Gramians for a New Class of Nonlinear Control Systems Using Koopman and a Novel Generalized SVD
This provides a method for certified model reduction in nonlinear control systems, addressing a bottleneck in control theory for applications like neural networks, but it is incremental as it builds on existing Koopman and GSVD approaches.
The paper tackled the challenge of certified model reduction for high-dimensional nonlinear control systems by introducing a Generalized Singular Value Decomposition (GSVD)-based construction that preserves input-energy metrics, enabling computable Hankel-singular-value-based error certificates in physical input norms. It demonstrated this on a 25-dimensional Hodgkin-Huxley network, achieving reduction to a single dominant mode with certified error bounds.
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.