From Nonlinear Identification to Linear Parameter Varying Models: Benchmark Examples

Linear parameter-varying (LPV) models form a powerful model class to analyze and control a (nonlinear) system of interest. Identifying a LPV model of a nonlinear system can be challenging due to the difficulty of selecting the scheduling variable(s) a priori, which is quite challenging in case a first principles based understanding of the system is unavailable. This paper presents a systematic LPV embedding approach starting from nonlinear fractional representation models. A nonlinear system is identified first using a nonlinear block-oriented linear fractional representation (LFR) model. This nonlinear LFR model class is embedded into the LPV model class by factorization of the static nonlinear block present in the model. As a result of the factorization a LPV-LFR or a LPV state-space model with an affine dependency results. This approach facilitates the selection of the scheduling variable from a data-driven perspective. Furthermore the estimation is not affected by measurement noise on the scheduling variables, which is often left untreated by LPV model identification methods. The proposed approach is illustrated on two well-established nonlinear modeling benchmark examples

Nonlinear system modeling based on constrained Volterra series estimates

A simple nonlinear system modeling algorithm designed to work with limited \emph{a priori }knowledge and short data records, is examined. It creates an empirical Volterra series-based model of a system using an $l_{q}$-constrained least squares algorithm with $q\geq 1$. If the system $m\left( \cdot \right)$ is a continuous and bounded map with a finite memory no longer than some known $\tau$, then (for a $D$ parameter model and for a number of measurements $N$) the difference between the resulting model of the system and the best possible theoretical one is guaranteed to be of order $\sqrt{N^{-1}\ln D}$, even for $D\geq N$. The performance of models obtained for $q=1,1.5$ and $2$ is tested on the Wiener-Hammerstein benchmark system. The results suggest that the models obtained for $q>1$ are better suited to characterize the nature of the system, while the sparse solutions obtained for $q=1$ yield smaller error values in terms of input-output behavior

Urysohn Forest for Aleatoric Uncertainty Quantification

This paper focuses on building models of stochastic systems with aleatoric uncertainty. The main novelty is an algorithm of boosted ensemble training of multiple models for obtaining a probability distribution of an individual output as a function of the system input. The second novel contribution is a new regression model to be used in the ensemble. The model is a multi-layered tree of hierarchically-connected discrete Urysohn operators (or generalised additive models, which are mathematically equivalent to the discrete Urysohn operators in this case). Since multiple models (trees) are trained in the ensemble, the authors refer them as an Urysohn forest. The source code is freely available online

Recurrent Equilibrium Networks: Flexible Dynamic Models with Guaranteed Stability and Robustness

This paper introduces recurrent equilibrium networks (RENs), a new class of nonlinear dynamical models for applications in machine learning, system identification and control. The new model class has ``built in'' guarantees of stability and robustness: all models in the class are contracting - a strong form of nonlinear stability - and models can satisfy prescribed incremental integral quadratic constraints (IQC), including Lipschitz bounds and incremental passivity. RENs are otherwise very flexible: they can represent all stable linear systems, all previously-known sets of contracting recurrent neural networks and echo state networks, all deep feedforward neural networks, and all stable Wiener/Hammerstein models. RENs are parameterized directly by a vector in R^N, i.e. stability and robustness are ensured without parameter constraints, which simplifies learning since generic methods for unconstrained optimization can be used. The performance and robustness of the new model set is evaluated on benchmark nonlinear system identification problems, and the paper also presents applications in data-driven nonlinear observer design and control with stability guarantees.Comment: Journal submission, extended version of conference paper (v1 of this arxiv preprint

the correlation model of head roll and lateral acceleration during curve driving via hammerstein-wiener

Generally, passengers are more prone to Motion Sickness (MS) than the drivers. The difference of their severity level of MS is due to their different head movement towards the direction of the lateral acceleration. During cornering, the passengers tend to tilt their heads according to the direction, while the drivers tends to tilt their head opposite to the direction. Based on this fact, the passengers are able to reduce their MS level if they can imitate the driver’s head movement or lessen their head tilt angle towards the direction of the lateral acceleration. However, it is easier to design MS mitigation method based on the head tilt movement strategy if the mathematical expression of their head behaviour is known beforehand. On way to derive the mathematical expression is by modelling the relationship between the occupant’s head tilt movements and the vehicle’s lateral acceleration during curve driving. Therefore, this study proposed the usage of Hammerstein-Wiener (H-W) method for the modelling purpose. Experiment is set up to obtain the naturalistic data for the modelling process. The modelling process is carried out by varying the input output nonlinearities estimators. The results show that the estimated output responses from the H-W models are similar with the real responses taken from the experiment.The derived models for both passenger and driver have 68.88% and 66.32% of Best Fit (BF) percentages. With further study, the passenger’s and driver’s models which are developed by the proposed H-W modelling strategy are expected to contribute in MS minimisation studies