A bibliography on nonlinear system identification
TL;DR: The present bibliography represents a comprehensive list of references on nonlinear system identification and its applications in signal processing, communications, and biomedical engineering.
About: This article is published in Signal Processing.The article was published on 2001-03-01. It has received 242 citations till now. The article focuses on the topics: Listing (computer).
Citations
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TL;DR: This paper investigates a new model reduction criterion that makes computationally demanding sparsification procedures unnecessary and incorporates the coherence criterion into a new kernel-based affine projection algorithm for time series prediction.
Abstract: Kernel-based algorithms have been a topic of considerable interest in the machine learning community over the last ten years. Their attractiveness resides in their elegant treatment of nonlinear problems. They have been successfully applied to pattern recognition, regression and density estimation. A common characteristic of kernel-based methods is that they deal with kernel expansions whose number of terms equals the number of input data, making them unsuitable for online applications. Recently, several solutions have been proposed to circumvent this computational burden in time series prediction problems. Nevertheless, most of them require excessively elaborate and costly operations. In this paper, we investigate a new model reduction criterion that makes computationally demanding sparsification procedures unnecessary. The increase in the number of variables is controlled by the coherence parameter, a fundamental quantity that characterizes the behavior of dictionaries in sparse approximation problems. We incorporate the coherence criterion into a new kernel-based affine projection algorithm for time series prediction. We also derive the kernel-based normalized LMS algorithm as a particular case. Finally, experiments are conducted to compare our approach to existing methods.
405 citations
Cites background from "A bibliography on nonlinear system ..."
...extensive bibliography [1] devoted to the theory of nonlinear systems....
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04 Oct 2012
TL;DR: This paper investigates a new model reduction criterion that makes computationally demanding sparsification procedures unnecessary and incorporates the coherence criterion into a new kernel-based affine projection algorithm for time series prediction.
Abstract: During the last few years, kernel methods have been very useful to solve nonlinear identification problems. The main drawback of these methods resides in the fact that the number of elements of the kernel development, i.e., the size of the dictionary, increases with the number of input data, making the solution not suitable for online problems especially time series applications. Recently, Richard, Bermudez and Honeine investigated a method where the size of the dictionary is controlled by a coherence criterion. In this paper, we extend this method by adjusting the dictionary elements in order to reduce the residual error and/or the average size of the dictionary. The proposed method is implemented for time series prediction using the kernel-based affine projection algorithm.
265 citations
TL;DR: This paper presents a meta-analysis of statistical errors in Nonlinear Estimates of Linear and Nonlinear Systems and their applications in Input/Output Relationships and Bilinear and Trilinear Systems.
Abstract: Linear Systems, Random Data, Spectra Zero-Memory Nonlinear Systems Bilinear and Trilinear Systems Nonlinear System Input/Output Relationships Square-Law and Cubic Nonlinear Systems Statistical Errors in Nonlinear Estimates Parallel Linear and Nonlinear Systems.
207 citations
TL;DR: It is shown that Volterra and Wiener series can be represented implicitly as elements of a reproducing kernel Hilbert space by using polynomial kernels.
Abstract: Volterra and Wiener series are perhaps the best-understood nonlinear system representations in signal processing. Although both approaches have enjoyed a certain popularity in the past, their application has been limited to rather low-dimensional and weakly nonlinear systems due to the exponential growth of the number of terms that have to be estimated. We show that Volterra and Wiener series can be represented implicitly as elements of a reproducing kernel Hilbert space by using polynomial kernels. The estimation complexity of the implicit representation is linear in the input dimensionality and independent of the degree of nonlinearity. Experiments show performance advantages in terms of convergence, interpretability, and system sizes that can be handled.
134 citations
Cites background from "A bibliography on nonlinear system ..."
...…that can be represented by Volterra operators and their application in such diverse fields as nonlinear differential equations, neuroscience, fluid dynamics and electrical engineering (overviews and bibliography in Schetzen, 1980; Rugh, 1981; Mathews & Sicuranza, 2000; Giannakis & Serpedin, 2001)....
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01 Jan 2005
TL;DR: 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.
Abstract: 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.
122 citations
References
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28,888 citations
Book•
01 Jan 1987
TL;DR: Das Buch behandelt die Systemidentifizierung in dem theoretischen Bereich, der direkte Auswirkungen auf Verstaendnis and praktische Anwendung der verschiedenen Verfahren zur IdentifIZierung hat.
Abstract: Das Buch behandelt die Systemidentifizierung in dem theoretischen Bereich, der direkte Auswirkungen auf Verstaendnis und praktische Anwendung der verschiedenen Verfahren zur Identifizierung hat. Da ...
20,436 citations
2,248 citations