Modeling and Control of Flexible Transporter System With Arbitrarily Time-Varying Cable Lengths

2003 ◽  
Author(s):  
Yuhong Zhang ◽  
Sunil K. Agrawal ◽  
Peter Hagedorn
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ritz Method ◽  
Nonlinear Partial Differential Equations ◽  
Time Varying ◽  
Governing Equations ◽  
Partial Differential ◽  
Modeling And Control ◽  
Lyapunov Controller ◽  
And Control

A systematic procedure for deriving the system model of a cable transporter system with arbitrarily time-varying lengths is presented. Two different approaches are used to develop the model, namely, Newton’s Law and Hamilton’s Principle. The derived governing equations are nonlinear partial differential equations. The same results are obtained using the two methods. The Rayleigh-Ritz method is used to obtain an approximate numerical solution of the governing equations by transforming the infinite order partial differential equations into a finite order discretized system. A Lyapunov controller which can both dissipate the vibratory energy and assure the attainment of the desired goal is derived. The validity of the proposed controller is verified by numerical simulation.

Longitudinal Vibration Modeling and Control of a Flexible Transporter System with Arbitrarily Varying Cable Lengths

2005 ◽  
Vol 11 (3) ◽  
pp. 431-456 ◽  
Author(s):  
Yuhong Zhang ◽  
Sunil K. Agrawal ◽  
Peter Hagedorn
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Longitudinal Vibration ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Governing Equations ◽  
Partial Differential ◽  
Modeling And Control ◽  
Assumed Mode Method ◽  
Mode Method

We present a systematic procedure for deriving the model of a cable transporter system with arbitrarily varying cable lengths. The Hamilton principle is applied to derive the governing equations of motion. The derived governing equations are nonlinear partial differential equations. The results are verified using the Newton law. The assumed mode method is used to obtain an approximate numerical solution of the governing equations by transforming the infinite-dimensional partial differential equations into a finite-dimensional discretized system. We propose a Lyapunov controller, based directly on the governing partial differential equations, which can both dissipate the vibratory energy during the motion of the transporter and guarantee the attainment of the desired goal point. The validity of the proposed controller is verified by numerical simulation.

scholarly journals Continuum Modeling and Control of Large Nonuniform Wireless Networks via Nonlinear Partial Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-16 ◽  
Author(s):  
Yang Zhang ◽  
Edwin K. P. Chong ◽  
Jan Hannig ◽  
Donald Estep
Keyword(s):  
Wireless Networks ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Continuum Limit ◽  
Nonlinear Partial Differential Equations ◽  
Mobile Nodes ◽  
Continuum Modeling ◽  
Partial Differential ◽  
Modeling And Control ◽  
And Control

We introduce a continuum modeling method to approximate a class of large wireless networks by nonlinear partial differential equations (PDEs). This method is based on the convergence of a sequence of underlying Markov chains of the network indexed byN, the number of nodes in the network. AsNgoes to infinity, the sequence converges to a continuum limit, which is the solution of a certain nonlinear PDE. We first describe PDE models for networks with uniformly located nodes and then generalize to networks with nonuniformly located, and possibly mobile, nodes. Based on the PDE models, we develop a method to control the transmissions in nonuniform networks so that the continuum limit is invariant under perturbations in node locations. This enables the networks to maintain stable global characteristics in the presence of varying node locations.

Modeling and Control of a Thermoelastic Beam

2013 ◽  
Author(s):  
Ilhan Tuzcu ◽  
Javier Gonzalez-Rocha
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Elastic Displacement ◽  
Linear Quadratic ◽  
Partial Differential ◽  
Numerical Application ◽  
Modeling And Control ◽  
Thermoelastic Beam ◽  
And Control

The objective of this paper is to model a thermoelastic beam and use thermoelectric heat actuators dispersed over the beam to control not only its vibration, but also its temperature. The model is represented by two coupled partial differential equations governing the elastic bending displacement and temperature variation over the length of the beam. The partial differential equations are replaced by a set of ordinary differential equations through discretization. The first-order ordinary differential equations are cast in the compact state-space form to be used in the thermoelastic analysis and control. The Linear Quadratic Gaussian (LQG) is used for control design. An numerical application to a uniform cantilever beam demonstrates the coupling between the temperature and the elastic displacement and feasibility of using thermoelectric actuators in controlling the vibration and temperature simultaneously.

Lyapunov-type inequalities for partial differential equations with 𝑝-Laplacian

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Robert Stegliński
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Nonlinear Partial Differential Equations ◽  
Dirichlet Boundary Conditions ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Lyapunov Inequalities ◽  
Funct Anal

Abstract The aim of this paper is to extend results from [A. Cañada, J. A. Montero and S. Villegas, Lyapunov inequalities for partial differential equations, J. Funct. Anal. 237 (2006), 1, 176–193] about Lyapunov-type inequalities for linear partial differential equations to nonlinear partial differential equations with 𝑝-Laplacian with zero Neumann or Dirichlet boundary conditions.

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