Control of Nonlinear PDEs based on Space Vectors Clustering reduced order systems ?

Nonlinear PDEs domain discretization yields finite but high dimensional nonlinear systems. Proper Orthogonal Decomposition (POD) is widely used to reduce the order of such systems but it assumes that data belongs to a linear space and therefore fails to capture the nonlinear degrees of freedom. To overcome this problem, we develop a Space Vector Clustering (SVC) POD and use the reduced order model to design the controller which will then be applied to the full order system. A space vector is the solution at a particular space location over all times. The solution space is grouped into clusters where the behavior has significantly different features, then local POD modes will be constructed based on these clusters. We apply our method to reduce and control a nonlinear convective PDE system governed by the Burgers’ equation over 1D and 2D domains and show a significant improvement over global POD. We also design a reference tracking controller and compare the controlled systems. We show that the controller based on our SVC local POD reduced system yields more accurate tracking results over the one based on global POD.