications to submarine detection and system identification

paper_content:

In this work we study the problem of estimating the parameters of a bilinear model describing, e.g., the amplitude modulation of extremely low frequency electromagnetic (ELFE) signatures of submarines. A similar problem arises in estimation of a nonlinear dynamic system using a Hammerstein-Wiener model, where two nonlinear static blocks surround a linear dynamic block. For these purposes a new method is derived. It is also shown in the same context that a two-stage method for parameter estimation of Hammerstein-Wiener models can be interpreted as an approximate least squares method. We also show the similarities with the problem of weighted low-rank approximation and the fact that these problems can be solved exactly in finite time using solvers for global optimization of systems of polynomials based on self dual optimization.
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paper_title: An optimal two stage identification algorithm for Hammerstein-Wiener nonlinear systems

paper_content:

An optimal two stage identification algorithm is presented for Hammerstein-Wiener systems where two static nonlinear elements surround a linear block. The proposed algorithm consists of two steps: The first one is the recursive least squares and the second one is the singular value decomposition of two matrices whose dimensions are fixed and do not increase as the number of the data point increases. Moreover, the algorithm is shown to be convergent in the absence of noise and convergent with probability one in the presence of white noise.
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paper_title: Identification and prediction of ionospheric dynamics using a Hammerstein-Wiener model with radial basis functions

paper_content:

To construct a model for ionospheric dynamics, a two step identification technique based on subspace algorithms is used. In the first step a Hammerstein model is identified using subspace algorithms and a basis function expansion for the input nonlinearities. In the second step the Wiener nonlinearity is identified as a standard least squares procedure. The inputs to the model are measurements made by the ACE satellite, which is located at the first Lagrangian point between the sun and the earth, while the outputs of the model are ground-based magnetometer readings. To avoid overfitting, the inputs are ranked in order of their effectiveness using an error search algorithm. Results for the ground-based magnetometer located at Thule in Greenland are presented.
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paper_title: Extended stochastic gradient identification algorithms for Hammerstein-Wiener ARMAX systems

paper_content:

An extended stochastic gradient algorithm is developed to estimate the parameters of Hammerstein-Wiener ARMAX models. The basic idea is to replace the unmeasurable noise terms in the information vector of the pseudo-linear regression identification model with the corresponding noise estimates which are computed by the obtained parameter estimates. The obtained parameter estimates of the identification model include the product terms of the parameters of the original systems. Two methods of separating the parameter estimates of the original parameters from the product terms are discussed: the average method and the singular value decomposition method. To improve the identification accuracy, an extended stochastic gradient algorithm with a forgetting factor is presented. The simulation results indicate that the parameter estimation errors become small by introducing the forgetting factor.
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paper_title: IDENTIFICATION OF HAMMERSTEIN-WIENER SYSTEMS WITH BACKLASH INPUT NONLINEARITY BORDERED BY STRAIGHT LINES

paper_content:

Abstract Standard Hammerstein-Wiener models consist of a linear subsystem sandwiched by two memoryless nonlinearities. Presently, the linear subsystem may be parametric or not, continuous- or discrete-time. The input nonlinearity is allowed to be a memory operator of backlash type bordered by straight lines. The output nonlinearity may be noninvertible and is only supposed to be welld approximated, within any subinterval belonging to the working interval, with a polynomial of unknown order and parameters. An optimal strategy is presented to identify the system nonlinearities and an identification approach is developed that provides estimates of the linear subsystem. The method involves easily generated excitation signals. Finally, all the suggested estimators are shown consistent.
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paper_title: AN ITERATIVE METHOD FOR HAMMERSTEIN-WIENER SYSTEMS PARAMETER IDENTIFICATION

paper_content:

The paper deals with parameter identification of nonlinear dynamic systems using Hammerstein-Wiener models. Multiple application of a decomposition technique provides special expressions for the model description that are linear in parameters. This allows iterative estimation of all model parameters based on the measured input/output data and estimates of internal variables. The proposed algorithm is illustrated by identification of systems with polynomial characteristics. K e y w o r d s: nonlinear systems, identification, Hammerstein-Wiener model
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paper_title: Wiener-Hammerstein systems modeling using diagonal Volterra kernels coefficients

paper_content:

In this letter, we first present explicit relations between block-oriented nonlinear representations and Volterra models. For an identification purpose, we show that the estimation of the diagonal coefficients of the Volterra kernels associated with the considered block-oriented nonlinear structures is sufficient to recover the overall model. An alternating least squares-type algorithm is provided to carry out this model identification.
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paper_title: Identification of Wiener-Hammerstein models using linear interpolation in the frequency domain (LIFRED)

paper_content:

A new method to identify the linear subsystems of a Wiener-Hammerstein model through the measurement of the second-order Volterra kernel is proposed. This technique makes use of the symmetry properties of the Volterra kernel and assumes that the frequency response gain and phase between estimated points can be reasonably well approximated by a straight line. The signal applied for the identification is a multisine with properties of no interharmonic distortion. Several advantages of the proposed method over existing ones are discussed, and two simulation examples are presented to illustrate the applicability of the technique. The method is also shown to be robust to noise and distortion in the input signal.
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paper_title: Non-Parametric Estimation of the Frequency Response Functions of the Linear Blocks of a Wienerhammerstein Model.

paper_content:

Abstract This paper proposes a method to estimate the non-parametric frequency response functions of the linear parts of a Wiener-Hammerstein model using special designed periodic excitation signals. No assumptions are made concerning the static non-linearity. The estimator is developed within a stochastic errors-in-variables framework. The applicability of the method is illustrated using simulations.
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paper_title: Wiener–Hammerstein system identification – an evolutionary approach

paper_content:

The problem of identifying parametric Wiener–Hammerstein WH systems is addressed within the evolutionary optimisation context. Specifically, a hybrid culture identification method is developed that involves model structure adaptation using genetic recombination and model parameter learning using particle swarm optimisation. The method enjoys three interesting features: 1 the risk of premature convergence of model parameter estimates to local optima is significantly reduced, due to the constantly maintained diversity of model candidates; 2 no prior knowledge is needed except for upper bounds on the system structure indices; 3 the method is fully autonomous as no interaction is needed with the user during the optimum search process. The performances of the proposed method will be illustrated and compared to alternative methods using a well-established WH benchmark.
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paper_title: Linear approximation and identification of MIMO Wiener-Hammerstein systems

paper_content:

This paper considers the linear approximation and identification of multi-input multi-output (MIMO) Wiener-Hammerstein systems, or LNL systems. Evaluating the input-output cross-covariance matrix of the MIMO LNL system for Gaussian inputs, we show that the best linear approximation of the MIMO LNL system in the mean square sense can be obtained by the orthogonal projection (ORT) subspace identification method. For each allocation of the poles of the best linear approximation between the two linear subsystems, the unknown parameters in the numerators of the linear subsystems and the coefficients of a basis function expansion of the nonlinearity are estimated by applying the separable least-squares. The best LNL system is the one that gives the minimum mean square output error. A numerical example is included to show the feasibility of the present approach.
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paper_title: Identification of a Wiener–Hammerstein system using an incremental nonlinear optimisation technique

paper_content:

A method for black-box identification of a Wiener–Hammerstein system is described and applied to a set of Benchmark data originally presented at the 15th IFAC Symposium on System Identification. An incremental nonlinear optimisation procedure is used, which is able to avoid local minima, thus enabling the solution to converge to the global minimum. The use of a dual-polynomial to describe the static nonlinearity allows the number of parameters needed to be significantly reduced compared with the case if a single polynomial is utilised; this also improves robustness against extrapolation errors. The overall approach requires a relatively small number of parameters.
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paper_title: Computing an initial estimate of a Wiener-Hammerstein system with a random phase multisine excitation

paper_content:

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.
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paper_title: Identification Of Nonlinear Physiological Systems

paper_content:

Preface. 1. Introduction. 1.1 Signals. 1.2 Systems and Models. 1.3 System Modeling. 1.4 System Identification. 1.5 How Common are Nonlinear Systems? 2. Background. 2.1 Vectors and Matrices. 2.2 Gaussian Random Variables. 2.3 Correlation Functions. 2.4 Mean-Square Parameter Estimation. 2.5 Polynomials. 2.6 Notes and References. 2.7 Problems. 2.8 Computer Exercises. 3. Models of Linear Systems. 3.1 Linear Systems. 3.2 Nonparametric Models. 3.3 Parametric Models. 3.4 State-Space Models. 3.5 Notes and References. 3.6 Theoretical Problems. 3.7 Computer Exercises. 4. Models of Nonlinear Systems. 4.1 The Volterra Series. 4.2 The Wiener Series. 4.3 Simple Block Structures. 4.4 Parallel Cascades. 4.5 The Wiener-Bose Model. 4.6 Notes and References. 4.7 Theoretical Problems. 4.8 Computer Exercises. 5. Identification of Linear Systems. 5.1 Introduction. 5.2 Nonparametric Time-Domain Models. 5.3 Frequency Response Estimation. 5.4 Parametric Methods. 5.5 Notes and References. 5.6 Computer Exercises. 6. Correlation-Based Methods. 6.1 Methods for Functional Expansions. 6.2 Block Structured Models. 6.3 Problems. 6.4 Computer Exercises. 7. Explicit Least-Squares Methods. 7.1 Introduction. 7.2 The Orthogonal Algorithms. 7.3 Expansion Bases. 7.4 Principal Dynamic Modes. 7.5 Problems. 7.6 Computer Exercises. 8. Iterative Least-Squares Methods. 8.1 Optimization Methods. 8.2 Parallel Cascade Methods. 8.3 Application: Visual Processing in the Light Adapted Fly Retina. 8.4 Problems 8.5 Computer Exercises. References. Index. IEEE Press Series in Biomedical Engineering.
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paper_title: Recursive identification of errors-in-variables Wiener–Hammerstein systems

paper_content:

Abstract 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.
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paper_title: Identification of Wiener-Hammerstein systems by a nonparametric separation of the best linear approximation

paper_content:

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.
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paper_title: A fractional approach to identify Wiener-Hammerstein systems

paper_content:

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.
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paper_title: A Time-Domain Fractional Approach for Wiener-Hammerstein Systems Identification

paper_content:

This paper describes a new approach to initialize Wiener-Hammerstein models for iterative prediction error minimization. The key idea is to parameterize the division of poles and zeros of the best linear approximation (BLA), between the two linear subsystems. Taylor expansion is used to handle the parameterization in the time-domain. Identifiability aspects are investigated on a low order example. Results regarding uniqueness of the estimation are proved. The initial estimate generated with this approach allows to avoid problems with local minima and the identification problem is solved performing only two iterative optimizations.
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paper_title: Initial estimates for Wiener-Hammerstein models using phase-coupled multisines

paper_content:

Block-oriented models are often used to model nonlinear systems. These models consist of linear dynamic (L) and nonlinear static (N) sub-blocks. This paper addresses the generation of initial estimates for a Wiener-Hammerstein model (LNL cascade). While it is easy to measure the product of the two linear blocks using a Gaussian excitation and linear identification methods, it is difficult to split the global dynamics over the individual blocks. This paper first proposes a well-designed multisine excitation with pairwise coupled random phases. Next, a modified best linear approximation is estimated on a shifted frequency grid. It is shown that this procedure creates a shift of the input dynamics with a known frequency offset, while the output dynamics do not shift. The resulting transfer function, which has complex coefficients due to the frequency shift, is estimated with a modified frequency domain estimation method. The identified poles and zeros can be assigned to either the input or output dynamics. Once this is done, it is shown in the literature that the remaining initialization problem can be solved much easier than the original one. The method is illustrated on experimental data obtained from the Wiener-Hammerstein benchmark system.
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paper_title: Generating initial estimates for Wiener-Hammerstein systems using phase coupled multisines

paper_content:

Abstract Block oriented nonlinear models capture the dynamics of a nonlinear system with linear dynamic sub-systems (L), the nonlinear behavior is modelled using static nonlinear sub-blocks (N). In this paper we study the generation of initial estimates for the linear dynamic blocks of a Wiener-Hammerstein system that has a cascaded LNL structure. While it is very easy to identify the product of the transfer functions of the first and last dynamic block using linear system identification methods, it turns out to be very difficult to split the global dynamics over these individual blocks. In this paper a method is proposed that allows the poles of the best linear approximation to be assigned to the first or second linear block. Once this split is made, it is shown in the literature that the remaining initialization problem can be solved much easier than the original one. The first step of the method is the design of a special random phase multisine excitation, using pair-wise coupled random phases. Next, a modfied best linear approximation will be estimated on a shifted frequency grid. It will be shown that this procedure shifts the poles and zeros of the first linear sub-block with a known frequency offset, while those of the second sub-block are not changed. The shifted poles and zeros result in a transfer function with complex coefficients that can be identified using a modified frequency domain estimation method. This results in a simple initialization method, based on a linear system identification step.
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paper_title: Recursive identification method for MISO Wiener-Hammerstein model

paper_content:

A simple technique for recursive identification of the Wiener-Hammerstein model with extension to the multi-input single-output (MISO) case is presented. We use a new transformation of the input-output difference equation where parameters to be estimated are those of each subsystem of the initial and unique realization. After that, a weighted extended least squares method is employed to estimate recursively and separately parameters of the linear subsystems and the static nonlinear element. The convergence analysis of the proposed procedure is also studied. Finally, a numerical example is provided to show the efficiency of the algorithm. >
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paper_title: Initial estimates of the linear subsystems of Wiener-Hammerstein models

paper_content:

The iterative optimizations often used to identify Wiener-Hammerstein models, pairs of linear filters separated by memoryless nonlinearities, require good initial estimates of the linear elements in order to avoid them getting caught in local minima. Previous work has shown that initial estimates of the two linear elements can be formed by splitting the poles and zeros of the best linear approximation of the Wiener-Hammerstein system between the two linear elements, an approach which can generate a large number of initializations. This paper develops a scanning technique that can efficiently evaluate each of the proposed initializations using estimates of some carefully constructed nonlinear characteristics of the system, estimates which can be formed using linear system identification techniques after some data pre-processing. This approach results in a much smaller number, often only one, of potential starting points for the optimization. The proposed algorithm is demonstrated using a Monte Carlo simulation using data from the SYSID 2009 Wiener-Hammerstein Benchmark system.
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paper_title: Identification of a class of nonlinear systems using correlation analysis

paper_content:

An algorithm is presented for the identification of nonlinear systems which can be described by a model consisting of a linear system in cascade with a nonlinear element followed by another linear system. Cross-correlation techniques are employed to decouple the identification of the linear dynamics from the characterisation of the nonlinear element when the input is a white Gaussian signal. Parameterisation of both the linear and nonlinear component subsystems is discussed and the results of a simulation study are included to illustrate the validity of the algorithm.
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paper_title: A subspace-based identification of Wiener–Hammerstein benchmark model

paper_content:

Abstract This paper develops a subspace-based method of identifying the Wiener–Hammerstein system, where a nonlinearity is sandwiched by two linear subsystems. First, a state space model of the best linear approximation of it is identified by using a subspace identification method and the poles of the best linear model are allocated between two linear subsystems by a state transformation. Unknown system parameters and coefficients of a basis function expansion of the nonlinearity are estimated by using the separable least-squares for all possible allocations of poles, so that there is a possibility that many iterative minimization problems should be solved. Finally, the best Wiener–Hammerstein system that yields the minimum mean square error is selected. Numerical results for a benchmark model show the applicability of the proposed method.
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paper_title: AN ITERATIVE METHOD FOR WIENER-HAMMERSTEIN SYSTEMS PARAMETER IDENTIFICATION

paper_content:

The paper deals with parameter identification of nonlinear dynamic systems using the Wiener-Hammerstein model. Application of an operator decomposition technique provides a special output equation that is linear in the parameters of all the model blocks. As two unmeasurable internal variables are included into the model description, an iterative parameter estimation algorithm is proposed. K e y w o r d s: parameter identification, Wiener-Hammerstein model
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paper_title: Combining the best linear approximation and dimension reduction to identify the linear blocks of parallel Wiener systems

paper_content:

Abstract A Wiener model is a fairly simple, well known, and often used nonlinear block-oriented black-box model. A possible generalization of the class of Wiener models lies in the parallel Wiener model class. This paper presents a method to estimate the linear time-invariant blocks of such parallel Wiener models from input/output data only. The proposed estimation method combines the knowledge obtained by estimating the best linear approximation of a nonlinear system with the MAVE dimension reduction method to estimate the linear time-invariant blocks present in the model. The estimation of the static nonlinearity boils down to a standard static nonlinearity estimation problem starting from input-output data once the linear blocks are known.
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paper_title: Parametric Identification of Parallel Wiener Systems

paper_content:

This paper proposes a parametric identification method for parallel Wiener systems. The linear dynamic parts of the Wiener system are modeled by a parametric rational function in the Laplace or z-domain. The static nonlinearity is represented by a linear combination of multiple-input single-output nonlinear basis functions. The identification method uses a three-step procedure to obtain initial estimates. In the first step, the frequency response function of the best linear approximation is estimated for different input excitation levels. In the second step, the power-dependent dynamics are decomposed over a number of parallel orthogonal branches. In the last step, the static nonlinearities are estimated using a linear least squares estimation. Furthermore, a nonlinear optimization method is implemented to refine the estimates. The method is illustrated on a simulation and a validation measurement example.
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paper_title: Efficient Power Amplifier Identification Using Modified Parallel Cascade Hammerstein Models

paper_content:

The automatic generation of behavioral amplifier models based on the measurement of the input and output signals imply demanding requirements for the identification algorithm. In addition to the computational efficiency it is very important that such an algorithm needs only a low number of predefined parameters for the fitting process. To fulfil these requirements we suggest the use of a modified parallel cascade Hammerstein model (MPCHM). To verify the performance of the presented algorithm the behavior of a 3-stage RF power amplifier (PA) was identified. The resulting model was validated using different measurements with similar statistical properties. Furthermore the static nonlinear contributions were deembedded from both the measured and the modeled output signals to visualize the capability of the MPCHM in predicting the amplifier memory effects
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paper_title: Identification and Control Using Volterra Models

paper_content:

1. Introduction.- 2. Qualitative Behavior.- 3. Restrietions & Extensions.- 4. Determination of Volterra Model Parameters.- 5. Practical Considerations in Volterra Model Identification.- 6. Model-Based Controller Synthesis.- 7. Advanced Direct Synthesis Controller Design.- 8. Model Predictive Control Using Volterra Series.- 9. Application Case Studies.- 10. Summary.
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paper_title: Inverse Regression for the Wiener Class of Systems

paper_content:

The concept of inverse regression has turned out to be quite useful for dimension reduction in regression analysis problems. Using methods like sliced inverse regression (SIR) and directional regression (DR), some high-dimensional nonlinear regression problems can be turned into more tractable low-dimensional problems. Here, the usefulness of inverse regression for identification of nonlinear dynamical systems will be discussed. In particular, it will be shown that the inverse regression methods can be used for identification of systems of the Wiener class, that is, systems consisting of a number of parallel linear subsystems followed by a static multiple-input single-output nonlinearity. For a particular class of input signals, including Gaussian signals, the inverse regression approach makes it possible to estimate the linear subsystems without knowing or estimating the nonlinearity.
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paper_title: Generalized eigenvector algorithm for nonlinear system identification with non-white inputs

paper_content:

Traditional methods for nonlinear system identification require a white, Gaussian, test input, a restriction that has limited their usability in many fields. In this study, we address the problem of identifying the dynamics of a nonlinear system when the input is highly colored-a restriction commonly encountered in the study of physiological systems. An extension of the parallel cascade method is developed that is optimal in a constrained minimum mean squared error sense and exactly corrects for the distortion induced by the non-white input spectrum. However, this correction is a deconvolution, which may become extremely ill-conditioned if the input spectrum departs significantly from whiteness; to confront this, we develop a low-rank projection operation that stabilizes the deconvolution. The overall algorithm is robust and places few requirements on the nature of the test input. Practical application of this new method is demonstrated by using it to identify a known analog nonlinear system from experimental data.
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paper_title: Parallel cascade identification and kernel estimation for nonlinear systems

paper_content:

We consider the representation and identification of nonlinear systems through the use of parallel cascades of alternating dynamic linear and static nonlinear elements. Building on the work of Palm and others, we show that any discrete-time finite-memory nonlinear system having a finite-order Volterra series representation can be exactly represented by a finite number of parallel LN cascade paths. Each LN path consists of a dynamic linear system followed by a static nonlinearity (which can be a polynomial). In particular, we provide an upper bound for the number of parallel LN paths required to represent exactly a discrete-time finite-memory Volterra functional of a given order. Next, we show how to obtain a parallel cascade representation of a nonlinear system from a single input-output record. The input is not required to be Gaussian or white, nor to have special autocorrelation properties. Next, our parallel cascade identification is applied to measure accurately the kernels of nonlinear systems (even those with lengthy memory), and to discover the significant terms to include in a nonlinear difference equation model for a system. In addition, the kernel estimation is used as a means of studying individual signals to distinguish deterministic from random behaviour, in an alternative to the use of chaotic dynamics. Finally, an alternate kernel estimation scheme is presented.
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paper_title: Identification of Parallel-Cascade Wiener Systems Using Joint Diagonalization of Third-Order Volterra Kernel Slices

paper_content:

This letter is concerned with the parameter estimation of linear and nonlinear subsystems of parallel-cascade Wiener systems (PCWS). We first present the relationship between a PCWS and its associated Volterra model. We show that the coefficients of the linear subsystems can be obtained using a joint diagonalization of the third-order Volterra kernel slices. Then, the coefficients of the nonlinear subsystems are estimated using the least square algorithm. The proposed parameter estimation method is illustrated by means of simulation results.
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paper_title: Identification and Control Using Volterra Models

paper_content:

1. Introduction.- 2. Qualitative Behavior.- 3. Restrietions & Extensions.- 4. Determination of Volterra Model Parameters.- 5. Practical Considerations in Volterra Model Identification.- 6. Model-Based Controller Synthesis.- 7. Advanced Direct Synthesis Controller Design.- 8. Model Predictive Control Using Volterra Series.- 9. Application Case Studies.- 10. Summary.
---
paper_title: Inverse Regression for the Wiener Class of Systems

paper_content:

The concept of inverse regression has turned out to be quite useful for dimension reduction in regression analysis problems. Using methods like sliced inverse regression (SIR) and directional regression (DR), some high-dimensional nonlinear regression problems can be turned into more tractable low-dimensional problems. Here, the usefulness of inverse regression for identification of nonlinear dynamical systems will be discussed. In particular, it will be shown that the inverse regression methods can be used for identification of systems of the Wiener class, that is, systems consisting of a number of parallel linear subsystems followed by a static multiple-input single-output nonlinearity. For a particular class of input signals, including Gaussian signals, the inverse regression approach makes it possible to estimate the linear subsystems without knowing or estimating the nonlinearity.
---
paper_title: Generalized eigenvector algorithm for nonlinear system identification with non-white inputs

paper_content:

Traditional methods for nonlinear system identification require a white, Gaussian, test input, a restriction that has limited their usability in many fields. In this study, we address the problem of identifying the dynamics of a nonlinear system when the input is highly colored-a restriction commonly encountered in the study of physiological systems. An extension of the parallel cascade method is developed that is optimal in a constrained minimum mean squared error sense and exactly corrects for the distortion induced by the non-white input spectrum. However, this correction is a deconvolution, which may become extremely ill-conditioned if the input spectrum departs significantly from whiteness; to confront this, we develop a low-rank projection operation that stabilizes the deconvolution. The overall algorithm is robust and places few requirements on the nature of the test input. Practical application of this new method is demonstrated by using it to identify a known analog nonlinear system from experimental data.
---
paper_title: Parallel cascade identification and kernel estimation for nonlinear systems

paper_content:

We consider the representation and identification of nonlinear systems through the use of parallel cascades of alternating dynamic linear and static nonlinear elements. Building on the work of Palm and others, we show that any discrete-time finite-memory nonlinear system having a finite-order Volterra series representation can be exactly represented by a finite number of parallel LN cascade paths. Each LN path consists of a dynamic linear system followed by a static nonlinearity (which can be a polynomial). In particular, we provide an upper bound for the number of parallel LN paths required to represent exactly a discrete-time finite-memory Volterra functional of a given order. Next, we show how to obtain a parallel cascade representation of a nonlinear system from a single input-output record. The input is not required to be Gaussian or white, nor to have special autocorrelation properties. Next, our parallel cascade identification is applied to measure accurately the kernels of nonlinear systems (even those with lengthy memory), and to discover the significant terms to include in a nonlinear difference equation model for a system. In addition, the kernel estimation is used as a means of studying individual signals to distinguish deterministic from random behaviour, in an alternative to the use of chaotic dynamics. Finally, an alternate kernel estimation scheme is presented.
---
paper_title: Combining the best linear approximation and dimension reduction to identify the linear blocks of parallel Wiener systems

paper_content:

Abstract A Wiener model is a fairly simple, well known, and often used nonlinear block-oriented black-box model. A possible generalization of the class of Wiener models lies in the parallel Wiener model class. This paper presents a method to estimate the linear time-invariant blocks of such parallel Wiener models from input/output data only. The proposed estimation method combines the knowledge obtained by estimating the best linear approximation of a nonlinear system with the MAVE dimension reduction method to estimate the linear time-invariant blocks present in the model. The estimation of the static nonlinearity boils down to a standard static nonlinearity estimation problem starting from input-output data once the linear blocks are known.
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paper_title: Support vector method for identification of Wiener models

paper_content:

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.
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paper_title: ITERATIVE OPTIMIZATION METHOD OF GOB-VOLTERRA FILTERS

paper_content:

Abstract An iterative method is proposed to optimize the Volterra kernels expansions on Generalized Orthonormal Bases (GOB). Each kernel is expanded on an independent GOB. The expansion coefficients, also called Fourier coefficients, are estimated in using an orthogonal formulation of the Least Squares (LS) algorithm. The proposed method allows optimization of both the Fourier coefficients and the GOBs poles. It can be seen as a good compromise between the exhaustive method for GOB poles optimization, costly by nature, and the analytical solution to Laguerre poles optimization that generally furnishes worse performances for system approximation than expansions on GOBs.
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paper_title: Identification of block-oriented nonlinear systems using orthonormal bases

paper_content:

In this paper, new noniterative algorithms for the identification of (multivariable) block-oriented nonlinear models consisting of the interconnection of linear time invariant systems and static nonlinearities are presented. The proposed algorithms are numerically robust, since they are based only on least squares estimation and singular value decomposition. Two different block-oriented nonlinear models are considered in this paper, viz., the Hammerstein model, and the Wiener model. For the Hammerstein model, the proposed algorithm provides consistent estimates even in the presence of colored output noise, under weak assumptions on the persistency of excitation of the inputs. For the Wiener model, consistency of the estimates can only be guaranteed in the noise free case. Key in the derivation of the results is the use of basis functions for the representation of the linear and nonlinear parts of the models. The performance of the proposed identification algorithms is illustrated through simulation examples of two benchmark problems drawn from the process control literature, viz., a binary distillation column and a pH neutralization process. � 2003 Elsevier Ltd. All rights reserved.
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paper_title: Fading Memory and the Problem of Approximating Nonlinear Operators with Volterra Series

paper_content:

Using the notion of fading memory we prove very strong versions of two folk theorems. The first is that any time-invariant (TI) continuous nonlinear operator can be approximated by a Volterra series operator, and the second is that the approximating operator can be realized as a finite-dimensional linear dynamical system with a nonlinear readout map. While previous approximation results are valid over finite time intervals and for signals in compact sets, the approximations presented here hold for all time and for signals in useful (noncompact) sets. The discretetime analog of the second theorem asserts that any TI operator with fading memory can be approximated (in our strong sense) by a nonlinear moving- average operator. Some further discussion of the notion of fading memory is given.
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paper_title: A Combined SVM/OB-based Wiener Model Identification Method

paper_content:

Abstract A novel method, combining Support Vector Machines and Least Squares Prediction Error techniques, for the identification of the linear and nonlinear blocks in a Wiener model is presented in this paper. The identification is carried out by minimizing an augmented cost function defined as the sum of the standard structural risk function appearing in Support Vector Regression and the quadratic criterion on the prediction errors associated to Least Squares estimation methods. The properties of the proposed method are illustrated through simulation examples.
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paper_title: Modelling and Identification with Rational Orthogonal Basis Functions

paper_content:

Abstract Decomposing dynamical systems in terms of orthogonal expansions enables the modelling/approximation of a system with a finite length expansion. By flexibly tuning the basis functions to underlying system characteristics, the rate of convergence of these expansions can be drastically increased, leading to highly accurate models (small bias) being represented by few parameters (small variance). Additionally algorithmic and numerical aspects are favourable. A recently developed general theory for basis construction will be presented, that is a generalization of the classical Laguerre theory. The basis functions are applied in problems of identification, approximation, realization, uncertainty modelling, and adaptive filtering, particularly exploiting the property that basis function models are linearly parametrized. Besides powerful algorithms, they also provide useful analysis tools for understanding the underlying identification/approximation algorithms.
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paper_title: Nonlinear Problems in Random Theory

paper_content:

A series of lectures on the role of nonlinear processes in physics, mathematics, electrical engineering, physiology, and communication theory.From the preface:"For some time I have been interested in a group of phenomena depending upon random processes. One the one hand, I have recorded the random shot effect as a suitable input for testing nonlinear circuits. On the other hand, for some of the work that Professor W. A. Rosenblith and I have been doing concerning the nature of the electroencephalogram, and in particular of the alpha rhythm, it has occurred to me to use the model of a system of random nonlinear oscillators excited by a random input...At the beginning we had contemplated a series of only four or five lectures. My ideas developed pari passu with the course, and by the end of the term we found ourselves with a set of fifteen lectures. The last few of these were devoted to the application of my ideas to problems in the statistical mechanics of gases. This work is both new and tentative, and I found that I had to supplement my course by the writing over of these with the help of Professer Y. W. Lee. "
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paper_title: Complete identification of a class of nonlinear systems from steady state frequency response

paper_content:

Measurement of the steady state response to a sinusoidal input has long been an important technique for the identification of linear systems. In this paper we consider this approach to identification for the class of nonlinear systems shown in Figure I.
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paper_title: Identification of parallel Wiener-Hammerstein systems with a decoupled static nonlinearity

paper_content:

Abstract 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.
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paper_title: Combining the best linear approximation and dimension reduction to identify the linear blocks of parallel Wiener systems

paper_content:

Abstract A Wiener model is a fairly simple, well known, and often used nonlinear block-oriented black-box model. A possible generalization of the class of Wiener models lies in the parallel Wiener model class. This paper presents a method to estimate the linear time-invariant blocks of such parallel Wiener models from input/output data only. The proposed estimation method combines the knowledge obtained by estimating the best linear approximation of a nonlinear system with the MAVE dimension reduction method to estimate the linear time-invariant blocks present in the model. The estimation of the static nonlinearity boils down to a standard static nonlinearity estimation problem starting from input-output data once the linear blocks are known.
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paper_title: On representation and approximation of nonlinear systems

paper_content:

Many methods for the analysis of nonlinear systems rely on a Volterra system-representation in terms of integral kernels. This paper considers two questions: 1) whether Volterra-like representations are possible for all smooth systems (i.e. analytic operators),-2) which classes of systems can be approximated by some interesting subclasses of the smooth systems, e.g. Volterra systems with separable kernels and "sandwich systems" (cascades of linear systems and nomemory nonlinearities). Whereas the answer to question 1 is positive, the answer to question 2 depends on the type of approximation, i.e. topology, that is used.
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paper_title: Identification of systems containing linear dynamic and static nonlinear elements

paper_content:

Identification of nonlinear systems which can be represented by combinations of linear dynamic and static nonlinear elements are considered. Previous results by the authors based on correlation analysis are combined to provide a unified treatment for this class of systems. It is shown that systems composed of cascade, feedforward, feedback and multiplicative connections of linear dynamic and zero memory nonlinear elements can be identified in terms of the individual component subsystems from measurements of the system input and output only.
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paper_title: Identification of parallel Wiener-Hammerstein systems with a decoupled static nonlinearity

paper_content:

Abstract 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.
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paper_title: Combining the best linear approximation and dimension reduction to identify the linear blocks of parallel Wiener systems

paper_content:

Abstract A Wiener model is a fairly simple, well known, and often used nonlinear block-oriented black-box model. A possible generalization of the class of Wiener models lies in the parallel Wiener model class. This paper presents a method to estimate the linear time-invariant blocks of such parallel Wiener models from input/output data only. The proposed estimation method combines the knowledge obtained by estimating the best linear approximation of a nonlinear system with the MAVE dimension reduction method to estimate the linear time-invariant blocks present in the model. The estimation of the static nonlinearity boils down to a standard static nonlinearity estimation problem starting from input-output data once the linear blocks are known.
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paper_title: From coupled to decoupled polynomial representations in parallel Wiener-Hammerstein models

paper_content:

A large variety of nonlinear systems can be approximated by parallel Wiener-Hammerstein models. These models consist of a multiple input multiple output (MIMO) nonlinear static block sandwiched between two linear dynamic blocks. One method is available for the identification of a general parallel Wiener-Hammerstein model. It represents the nonlinear block as a multivariate polynomial, which typically contains cross-terms. These make it harder to interpret and to invert the model.We want to eliminate the cross-terms, and thus come to a decoupled polynomial representation. In this paper, the simultaneous decoupling of quadratic and cubic polynomials is formulated as a standard tensor decomposition. A simulation example shows that the simultaneous decoupling can result in a model with less parallel branches than a decoupling of all polynomials separately.
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paper_title: Decoupling static nonlinearities in a parallel Wiener-Hammerstein system: A first-order approach

paper_content:

We present a method to decompose a static MIMO (multiple-input-multiple-output) nonlinearity into a set of SISO (single-input-single-output) polynomials acting on internal variables that are related to the inputs and outputs of the MIMO nonlinearity by linear transformations. The method is inspired on the small-signal analysis of nonlinear circuits and proceeds by collecting first-order information of the MIMO function into a set of Jacobian matrices. A simultaneous diagonalization of the set of Jacobian matrices is computed using a tensor decomposition, providing the required linear transformations, after which also the coefficients of the internal SISO polynomials can be computed. The method is validated on measurements of a parallel two-branch Wiener-Hammerstein identification setup.
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paper_title: Recursive Discrete-Time Models for Continuous-Time Systems Under Band-Limited Assumptions

paper_content:

Discrete-time models are very convenient to simulate a nonlinear system on a computer. In order to build the discrete-time simulation models for the nonlinear feedback systems (which is a very important class of systems in many applications) described as y(t) = g1(u(t), y(t)), one has to solve at each time step a nonlinear algebraic loop for y(t). If a delay is present in the loop, i.e., y(t) = g2(u(t), y(t -1)), fast recursive simulation models can be developed and the need to solve the nonlinear differential-algebraic equations is removed. In this paper, we use the latter to model the nonlinear feedback system using recursive discrete-time models. Theoretical error bounds for such kind of approximated models are provided in the case of band-limited signals, and furthermore, a measurement methodology is proposed for quantifying and validating the output error bounds experimentally.
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paper_title: Initial estimates for block structured nonlinear systems with feedback

paper_content:

This paper studies the identification of a block structured nonlinear Wiener-Hammerstein system that is captured in the feedforward or the feedback path of a feed- back loop. Nonparametric initial estimates are generated for the three dynamic blocks, modelled by their frequency response function, and the static nonlinear system. The method can be applied to input/output data resulting from random or periodic excitations.
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paper_title: Fast identification of systems with nonlinear feedback

paper_content:

Abstract In this paper a method is proposed to identify LTI systems combined with a static nonlinearity in the feedback loop. By rearranging this structure and imposing one delay tab for the linear system, the identification is reduced to a linear problem which allows a fast estimation of the feedback parameters. The results are applied on a physical system.
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paper_title: Convex relaxation approach to the identification of the Wiener-Hammerstein model

paper_content:

In this paper, an input/output system identification technique for the Wiener-Hammerstein model and its feedback extension is proposed. In the proposed framework, the identification of the nonlinearity is non-parametric. The identification problem can be formulated as a non-convex quadratic program (QP). A convex semidefinite programming (SDP) relaxation is then formulated and solved to obtain a sub-optimal solution to the original non-convex QP. The convex relaxation turns out to be tight in most cases. Combined with the use of local search, high quality solutions to the Wiener-Hammerstein identification can frequently be found. As an application example, randomly generated Wiener-Hammerstein models are identified.
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paper_title: Initial estimates for block structured nonlinear systems with feedback

paper_content:

This paper studies the identification of a block structured nonlinear Wiener-Hammerstein system that is captured in the feedforward or the feedback path of a feed- back loop. Nonparametric initial estimates are generated for the three dynamic blocks, modelled by their frequency response function, and the static nonlinear system. The method can be applied to input/output data resulting from random or periodic excitations.
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paper_title: Nonlinear LFR Block-Oriented Model: Potential Benefits and Improved, User-Friendly Identification Method

paper_content:

Nowadays, there is a high need for accurate, parsimonious nonlinear dynamic models. Block-oriented nonlinear model structures are known to be excellent candidates for this task. The nonlinear linear fractional representation model, composed of a static nonlinearity (SNL) and a multiple-input-multiple-output (MIMO) linear time-invariant (LTI) part, is highly flexible since it creates an arbitrary MIMO-LTI interconnection between the model's input and output and the SNL's input and output. First of all, it can cope with the nonlinear feedback (which is very important in oscillators and mechanical applications). Secondly, it incorporates certain classical block-oriented models as special cases. Finally, it does not postulate the SNL's location prior to the identification. Starting from two classical frequency response measurements of the system, the method generates the best possible MIMO-LTI configuration and estimates the SNL in an automated, user-friendly, and efficient (noniterative) way. The method will be illustrated on simulation examples and experimental data.
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paper_title: Measurement and identification of nonlinear systems consisting of linear dynamic blocks and one static nonlinearity

paper_content:

This paper considers the measurement and the identification of nonlinear time-invariant single-input/single-output (SISO) systems, consisting of a multivariable linear dynamic system and one static nonlinear SISO system. This includes Wiener-Hammerstein systems in a linear feedback loop. The nonparametric identification of the frequency response functions of the linear parts are obtained without measuring the signals over the static nonlinearity. Measurements on an electronic circuit demonstrate the usability of this identification scheme.
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paper_title: Identification of systems with localised nonlinearity: From state-space to block-structured models

paper_content:

This paper presents a method that generates initial estimates for a rather general block-structured model, starting from the (more general) polynomial nonlinear state-space model. The considered block-structure, sometimes referred to as Linear Fractional Transformation (LFT) or Linear Fractional Representation (LFR), encompasses several simpler structures. It can e.g. describe Wiener, Hammerstein, Wiener-Hammerstein and nonlinear feedback structures. In fact, the chosen block-structure is the most general representation of a system with one Single-Input Single-Output (SISO) static nonlinearity. As is quite common in block-structure identification, the states and internal signals are assumed to be unknown. The method gradually imposes the structure of the LFR system, and at the same time finds an estimate of the Multiple-Input Multiple-Output (MIMO) linear dynamic part and the static nonlinearity (SNL). The method is illustrated via an experimental-data example.
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paper_title: Nonlinear LFR Block-Oriented Model: Potential Benefits and Improved, User-Friendly Identification Method

paper_content:

Nowadays, there is a high need for accurate, parsimonious nonlinear dynamic models. Block-oriented nonlinear model structures are known to be excellent candidates for this task. The nonlinear linear fractional representation model, composed of a static nonlinearity (SNL) and a multiple-input-multiple-output (MIMO) linear time-invariant (LTI) part, is highly flexible since it creates an arbitrary MIMO-LTI interconnection between the model's input and output and the SNL's input and output. First of all, it can cope with the nonlinear feedback (which is very important in oscillators and mechanical applications). Secondly, it incorporates certain classical block-oriented models as special cases. Finally, it does not postulate the SNL's location prior to the identification. Starting from two classical frequency response measurements of the system, the method generates the best possible MIMO-LTI configuration and estimates the SNL in an automated, user-friendly, and efficient (noniterative) way. The method will be illustrated on simulation examples and experimental data.
---
paper_title: Nonlinear LFR Block-Oriented Model: Potential Benefits and Improved, User-Friendly Identification Method

paper_content:

Nowadays, there is a high need for accurate, parsimonious nonlinear dynamic models. Block-oriented nonlinear model structures are known to be excellent candidates for this task. The nonlinear linear fractional representation model, composed of a static nonlinearity (SNL) and a multiple-input-multiple-output (MIMO) linear time-invariant (LTI) part, is highly flexible since it creates an arbitrary MIMO-LTI interconnection between the model's input and output and the SNL's input and output. First of all, it can cope with the nonlinear feedback (which is very important in oscillators and mechanical applications). Secondly, it incorporates certain classical block-oriented models as special cases. Finally, it does not postulate the SNL's location prior to the identification. Starting from two classical frequency response measurements of the system, the method generates the best possible MIMO-LTI configuration and estimates the SNL in an automated, user-friendly, and efficient (noniterative) way. The method will be illustrated on simulation examples and experimental data.
---
paper_title: Fading Memory and the Problem of Approximating Nonlinear Operators with Volterra Series

paper_content:

Using the notion of fading memory we prove very strong versions of two folk theorems. The first is that any time-invariant (TI) continuous nonlinear operator can be approximated by a Volterra series operator, and the second is that the approximating operator can be realized as a finite-dimensional linear dynamical system with a nonlinear readout map. While previous approximation results are valid over finite time intervals and for signals in compact sets, the approximations presented here hold for all time and for signals in useful (noncompact) sets. The discretetime analog of the second theorem asserts that any TI operator with fading memory can be approximated (in our strong sense) by a nonlinear moving- average operator. Some further discussion of the notion of fading memory is given.
---
paper_title: Parallel cascade identification and kernel estimation for nonlinear systems

paper_content:

We consider the representation and identification of nonlinear systems through the use of parallel cascades of alternating dynamic linear and static nonlinear elements. Building on the work of Palm and others, we show that any discrete-time finite-memory nonlinear system having a finite-order Volterra series representation can be exactly represented by a finite number of parallel LN cascade paths. Each LN path consists of a dynamic linear system followed by a static nonlinearity (which can be a polynomial). In particular, we provide an upper bound for the number of parallel LN paths required to represent exactly a discrete-time finite-memory Volterra functional of a given order. Next, we show how to obtain a parallel cascade representation of a nonlinear system from a single input-output record. The input is not required to be Gaussian or white, nor to have special autocorrelation properties. Next, our parallel cascade identification is applied to measure accurately the kernels of nonlinear systems (even those with lengthy memory), and to discover the significant terms to include in a nonlinear difference equation model for a system. In addition, the kernel estimation is used as a means of studying individual signals to distinguish deterministic from random behaviour, in an alternative to the use of chaotic dynamics. Finally, an alternate kernel estimation scheme is presented.
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paper_title: Identification of Linear Systems with Nonlinear Distortions

paper_content:

Abstract In this paper the impact of nonlinear distortions on the linear system identification framework is studied. In the first part the nonlinear system is replaced by a linear model plus a nonlinear noise source. The properties of this representation are studied. Next a method to detect, qualify and quantify the nonlinear distortions is presented. In the second part, the (non)-parametric identification of the best linear approximation is studied. In the last part, the linear modelling approach is extended towards nonlinear modelling. A fast approximate nonlinear modelling framework is set up that is a natural extension of the linear framework, and bridges the gap between the linear and the nonlinear identification approaches.
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paper_title: Structure Discrimination in Block-Oriented Models Using Linear Approximations: A Theoretic Framework

paper_content:

In this paper we show that it is possible to retrieve structural information about complex block-oriented nonlinear systems, starting from linear approximations of the nonlinear system around different setpoints. The key idea is to monitor the movements of the poles and zeros of the linearized models and to reduce the number of candidate models on the basis of these observations. Besides the well known open loop single branch Wiener-, Hammerstein-, and Wiener-Hammerstein systems, we also cover a number of more general structures like parallel (multi branch) Wiener-Hammerstein models, and closed loop block oriented models, including linear fractional representation (LFR) models.
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paper_title: Structure identification of nonlinear dynamic systems - A survey on input/output approaches

paper_content:

Abstract In the past several methods have been elaborated for the identification of nonlinear dynamic systems. Most of the methods assume that the structure of the system is given a priori . Therefore they are in reality parameter estimation algorithms and structure identification is thus usually performed by repeated parameter estimation. However in nonlinear system theory several methods are known to determine the structure of a system. In this paper structure identification of block-oriented (especially cascade) models, of semi-linear dynamic models with signal-dependent parameters and of nonlinear dynamic models being linear in the parameters will be considered. Different structure selection methods are summarized based on step and impulse tests, frequency response measurements, correlation analysis, repeated reproducible tests and normal operating data.
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paper_title: System Identification: A Frequency Domain Approach

paper_content:

System identification is a general term used to describe mathematical tools and algorithms that build dynamical models from measured data. Used for prediction, control, physical interpretation, and the designing of any electrical systems, they are vital in the fields of electrical, mechanical, civil, and chemical engineering. Focusing mainly on frequency domain techniques, System Identification: A Frequency Domain Approach, Second Edition also studies in detail the similarities and differences with the classical time domain approach. It high??lights many of the important steps in the identification process, points out the possible pitfalls to the reader, and illustrates the powerful tools that are available.
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paper_title: Selecting nonlinear model structures for computer control

paper_content:

Many authors have noted the difficulty of developing the models required for nonlinear model predictive control (NMPC) and other nonlinear, model-based control strategies. One reason this task is difficult is that success depends strongly on initially selecting a reasonable structure for this nonlinear model. Unfortunately, this selection is extremely difficult because most of our intuition about structure/behavior relations (e.g., if the step response exhibits overshoot, a model of at least second order is required) is based on experience with relatively low-order linear models and often fails completely when confronted with comparably simple nonlinear models. To help bridge this chasm between nonlinear model behavior and our linear intuition, this paper describes some broad classes of nonlinear model structures, which may be approximately characterized as mildly nonlinear, strongly nonlinear, or of intermediate nonlinearity, depending on the different ways they violate linear intuition. It is hoped that these results will be useful in selecting simple nonlinear model structures for use in model-based control.
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paper_title: A Nonlinear Block Structure Identification Procedure Using Frequency Response Function Measurements

paper_content:

Based on simple frequency response function (FRF) measurements, we give the user some guidance in the selection of an appropriate nonlinear block structure for the system to be modeled. The method consists in measuring the FRF using a Gaussian-like input signal and varying in a first experiment the root-mean-square (rms) value of this signal while maintaining the coloring of the power spectrum. Next, in a second experiment, the coloring of the power spectrum is varied while keeping the rms value constant. Based on the resulting behavior of the FRF, an appropriate nonlinear block structure can be selected to approximate the real system. The identification of the selected block-oriented model itself is not addressed in this paper. A theoretical analysis and two practical applications of this structure identification method are presented for some nonlinear block structures.
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paper_title: Structure Discrimination in Block-Oriented Models Using Linear Approximations: A Theoretic Framework

paper_content:

In this paper we show that it is possible to retrieve structural information about complex block-oriented nonlinear systems, starting from linear approximations of the nonlinear system around different setpoints. The key idea is to monitor the movements of the poles and zeros of the linearized models and to reduce the number of candidate models on the basis of these observations. Besides the well known open loop single branch Wiener-, Hammerstein-, and Wiener-Hammerstein systems, we also cover a number of more general structures like parallel (multi branch) Wiener-Hammerstein models, and closed loop block oriented models, including linear fractional representation (LFR) models.
---
paper_title: System Identification: A Frequency Domain Approach

paper_content:

System identification is a general term used to describe mathematical tools and algorithms that build dynamical models from measured data. Used for prediction, control, physical interpretation, and the designing of any electrical systems, they are vital in the fields of electrical, mechanical, civil, and chemical engineering. Focusing mainly on frequency domain techniques, System Identification: A Frequency Domain Approach, Second Edition also studies in detail the similarities and differences with the classical time domain approach. It high??lights many of the important steps in the identification process, points out the possible pitfalls to the reader, and illustrates the powerful tools that are available.
---
paper_title: Using the Best Linear Approximation With Varying Excitation Signals for Nonlinear System Characterization

paper_content:

Block oriented model structure detection is quite desirable since it helps to imagine the system with real physical elements. In this work we explore experimental methods to detect the internal structure of the system, using a black box approach. Two different strategies are compared and the best combination of these is introduced. The methods are applied on two real systems with a static nonlinear block in the feedback path. The main goal is to excite the system in a way that reduces the total distortion in the measured frequency response functions to have more precise measurements and more reliable decision about the structure of the system.
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paper_title: Nonlinear system-identification of the filling phase of a wet-clutch system

paper_content:

The work presented illustrates how the choice of input perturbation signal and experimental design improves the derived model of a nonlinear system, in particular the dynamics of a wet-clutch system. The relationship between the applied input current signal and resulting output pressure in the filling phase of the clutch is established based on bandlimited periodic signals applied at different current operating points and signals approximating the desired filling current signal. A polynomial nonlinear state space model is estimated and validated over a range of measurements and yields better fits over a linear model, while the performance of either model depends on the perturbation signal used for model estimation.
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paper_title: Three free data sets for development and benchmarking in nonlinear system identification

paper_content:

System identification is a fundamentally experimental field of science in that it deals with modeling of system dynamics using measured data. Despite this fact many algorithms and theoretical results are only tested with simulations at the time of publication. One reason for this may be a lack of easily available live data. This paper therefore presents three sets of data, suitable for development, testing and benchmarking of system identification algorithms for nonlinear systems. The data sets are collected from laboratory processes that can be described by block - oriented dynamic models, and by more general nonlinear difference and differential equation models. All data sets are available for free download.
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paper_title: Initial estimates for block structured nonlinear systems with feedback

paper_content:

This paper studies the identification of a block structured nonlinear Wiener-Hammerstein system that is captured in the feedforward or the feedback path of a feed- back loop. Nonparametric initial estimates are generated for the three dynamic blocks, modelled by their frequency response function, and the static nonlinear system. The method can be applied to input/output data resulting from random or periodic excitations.
---
paper_title: Parametric Identification of Parallel Wiener Systems

paper_content:

This paper proposes a parametric identification method for parallel Wiener systems. The linear dynamic parts of the Wiener system are modeled by a parametric rational function in the Laplace or z-domain. The static nonlinearity is represented by a linear combination of multiple-input single-output nonlinear basis functions. The identification method uses a three-step procedure to obtain initial estimates. In the first step, the frequency response function of the best linear approximation is estimated for different input excitation levels. In the second step, the power-dependent dynamics are decomposed over a number of parallel orthogonal branches. In the last step, the static nonlinearities are estimated using a linear least squares estimation. Furthermore, a nonlinear optimization method is implemented to refine the estimates. The method is illustrated on a simulation and a validation measurement example.
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Title: Identification of Nonlinear Block-Oriented Systems starting from Linear Approximations: A Survey

Section 1: Introduction
Description 1: Provide an overview of nonlinear models, particularly block-oriented models, and the rationale for using linear approximations as a starting point for their identification.

Section 2: Block-Oriented Models: Structures, Representation and Noise Framework
Description 2: Discuss the different block-oriented structures, their representation, issues related to model parametrization, and the noise framework applied in this paper.

Section 3: Bussgang's Theorem
Description 3: Explain Bussgang's Theorem and its relevance in analyzing block-oriented structures using linear approximations.

Section 4: Linear Approximations of Nonlinear Systems
Description 4: Introduce the theoretical framework of the Best Linear Approximation (BLA) and the epsilon-approximation for estimating linear approximations of nonlinear systems.

Section 5: Maximum Likelihood
Description 5: Discuss the use of a least squares cost function for parameter estimation and the role of the maximum likelihood estimation in achieving efficient estimates.

Section 6: Single Branch Models
Description 6: Detail the identification algorithms for single branch models, including Hammerstein, Wiener, Hammerstein-Wiener, and Wiener-Hammerstein structures.

Section 7: Parallel Branch Model Structures
Description 7: Describe identification methods for parallel branch structures, including parallel Hammerstein, parallel Wiener, and parallel Wiener-Hammerstein systems.

Section 8: Feedback Model Structures
Description 8: Present the identification techniques for feedback model structures such as simple feedback structures, Wiener-Hammerstein feedback, and Linear Fractional Representation (LFR) models.

Section 9: Pros and Cons of the Model Structures
Description 9: Compare the various model structures in terms of their descriptive power, number of parameters, and the general difficulty of their identification.

Section 10: Guidelines for the User
Description 10: Offer practical guidelines for model structure selection and input design to assist users in applying the identification techniques discussed.

Section 11: Benchmark and Practical Results
Description 11: Provide an overview of benchmark setups and real-life systems where the discussed identification methods have been successfully applied.

Section 12: Conclusion
Description 12: Summarize the key points of the survey, emphasizing the usefulness of the Best Linear Approximation framework in identifying a wide range of block-oriented nonlinear systems.