scholarly journals Distributed Control of the Generalized Korteweg-de Vries-Burgers Equation

2008 ◽  
Vol 2008 ◽  
pp. 1-19 ◽  
Author(s):  
Nejib Smaoui ◽  
Rasha H. Al-Jamal
Keyword(s):  
Distributed Control ◽  
Feedback Linearization ◽  
Burgers Equation ◽  
Galerkin Projection ◽  
Decomposition Procedure ◽  
Control Schemes ◽  
Feedback Linearization Control ◽  
De Vries ◽  
The Stability ◽  
Model Reduction Technique

The paper deals with the distributed control of the generalized Kortweg-de Vries-Burgers equation (GKdVB) subject to periodic boundary conditions via the Karhunen-Loève (K-L) Galerkin method. The decomposition procedure of the K-L method is presented to illustrate the use of this method in analyzing the numerical simulations data which represent the solutions to the GKdVB equation. The K-L Galerkin projection is used as a model reduction technique for nonlinear systems to derive a system of ordinary differential equations (ODEs) that mimics the dynamics of the GKdVB equation. The data coefficients derived from the ODE system are then used to approximate the solutions of the GKdVB equation. Finally, three state feedback linearization control schemes with the objective of enhancing the stability of the GKdVB equation are proposed. Simulations of the controlled system are given to illustrate the developed theory.

scholarly journals Modelling and nonlinear boundary stabilization of the modified generalized Korteweg–de Vries–Burgers equation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
N. Smaoui ◽  
B. Chentouf ◽  
A. Alalabi
Keyword(s):  
Nonlinear Boundary ◽  
Burgers Equation ◽  
Stabilization Problem ◽  
Boundary Stabilization ◽  
Wave Approximation ◽  
Long Wave ◽  
Control Schemes ◽  
De Vries ◽  
Long Wave Approximation ◽  
Stability Of The Solution

Abstract In this paper, we study the modelling and nonlinear boundary stabilization problem of the modified generalized Korteweg–de Vries–Burgers equation (MGKdVB) when the spatial domain is $[0,1]$ [ 0 , 1 ] . First, the MGKdVB equation is derived using the long-wave approximation and perturbation method. Then, two nonlinear boundary controllers are proposed for this equation and the $L^{2} $ L 2 -global exponential stability of the solution is shown. Numerical simulations are given to illustrate the efficiency of the developed control schemes.

Painlevé analysis and new analytical solutions for compound KdV–Burgers equation with variable coefficients

2010 ◽  
Vol 88 (3) ◽  
pp. 211-221 ◽  
Author(s):  
A. M. Abourabia ◽  
K. M. Hassan ◽  
E. S. Selima
Keyword(s):  
Dispersion Relation ◽  
Shallow Water ◽  
Phase Portrait ◽  
Analytical Solutions ◽  
Burgers Equation ◽  
Variable Coefficients ◽  
Dissipative Effects ◽  
De Vries ◽  
The Stability ◽  
Truncated Painlevé Expansion

We consider the solutions of the compound Korteweg–de Vries (KdV)–Burgers equation with variable coefficients (vccKdV–B) that describe the propagation of undulant bores in shallow water with certain dissipative effects. The Weiss–Tabor–Carnevale (WTC)–Kruskal algorithm is applied to study the integrability of the vccKdV–B equation. We found that the vccKdV–B equation is not Painlevé integrable unless the variable coefficients satisfy certain constraints. We used the outcome of the truncated Painlevé expansion to construct the Bäcklund transformation, and three families of new analytical solutions for the vccKdV –B equation are obtained. The dispersion relation and its characteristics are illustrated. The stability for the vccKdV–B equation is analyzed by using the phase portrait method.

scholarly journals Nonlinear Adaptive Boundary Control of the Modified Generalized Korteweg–de Vries–Burgers Equation

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-18 ◽  
Author(s):  
B. Chentouf ◽  
N. Smaoui ◽  
A. Alalabi
Keyword(s):  
Boundary Control ◽  
Burgers Equation ◽  
Lyapunov Theory ◽  
Physical Parameters ◽  
Control Laws ◽  
Nonlinear Adaptive Control ◽  
Control Schemes ◽  
De Vries ◽  
Adaptive Boundary Control ◽  
Stability Of The Solution

In this paper, we study the nonlinear adaptive boundary control problem of the modified generalized Korteweg–de Vries–Burgers equation (MGKdVB) when the spatial domain is 0,1. Four different nonlinear adaptive control laws are designed for the MGKdVB equation without assuming the nullity of the physical parameters ν, μ, γ1, and γ2 and depending whether these parameters are known or unknown. Then, using Lyapunov theory, the L2-global exponential stability of the solution is proven in each case. Finally, numerical simulations are presented to illustrate the developed control schemes.

Stability Analysis of a Bilateral Teleoperating System

1993 ◽  
Vol 115 (3) ◽  
pp. 419-426 ◽  
Author(s):  
Y. Strassberg ◽  
A. A. Goldenberg ◽  
J. K. Mills
Keyword(s):  
Feedback Linearization ◽  
Closed Loop ◽  
Tracking Error ◽  
Degree Of Freedom ◽  
Closed Loop System ◽  
Control Scheme ◽  
Feedback Linearization Control ◽  
Small Gain ◽  
The Stability ◽  
Three Degree Of Freedom

In this paper the stability of a control scheme for bilateral master-slave teleoperation is investigated. Given the nominal models of the master and slave dynamics, and using an approximate feedback linearization control, based on the earlier work of Spong and Vidyasagar, 1987, a robust closed-loop system (position and force) can be obtained with a multiloop version of the small gain theorem. It is shown that stable bilateral teleoperating systems can be achieved under the assumption that the deviation of the models from the actual systems satisfies certain norm inequalities. We also show that, using the proposed scheme, the tracking error (position/velocity and force/torque) is bounded and it can be made arbitrarily small. The control scheme is illustrated using the simulation of a three-degree-of-freedom master-slave teleoperator (three-degree-of-freedom master and three-degree-of-freedom slave).

A computationally efficient adaptive robust control scheme for a quad-rotor transporting cable-suspended payloads

2021 ◽  
pp. 095441002110136
Author(s):  
Nasim Ullah ◽  
Irfan Sami ◽  
Wang Shaoping ◽  
Hamid Mukhtar ◽  
Xingjian Wang ◽  
...  
Keyword(s):  
Robust Control ◽  
Fractional Order ◽  
Sliding Mode ◽  
Adaptive Robust Control ◽  
Computationally Efficient ◽  
Control Scheme ◽  
Control Schemes ◽  
Adaptive Laws ◽  
Unknown Disturbances ◽  
The Stability

This article proposes a computationally efficient adaptive robust control scheme for a quad-rotor with cable-suspended payloads. Motion of payload introduces unknown disturbances that affect the performance of the quad-rotor controlled with conventional schemes, thus novel adaptive robust controllers with both integer- and fractional-order dynamics are proposed for the trajectory tracking of quad-rotor with cable-suspended payload. The disturbances acting on quad-rotor due to the payload motion are estimated by utilizing adaptive laws derived from integer- and fractional-order Lyapunov functions. The stability of the proposed control systems is guaranteed using integer- and fractional-order Lyapunov theorems. Overall, three variants of the control schemes, namely adaptive fractional-order sliding mode (AFSMC), adaptive sliding mode (ASMC), and classical Sliding mode controllers (SMC)s) are tested using processor in the loop experiments, and based on the two performance indicators, namely robustness and computational resource utilization, the best control scheme is evaluated. From the results presented, it is verified that ASMC scheme exhibits comparable robustness as of SMC and AFSMC, while it utilizes less sources as compared to AFSMC.

Stability and instability of some nonlinear dispersive solitary waves in higher dimension

1996 ◽  
Vol 126 (1) ◽  
pp. 89-112 ◽  
Author(s):  
Anne de Bouard
Keyword(s):  
Solitary Waves ◽  
Acoustic Waves ◽  
Vries Equation ◽  
Three Dimensional ◽  
Higher Dimension ◽  
Ion Acoustic Waves ◽  
Stability And Instability ◽  
Ion Acoustic ◽  
De Vries ◽  
The Stability

We study the stability of positive radially symmetric solitary waves for a three dimensional generalisation of the Korteweg de Vries equation, which describes nonlinear ion-acoustic waves in a magnetised plasma, and for a generalisation in dimension two of the Benjamin–Bona–Mahony equation.

Sign in / Sign up

Export Citation Format

Share Document