Latent Representations for Control Design with Provable Stability and Safety Guarantees

TL;DR

This paper develops a theoretical framework for using low-dimensional latent representations of dynamical systems to enable verifiable control synthesis with stability and safety guarantees, addressing computational challenges.

Contribution

It introduces conjugacy conditions for dynamics-aware latent spaces and links stability guarantees from latent to original systems, with practical loss functions for learning such representations.

Findings

Validated on cartpole stabilization

Demonstrated collision avoidance in vehicle systems

Provided guidelines for geometric properties in latent space learning

Abstract

We initiate a formal study on the use of low-dimensional latent representations of dynamical systems for verifiable control synthesis. Our main goal is to enable the application of verification techniques -- such as Lyapunov or barrier functions -- that might otherwise be computationally prohibitive when applied directly to the full state representation. Towards this goal, we first provide dynamics-aware approximate conjugacy conditions which formalize the notion of reconstruction error necessary for systems analysis. We then utilize our conjugacy conditions to transfer the stability and invariance guarantees of a latent certificate function (e.g., a Lyapunov or barrier function) for a latent space controller back to the original system. Importantly, our analysis contains several important implications for learning latent spaces and dynamics, by highlighting the necessary geometric properties which need to be preserved by the latent space, in addition to providing concrete loss functions for dynamics reconstruction that are directly related to control design. We conclude by demonstrating the applicability of our theory to two case studies: (1) stabilization of a cartpole system, and (2) collision avoidance for a two vehicle system.