Data-Driven Observability Analysis for Nonlinear Stochastic Systems

Distinguishability and, by extension, observability are key properties of dynamical systems. Establishing these properties is challenging, especially when no analytical model is available and they are to be inferred directly from measurement data. The presence of noise further complicates this analysis, as standard notions of distinguishability are tailored to deterministic systems. We build on distributional distinguishability, which extends the deterministic notion by comparing distributions of outputs of stochastic systems. We first show that both concepts are equivalent for a class of systems that includes linear systems. We then present a method to assess and quantify distributional distinguishability from output data. Specifically, our quantification measures how much data is required to tell apart two initial states, inducing a continuous spectrum of distinguishability. We propose a statistical test to determine a threshold above which two states can be considered distinguishable with high confidence. We illustrate these tools by computing distinguishability maps over the state space in simulation, then leverage the test to compare sensor configurations on hardware.

Safe Value Functions

The relationship between safety and optimality in control is not well understood, and they are often seen as important yet conflicting objectives. There is a pressing need to formalize this relationship, especially given the growing prominence of learning-based methods. Indeed, it is common practice in reinforcement learning to simply modify reward functions by penalizing failures, with the penalty treated as a mere heuristic. We rigorously examine this relationship, and formalize the requirements for safe value functions: value functions that are both optimal for a given task, and enforce safety. We reveal the structure of this relationship through a proof of strong duality, showing that there always exists a finite penalty that induces a safe value function. This penalty is not unique, but upper-unbounded: larger penalties do not harm optimality. Although it is often not possible to compute the minimum required penalty, we reveal clear structure of how the penalty, rewards, discount factor, and dynamics interact. This insight suggests practical, theory-guided heuristics to design reward functions for control problems where safety is important.