Koopman-Hopf Hamilton-Jacobi Reachability and Control

Will Sharpless, Nikhil Uday Shinde, Matthew Kim, Yat Tin Chow, Sylvia Lee Herbert

Abstract:

Abstract— The Hopf formula for Hamilton-Jacobi reachability (HJR) analysis has been proposed to solve high-dimensional
differential games, producing the set of initial states and
corresponding controller required to reach (or avoid) a target despite bounded disturbances. As a space-parallelizable
method, the Hopf formula avoids the curse of dimensionality
that afflicts standard dynamic-programming HJR, but is restricted to linear time-varying systems. To compute reachable
sets for high-dimensional nonlinear systems, we pair the Hopf
solution with Koopman theory for global linearization. By
first lifting a nonlinear system to a linear space and then
solving the Hopf formula, approximate reachable sets can
be efficiently computed that are much more accurate than
local linearizations. Furthermore, we construct a KoopmanHopf disturbance-rejecting controller, and test its ability to
drive a 10-dimensional nonlinear glycolysis model. We find
that it significantly out-competes expectation-minimizing and
game-theoretic model predictive controllers with the same
Koopman linearization in the presence of bounded stochastic
disturbance. In summary, we demonstrate a dimension-robust
method to approximately solve HJR, allowing novel application
to analyze and control high-dimensional, nonlinear systems with
disturbance. An open-source toolbox in Julia is introduced for
both Hopf and Koopman-Hopf reachability and control.