Ensemble Control of Finite-Dimensional Time-Varying Linear Systems

• Published in: IEEE transactions on automatic control Vol. 56; no. 2; pp. 345 - 357
• Main Author: Li, Jr-Shin
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.02.2011; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Applied sciences; Computer science; control theory; systems; Control system analysis; Control systems; Control theory; Control theory. Systems; Controllability; Dynamical systems; Eigensystems; ensemble control; ensemble controllability; Exact sciences and technology; Imaging; Instruments, apparatus, components and techniques common to several branches of physics and astronomy; lie brackets; linear operators; Linear systems; Magnetic components, instruments and techniques; Magnetic resonance imaging; magnetic resonance imaging (MRI); Mathematical analysis; NMR; Nuclear magnetic resonance; nuclear magnetic resonance (NMR); Optimal control; Optimization; Physics; singular systems; Spectroscopy; Studies; System theory; Time varying systems; Wave functions; Singularity; singular systems; Time frequency domain method; Necessary and sufficient condition; Control program; lie brackets; Set computation; Dynamical system; Modeling; Linear operator; Time varying system; ensemble controllability; Uncertain system; nuclear magnetic resonance (NMR); Controllability; Nuclear magnetic resonance; linear operators; Wave function; ensemble control; Systems theory; Nuclear magnetic resonance imaging; magnetic resonance imaging (MRI); Quantum computation; Optimal control; Eigensystems; Ellipsoid; Harmonic oscillator
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2010.2060259

In this article, we investigate the problem of simultaneously steering an uncountable family of finite-dimensional time-varying linear systems with the same control signal. This class of control problems motivates further research in the new subject of control theory called Ensemble Control, a notion coming from the study of complex spin dynamics in nuclear magnetic resonance spectroscopy and imaging. We derive the necessary and sufficient controllability conditions and an accompanying analytical optimal control law for an ensemble of finite-dimensional time-varying linear systems. We show that ensemble controllability is in connection with singular values of the operator characterizing the system dynamics. In addition, the problem of optimal ensemble control of harmonic oscillators is studied to demonstrate the controllability results. We show that the optimal solutions are pertinent to the study of time-frequency limited signals and prolate spheroidal wave functions. A systematic study of ensemble control systems has immediate applications to dynamical systems with parameter uncertainty as well as to wide-ranging areas such as neuroscience and quantum control. The work in ensemble control will foster further developments in mathematical control and systems theory.