Regularized Nonparametric Volterra Kernel Estimation

In this paper, the regularization approach introduced recently for nonparametric estimation of linear systems is extended to the estimation of nonlinear systems modelled as Volterra series. The kernels of order higher than one, representing higher dimensional impulse responses in the series, are considered to be realizations of multidimensional Gaussian processes. Based on this, prior information about the structure of the Volterra kernel is introduced via an appropriate penalization term in the least squares cost function. It is shown that the proposed method is able to deliver accurate estimates of the Volterra kernels even in the case of a small amount of data points

Efficient Multidimensional Regularization for Volterra Series Estimation

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

Gaussian process regression for the estimation of stable univariate time-series processes

In this paper, estimation of AutoRegressive (AR) and AutoRegressive Moving Average (ARMA) models is proposed in a Bayesian framework using a Gaussian Process Regression (GPR) approach. Impulse response properties of the underlying process to be modeled are exploited during the parameter estimation. As such, models of enhanced predictability can be consistently obtained, even in the case of large model orders. It is also proved that the proposed approach is strongly linked with the Prediction Error (PE) model estimation approaches, if the estimated parameters are regularized. Simulations are provided to illustrate the efficiency of the proposed approach

The Effect of Prior Knowledge on the Least Costly Identification Experiment

International audienceDesign of optimal input excitations is one of the most challenging problems in the field of system identification. The main difficulty lies in the fact that the optimization problem cannot always be formulated to be convex, therefore a globally optimal excitation for the dynamic system of interest cannot be guaranteed. In this paper, optimal input design (OID) for linear systems in the presence of prior knowledge is studied. Information related to exponential decay and smoothness is incorporated in the optimal input design problem by making use of the Bayes rule of information. Three different cases of modeling the linear dynamics are considered, namely Finite Impulse Response (FIR) model with and without prior knowledge, as well as the rational transfer function case. It is shown that the prior information affects the spectrum of the minimum power optimal input. The input with the least power is always obtained for the transfer function model case

Nonparametric Volterra Series Estimate of the Cascaded Water Tanks Using Multidimensional Regularization

This paper presents an efficient nonparametric time domain nonlinear system identification method applied to the measurement benchmark data of the cascaded water tanks. In this work a method to estimate efficiently finite Volterra kernels without the need of long records is presented. This work is a novel extension of the regularization methods that have been developed for impulse response estimates of linear time invariant systems. Due to the limited number of available data samples, the highest considered Volterra order is limited. In the paper the results for different scenarios varying from a simple Finite Impulse Response (FIR) model to a 3rd degree Volterra series are compared and studied. In each case, the transients are removed by a special regularization method based on the novel ideas of transient removal for Linear Time-Varying (LTV) systems. Using the proposed methodologies, the nonparametric Volterra models provide a very good data-fit, and their performance is comparable with the white-box (physical) model

Gaussian process regression for the estimation of generalized frequency response functions

Bayesian learning techniques have recently garnered significant attention in the system identification community. Originally introduced for low variance estimation of linear impulse response models, the concept has since been extended to the nonlinear setting for Volterra series estimation in the time domain. In this paper, we approach the estimation of nonlinear systems from a frequency domain perspective, where the Volterra series has a representation comprised of Generalized Frequency Response Functions (GFRFs). Inspired by techniques developed for the linear frequency domain case, the GFRFs are modelled as real/complex Gaussian processes with prior covariances related to the time domain characteristics of the corresponding Volterra series. A Gaussian process regression method is developed for the case of periodic excitations, and numerical examples demonstrate the efficacy of the proposed method, as well as its advantage over time domain methods in the case of band-limited excitations

Gaussian process regression for the estimation of generalized frequency response functions

Bayesian learning techniques have recently garnered significant attention in the system identification community. Originally introduced for low variance estimation of linear impulse response models, the concept has since been extended to the nonlinear setting for Volterra series estimation in the time domain. In this paper, we approach the estimation of nonlinear systems from a frequency domain perspective, where the Volterra series has a representation comprised of Generalized Frequency Response Functions (GFRFs). Inspired by techniques developed for the linear frequency domain case, the GFRFs are modelled as real/complex Gaussian processes with prior covariances related to the time domain characteristics of the corresponding Volterra series. A Gaussian process regression method is developed for the case of periodic excitations, and numerical examples demonstrate the efficacy of the proposed method, as well as its advantage over time domain methods in the case of band-limited excitations

Nonparametric volterra kernel estimation using regularization

Modeling of nonlinear dynamic systems constitutes one of the most challenging topics in the field of system identifi- cation. One way to describe the nonlinear behavior of a process is by use of the nonparametric Volterra Series representation. The drawback of this method lies in the fact that the number of parameters to be estimated increases fast with the number of lags considered for the description of the several impulse responses. The result is that the estimated parameters admit a very large variance leading to a very uncertain description of the nonlinear system. In this paper, inspired from the regularization techniques that have been applied to one-dimensional (1-D) impulse responses for a linear time invariant (LTI) system, we present a method to estimate efficiently finite Volterra kernels. The latter is achieved by constraining the estimated parameters appropriately during the identification step in a way that prior knowledge about the to-be-estimated kernels is reflected on the resulting model. The enormous benefit for the identification of Volterra kernels due to the regularization is illustrated with a numerical example