Safely Learning Dynamical Systems
Amir Ali Ahmadi Abraar Chaudhry Vikas Sindhwani Stephen Tu
A fundamental challenge in learning an unknown dynamical system is to reduce
model uncertainty by making measurements while maintaining safety. In this
work, we formulate a mathematical definition of what it means to safely learn a
dynamical system by sequentially deciding where to initialize the next
trajectory. In our framework, the state of the system is required to 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 the information gathered so far.
  For our first set of results, we consider the setting of safely learning a
linear dynamical system involving $n$ states. For the case $T=1$, we present a
linear programming-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 a semidefinite representation of the set of safe
initial conditions and show that $\lceil n/2 \rceil$ trajectories generically
suffice for safe learning. Finally, for $T = \infty$, we provide semidefinite
representable inner approximations of the set of safe initial conditions and
show that one trajectory generically suffices for safe learning.
  Our second set of results concerns the problem of safely learning a general
class of nonlinear dynamical systems. For the case $T=1$, we give a
second-order cone programming 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.