In this paper, we propose two novel p-norm penalty least mean square (Lp-LMS) algorithms as supplements of the conventional Lp-LMS algorithm established for sparse adaptive filtering recently. A gradient comparator is employed to selectively apply the zero attractor of p-norm constraint for only those taps that have the same polarity as that of the gradient of the squared instantaneous error, which leads to the new proposed gradient compared p-norm constraint LMS algorithm (LpGC-LMS). We explain that the LpGC-LMS can achieve lower mean square error than the standard Lp-LMS algorithm theoretically and experimentally. To further improve the performance of the filter, the LpNGC-LMS algorithm is derived using a new gradient comparator which takes the sign-smoothed version of the previous one. The performance of the LpNGC-LMS is superior to that of the LpGC-LMS in theory and in simulations. Moreover, these two comparators can be easily applied to other norm constraint LMS algorithms to derive some new approaches for sparse adaptive filtering. The numerical simulation results show that the two proposed algorithms achieve better performance than the standard LMS algorithm and Lp-LMS algorithm in terms of convergence rate and steady-state behavior in sparse system identification settings.
The purpose of this article is to discuss some aspects of the convergence analysis performed in the paper [Design of momentum fractional LMS for Hammerstein nonlinear system identification with application to electrically stimulated muscle model, Eur. Phys. J. Plus (2019) \textbf{134}: 407]. It is highlighted that the way the authors prove convergence suffers a lack of correct and valid mathematical justifications.
Low complexity of a system model is essential for its use in real-time applications. However, sparse identification methods commonly have stringent requirements that exclude them from being applied in an industrial setting. In this paper, we introduce a flexible method for the sparse identification of dynamical systems described by ordinary differential equations. Our method relieves many of the requirements imposed by other methods that relate to the structure of the model and the data set, such as fixed sampling rates, full state measurements, and linearity of the model. The Levenberg-Marqua…
Korea Advanced Institute of Science and Technology (KAIST)（韩国科学技术院）；Karachi Institute of Economics and Technology（卡拉奇经济技术学院）；The University of Western Australia（西澳大学）
Forced oscillations may jeopardize the secure operation of power systems. To mitigate forced oscillations, locating the sources is critical. In this paper, leveraging on Sparse Identification of Nonlinear Dynamics (SINDy), an online purely data-driven method to locate the forced oscillation is developed. Validations in all simulated cases (in the WECC 179-bus system) and actual oscillation events (in ISO New England system) in the IEEE Task Force test cases library are carried out, which demonstrate that the proposed algorithm, requiring no model information, can accurately locate sources in m…
This paper considers the distributed sparse identification problem over wireless sensor networks such that all sensors cooperatively estimate the unknown sparse parameter vector of stochastic dynamic systems by using the local information from neighbors. A distributed sparse least squares algorithm is proposed by minimizing a local information criterion formulated as a linear combination of accumulative local estimation error and L_1-regularization term. The upper bounds of the estimation error and the regret of the adaptive predictor of the proposed algorithm are presented. Furthermore, by de…
Focusing on identification, this paper develops techniques to reconstruct zero and nonzero elements of a sparse parameter vector of a stochastic dynamic system under feedback control, for which the current input may depend on the past inputs and outputs, system noises as well as exogenous dithers. First, a sparse parameter identification algorithm is introduced based on L2 norm with L1 regularization, where the adaptive weights are adopted in the optimization variables of L1 term. Second, estimates generated by the algorithm are shown to have both set and parameter convergence. That is, sets o…
This paper deals with the error processing problem of sparse identification of nonlinear dynamical systems(SINDy) through introducing the $L_\infty$ approximation to take place of the former $L_2$ approximation. The motivation is that the $L_\infty$ approximation could better describe the error phenomenon in the SINDy, which consists of the derivative approximation error and the measurement noise. Then, an iterative thresholding algorithm is proposed to solve the reformulated problem. 3 scenarios of possible errors are considered in the experiment. The results show that the $L_\infty$ approxim…
Delft University of Technology（代尔夫特理工大学）；Western Sydney University（西悉尼大学）；The University of Manchester（曼彻斯特大学）
This paper proposes a sparse Bayesian treatment of deep neural networks (DNNs) for system identification. Although DNNs show impressive approximation ability in various fields, several challenges still exist for system identification problems. First, DNNs are known to be too complex that they can easily overfit the training data. Second, the selection of the input regressors for system identification is nontrivial. Third, uncertainty quantification of the model parameters and predictions are necessary. The proposed Bayesian approach offers a principled way to alleviate the above challenges by …
In recent years, identification of nonlinear dynamical systems from data has become increasingly popular. Sparse regression approaches, such as Sparse Identification of Nonlinear Dynamics (SINDy), fostered the development of novel governing equation identification algorithms assuming the state variables are known a priori and the governing equations lend themselves to sparse, linear expansions in a (nonlinear) basis of the state variables. In the context of the identification of governing equations of nonlinear dynamical systems, one faces the problem of identifiability of model parameters whe…
Online sparse Bayesian identification of nonlinear time-varying systems
College of Electrical and Power Engineering, Taiyuan University of Technology, Taiyuan, 030024, China（电气与电力工程学院，太原理工大学，太原，030024，中国）；Department of Automation, Taiyuan Institute of Technology, Taiyuan, 030024, China（自动化系，太原理工大学，太原，030024，中国）
University of Washington（华盛顿大学）；Arts et Métiers Institute of Technology（国立高等工程技术学院）
Automated data-driven modeling, the process of directly discovering the governing equations of a system from data, is increasingly being used across the scientific community. PySINDy is a Python package that provides tools for applying the sparse identification of nonlinear dynamics (SINDy) approach to data-driven model discovery. In this major update to PySINDy, we implement several advanced features that enable the discovery of more general differential equations from noisy and limited data. The library of candidate terms is extended for the identification of actuated systems, partial differ…
Regularized methods have been widely applied to system identification problems without known model structures. This paper proposes an infinite-dimensional sparse learning algorithm based on atomic norm regularization. Atomic norm regularization decomposes the transfer function into first-order atomic models and solves a group lasso problem that selects a sparse set of poles and identifies the corresponding coefficients. The difficulty in solving the problem lies in the fact that there are an infinite number of possible atomic models. This work proposes a greedy algorithm that generates new can…
University of Maryland（马里兰大学）；University of Washington（华盛顿大学）；Imperial College（帝国理工学院）
Sparse system identification is the data-driven process of obtaining parsimonious differential equations that describe the evolution of a dynamical system, balancing model complexity and accuracy. There has been rapid innovation in system identification across scientific domains, but there remains a gap in the literature for large-scale methodological comparisons that are evaluated on a variety of dynamical systems. In this work, we systematically benchmark sparse regression variants by utilizing the dysts standardized database of chaotic systems. In particular, we demonstrate how this open-so…
Sparse Identification of Nonlinear Dynamics (SINDy) has been shown to successfully recover governing equations from data; however, this approach assumes the initial condition to be exactly known in advance and is sensitive to noise. In this work we propose an integral SINDy (ISINDy) method to simultaneously identify model structure and parameters of nonlinear ordinary differential equations (ODEs) from noisy time-series observations. First, the states are estimated via penalized spline smoothing and then substituted into the integral-form numerical discretization solver, leading to a pseudo-li…
In this document, some general results in approximation theory and matrix analysis with applications to sparse identification of time series models and nonlinear discrete-time dynamical systems are presented. The aforementioned theoretical methods are translated into algorithms that can be used for sparse model identification of discrete-time dynamical systems, based on structured data measured from the systems. The approximation of the state-transition operators that are determined primarily by matrices of parameters to be identified based on data measured from a given system, is approached b…
Northeastern University（东北大学）；Massachusetts Institute of Technology（麻省理工学院）
Identifying damage of structural systems is typically characterized as an inverse problem which might be ill-conditioned due to aleatory and epistemic uncertainties induced by measurement noise and modeling error. Sparse representation can be used to perform inverse analysis for the case of sparse damage. In this paper, we propose a novel two-stage sensitivity analysis-based framework for both model updating and sparse damage identification. Specifically, an $\ell_2$ Bayesian learning method is firstly developed for updating the intact model and uncertainty quantification so as to set forward …
