– Titre : Nonlinear k-Moments: New Tools for Analysis and Design of Nonlinear Systems
– Résumé : Despite the central role of moments in nonlinear model reduction, the interpretation of the k-moment in nonlinear moment matching remains elusive, unlike the well-established linear time-invariant case. We address this fundamental gap by providing, for the first time, a concrete and rigorous interpretation of the k-moment for nonlinear systems. Our approach builds on a regular perturbation expansion of the solution to the partial differential equation that characterizes the standard nonlinear moment function. By introducing a perturbed signal generator—-comprising a nominal vector field, a dimensionless perturbation parameter, and a departure mapping—-we characterize k-moments as higher-order perturbation terms, offering a time-domain interpretation for nonlinear systems. This approach enables more accurate moment matching and improved robustness to small perturbations in the matching signal. Then, with this machinery at hand, we introduce a novel family of reduced-order models capable of matching the k-moments of the underlying system, in both linear and nonlinear contexts. To validate our theory and demonstrate the advantages of matching k-moments, we provide examples that construct reduced-order models with reduced sensitivity to small input perturbations. Furthermore, we illustrate how the notion of k-moment can be extended beyond model reduction, for instance to support the design of robust regulators.
– Biographie : Fahim Shakib received his M.Sc. degree (cum laude) in Mechanical Engineering in 2017 from the Eindhoven University of Technology, Eindhoven, the Netherlands, where he also received his Ph.D. degree (cum laude) in 2022 for his thesis entitled ‘Data-driven modeling and complexity reduction for nonlinear systems with stability guarantees’. In 2023, he worked at Thermo Fisher Scientific in Eindhoven, the Netherlands, before joining the Control and Power Group in the Department of Electrical and Electronic Engineering at Imperial College London, UK, as a Research Associate. Since 2025, he holds an Assistant Professor position in the Department of Mechanical Engineering at TU/e. His research interests include system identification, model reduction, and control of nonlinear systems with a core focus on the integration of physics-based approaches and data-based approaches.
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