A Cluster-POD Based Model Reduction Method for Boundary Control of Fluid Flows

Efficient reduced-order models are needed for real-time simulation and control of high-dimensional nonlinear partial differential equations (PDEs). Global Proper Orthogonal Decomposition (POD) often underperforms because it assumes a single linear subspace, which is inadequate for nonlinear manifolds. We propose a Cluster-POD approach that partitions snapshot data and computes local POD bases, with manifold distances approximated locally using Euclidean metrics. Applied to the two-dimensional (2D) Burgers' equations, Cluster-POD achieves lower reconstruction error than standard POD for the same number of modes. A linear quadratic regulator (LQR) is designed using the reduced-order model and validated in closed loop on the full-order system, demonstrating improved tracking and stability with fewer modes and comparable control cost. The Burgers' equations are used as a proxy for Navier–Stokes due to their shared quadratic nonlinearity and lower computational cost. Overall, Cluster-POD offers an effective balance between accuracy and computational efficiency for data-driven control of nonlinear flows.