GPU-Parallel Linearization Error Bounds for Real-Time Robust Optimal Control of Nonlinear and Neural Network Dynamics

Sep 21, 2026

This paper develops real-time robust optimal control for uncertain nonlinear systems by using linear time-varying (LTV) approximations that are only valid when sound linearization error bounds (LEBs) exist. It introduces tight, differentiable, GPU-parallel LEBs for both analytic nonlinear dynamics and neural network (NN) dynamics, using path-based Hessian bounds for analytic models and certified verifier-based affine relaxations plus local Jacobian corrections for NNs. The resulting GPUSLS-LEO solver extends system-level synthesis to handle right-invertible disturbance matrices and nonzero-centered disturbance sets, enabling online optimization with formally verified reachable tubes and up to 67 Hz performance on systems with up to 168 states.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages13

Reading Time33 min

AccessFree to download

Introduction

Linearization Error Bounds (LEBs)

Analytic dynamics via path-based Hessian bounds

Neural network dynamics via certified affine relaxations

GPUSLS-LEO GPU-parallel solver

Right-invertible disturbance matrices and zonotopic propagation

Nonzero-centered disturbance sets

Evaluation and results

GPU-Parallel Linearization Error Bounds for Real-Time Robust Optimal Control of Nonlinear and Neural Network Dynamics arXiv :2607 .01203v1 [ ee ss . SY] 1 Jul 2026 Jeffrey Fang⋆ Keyi Shen⋆ Abstract—This paper studies real-time robust optimal control for uncertain nonlinear systems, where linear time-varying (LTV) approximations make planning tractable but require sound linearization error bounds (LEBs) to guarantee robust constraint satisfaction. We develop tight, differentiable, GPUparallel LEBs for LTV approximations of nonlinear and neural network (NN) dynamics. For analytic dynamics, we introduce path-based Hessian bounds that are tighter than standard interval methods. For NN dynamics, we derive certified LEBs using NN verifier-generated affine relaxations and local Jacobian corrections. We adapt a GPU-parallel system-level synthesis LTV-based robust control solver to be compatible with these LEBs by extending it to handle right-invertible disturbance matrices and non-zero-centered disturbance sets for tight zonotopic uncertainty propagation. Our method, GPUSLS-LEO, enables online optimization of robust feedback policies that account for linearization error, producing tight, formally verified reachable tubes. On complex nonlinear and NN dynamics up to 168 state dimensions, our method can compute robust control policies on the GPU at rates up to 67 Hz, reducing solve times and conservativeness relative to baselines while preserving formal guarantees and real-time performance. 􀂀 Project Website § Code (GitHub) I. INTRODUCTION Safe real-time nonlinear control under uncertainty is essential for the resilient operation of robots, drones, and spacecraft. This motivates methods for solving the robust nonlinear optimal control problem (RNOCP), which seeks a control policy that robustly guarantees constraint satisfaction under worst-case disturbances. In practice, RNOCP solvers typically optimize (1) a nominal trajectory and (2) a stabilizing feedback controller to track it under disturbance. Robust constraint satisfaction is enforced by tightening the nominal constraints using an overapproximation of the closed-loop tracking-error reachable tube, ensuring that the closed-loop dynamics remain in a safe subset of the state and input space. However, exact reachability via error propagation through nonlinear dynamics [1]–[3] is generally intractable in realtime. To address this, [4]–[7] approximate the nonlinear dynamics along a nominal trajectory with a linear time-varying (LTV) system and compute reachable sets using the LTV dynamics. System-level synthesis (SLS) [4], [8] is one such LTV robust control framework that can be solved in real time over long horizons for high-dimensional systems [9], [10] . Despite strong performance, LTV-based methods only guarantee robust constraint satisfaction given a bound on the linearization error, which captures the deviation between the ⋆ Equal contribution. All authors are with the Georgia Institute of Technology, Atlanta, GA, USA, 30308 . {jfang301, kshen84, asrinivasan350, [chou](chou}@gatech.edu)[}](chou}@gatech.edu)[@gatech.edu](chou}@gatech.edu) Anutam Srinivasan Glen Chou (a) (b) 􀀅 􀀆 􀀈 􀀊 􀀋􀀐􀀄􀀇􀀒 􀀑􀀌􀀉 Fig. 1. (a): Robust tubes from GPUSLS-LEO for a neural T pusher system and disturbed rollouts for a rotation and a push trajectory against random disturbances. (b): MPC rollout using our method on real dynamics, successfully moving the T to the goal. (c): Robust tubes of x and y position showing tight tubes and all simulated rollouts staying within the tubes. true nonlinear dynamics and their LTV approximation. Existing methods often use global bounds [4], [11], which yield conservative (loose) reachable tubes. Tighter local bounds around the nominal trajectory are often possible, but must be computed efficiently for real-time planning and informative enough to guide the optimizer toward safe trajectories with minimal linearization error. Thus, the challenge is to

FAQs

What is the main problem addressed by this paper?

The paper addresses real-time robust optimal control under uncertainty, where LTV approximations must be supported by accurate linearization error bounds to ensure robust constraint satisfaction.

How are linearization error bounds computed for analytic nonlinear dynamics?

For analytic dynamics, the paper introduces path-based Hessian bounds that are tighter than standard interval methods, producing differentiable bounds suitable for online optimization.

How does GPUSLS-LEO enable real-time verified control?

GPUSLS-LEO adapts an SLS-based LTV robust control solver to work with the proposed differentiable GPU-parallel LEBs by handling right-invertible disturbance matrices and nonzero-centered disturbance sets, yielding formally verified reachable tubes at high GPU rates.

Sep 21, 2026

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