{"ID":2898218,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2507.04188","arxiv_id":"2507.04188","title":"Gramians for a New Class of Nonlinear Control Systems Using Koopman and a Novel Generalized SVD","abstract":"Certified model reduction for high-dimensional nonlinear control systems remains challenging: unlike balanced truncation for LTI systems, most nonlinear reduction methods either lack computable worst-case error bounds or rely on intractable PDEs. Data-driven Koopman/DMDc surrogates improve tractability, but standard \\emph{input lifting} can distort the physical input-energy metric, so $H_\\infty$ and Hankel-based bounds computed on the lifted model may be valid only in a lifted-input norm and need not certify the original system. We address this metric mismatch by a Generalized Singular Value Decomposition (GSVD)-based construction that represents general (including non-affine) input nonlinearities in an LTI-like lifted form with a \\emph{pointwise norm-preserving} input map $v(x,u)$ satisfying $\\|v(x,u)\\|_2=\\|u\\|_2$ and constant matrices $A,B$. This preserves strict causality (constant $B$, no input-history augmentation) and yields computable Hankel-singular-value-based $H_\\infty$ error certificates in the physical input norm for reduced-order surrogates. We illustrate the method on a 25-dimensional Hodgkin--Huxley network with saturating optogenetic actuation, reducing to a single dominant mode while retaining certified error bounds.","short_abstract":"Certified model reduction for high-dimensional nonlinear control systems remains challenging: unlike balanced truncation for LTI systems, most nonlinear reduction methods either lack computable worst-case error bounds or rely on intractable PDEs. Data-driven Koopman/DMDc surrogates improve tractability, but standard \\e...","url_abs":"https://arxiv.org/abs/2507.04188","url_pdf":"https://arxiv.org/pdf/2507.04188v2","authors":"[\"Brian Brown\",\"Michael King\"]","published":"2025-07-05T23:44:19Z","proceeding":"math.OC","tasks":"[\"math.OC\",\"eess.SY\"]","methods":"[]","has_code":false}