Distributed Control of Two-Dimensional Navier–Stokes Equations in Fourier Spectral Simulations

A method for distributed control of nonlinear flow equations is proposed. In this method, first, Takagi–Sugeno (T–S) fuzzy model is used to substitute the nonlinear partial differential equations (PDEs) governing the system by a set of linear PDEs, such that their fuzzy composition exactly recovers the original nonlinear equations. This is done to alleviate the mode-interaction phenomenon occurring in spectral treatment of nonlinear equations. Then, each of the so-obtained linear equations is converted to a set of ordinary differential equations (ODEs) using the fast Fourier transform (FFT) technique. Thus, the combination of T–S method and FFT technique leads to a number of ODEs for each grid point. For the stabilization of the dynamics of each grid point, the use is made of the parallel distributed compensation (PDC) method. The stability of the proposed control method is proved using the second Lyapunov theorem for fuzzy systems. In order to solve the nonlinear flow equation, a combination of FFT and Runge–Kutta methodologies is implemented. Simulation studies show the performance of the proposed method, for example, the smaller settling time and overshoot and also its relatively robustness with respect to the measurement noises.