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.

Figures (2)

• Figure 1: (a) The mean trajectory and the safety envelope over $50$ simulation runs with arbitrary initial conditions. The blue curve represents the mean state trajectory, while the black curves denote the pointwise minimum and maximum across all trajectories at each time instant. The results show that the system trajectories remain within the prescribed safety bounds for all runs. (b) The error between the output trajectories of the two systems in each simulation run, showing that the obtained closeness guarantee is not violated.
• Figure 2: (a) The trajectories of the original system for $100$ outbound trips starting from different initial conditions to the target and $100$ return trips from to , while avoiding the obstacles . One representative outbound trip and one representative return trip are depicted in bold, whereas the rest are shown in a faded style. (b) The error between the output trajectories of the two systems in each simulation run, illustrating that the obtained closeness guarantee is not violated. (c) A representative system trajectory after applying a smoothing filter to attenuate the zigzag behavior observed in (a), stemming from the SCOTS synthesis tool, due primarily to the chosen discretization parameter. This simulation's video is available at https://youtu.be/3N1B1E1GQIs.

Theorems & Definitions (23)

• Definition 1: ct-NCS
• Remark 1: On Assumption \ref{['assump-on-D']}
• Remark 2: On the Lifting of $f(x)$ to $\mathcal{D}(x)$
• Definition 2: ct-ROM
• Definition 3: SF
• Remark 3: Interface Function
• Theorem 1: Closeness Guarantee
• proof
• Remark 4: On Theorem \ref{['thm: cont: model-based']}
• Remark 5: Complex Specifications Enforcement