Distributed control of nonlinear conductivity heat transfer equation in a thick functionally graded plate. (January 2022) - Shafie, Shojaat; Rahmani, Behrooz; Moosaie, Amin; Panahi-Kalus, Hamed
Abstract: In this paper, distributed control of nonhomogeneous nonlinear conductivity heat transfer equation on a single plate is studied. The inhomogeneity of the equation is due to the use of functionally graded materials. In the proposed method, the governing equation of the system, which is a nonlinear partial differential equation, is represented as some linear partial differential equation using the Takagi-Sugeno linearization technique; so that their fuzzy combination accurately retrieve the original nonlinear equation. This process is done to prevent the phenomenon of frequency disorder or aliasing. Then, the modified wavenumber method is utilized to convert the obtained linear partial differential equations to a set of ordinary differential equations. In this way, a combination of fast Fourier transform and finite difference methods is implemented. Consequently, by combining Takagi-Sugeno and the modified wavenumber strategies, several ordinary differential equations are obtained for each grid point. In this regard, the parallel distributed compensation method is used to stabilize each grid point. The second method of Lyapunov is implemented to study the stability of the proposed control strategy. Numerical Simulations have been used to demonstrate the efficacy of the proposed control scheme and also its relative robustness with respect to measurement noises and parametric uncertainties.