Constrained optimized dynamic mode decomposition with control for physically stable systems with exogeneous inputs

Reduced order models (ROMs) generated by the dynamic mode decomposition (DMD) are sensitive to the selection of DMD hyperparameters and suffer from spurious eigenvalue issues. To address these challenges, this paper presents a novel method named constrained optimized DMD with Control (cOptDMDc) that extends optimized DMD to systems with exogenous inputs and can optionally enforce the stability of the resulting ROM. The variable projection and the exponential data fitting approaches are reformulated to consider temporal effects of exogenous inputs on state evolution. A new process combining iterative exponential data fitting and a linear inequality constraint is proposed to optimally place eigenvalues within the stable region, yielding a suboptimal spectral structure, which is more robust and effective to mitigate spurious eigenvalue issues and alleviate the brittleness of DMDc. Three case studies are conducted to compare the proposed cOptDMDc with the native Dynamic Mode Decomposition with Control (DMDc) and Sparsity-Promoting DMDc (SPDMDc). The results prove that the cOptDMDc method is accurate with maximum relative error < 0.5% and more robust to the choice of ROM subspace dimensions, thus eliminating the need for a trial-and-error process and additional validation data for ROM construction.

Automated summary

This paper presents a novel method called constrained optimized DMD with Control (cOptDMDc), which extends the optimized DMD method to systems with exogenous inputs and can enforce the stability of the resulting reduced order model (ROM). The proposed method optimally places eigenvalues within the stable region, thus mitigating spurious eigenvalue issues. Comparative studies show that cOptDMDc achieves high accuracy and robustness.