6 research outputs found
Robust Output Tracking for a Room Temperature Model with Distributed Control and Observation
We consider robust output regulation of a partial differential equation model
describing temperature evolution in a room. More precisely, we examine a
two-dimensional room model with the velocity field and temperature evolution
governed by the incompressible steady state Navier-Stokes and
advection-diffusion equations, respectively, which coupled together form a
simplification of the Boussinesq equations. We assume that the control and
observation operators of our system are distributed, whereas the disturbance
acts on a part of the boundary of the system. We solve the robust output
regulation problem using a finite-dimensional low-order controller, which is
constructed using model reduction on a finite element approximation of the
model. Through numerical simulations, we compare performance of the
reduced-order controller to that of the controller without model reduction as
well as to performance of a low-gain robust controller.Comment: 12 pages, 5 figures. Accepted for publication in the Proceedings of
the 24th International Symposium on Mathematical Theory of Networks and
Systems, 23-27 August, 202
The Boussinesq system with mixed non-smooth boundary conditions and ``do-nothing'' boundary flow
A stationary Boussinesq system for an incompressible viscous fluid in a bounded domain with a nontrivial condition at an open boundary is studied. We consider a novel non-smooth boundary condition associated to the heat transfer on the open boundary that involves the temperature at the boundary, the velocity of the fluid, and the outside temperature. We show that this condition is compatible with two approaches at dealing with the do-nothing boundary condition for the fluid: 1) the directional do-nothing condition and 2) the do-nothing condition together with an integral bound for the backflow. Well-posedness of variational formulations is proved for each problem
Robust Output Regulation of the Linearized Boussinesq Equations with Boundary Control and Observation
We study temperature and velocity output tracking problem for a
two-dimensional room model with the fluid dynamics governed by the linearized
translated Boussinesq equations. Additionally, the room model includes
finite-dimensional models for actuation and sensing dynamics, thus the complete
model dynamics are governed by an ODE-PDE-ODE system. As the main result, we
design a low-dimensional internal model based controller for robust output
racking of the room model. Efficiency of the controller is demonstrated through
a numerical example of velocity and temperature tracking.Comment: 26 pages, 9 figures, submitte
A numerical approach to the optimal control of thermally convective flows
The optimal control of thermally convective flows is usually modeled by an
optimization problem with constraints of Boussinesq equations that consist of
the Navier-Stokes equation and an advection-diffusion equation. This optimal
control problem is challenging from both theoretical analysis and algorithmic
design perspectives. For example, the nonlinearity and coupling of fluid flows
and energy transports prevent direct applications of gradient type algorithms
in practice. In this paper, we propose an efficient numerical method to solve
this problem based on the operator splitting and optimization techniques. In
particular, we employ the Marchuk-Yanenko method leveraged by the
projection for the time discretization of the Boussinesq equations so
that the Boussinesq equations are decomposed into some easier linear equations
without any difficulty in deriving the corresponding adjoint system.
Consequently, at each iteration, four easy linear advection-diffusion equations
and two degenerated Stokes equations at each time step are needed to be solved
for computing a gradient. Then, we apply the Bercovier-Pironneau finite element
method for space discretization, and design a BFGS type algorithm for solving
the fully discretized optimal control problem. We look into the structure of
the problem, and design a meticulous strategy to seek step sizes for the BFGS
efficiently. Efficiency of the numerical approach is promisingly validated by
the results of some preliminary numerical experiments
Feedback Stabilization of a Thermal Fluid System with Mixed Boundary Control
We consider the problem of local exponential stabilization of the nonlinear Boussinesq equations with control acting on portion of the boundary. In particular, given a steady state solution on an bounded and connected domain Ω Ïč R2, we show that a finite number of controls acting on a part of the boundary through Neumann/Robin boundary conditions is sufficient to stabilize the full nonlinear equations in a neighborhood of this steady state solution. Dirichlet boundary conditions are imposed on the rest of the boundary. We prove that a stabilizing feedback control law can be obtained by solving a Linear Quadratic Regulator (LQR) problem for the linearized Boussinesq equations. Numerical result are provided for a 2D problem to illustrate the ideas