Residual dynamic mode decomposition with control for nonlinear dynamic systems

Abstract The Koopman theory has received increasing attention for constructing reduced-order models (ROMs) as it converts a nonlinear dynamical system to a linear one in a higher-dimensional space. This ability to rapidly and unintrusively model nonlinear dynamics yields a promising system identification method for digital twins (DTs), which often handle large amounts of highly nonlinear data. However, conventional ROMs based on Koopman are often made more complicated and sometimes inaccurate by unnecessary spurious eigenvalue-eigenvector pairs. This paper leverages the recent method of residual dynamic mode decomposition (ResDMD) for autonomous systems, and extends its $$\epsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϵ</mml:mi> </mml:math> -pseudospectrum residual computation to Koopman with inputs and control (KIC), for systems with control. This novel ResDMD with control (ResDMDc) procedure eliminates spurious eigenpairs and results in low-rank and accurate linear ROMs for nonlinear dynamics. The method was tested on a nonlinear aeroelastic problem with exogenous inputs, a nonlinear Van der Pol oscillator problem, and a nonlinear Burgers’ equation problem and demonstrated high accuracy compared to conventional mode truncation strategies.