Anna Marconato

Parametric identification of parallel Wiener-Hammerstein systems

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

Improved Initialization for Nonlinear State-Space Modeling

This paper discusses a novel initialization algorithm for the estimation of nonlinear state-space models. Good initial values for the model parameters are obtained by identifying separately the linear dynamics and the nonlinear terms in the model. In particular, the nonlinear dynamic problem is transformed into an approximate static formulation, and simple regression methods are applied to obtain the solution in a fast and efficient way. The proposed method is validated by means of two measurement examples: the Wiener-Hammerstein benchmark problem and the identification of a crystal detector.

Study of the effective number of parameters in nonlinear identification benchmarks

This paper discusses the importance of the notion of effective number of parameters as a measure of model complexity. Exploiting this concept allows a fair comparison of models obtained from different model classes. Several illustrative examples of linear and nonlinear models are presented to provide more insight in the problem. As one possible way of showing that model complexity can be reduced without having to pull any parameters to zero, an approach for rank reduced estimation based on the truncated SVD is also discussed. These ideas are then applied to two nonlinear real world problems: the Wiener-Hammerstein and the Silverbox benchmarks.

Identification of the Silverbox Benchmark Using Nonlinear State-Space Models

This work presents the application of an initialization scheme for nonlinear state-space models on a real data benchmark example: the Silverbox problem. The goal of the proposed approach is to transform the identification of a nonlinear dynamic system into an approximate static problem, so that system dynamics and nonlinear terms are identified separately. Classic identification techniques are used to handle dynamics, while regression methods from the statistical learning community are introduced to estimate the nonlinearities in the model. Results obtained on the Silverbox problem are discussed and compared with the performance of other related methods.

Filter-based regularisation for impulse response modelling

In the last years, the success of kernel-based regularisation techniques in solving impulse response modelling tasks has revived the interest on linear system identification. In this work, an alternative perspective on the same problem is introduced. Instead of relying on a Bayesian framework to include assumptions about the system in the definition of the covariance matrix of the parameters, here the prior knowledge is injected at the cost function level. The key idea is to define the regularisation matrix as a filtering operation on the parameters, which allows for a more intuitive formulation of the problem from an engineering point of view. Moreover, this results in a unified framework to model low-pass, band-pass and high-pass systems, and systems with one or more resonances. The proposed filter-based approach outperforms the existing regularisation method based on the TC and DC kernels, as illustrated by means of Monte Carlo simulations on several linear modelling examples.

Regularized nonparametric Volterra kernel estimation

Abstract In this paper, the regularization approach introduced recently for nonparametric estimation of linear systems is extended to the estimation of nonlinear systems modeled 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.

Identification of Wiener-Hammerstein Benchmark Data by Means of Support Vector Machines

Abstract This work presents the identification of a Wiener-Hammerstein system by a learning-from-examples approach, namely the Support Vector Machines for Regression, on the basis of a set of real-life benchmark data. A multi-objective optimization procedure based on genetic algorithms is employed in order to select the best model that describes the input-output relationship of the considered system. Training sets of reduced size are employed to analyze the effect on the accuracy performance.

Comparison of several data-driven non-linear system identification methods on a simplified glucoregulatory system example

In this paper, several advanced data-driven nonlinear identification techniques are compared on a specific problem: a simplified glucoregulatory system modeling example. This problem represents a challenge in the development of an artificial pancreas for T1DM treatment, since for this application good nonlinear models are needed to design accurate closed-loop controllers to regulate the glucose level in the blood. Block-oriented as well as state-space models are used to describe both the dynamics and the nonlinear behavior of the insulin-glucose system, and the advantages and drawbacks of each method are pointed out. The obtained nonlinear models are accurate in simulating the patients behavior, and some of them are also sufficiently simple to be considered in the implementation of a model-based controller to develop the artificial pancreas.

Linking regularization and low-rank approximation for impulse response modeling

Abstract In the last years, nonparametric linear dynamical systems modeling has regained attention in the system identification world. In particular, the application of regularization techniques that were already widely used in statistics and machine learning, has proven beneficial for the estimation of the impulse response of linear systems. The low-rank approximation of the impulse response obtained by the truncated singular value decomposition (SVD) also leads to reduced complexity estimates. In this paper, the link between regularization and SVD truncation for finite impulse response (FIR) model estimation is made explicit. The SVD truncation is reformulated as a regularization problem with a specific choice of the regularization matrix. Both approaches (regularization and SVD truncation) are applied on a FIR modeling example and compared with the classic prediction error method/maximum likelihood approach. The results show the advantage of these techniques for impulse response estimation.

System identification in a real world

In this paper we discuss how to identify a mathematical model for a (non)linear dynamic system starting from experimental data. In the initial step, the frequency response function is measured, together with the properties of the disturbing noise and the nonlinear distortions. This uses nonparametric preprocessing techniques that require very little user interaction. On the basis of this information, the user can decide on an objective basis, in an early phase of the modelling process, to use either a simple linear approximation framework, or to build a more involved nonlinear model. We discuss both options here: i) Identification of linear models in the presence of nonlinear distortions, including the generation of error bounds; and ii) Identification of a nonlinear model. For the latter, a double approach is proposed, using either unstructured nonlinear state space models, or highly structured block oriented nonlinear models. The paper is written from a users perspective.