Legendre-Moment Transform for Linear Ensemble Control and Computation

TL;DR

This paper introduces a novel Legendre-moment transform approach for controlling linear ensemble systems, enabling efficient control design and error analysis without sampling, applicable to large-scale underactuated systems.

Contribution

The paper develops a new Legendre-moment transform that maps ensemble systems to a moment space, establishing controllability equivalence and enabling a sampling-free control design method.

Findings

Control of ensemble systems via Legendre-moments is feasible.

The proposed method provides error bounds for truncated moment control.

Numerical examples validate the effectiveness of the approach.

Abstract

Ensemble systems, pervasive in diverse scientific and engineering domains, pose challenges to existing control methods due to their massive scale and underactuated nature. This paper presents a dynamic moment approach to addressing theoretical and computational challenges in systems-theoretic analysis and control design for linear ensemble systems. We introduce the Legendre-moments and Legendre-moment transform, which maps an ensemble system defined on the $L^2$-space to a Legendre-moment system defined on the $\ell^2$-space. We show that this pair of systems is of one-to-one correspondence and shares the same controllability property. This equivalence admits the control of an ensemble system through the control of the corresponding Legendre-moment system and inspires a unified control design scheme for linear ensemble systems using structured truncated moment systems. In particular, we develop a sampling-free ensemble control design algorithm, then conduct error analysis for control design using truncated moment systems and derive error bounds with respect to the truncation orders, which are illustrated with numerical examples.