Researchers present a new method that computes tight error bounds for linear approximations of nonlinear and neural network dynamics on a GPU. Those bounds let a real-time robust controller account for the mismatch between a nonlinear system and the simpler linear model used for planning. The code, called GPUSLS-LEO, is written in JAX and is designed so the optimizer can use the bounds directly while running on the GPU.

The problem they address is common in robotics, drones, and spacecraft: planners often use a linear time-varying (LTV) approximation of a nonlinear system because it is much easier to optimize. But an LTV model is only useful if one can reliably bound the linearization error — the difference between the true nonlinear dynamics and the linear approximation. If those bounds are loose, the planner must be conservative. If they are slow or not differentiable, they cannot be used inside a real-time optimizer.

To fix this, the authors develop several new kinds of linearization error bounds (LEBs). For analytic dynamics they introduce path-based Hessian bounds, which are tighter than standard interval methods. For neural-network dynamics they derive certified bounds using neural-network verifier tools to make affine relaxations, then add local Jacobian corrections. They also make these bounds differentiable and evaluate them in parallel on the GPU to keep computation fast enough for online use.

The paper also adapts a system-level synthesis (SLS) solver that designs disturbance-feedback controllers from LTV models. The authors extended that solver to handle right-invertible disturbance matrices and disturbance sets that are not centered at zero. Those extensions let them propagate uncertainty as zonotopes — a compact geometric representation of reachable error — without conservative approximations. Together, the bounds and solver produce formally verified “reachable tubes” that enclose all possible states under worst-case disturbances and linearization error.