13 research outputs found
Ensembles of Hyperbolic PDEs: Stabilization by Backstepping
For the quite extensively developed PDE backstepping methodology for coupled
linear hyperbolic PDEs, we provide a generalization from finite collections of
such PDEs, whose states at each location in space are vector-valued, to
previously unstudied infinite (continuum) ensembles of such hyperbolic PDEs,
whose states are function-valued. The motivation for studying such systems
comes from traffic applications (where driver and vehicle characteristics are
continuously parametrized), fluid and structural applications, and future
applications in population dynamics, including epidemiology. Our design is of
an exponentially stabilizing scalar-valued control law for a PDE system in two
independent dimensions, one spatial dimension and one ensemble dimension. In
the process of generalizing PDE backstepping from finite to infinite
collections of PDE systems, we generalize the results for PDE backstepping
kernels to the continuously parametrized Goursat-form PDEs that govern such
continuously parametrized kernels. The theory is illustrated with a simulation
example, which is selected so that the kernels are explicitly solvable, to lend
clarity and interpretability to the simulation results.Comment: 16 pages, 4 figures, to be publishe
Controllability Canonical Forms of Linear Ensemble Systems
Ensemble control, an emerging research field focusing on the study of large
populations of dynamical systems, has demonstrated great potential in numerous
scientific and practical applications. Striking examples include pulse design
for exciting spin ensembles in quantum physics, neurostimulation for relieving
neurological disorder symptoms, and path planning for steering robot swarms.
However, the control targets in such applications are generally large-scale
complex and severely underactuated ensemble systems, research into which
stretches the capability of techniques in classical control and dynamical
systems theory to the very limit. This paper then devotes to advancing our
knowledge about controllability of linear ensemble systems by integrating tools
in modern algebra into the technique of separating points developed in our
recent work. In particular, we give an algebraic interpretation of the dynamics
of linear systems in terms of actions of polynomials on vector spaces, and this
leads to the development of the functional canonical form of matrix-valued
functions, which can also be viewed as the generalization of the rational
canonical form of matrices in linear algebra. Then, leveraging the technique of
separating points, we achieve a necessary and sufficient characterization of
uniform ensemble controllability for time-invariant linear ensemble systems as
the ensemble controllability canonical form, in which the system and control
matrices are in the functional canonical and block diagonal form, respectively.
This work successfully launches a new research scheme by adopting and tailoring
finite-dimensional methods to tackle control problems involving
infinite-dimensional ensemble systems, and lays a solid foundation for a more
inclusive ensemble control theory targeting a much broader spectrum of control
and learning problems in both scientific research and practice