We address the computational bottleneck of linearization error bound (LEB) computation for robust model predictive control (RMPC) of systems with nonlinear and neural network (NN) dynamics. By exploiting GPU parallelism, we show that tight, local LEBs can be computed online while providing both real-time performance on high-dimensional systems and hard guarantees.
Related Work
As LEBs can be slow to compute, they are often generated once offline but are loose if we wish to bound error around one trajectory, as in RMPC. Thus, practitioners often omit linearization error, yielding heuristic reachable tubes that trade formal guarantees for performance. To address this, we show that tight, local, GPU-parallel LEBs can be computed online while providing both real-time performance on high-dimensional systems and hard guarantees.
Differentiable, Parallel Bound Implementation
We implement our LEBs in JAX, with the PBH using immrax for interval arithmetic and jax_verify providing CROWN-based bounds for NN dynamics. JAX offers differentiability and GPU parallelization, enabling gradient-based refinement of the nominal point the linearization point and scalable input domain partitioning to reduce conservativeness for SLS.
Using Linearization Error Bounds in GPUSLS
We discuss the use of our LEBs in SLS and describe an efficient GPU-based implementation.
Using Linearization Bounds for RNOCP
In this section, we evaluate GPUSLS-LEO on a suite of analytic and neural dynamical systems against various RNOCP baselines. We further assess the impact of key ablations, including the effect of optimizing over the linearization error gradients and the use of nonzero-centered zonotopes. All benchmarked methods are evaluated under the same disturbance set for the given dynamical system and, where applicable, are run for a maximum of 100 SQP iterations.
Satellite
We use the same satellite dynamics and experimental setup as previous work. Specifically, we define the dynamics as standard satellite orbital dynamics with nonlinear gravitational effects.
Planar Quadrotor
We consider a planar quadrotor with the following dynamics, a standard model of a quadrotor constrained to motion in the vertical plane.
Frequently Asked Questions
How does GPU parallelism accelerate linearization error bound computation? GPU parallelism allows simultaneous computation of LEBs across many points in the state-input space, enabling real-time performance even for high-dimensional systems with nonlinear or neural network dynamics.
What guarantees do these linearization error bounds provide? The bounds provide hard, formal guarantees on the linearization error, making them suitable for robust model predictive control with provable safety and performance certificates.
How do these bounds compare to traditional offline-computed bounds? Traditional offline bounds are often loose for trajectory-specific applications, whereas our local, online-computed bounds are tighter and more suitable for real-time robust optimal control.
What types of dynamical systems are supported? The method supports both analytic nonlinear dynamics and neural network dynamics, leveraging interval arithmetic and CROWN-based bound propagation techniques.