Time-varying model identification for time-frequency feature extraction from EEG data

A novel modelling scheme that can be used to estimate and track time-varying properties of nonstationary signals is investigated. This scheme is based on a class of time-varying AutoRegressive with an eXogenous input (ARX) models where the associated time-varying parameters are represented by multi-wavelet basis functions. The orthogonal least square (OLS) algorithm is then applied to refine the model parameter estimates of the time-varying ARX model. The main features of the multi-wavelet approach is that it enables smooth trends to be tracked but also to capture sharp changes in the time-varying process parameters. Simulation studies and applications to real EEG data show that the proposed algorithm can provide important transient information on the inherent dynamics of nonstationary processes

Linear identification of nonlinear systems: A lifting technique based on the Koopman operator

We exploit the key idea that nonlinear system identification is equivalent to linear identification of the socalled Koopman operator. Instead of considering nonlinear system identification in the state space, we obtain a novel linear identification technique by recasting the problem in the infinite-dimensional space of observables. This technique can be described in two main steps. In the first step, similar to the socalled Extended Dynamic Mode Decomposition algorithm, the data are lifted to the infinite-dimensional space and used for linear identification of the Koopman operator. In the second step, the obtained Koopman operator is "projected back" to the finite-dimensional state space, and identified to the nonlinear vector field through a linear least squares problem. The proposed technique is efficient to recover (polynomial) vector fields of different classes of systems, including unstable, chaotic, and open systems. In addition, it is robust to noise, well-suited to model low sampling rate datasets, and able to infer network topology and dynamics.Comment: 6 page

Non-linear estimation is easy

Non-linear state estimation and some related topics, like parametric estimation, fault diagnosis, and perturbation attenuation, are tackled here via a new methodology in numerical differentiation. The corresponding basic system theoretic definitions and properties are presented within the framework of differential algebra, which permits to handle system variables and their derivatives of any order. Several academic examples and their computer simulations, with on-line estimations, are illustrating our viewpoint

A comparative study on global wavelet and polynomial models for nonlinear regime-switching systems

A comparative study of wavelet and polynomial models for non-linear Regime-Switching (RS) systems is carried out. RS systems, considered in this study, are a class of severely non-linear systems, which exhibit abrupt changes or dramatic breaks in behaviour, due to RS caused by associated events. Both wavelet and polynomial models are used to describe discontinuous dynamical systems, where it is assumed that no a priori information about the inherent model structure and the relative regime switches of the underlying dynamics is known, but only observed input-output data are available. An Orthogonal Least Squares (OLS) algorithm interfered with by an Error Reduction Ratio (ERR) index and regularised by an Approximate Minimum Description Length (AMDL) criterion, is used to construct parsimonious wavelet and polynomial models. The performance of the resultant wavelet models is compared with that of the relative polynomial models, by inspecting the predictive capability of the associated representations. It is shown from numerical results that wavelet models are superior to polynomial models, in respect of generalisation properties, for describing severely non-linear RS systems

Towards Efficient Maximum Likelihood Estimation of LPV-SS Models

How to efficiently identify multiple-input multiple-output (MIMO) linear parameter-varying (LPV) discrete-time state-space (SS) models with affine dependence on the scheduling variable still remains an open question, as identification methods proposed in the literature suffer heavily from the curse of dimensionality and/or depend on over-restrictive approximations of the measured signal behaviors. However, obtaining an SS model of the targeted system is crucial for many LPV control synthesis methods, as these synthesis tools are almost exclusively formulated for the aforementioned representation of the system dynamics. Therefore, in this paper, we tackle the problem by combining state-of-the-art LPV input-output (IO) identification methods with an LPV-IO to LPV-SS realization scheme and a maximum likelihood refinement step. The resulting modular LPV-SS identification approach achieves statical efficiency with a relatively low computational load. The method contains the following three steps: 1) estimation of the Markov coefficient sequence of the underlying system using correlation analysis or Bayesian impulse response estimation, then 2) LPV-SS realization of the estimated coefficients by using a basis reduced Ho-Kalman method, and 3) refinement of the LPV-SS model estimate from a maximum-likelihood point of view by a gradient-based or an expectation-maximization optimization methodology. The effectiveness of the full identification scheme is demonstrated by a Monte Carlo study where our proposed method is compared to existing schemes for identifying a MIMO LPV system