2 research outputs found
Ensemble Observability of Bloch Equations with Unknown Population Density
We introduce in the paper a novel observability problem for a continuum
ensemble of nonholonomic control systems with unknown population density. We
address the problem by focussing on a prototype of such ensemble system,
namely, the ensemble of Bloch equations. The dynamics of the equations are
structurally identical, but show variations in Larmor dispersion and radio
frequency (rf) inhomogeneity. We assume that the initial state of every
individual system is unknown and, moreover, the population density of these
individual systems is also unknown. Furthermore, we assume that at any time,
there is only one scalar measurement output at our disposal. The measurement
output integrates a certain observation function, common to all individual
systems, over the continuum ensemble. The observability problem we pose in the
paper is thus the following: Whether one is able to use the common control
input (i.e., the rf field) and the single measurement output to estimate the
initial states of the individual systems and, moreover, to identify the
population density? Amongst other things, we establish a sufficient condition
for the ensemble system to be observable: We show that if the common
observation function is any harmonic homogeneous polynomial of positive degree,
then the ensemble system is observable. The main focus of the paper is to
demonstrate how to leverage tools from representation theory of Lie algebras to
address the observability problem. Although the results we establish in the
paper are for the specific ensemble of Bloch equations, the approach we develop
along the analysis can be generalized to investigate observability of other
general ensembles of nonholonomic control systems with a single, integrated
measurement output
Moment-Based Ensemble Control
Controlling a large population, in the limit, a continuum, of structurally
identical dynamical systems with parametric variations is a pervasive task in
diverse applications in science and engineering. However, the severely
underactuated nature and the inability to avail comprehensive state feedback
information of such ensemble systems raise significant challenges in analysis
and design of ensemble systems. In this paper, we propose a moment-based
ensemble control framework, which incorporates and expands the method of
moments in probability theory to control theory. In particular, we establish an
equivalence between ensemble systems and their moment systems in terms of
control and their controllability properties by extending the Hausdorff moment
problem from the perspectives of differential geometry and dynamical systems.
The developments enable the design of moment-feedback control laws for closing
the loop in ensemble systems using the aggregated type of measurements. The
feasibility of this closed-loop control design procedure is validated both
mathematically and numerically.Comment: Keywords: Ensemble systems, Hausdorff moment problem, Aggregated
measurements, Aggregated feedbac