Smart abstraction based on iterative cover and non-uniform cells

TL;DR

This paper introduces a multi-scale abstraction method for dynamical systems that combines local and global optimal control, utilizing non-uniform cells and an RRT*-based approach to efficiently handle high-dimensional problems.

Contribution

It presents a novel approach for constructing goal-specific, non-uniform state space abstractions using iterative local and global optimization, improving scalability for complex systems.

Findings

Reduces abstraction complexity with non-uniform cells.

Effectively handles high-dimensional systems.

Provides a practical implementation for nonlinear dynamics.

Abstract

We propose a multi-scale approach for computing abstractions of dynamical systems, that incorporates both local and global optimal control to construct a goal-specific abstraction. For a local optimal control problem, we not only design the controller ensuring the transition between every two subsets (cells) of the state space but also incorporate the volume and shape of these cells into the optimization process. This integrated approach enables the design of non-uniform cells, effectively reducing the complexity of the abstraction. These local optimal controllers are then combined into a digraph, which is globally optimized to obtain the entire trajectory. The global optimizer attempts to lazily build the abstraction along the optimal trajectory, which is less affected by an increase in the number of dimensions. Since the optimal trajectory is generally unknown in practice, we propose a methodology based on the RRT* algorithm to determine it incrementally. Finally, we provide a tractable implementation of this algorithm for the optimal control of L-smooth nonlinear dynamical systems.