Kalman filter-based dynamic mode decomposition for closed-loop flow control with noise-corrupted sensing

The increasing adoption of dynamic mode decomposition with control (DMDc) in closed-loop flow control highlights a critical gap, with success in idealized conditions contrasting sharply with compromised accuracy in noise-prone scenarios. This study responds by presenting a rigorous analysis of the systematic bias induced by measurement noise and proposing the Kalman filter-based DMDc (KFDMDc) as a tailored solution. This analysis reveals the quantitative dependence of the bias on both data size and noise level, and demonstrates how noise confined to state measurements propagates nonlinearly to the control matrix. Moreover, the proposed KFDMDc algorithm directly alleviates systematic bias by employing a Kalman filter to estimate the true underlying states from noisy measurements, resulting in a significantly more accurate system identification. Numerical investigations on synthetic systems reveal a critical trade-off: the well-recognized accuracy of established diagnostic algorithms is often achieved at the expense of robustness and computational efficiency. As a result, these algorithms perform poorly when applied to time-varying controlled flow systems. In contrast, KFDMDc achieves a more favorable balance between robustness and satisfactory accuracy. The practical effectiveness of the proposed method is confirmed through closed-loop flow control simulations, achieving a $22.47\%$ reduction in convergence time at low noise levels while suppressing input perturbations by $74.63\%$ in high-noise regimes. Given the ubiquity of sensor noise in physical systems, the proposed KFDMDc provides a promising bridge between noise-free scenarios and the practical application of data-driven control.