Polynomial-Time Reachability for LTI Systems With Two-Level Lattice Neural Network Controllers

In this letter, we consider the computational complexity of bounding the reachable set of a Linear Time-Invariant (LTI) system controlled by a Rectified Linear Unit (ReLU) Two-Level Lattice (TLL) Neural Network (NN) controller. In particular, we show that for such a system and controller, it is possible to compute the exact one-step reachable set in polynomial time in the size of the TLL NN controller (number of neurons). Additionally, we show that a tight bounding box of the reachable set is computable via two polynomial-time methods: one with polynomial complexity in the size of the TLL and the other with polynomial complexity in the Lipschitz constant of the controller and other problem parameters. Finally, we propose a pragmatic algorithm that adaptively combines the benefits of (semi-)exact reachability and approximate reachability, which we call L-TLLBox. We evaluate L-TLLBox with an empirical comparison to a state-of-the-art NN controller reachability tool. In our experiments, L-TLLBox completed reachability analysis as much as
$5000\times$
faster than this tool on the same network/system, while producing reach boxes that were from 0.08 to 1.42 times the area.