Second-order bounds on the domain of attraction and the rate of convergence of nonlinear dynamical systems and neural networks

1994 ◽  
Vol 5 (4) ◽  
pp. 551-560 ◽  
Author(s):  
N. Peterfreund ◽  
Y. Baram
Keyword(s):  
Neural Networks ◽  
Dynamical Systems ◽  
Rate Of Convergence ◽  
Nonlinear Dynamical Systems ◽  
Domain Of Attraction ◽  
Second Order ◽  
Nonlinear Dynamical ◽  
Order Bounds

scholarly journals Long-Time Predictive Modeling of Nonlinear Dynamical Systems Using Neural Networks

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-26 ◽  
Author(s):  
Shaowu Pan ◽  
Karthik Duraisamy
Keyword(s):  
Neural Networks ◽  
Dynamical Systems ◽  
Polynomial Regression ◽  
Nonlinear Dynamical Systems ◽  
Feedforward Neural Networks ◽  
Data Availability ◽  
Training Data ◽  
Local Error ◽  
Nonlinear Dynamical ◽  
Sparse Polynomial

We study the use of feedforward neural networks (FNN) to develop models of nonlinear dynamical systems from data. Emphasis is placed on predictions at long times, with limited data availability. Inspired by global stability analysis, and the observation of strong correlation between the local error and the maximal singular value of the Jacobian of the ANN, we introduce Jacobian regularization in the loss function. This regularization suppresses the sensitivity of the prediction to the local error and is shown to improve accuracy and robustness. Comparison between the proposed approach and sparse polynomial regression is presented in numerical examples ranging from simple ODE systems to nonlinear PDE systems including vortex shedding behind a cylinder and instability-driven buoyant mixing flow. Furthermore, limitations of feedforward neural networks are highlighted, especially when the training data does not include a low dimensional attractor. Strategies of data augmentation are presented as remedies to address these issues to a certain extent.

A Terminal Sliding Mode Control of Disturbed Nonlinear Second-Order Dynamical Systems

2016 ◽  
Vol 11 (5) ◽  
Author(s):  
Pawel Skruch
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Sliding Mode ◽  
Second Order ◽  
Control Law ◽  
Sliding Surface ◽  
Terminal Sliding Mode ◽  
Nonlinear Dynamical ◽  
Second Order Differential Equations ◽  
Bounded Disturbance

The paper presents a terminal sliding mode controller for a certain class of disturbed nonlinear dynamical systems. The class of such systems is described by nonlinear second-order differential equations with an unknown and bounded disturbance. A sliding surface is defined by the system state and the desired trajectory. The control law is designed to force the trajectory of the system from any initial condition to the sliding surface within a finite time. The trajectory of the system after reaching the sliding surface remains on it. A computer simulation is included as an example to verify the approach and to demonstrate its effectiveness.

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