Exponential Stabilization of a Transversely Vibrating Beam by Boundary Control Via Lyapunov’s Direct Method

1999 ◽  
Vol 123 (2) ◽  
pp. 195-200 ◽  
Author(s):  
Mehrdad P. Fard ◽  
Svein I. Sagatun
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Control ◽  
Direct Method ◽  
Nonlinear Partial Differential Equation ◽  
Nonlinear Function ◽  
Control Law ◽  
Boundary Stabilization ◽  
Free Transverse Vibration ◽  
Nonlinear Control Law

This paper discusses the boundary stabilization of a beam in free transverse vibration. The dynamics of the beam is presented by a nonlinear partial differential equation (PDE). Based on this model a nonlinear control law is constructed to stabilize the system. The control law is a nonlinear function of the slopes and velocity at the boundary of the beam. The novelty of this article is that it has been possible to exponentially stabilize a free transversely vibrating beam via boundary control without restoring to truncation of the model. This result is achieved while the coupling between longitudinal and transversal displacements has been taken into account.

scholarly journals EVOLUTION OF MODULATIONAL INSTABILITY IN TRAVELLING WAVE SOLUTION OF NON-LINEAR PARTIAL DIFFERENTIAL EQUATION

2020 ◽  
Vol 5 (1) ◽  
pp. 1-7
Author(s):  
Ram Dayal Pankaj ◽  
Arun Kumar ◽  
Chandrawati Sindhi
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Wave Solution ◽  
Modulational Instability ◽  
Nonlinear Partial Differential Equation ◽  
Linear Partial Differential Equation ◽  
Travelling Wave ◽  
Travelling Wave Solution ◽  
Partial Differential ◽  
Jacobian Elliptic Functions

The Ritz variational method has been applied to the nonlinear partial differential equation to construct a model for travelling wave solution. The spatially periodic trial function was chosen in the form of combination of Jacobian Elliptic functions, with the dependence of its parameters

Explorations and Expectations of Equidistribution Adaptations for Nonlinear Quenching Problems

2013 ◽  
Vol 5 (04) ◽  
pp. 407-422 ◽  
Author(s):  
Matthew A. Beauregard ◽  
Qin Sheng
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Finite Difference ◽  
Nonlinear Partial Differential Equation ◽  
Partial Differential ◽  
Spatial Adaptation ◽  
Monitor Function ◽  
Monitoring Strategies ◽  
Highly Nonlinear ◽  
Equidistribution Principle

AbstractFinite difference computations that involve spatial adaptation commonly employ an equidistribution principle. In these cases, a new mesh is constructed such that a given monitor function is equidistributed in some sense. Typical choices of the monitor function involve the solution or one of its many derivatives. This straightforward concept has proven to be extremely effective and practical. However, selections of core monitoring functions are often challenging and crucial to the computational success. This paper concerns six different designs of the monitoring function that targets a highly nonlinear partial differential equation that exhibits both quenching-type and degeneracy singularities. While the first four monitoring strategies are within the so-calledprimitiveregime, the rest belong to a later category of themodifiedtype, which requires the priori knowledge of certain important quenching solution characteristics. Simulated examples are given to illustrate our study and conclusions.

scholarly journals Weighted multiple model adaptive boundary control for a flexible manipulator

2019 ◽  
Vol 103 (1) ◽  
pp. 003685041988646
Author(s):  
Weicun Zhang ◽  
Qing Li ◽  
Yuzhen Zhang ◽  
Ziyi Lu ◽  
Cheng Nian
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Control ◽  
Control Strategy ◽  
Flexible Manipulator ◽  
Multiple Model ◽  
Parameter Uncertainties ◽  
Partial Differential ◽  
Local Boundary ◽  
Adaptive Boundary Control

In this article, a weighted multiple model adaptive boundary control scheme is proposed for a flexible manipulator with unknown large parameter uncertainties. First, the uncertainties are approximatively covered by a finite number of constant models. Second, based on Euler–Bernoulli beam theory and Hamilton principle, the distributed parameter model of the flexible manipulator is constructed in terms of partial differential equation for each local constant model. Correspondingly, local boundary controllers are designed to control the manipulator movement and suppress its vibration for each partial differential equation model, which are based on Lyapunov stability theory. Then, a novel weighted multiple model adaptive control strategy is developed based on an improved weighting algorithm. The stability of the overall closed-loop system is ensured by virtual equivalent system theory. Finally, numerical simulations are provided to illustrate the feasibility and effectiveness of the proposed control strategy.

Sign in / Sign up

Export Citation Format

Share Document