Closed - Loop Control of Plate Temperature using Inverse Problem
Dan Necsulescu
1
, Bilal Jarrah
1
and Jurek Sasiadek
2
1
University of Ottawa/Department of Mechanical Engineering, Ottawa, ON, Canada
2
University of Carleton/Department of Mechanical and Aerospace Engineering, Ottawa, ON, Canada
Keywords: Closed Loop Control, Inverse Problem, Plate Temperature Control, Quadrupole Model.
Abstract: In this paper the temperature at one side of a plate is used to control in closed loop the temperature on the
opposite side of the plate. To solve this problem, Laplace transform is used to obtain the quadrupole model
of the direct heat equation and the analytical solution for the transfer function for the inverse problem. The
resulting hyperbolic functions are approximated by Taylor expansions to facilitate the real-time closed loop
temperature control formulation. Simulation results illustrate the advantages and permit to identify the
limitations of using inverse problem to closed loop control temperature of a plate.
1 INTRODUCTION
In a metal plate the temperature distribution is
characterized by the fast decay with regards to
frequency. The goal of the paper consists in applying
input temperature at one side in order to modify the
temperature on the other side of the plate; in closed
loop control this is approached using the inverse
problem solution, known to lead to an ill-posed
problem (
Maillet, et al., 2000), (Beck et al., 1985).
There are many methods to address this ill-posed
problem and an investigation is required to find out a
suitable one for each application. In this paper is
searched a suitable solution for closed loop control
of a plate temperature. The books (
Maillet, et al.,
2000)
, (Beck, et al., 1985) and (Necsulescu, 2009)
presented a variety of solutions for solving inverse
heat transfer problems in case of temperature
monitoring for plates. Feng et al in 2010 solved the
problem of heat conduction over a finite slab to
estimate temperature and heat flux on the front
surface of a plate from the back surface
measurement, (
Feng, et al., 2010) and (Feng, et al.,
2010
). Feng et al. in 2011 solved the same problem
using a 1-Dimensional (1D) modal expansion (
Feng,
et al., 2010). Fan et al. obtained temperature
distribution on one side of a flat plate by solving the
inverse problem based on the temperature
measurement on the other side of the plate, using the
modified 1D correction and the finite volume
methods, (
Fan, et al., 2009). Monde developed an
analytical method to solve inverse heat conduction
problem using Laplace transform technique (
Monde,
2000
). Piazzi and Visioli investigated dynamic
inversion using transfer functions (
Piazzi, Visioli,
2001
).
In this paper the 1D heat conduction equation is
formulated in the Laplace domain to determine the
hyperbolic transfer functions relating input and
output temperature of a thin plate for both direct and
inverse problems. In this case, hyperbolical
functions depend on square root of complex variable
s and this does not facilitate real-time applications.
Closed loop control problem differs from the known
monitoring problems and from open-loop control
problems (
Necsulescu, Jarrah, 2016). For real-time
applications, this is approached using finite Taylor
expansions of the hyperbolic functions that permit to
obtain transfer functions that approximate
hyperbolic functions for a given frequency domain.
Temperature control of metal plates is
investigated for the case of heating one side to bring
the temperature on the other side at a desired value.
Earlier attempts to solve the ill-posed inverse
problem of indirect temperature estimation referred
to the study of overheating of the outer shell of a
rocket entering the atmosphere using temperature
measurement from inside (
Beck, et al., 1985). Closed-
loop control of plate temperature is applicable to
achieving accurate temperature output of heating
plates and to inside tanks temperature control using
outside heating.