Safely Learning Dynamical Systems

TL;DR

A mathematical definition of what it means to safely learn a dynamical system by sequentially deciding where to initialize trajectories is formulated and it is shown how one can safely collect trajectories and fit a polynomial model of the nonlinear dynamics that is consistent with the initial uncertainty set and best agrees with the observations.

Abstract

A fundamental challenge in learning an unknown dynamical system is to reduce model uncertainty by making measurements while maintaining safety. We formulate a mathematical definition of what it means to safely learn a dynamical system by sequentially deciding where to initialize trajectories. The state of the system must stay within a safety region for a horizon of $T$ time steps under the action of all dynamical systems that (i) belong to a given initial uncertainty set, and (ii) are consistent with information gathered so far. First, we consider safely learning a linear dynamical system involving $n$ states. For the case $T=1$, we present an LP-based algorithm that either safely recovers the true dynamics from at most $n$ trajectories, or certifies that safe learning is impossible. For $T=2$, we give an SDP representation of the set of safe initial conditions and show that $\lceil n/2 \rceil$ trajectories generically suffice for safe learning. For $T = \infty$, we provide SDP-representable inner approximations of the set of safe initial conditions and show that one trajectory generically suffices for safe learning. We extend a number of our results to the cases where the initial uncertainty set contains sparse, low-rank, or permutation matrices, or when the system has a control input. Second, we consider safely learning a general class of nonlinear dynamical systems. For the case $T=1$, we give an SOCP-based representation of the set of safe initial conditions. For $T=\infty$, we provide semidefinite representable inner approximations to the set of safe initial conditions. We show how one can safely collect trajectories and fit a polynomial model of the nonlinear dynamics that is consistent with the initial uncertainty set and best agrees with the observations. We also present some extensions to cases where the measurements are noisy or the dynamical system involves disturbances.

Figures (9)

• Figure 1: A conceptual illustration of the safe learning problem.
• Figure 2: One-step safe learning associated with the numerical example in \ref{['sec:one-step example']}.
• Figure 3: Initialization cost of the iterates chosen by alg:one-step (the online algorithm) for the distribution of four-dimensional problems described at the end of \ref{['sec:one-step example']}. The initialization cost for alg:offline-one-step (the offline algorithm) would be $4c^\mathsf{T} x_1$.
• Figure 4: One-step safe learning associated with the numerical example in \ref{['sec:dim increase']}.
• Figure 5: Two-step safe learning associated with the numerical example in \ref{['sec:two-step example']}.

Theorems & Definitions (61)

• proposition 1
• proof
• remark 1
• remark 2
• definition 1: One-Step Safe Learning
• remark 3
• lemma 1
• proof
• lemma 2
• theorem 1