An adaptive data sampling strategy for stabilizing dynamical systems via controller inference

TL;DR

The paper tackles data-driven stabilization of nonlinear dynamical systems by introducing an adaptive sampling framework that stabilizes the system during data collection and yields informative data sets of minimal size. It builds a sequence of low-dimensional controllers on nested subspaces (ICI) using stabilizing inputs and projected data informativity to progressively stabilize the full system. Across several high-dimensional benchmarks, the approach achieves stabilization with up to an order of magnitude fewer data samples than unguided data generation, demonstrating robustness in edge and limit-state scenarios. This work offers a practical, data-efficient route to stabilizing complex systems and suggests future directions for integrating adjoint information to further reduce data requirements.

Abstract

Learning stabilizing controllers from data is an important task in engineering applications; however, collecting informative data is challenging because unstable systems often lead to rapidly growing or erratic trajectories. In this work, we propose an adaptive sampling scheme that generates data while simultaneously stabilizing the system to avoid instabilities during the data collection. Under mild assumptions, the approach provably generates data sets that are informative for stabilization and have minimal size. The numerical experiments demonstrate that controller inference with the novel adaptive sampling approach learns controllers with up to one order of magnitude fewer data samples than unguided data generation. The results show that the proposed approach opens the door to stabilizing systems in edge cases and limit states where instabilities often occur and data collection is inherently difficult.

Figures (9)

• Figure 1: In all experiments, the new iterative controller inference (ICI) method needs less system queries to generate data sets informative enough for stabilization of the underlying system, with improvement factors up to $25$. In the case of the power network example, we were not able to find a setup in which classical low-dimensional controller inference (ContInf) with unguided data sampling was able to stabilize the system.
• Figure 2: Continuous-time unstable heat flow: ICI uses $28$ data samples at discrete-time points to construct a controller that quickly stabilizes the simulation of the unstable heat flow.
• Figure 3: Discrete-time unstable heat flow: The new ICI approach stabilizes the system after only $14$ system evaluations, which is around $1.6$ times less evaluations needed by either of the non-adaptive approaches. ContInf from a single or four trajectories using comparable numbers of system evaluations fail to stabilize the system.
• Figure 4: ICI convergence behavior for discrete-time unstable heat flow: The convergence of ICI is illustrated in terms of the approximation error of new states in the current projection space $\mathcal{V}_{j}$ and the relative change of the controller between steps. The peaks of the projection error indicate re-projections of the data, which are regularly needed for accurate orthogonalizations, but not if the data is more aggressively truncated with a large orthogonalization tolerance. In both cases, the distance of the trajectory to the steady state decreases, indicating stabilization.
• Figure 5: Discrete-time power network: In the uncontrolled simulation, the output trajectories tend to infinity due to the instability of the power network. ICI provides a stabilizing controller after $20$ queries. The comparing low-dimensional controller inference setups did not provide a stabilizing controller within our limit of $300$ model queries.

Theorems & Definitions (15)

• Proposition 1: Data informativity for stabilization VanETetal20WerP24
• Proposition 2: Low-dimensional data informativity WerP24
• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 1
• proof
• Corollary 1
• proof