Filter‐based regularisation for impulse response modelling

Marconato, Anna; Schoukens, Maarten; Schoukens, Johan

doi:10.1049/iet-cta.2016.0908

Cited by 45 publications

(34 citation statements)

References 15 publications

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“…The covariance matrices are built by using so-called regularization kernel functions. [36] presents a possibility to directly inject prior information at the cost function level, instead of creating covariance matrices and inverting them. The key idea is to include the regularization term in the cost function as a FIR filtering operation on the parameters to be estimated (see (17)).…”

Section: Avoid the Inversion In The Penalization Matricesmentioning

confidence: 99%

Efficient multidimensional regularization for Volterra series estimation

Birpoutsoukis

Csurcsia

Schoukens

2018

Mechanical Systems and Signal Processing

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This paper presents an efficient nonparametric time domain nonlinear system identification method. It is shown how truncated Volterra series models can be efficiently estimated without the need of long, transient-free measurements. The method is a novel extension of the regularization methods that have been developed for impulse response estimates of linear time invariant systems. To avoid the excessive memory needs in case of long measurements or large number of estimated parameters, a practical gradient-based estimation method is also provided, leading to the same numerical results as the proposed Volterra estimation method. Moreover, the transient effects in the simulated output are removed by a special regularization method based on the novel ideas of transient removal for Linear Time-Varying (LTV) systems. Combining the proposed methodologies, the nonparametric Volterra models of the cascaded water tanks benchmark are presented in this paper. The results for different scenarios varying from a simple Finite Impulse Response (FIR) model to a 3rd degree Volterra series with and without transient removal are compared and studied. It is clear that the obtained models capture the system dynamics when tested on a validation dataset, and their performance is comparable with the white-box (physical) models.

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Section: Avoid the Inversion In The Penalization Matricesmentioning

confidence: 99%

Efficient multidimensional regularization for Volterra series estimation

Birpoutsoukis

Csurcsia

Schoukens

2018

Mechanical Systems and Signal Processing

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“…The kernel design plays a similar role as the model structure design for ML/PEM and determines the underlying model structure for KRM. In the past few years, many efforts have been spent on this issue and several kernels have been invented to embed various types of prior knowledge, e.g., Carli et al (2017); Chen et al (2014Chen et al ( , 2012; Dinuzzo (2015); Marconato et al (2016); Pillonetto et al (2016Pillonetto et al ( , 2011; Pillonetto & De Nicolao (2010); Zorzi & Chiuso (2017). In particular, two systematic kernel design methods (one is from a machine learning perspective and the other one is from a system theory perspective) were developed in by embedding the corresponding type of prior knowledge.…”

Section: Introductionmentioning

confidence: 99%

On asymptotic properties of hyperparameter estimators for kernel-based regularization methods

Chen

Ljung

2018

Automatica

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The kernel-based regularization method has two core issues: kernel design and hyperparameter estimation. In this paper, we focus on the second issue and study the properties of several hyperparameter estimators including the empirical Bayes (EB) estimator, two Stein's unbiased risk estimators (SURE) and their corresponding Oracle counterparts, with an emphasis on the asymptotic properties of these hyperparameter estimators. To this goal, we first derive and then rewrite the first order optimality conditions of these hyperparameter estimators, leading to several insights on these hyperparameter estimators. Then we show that as the number of data goes to infinity, the two SUREs converge to the best hyperparameter minimizing the corresponding mean square error, respectively, while the more widely used EB estimator converges to another best hyperparameter minimizing the expectation of the EB estimation criterion. This indicates that the two SUREs are asymptotically optimal but the EB estimator is not. Surprisingly, the convergence rate of two SUREs is slower than that of the EB estimator, and moreover, unlike the two SUREs, the EB estimator is independent of the convergence rate of Φ T Φ/N to its limit, where Φ is the regression matrix and N is the number of data. A Monte Carlo simulation is provided to demonstrate the theoretical results.

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“…An alternative method relies on Volterra series analytical method and wavelet balance method under multilevel excitations [3]. In that area, this article introduces a regularized LS identification algorithm inspired on the recently developed regularized impulse response estimation techniques [21,14].…”

Section: Introductionmentioning

confidence: 99%

“…This type of prior knowledge can be imposed using a regularized linear least-squares approach. This approach has been successfully developed and applied in the linear time-invariant case [21,14], and has already been extended towards the identification of Hammerstein systems [26].…”

Section: Introducing Prior Knowledge Through Regularizationmentioning

confidence: 99%

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Comparison of least squares and exponential sine sweep methods for Parallel Hammerstein Models estimation

Rébillat

Schoukens

2018

Mechanical Systems and Signal Processing

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is an open access repository that collects the work of Arts et Métiers ParisTech researchers and makes it freely available over the web where possible. Linearity is a common assumption for many real-life systems, but in many cases the non-linear behavior of systems cannot be ignored and must be modeled and estimated. Among the various existing classes of nonlinear models, Parallel Hammerstein Models (PHM) are interesting as they are at the same time easy to interpret as well as to estimate. One way to estimate PHM relies on the fact that the estimation problem is linear in the parameters and thus that classical least squares (LS) estimation algorithms can be used. In that area, this article introduces a regularized LS estimation algorithm inspired on some of the recently developed regularized impulse response estimation techniques. Another mean to estimate PHM consists in using parametric or non-parametric exponential sine sweeps (ESS) based methods. These methods (LS and ESS) are founded on radically different mathematical backgrounds but are expected to tackle the same issue. A methodology is proposed here to compare them with respect to (i) their accuracy, (ii) their computational cost, and (iii) their robustness to noise. Tests are performed on simulated systems for several values of methods respective parameters and of signal to noise ratio. Results show that, for a given set of data points, the ESS method is less demanding in computational resources than the LS method but that it is also less accurate. Furthermore, the LS method needs parameters to be set in advance whereas the ESS method is not subject to conditioning issues and can be fully non-parametric. In summary, for a given set of data points, ESS method can provide a first, automatic, and quick overview of a nonlinear system than can guide more computa-tionally demanding and precise methods, such as the regularized LS one proposed here.

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Filter‐based regularisation for impulse response modelling

Cited by 45 publications

References 15 publications

Efficient multidimensional regularization for Volterra series estimation

Efficient multidimensional regularization for Volterra series estimation

On asymptotic properties of hyperparameter estimators for kernel-based regularization methods

Comparison of least squares and exponential sine sweep methods for Parallel Hammerstein Models estimation

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