Data-Driven Model Order Reduction of Nonlinear Systems with Noisy Data

Abstract

Model order reduction techniques simplify high-dimensional dynamical systems by deriving lower-dimensional models that retain essential system characteristics. These techniques are crucial for the controller design of complex systems while significantly reducing computational costs. Nevertheless, constructing effective reduced-order models (ROMs) poses considerable challenges, particularly for nonlinear dynamical systems. These challenges are further exacerbated when the actual system model is unavailable, a scenario frequently encountered in real-world applications. In this work, we propose a data-driven framework for constructing ROMs of nonlinear dynamical systems with unknown mathematical models, enabling controller synthesis directly from the resulting ROMs. We establish similarity relations between the output trajectories of the original systems and those of their ROMs by employing the notion of simulation functions (SFs), thereby enabling a formal characterization of their closeness. To achieve this, we collect one set of noise-corrupted input-state data from the system during a finite-time experiment, upon which we propose conditions to construct both ROMs and SFs simultaneously. These conditions are formulated as data-dependent semidefinite programs. We demonstrate that the data-driven ROMs obtained can be employed to synthesize controllers for the original unknown systems, ensuring that they satisfy high-level logic specifications. This is accomplished by first designing controllers for the data-driven ROMs and then translating the results back to the original systems via interface functions, designed directly from the proposed data-dependent conditions. We evaluate the efficacy of our data-driven framework through two case studies, including a challenging benchmark from the model reduction literature: a circuit of chained inverter gates with 20 state variables.