1,860 research outputs found
Delay-Adaptive Control of First-order Hyperbolic PIDEs
We develop a delay-adaptive controller for a class of first-order hyperbolic
partial integro-differential equations (PIDEs) with an unknown input delay. By
employing a transport PDE to represent delayed actuator states, the system is
transformed into a transport partial differential equation (PDE) with unknown
propagation speed cascaded with a PIDE. A parameter update law is designed
using a Lyapunov argument and the infinite-dimensional backstepping technique
to establish global stability results. Furthermore, the well-posedness of the
closed-loop system is analyzed. Finally, the effectiveness of the proposed
method was validated through numerical simulation
Machine Learning Accelerated PDE Backstepping Observers
State estimation is important for a variety of tasks, from forecasting to
substituting for unmeasured states in feedback controllers. Performing
real-time state estimation for PDEs using provably and rapidly converging
observers, such as those based on PDE backstepping, is computationally
expensive and in many cases prohibitive. We propose a framework for
accelerating PDE observer computations using learning-based approaches that are
much faster while maintaining accuracy. In particular, we employ the
recently-developed Fourier Neural Operator (FNO) to learn the functional
mapping from the initial observer state and boundary measurements to the state
estimate. By employing backstepping observer gains for previously-designed
observers with particular convergence rate guarantees, we provide numerical
experiments that evaluate the increased computational efficiency gained with
FNO. We consider the state estimation for three benchmark PDE examples
motivated by applications: first, for a reaction-diffusion (parabolic) PDE
whose state is estimated with an exponential rate of convergence; second, for a
parabolic PDE with exact prescribed-time estimation; and, third, for a pair of
coupled first-order hyperbolic PDEs that modeling traffic flow density and
velocity. The ML-accelerated observers trained on simulation data sets for
these PDEs achieves up to three orders of magnitude improvement in
computational speed compared to classical methods. This demonstrates the
attractiveness of the ML-accelerated observers for real-time state estimation
and control.Comment: Accepted to the 61st IEEE Conference on Decision and Control (CDC),
202
High order direct Arbitrary-Lagrangian-Eulerian schemes on moving Voronoi meshes with topology changes
We present a new family of very high order accurate direct
Arbitrary-Lagrangian-Eulerian (ALE) Finite Volume (FV) and Discontinuous
Galerkin (DG) schemes for the solution of nonlinear hyperbolic PDE systems on
moving 2D Voronoi meshes that are regenerated at each time step and which
explicitly allow topology changes in time.
The Voronoi tessellations are obtained from a set of generator points that
move with the local fluid velocity. We employ an AREPO-type approach, which
rapidly rebuilds a new high quality mesh rearranging the element shapes and
neighbors in order to guarantee a robust mesh evolution even for vortex flows
and very long simulation times. The old and new Voronoi elements associated to
the same generator are connected to construct closed space--time control
volumes, whose bottom and top faces may be polygons with a different number of
sides. We also incorporate degenerate space--time sliver elements, needed to
fill the space--time holes that arise because of topology changes. The final
ALE FV-DG scheme is obtained by a redesign of the fully discrete direct ALE
schemes of Boscheri and Dumbser, extended here to moving Voronoi meshes and
space--time sliver elements. Our new numerical scheme is based on the
integration over arbitrary shaped closed space--time control volumes combined
with a fully-discrete space--time conservation formulation of the governing PDE
system. In this way the discrete solution is conservative and satisfies the GCL
by construction.
Numerical convergence studies as well as a large set of benchmarks for
hydrodynamics and magnetohydrodynamics (MHD) demonstrate the accuracy and
robustness of the proposed method. Our numerical results clearly show that the
new combination of very high order schemes with regenerated meshes with
topology changes lead to substantial improvements compared to direct ALE
methods on conforming meshes
- …