Learning Expected Reward for Switched Linear Control Systems: A Non-Asymptotic View

TL;DR

This paper establishes non-asymptotic bounds for learning the expected reward in switched linear dynamical systems under a stationary control policy, based on invariant ergodic measures and ergodic theorems.

Contribution

It introduces a non-asymptotic analysis framework for average reward control in SLDSs using invariant measures and ergodic theory, which was previously lacking.

Findings

Existence of invariant ergodic measure under norm-stability.

Non-asymptotic bounds for reward learning from time-averages.

Validation through two case studies.

Abstract

In this work, we show existence of invariant ergodic measure for switched linear dynamical systems (SLDSs) under a norm-stability assumption of system dynamics in some unbounded subset of $\mathbb{R}^{n}$. Consequently, given a stationary Markov control policy, we derive non-asymptotic bounds for learning expected reward (w.r.t the invariant ergodic measure our closed-loop system mixes to) from time-averages using Birkhoff's Ergodic Theorem. The presented results provide a foundation for deriving non-asymptotic analysis for average reward-based optimal control of SLDSs. Finally, we illustrate the presented theoretical results in two case-studies.