Robust Direct Data-Driven Control for Probabilistic Systems

TL;DR

This work derives lower bounds on the amount of data required to achieve quadratic stability for probabilistic systems with aleatoric uncertainty, and finds that the learned controllers generalize well to high variations in the dynamics even when based on only a few short open-loop trajectories.

Abstract

We propose a data-driven control method for systems with aleatoric uncertainty, for example, robot fleets with variations between agents. Our method leverages shared trajectory data to increase the robustness of the designed controller and thus facilitate transfer to new variations without the need for prior parameter and uncertainty estimations. In contrast to existing work on experience transfer for performance, our approach focuses on robustness and uses data collected from multiple realizations to guarantee generalization to unseen ones. Our method is based on scenario optimization combined with recent formulations for direct data-driven control. We derive lower bounds on the amount of data required to achieve quadratic stability for probabilistic systems with aleatoric uncertainty and demonstrate the benefits of our data-driven method through a numerical example. We find that the learned controllers generalize well to high variations in the dynamics even when based on only a few short open-loop trajectories. Robust experience transfer enables the design of safe and robust controllers that work out of the box without any additional learning during deployment.

Figures (5)

• Figure 1: Sketch of the proposed method: A controller is based on a set of collected trajectory data from $N$ systems with variations (here symbolized as quadcopters). The resulting controller is guaranteed to work on any unseen system drawn from the same probability distribution with high probability.
• Figure 2: Illustrative example of scenario optimization for a one-dimensional probabilistic system: Resulting controller robustly stabilizes the union of the sets $\Sigma_{\tau_{1}}\dots \Sigma_{\tau_{4}}$. The more samples are available, the more likely it is that the controller stabilizes a sample from $\Theta$ (cf. Theorem \ref{['th:scenario_rddc']}). The fraction of $\Theta$ that remains unstable corresponds to $\alpha$.
• Figure 3: Synthetic example with different number of observed systems and increasing variance of the probabilistic system: The probability that the controller stabilizes a sample from the probabilistic system increases as more samples become available. For high variance the uncertainty set becomes too large and the synthesis returns no controller ($\times$). The frequency of stability is averaged over $50$ different data sets and the size of the dots indicates the average frequency at which the data is informative.
• Figure 4: Synthetic example with different number of observed systems and varying trajectory length ($\sigma^2=0.1$): The robustness is largely unaffected by the trajectory length but longer trajectories increase data informativity and the method is able to find controllers more often.
• Figure 5: Uncertainty sets for varying trajectory lengths: The columns represent rollouts with different noise realizations. For short trajectories, the form, the position, and the size of the uncertainty set vary significantly across different rollouts. Longer trajectories provide more consistent uncertainty sets.

Theorems & Definitions (11)

• Definition 1: Probabilistic linear system
• Definition 2: $\alpha$-probabilistic robust controller
• Definition 3: Generalized Slater's condition
• Remark 1
• Lemma 1: waarde2022noisy
• Definition 4: Informativity for quadratic stabilization of finite sets of systems
• Theorem 1
• proof
• Definition 5: Informativity for $\alpha$-probabilistic quadratic stabilization
• Theorem 2