Effect of Choice of Basis Functions in Neural Network for Capturing Unknown Function for Dynamic Inversion Control

2013 ◽  
pp. 327-341
Author(s):  
Gandham Ramesh ◽  
P. N. Dwivedi ◽  
P. Naveen Kumar ◽  
R. Padhi
Keyword(s):  
Neural Network ◽  
Unknown Function ◽  
Basis Functions ◽  
Dynamic Inversion

scholarly journals Theory of Functional Connections Applied to Linear ODEs Subject to Integral Constraints and Linear Ordinary Integro-Differential Equations

2021 ◽  
Vol 26 (3) ◽  
pp. 65
Author(s):  
Mario De De Florio ◽  
Enrico Schiassi ◽  
Andrea D’Ambrosio ◽  
Daniele Mortari ◽  
Roberto Furfaro
Keyword(s):  
Neural Network ◽  
Differential Equations ◽  
Unknown Function ◽  
Linear Constraints ◽  
Basis Functions ◽  
Mathematical Procedure ◽  
Functional Connections ◽  
Free Function ◽  
Linear Odes ◽  
Numerical Approaches

This study shows how the Theory of Functional Connections (TFC) allows us to obtain fast and highly accurate solutions to linear ODEs involving integrals. Integrals can be constraints and/or terms of the differential equations (e.g., ordinary integro-differential equations). This study first summarizes TFC, a mathematical procedure to obtain constrained expressions. These are functionals representing all functions satisfying a set of linear constraints. These functionals contain a free function, g(x), representing the unknown function to optimize. Two numerical approaches are shown to numerically estimate g(x). The first models g(x) as a linear combination of a set of basis functions, such as Chebyshev or Legendre orthogonal polynomials, while the second models g(x) as a neural network. Meaningful problems are provided. In all numerical problems, the proposed method produces very fast and accurate solutions.

An Algorithm for the Approximate Solution of the Fractional Riccati Differential Equation

2019 ◽  
Vol 20 (6) ◽  
pp. 661-674
Author(s):  
S. S. Ezz-Eldien ◽  
J. A. T. Machado ◽  
Y. Wang ◽  
A. A. Aldraiweesh
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Chebyshev Polynomials ◽  
Unknown Function ◽  
Numerical Approach ◽  
Continuous Functions ◽  
Basis Functions ◽  
Real Constant ◽  
Riccati Differential Equation ◽  
Suggested Approach

AbstractThis manuscript develops a numerical approach for approximating the solution of the fractional Riccati differential equation (FRDE): $$\begin{align*}D^{\mu}&u(x)+a(x) u^2(x)+b(x) u(x)= g(x),\quad 0\leq \mu \leq 1,\quad 0\leq x \leq t,\\&u(0)=d,\end{align*}$$where u(x) is the unknown function, a(x), b(x) and g(x) are known continuous functions defined in [0,t] and d is a real constant. The proposed method is applied for solving the FRDE with shifted Chebyshev polynomials as basis functions. In addition, the convergence analysis of the suggested approach is investigated. The efficiency of the algorithm is demonstrated by means of several examples and the results compared with those given using other numerical schemes.

scholarly journals AI Feynman: A physics-inspired method for symbolic regression

2020 ◽  
Vol 6 (16) ◽  
pp. eaay2631 ◽  
Author(s):  
Silviu-Marian Udrescu ◽  
Max Tegmark
Keyword(s):  
Neural Network ◽  
Artificial Intelligence ◽  
Success Rate ◽  
Unknown Function ◽  
State Of The Art ◽  
Practical Interest ◽  
Symbolic Regression ◽  
The State ◽  
Np Hard ◽  
Test Set

A core challenge for both physics and artificial intelligence (AI) is symbolic regression: finding a symbolic expression that matches data from an unknown function. Although this problem is likely to be NP-hard in principle, functions of practical interest often exhibit symmetries, separability, compositionality, and other simplifying properties. In this spirit, we develop a recursive multidimensional symbolic regression algorithm that combines neural network fitting with a suite of physics-inspired techniques. We apply it to 100 equations from the Feynman Lectures on Physics, and it discovers all of them, while previous publicly available software cracks only 71; for a more difficult physics-based test set, we improve the state-of-the-art success rate from 15 to 90%.

New sinusoidal basis functions and a neural network approach to solve nonlinear Volterra–Fredholm integral equations

2019 ◽  
Vol 31 (9) ◽  
pp. 4865-4878 ◽  
Author(s):  
Stefania Tomasiello ◽  
Jorge E. Macías-Díaz ◽  
Alireza Khastan ◽  
Zahra Alijani
Keyword(s):  
Neural Network ◽  
Integral Equations ◽  
Basis Functions ◽  
Fredholm Integral Equations ◽  
Network Approach ◽  
Neural Network Approach

Autopilot design for tilt-rotor unmanned aerial vehicle with nacelle mounted wing extension using single hidden layer perceptron neural network

2016 ◽  
Vol 231 (11) ◽  
pp. 1979-1992 ◽  
Author(s):  
Youngshin Kang ◽  
Nakwan Kim ◽  
Byoung-Soo Kim ◽  
Min-Jea Tahk
Keyword(s):  
Neural Network ◽  
Unmanned Aerial Vehicle ◽  
Lyapunov Equation ◽  
Pole Placement ◽  
Dynamic Inversion ◽  
North West ◽  
Tilt Rotor ◽  
Neural Network Controller ◽  
Aerial Vehicle ◽  
Hidden Layer

Single hidden layer perceptron neural network controllers combined with dynamic inversion are applied to the tilt-rotor unmanned aerial vehicle and its variant model with the nacelle mounted wing extension. The bandwidths of the inner loop and outer loop of the controller are designed using the timescale separation approach, which uses the combined analysis of the two loops. The bandwidth of each loop is selected to be close to each other using a combination of the pseudo-control-hedging and the pole-placement method. Similar to the previous studies on sigma-pi neural network, the dynamic inversion at hover conditions of the original tilt-rotor model is used as a baseline for both aircraft, and the compatible solution to the Lyapunov equation is suggested. The single hidden layer perceptron neural network minimizes the error of the inversion model through the back-propagation adaptation. The waypoint guidance is applied to the outermost loop of the neural network controller for autonomous flight which includes vertical take-off and landing as well as nacelle conversion. The simulation results under the two wind conditions for the tilt-rotor aircraft and its variant are presented. The south and north-west wind directions are simulated in order to compare with the results from the existing sigma-pi neural network, and the estimation results of the wind are presented.

scholarly journals Adaptive Neural Network-Based Satellite Attitude Control by Using the Dynamic Inversion Technique and a VSCMG Pyramidal Cluster

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-16 ◽  
Author(s):  
Mihai Lungu ◽  
Romulus Lungu
Keyword(s):  
Neural Network ◽  
Attitude Control ◽  
Approximation Error ◽  
Inversion Method ◽  
Dynamic Inversion ◽  
Control Law ◽  
Feed Forward Neural Network ◽  
Small Satellites ◽  
Control Moment ◽  
Satellite Attitude Control

The paper presents an adaptive system for the control of small satellites’ attitude by using a pyramidal cluster of four variable-speed control moment gyros as actuators. Starting from the dynamic model of the pyramidal cluster, an adaptive control law is designed by means of the dynamic inversion method and a feed-forward neural network-based nonlinear subsystem; the control law has a proportional-integrator component (for the control of the reduced-order linear subsystem) and an adaptive component (for the compensation of the approximation error associated with the function describing the dynamics of the nonlinear system). The software implementation and validation of the new control architecture are achieved by using the Matlab/Simulink environment.

Sign in / Sign up

Export Citation Format

Share Document