Legendre-Moment Transform for Linear Ensemble Control and Computation
arxiv(2024)
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
ℓ^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.
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