Scalable Underapproximation for the Stochastic Reach-Avoid Problem for High-Dimensional LTI Systems using Fourier Transforms
It addresses the scalability bottleneck in stochastic reach-avoid verification for high-dimensional systems, which is crucial for safety-critical applications.
The paper presents a scalable underapproximation method for the stochastic reach-avoid probability in high-dimensional LTI systems using Fourier transforms, enabling verification of systems with up to 40 states.
We present a scalable underapproximation of the terminal hitting time stochastic reach-avoid probability at a given initial condition, for verification of high-dimensional stochastic LTI systems. While several approximation techniques have been proposed to alleviate the curse of dimensionality associated with dynamic programming, these techniques are limited and cannot handle larger, more realistic systems. We present a scalable method that uses Fourier transforms to compute an underapproximation of the reach-avoid probability for systems with disturbances with arbitrary probability densities. We characterize sufficient conditions for Borel-measurability of the value functions. We exploit fixed control sequences parameterized by the initial condition (an open-loop control policy) to generate the underapproximation. For Gaussian disturbances, the underapproximation can be obtained using existing efficient algorithms by solving a convex optimization problem. Our approach produces non-trivial lower bounds and is demonstrated on a chain of integrators with 40 states.