590 research outputs found
Control and State Estimation of the One-Phase Stefan Problem via Backstepping Design
This paper develops a control and estimation design for the one-phase Stefan
problem. The Stefan problem represents a liquid-solid phase transition as time
evolution of a temperature profile in a liquid-solid material and its moving
interface. This physical process is mathematically formulated as a diffusion
partial differential equation (PDE) evolving on a time-varying spatial domain
described by an ordinary differential equation (ODE). The state-dependency of
the moving interface makes the coupled PDE-ODE system a nonlinear and
challenging problem. We propose a full-state feedback control law, an observer
design, and the associated output-feedback control law via the backstepping
method. The designed observer allows estimation of the temperature profile
based on the available measurement of solid phase length. The associated
output-feedback controller ensures the global exponential stability of the
estimation errors, the H1- norm of the distributed temperature, and the moving
interface to the desired setpoint under some explicitly given restrictions on
the setpoint and observer gain. The exponential stability results are
established considering Neumann and Dirichlet boundary actuations.Comment: 16 pages, 11 figures, submitted to IEEE Transactions on Automatic
Contro
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
Backstepping PDE Design: A Convex Optimization Approach
Abstract\u2014Backstepping design for boundary linear PDE is
formulated as a convex optimization problem. Some classes of
parabolic PDEs and a first-order hyperbolic PDE are studied,
with particular attention to non-strict feedback structures. Based
on the compactness of the Volterra and Fredholm-type operators
involved, their Kernels are approximated via polynomial
functions. The resulting Kernel-PDEs are optimized using Sumof-
Squares (SOS) decomposition and solved via semidefinite
programming, with sufficient precision to guarantee the stability
of the system in the L2-norm. This formulation allows optimizing
extra degrees of freedom where the Kernel-PDEs are included
as constraints. Uniqueness and invertibility of the Fredholm-type
transformation are proved for polynomial Kernels in the space
of continuous functions. The effectiveness and limitations of the
approach proposed are illustrated by numerical solutions of some
Kernel-PDEs
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
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