A High-Efficient Finite Difference Method for Flexible Manipulator with Boundary Feedback Control

The paper presents a high-efficient finite difference method for solving the PDE model of the single-link flexible manipulator system with boundary feedback control. Firstly, an abstract state-space model of the manipulator is derived from the original PDE model and the associated boundary conditions of the manipulator by using the velocity and bending curvature of the flexible link as the state variables. Then, the second-order implicit Crank-Nicolson scheme is adopted to discretize the state-space equation, and the second-order one-sided approximation is used to discretize the boundary conditions with excitations and feedback control. At last, the state-space equation combined with the boundary conditions of the flexible manipulator is transformed to a system of linear algebraic equations, from which the response of the flexible manipulator can be easily solved. Numerical simulations are carried out to simulate the manipulator under various excitations and boundary feedback control. The results are compared with ANSYS to demonstrate the accuracy and high efficiency of the presented method.

Global Stabilization of Two Dimensional Viscous Burgers’ Equation by Nonlinear Neumann Boundary Feedback Control and Its Finite Element Analysis

In this article, global stabilization results for the two dimensional viscous Burgers’ equation, that is, convergence of unsteady solution to its constant steady state solution with any initial data, are established using a nonlinear Neumann boundary feedback control law. Then, applying $$C^0$$ C 0 -conforming finite element method in spatial direction, optimal error estimates in $$L^\infty (L^2)$$ L ∞ ( L 2 ) and in $$L^\infty (H^1)$$ L ∞ ( H 1 ) -norms for the state variable and convergence result for the boundary feedback control law are derived. All the results preserve exponential stabilization property. Finally, several numerical experiments are conducted to confirm our theoretical findings.