Safe learning of regions of attraction for uncertain, nonlinear systems with Gaussian processes

Abstract

Abstract-Control theory can provide useful insights into the properties of controlled, dynamic systems. One important property of nonlinear systems is the region of attraction (ROA), a safe subset of the state space in which a given controller renders an equilibrium point asymptotically stable. The ROA is typically estimated based on a model of the system. However, since models are only an approximation of the real world, the resulting estimated safe region can contain states outside the ROA of the real system. This is not acceptable in safety-critical applications. In this paper, we consider an approach that learns the ROA from experiments on a real system, without ever leaving the true ROA and, thus, without risking safety-critical failures. Based on regularity assumptions on the model errors in terms of a Gaussian process prior, we use an underlying Lyapunov function in order to determine a region in which an equilibrium point is asymptotically stable with high probability. Moreover, we provide an algorithm to actively and safely explore the state space in order to expand the ROA estimate. We demonstrate the effectiveness of this method in simulation.

Feedback Linearization Based on Gaussian Processes With Event-Triggered Online Learning

“…However, none of these techniques updates the model while controlling the system. The work in [28] proposes a safe exploration by sequentially adding training points to the dataset, but it only stays within the region of attraction and cannot track an arbitrary trajectory in the state space.…”

“…In practice, it is often sufficient to set m(x) to a positive constant. To verify that µ(x) > 0 holds, the techniques in [28] can be utilized. The suitable prior mean function according to Lemma 2 will be denoted by m g (x).…”

“…Additionally, we set β κ constant ∀ κ and refer to the discussion in Sec. V-D and [28]. Additionally, we enforce a lower bound r > r min to avoid numerical difficulties.…”

Feedback linearization using Gaussian processes

“…There surely exist better methods for the exploration-exploitation trade-off, than the used PD controller, see [16], yet this is out of the scope of this work.…”

“…There exist methods to compute arbitrarily tight bounds for µ m=0 (x) on a compact set [16]. Thus, without elaborating further details, we assume a b µ is known, which is set as constant prior mean, for which µ(x) > 0, ∀x ∈ X .…”