Stabilizability of Vector Systems with Uniform Actuation Unpredictability

TL;DR

This paper investigates the fundamental limits of stabilizing underactuated vector systems with uniformly random actuation directions, providing tight conditions based on eigenvalues for system stabilizability.

Contribution

It introduces a novel analysis of stabilizability for vector systems with uniform actuation unpredictability, using a new proof technique based on weighted two-norms.

Findings

Derived necessary and sufficient conditions for stabilizability

Characterized stabilizability in terms of eigenvalues

Provided insights into control of underactuated systems

Abstract

This paper explores the fundamental limits of a simple system, inspired by the intermittent Kalman filtering model, where the actuation direction is drawn uniformly from the unit hypersphere. The model allows us to focus on a fundamental tension in the control of underactuated vector systems -- the need to balance the growth of the system in different dimensions. We characterize the stabilizability of $d$-dimensional systems with symmetric gain matrices by providing tight necessary and sufficient conditions that depend on the eigenvalues of the system. The proof technique is slightly different from the standard dynamic programming approach and relies on the fact that the second moment stability of the system can also be understood by examining any arbitrary weighted two-norm of the state.