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 $L_\infty$ approximation performs better or at least equal than the $L_2$ approximation in face of different error cases. Hence, it is reasonable to consider the $L_\infty$ approximation in the applications of the SINDy.
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…
We present a framework which incorporates three aspects of the estimation problem, namely, sparse sensor configuration, optimal precision, and robustness in the presence of model uncertainty. The problem is formulated in the $\mathcal{H}_{\infty}$ optimal observer design framework. We consider two types of uncertainties in the system, i.e. structured affine and unstructured uncertainties. The objective is to design an observer with a given $\mathcal{H}_{\infty}$ performance index with minimal number of sensors and minimal precision values, while guaranteeing the performance for all admissible …
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…
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…
In this paper, we simultaneously determine the optimal sensor precision and the observer gain, which achieves the specified accuracy in the state estimates. Along with the unknown observer gain, the formulation parameterizes the scaling of the exogenous inputs that correspond to the sensor noise. Reciprocal of this scaling is defined as the sensor precision, and sparseness is achieved by minimizing the $l_1$ norm of the precision vector. The optimization is performed with constraints guaranteeing specified accuracy in state estimates, which are defined in terms of $\mathcal{H}_2$ or $\mathcal{…
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…
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…
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…
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 …
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…
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…
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…
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）
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…
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 …
Distilling physical laws autonomously from data has been of great interest in many scientific areas. The sparse identification of nonlinear dynamics (SINDy) and its variations have been developed to extract the underlying governing equations from observation data. However, SINDy faces certain difficulties when the dynamics contain rational functions. The principle of the least action governs many mechanical systems, mathematically expressed in the Lagrangian formula. Compared to the actual equation of motions, the Lagrangian is much more concise, especially for complex systems, and does not us…
Nonlinear system identification is important with a wide range of applications. The typical approaches for nonlinear system identification include Volterra series models, nonlinear autoregressive with exogenous inputs models, block-structured models, state-space models and neural network models. Among them, neural networks (NN) is an important black-box method thanks to its universal approximation capability and less dependency on prior information. However, there are several challenges associated with NN. The first one lies in the design of a proper neural network structure. A relatively simp…
Dynamic financial processes identification using sparse regressive reservoir computers
Center for Innovation in Scientific Computing, Universidad Nacional Aut\'onoma de Honduras, Honduras；Department of Chemical Engineering, Norwegian University of Science；Department of Statistics；Research, National Commission of Banks
SINDyG: Sparse Identification of Nonlinear Dynamical Systems from Graph-Structured Data, with Applications to Stuart-Landau Oscillator Networks
Data Science and Analytics Institute, University of Oklahoma, Norman, OK, USA.（数据科学与分析研究所，俄克拉荷马大学，诺曼，OK，美国。）；Analytics Institute, University of Oklahoma, Norman, OK, USA. School of Computer Science, University of Oklahoma, Norman, OK, USA.（分析研究所，俄克拉荷马大学，诺曼，OK，美国。计算机科学学院，俄克拉荷马大学，诺曼，OK，美国。）
