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Real-Time Temperature Control of Thin Plates

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Proceedings of the 3rd International Conference on Control, Dynamic Systems, and Robotics (CDSR’16)
Ottawa, Canada May 9 10, 2016
Paper No. XXX (The number assigned by the OpenConf System)
Real-Time Temperature Control of Thin Plates
Dan Necsulescu1, Bilal Jarrah2
1,2University of Ottawa/Department of Mechanical Engineering
770 King Edward ave., Ottawa, ON, Canada
Dan.Necsulescu@uottawa.ca; bjarr031@uottawa.ca
Abstract-In this paper is investigated the control of the temperature on one side of the plate from the opposite side.
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 and the real-time open loop temperature control to a desired value is formulated.
This approach is different from previous regularization methods of ill-posed problem and is suitable for real-time
temperature control. Simulation results show the advantages and limitations of using inverse problem to control
temperature of a plate.
Keywords: open loop control, inverse problem, plate temperature control, quadrupole model.
1. Introduction
The problem of temperature control of a thin metal plate is affected by the fast temperature amplitude
decrease with regards to frequency and thickness. In most cases this consists in applying input
temperature at one side in order to modify the temperature on the other side of the plate and in open loop
control this requires to use inverse problem solution, known to lead to an ill-posed problem [1], [2]. There
are many methods to address the ill-posed problem and an investigation is required to find out a suitable
one for each application.
Maillet [1], Beck [2] and Necsulescu [3], in their books presented a wide 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. To do this, they linked the temperature and heat flux on both
sides by transfer functions and used these functions to solve the problem for heat input on the front
surface of a plate, [4] and [5]. Feng and al. In 2011solvedthesame problem using a1-Dimensional
(1D)modal expansion [6].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 a plate, using the
modified one-dimensional correction and the finite volume methods, [7]. Monde developed an analytical
method to solve inverse heat conduction problem using Laplace transform technique[8].
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. Real-time open loop control problem, which differs from the known monitoring
problem, is approached using finite Taylor expansions of the hyperbolic functions. Internet-of-Things
provides new interest in remote open loop control of systems.
2. Problem Formulation
The 1D heat conduction equation is given by:


 (1)
Boundary conditions were considered the following:
󰇛󰇜 󰂐 󰇛󰇜 (2)
󰇛󰇜  󰇛󰇜 (3)
where is the temperature and is the heat flux.
These boundary conditions were chosen for the investigation of temperature control with sinusoidal input
󰇛󰇜 resulting in the temperature output 󰇛󰇜on the opposite side of a plate of thickness L. The heat
flux 󰇛󰇜 results from the imposed 󰇛󰇜, while heat flux 󰇛󰇜 corresponds to isolated side of the
plate.
Applying Laplace transform, this equation becomes:
󰇛󰇜

󰇛󰇜,where α is the thermal diffusivity. (4)
The solution of this equation is [2, 3]:
󰇛󰇜 󰇛󰇜󰇛󰇜(5)
where θ is the temperature.
The heat flux is given by:
󰇛󰇜
(6)

(7)
These equations define the thermal quadrupole given the defined boundary conditions [1], in this case:
󰇛󰇜 󰂐
󰂐󰇛󰇜,(8)
where A is the input temperature amplitude.
󰇛󰇜󰇛󰇜 ,where L is the plate thickness.
Substituting boundary conditions for temperature equation give the following results:
󰂐
󰂐 󰂐
󰂐󰇛󰇜 (9)
For the abovethe solutions become:
󰇛󰇜 󰂐
󰂐󰇟󰇛󰇜󰇛󰇜󰇛󰇜󰇠 (10)
󰇛󰇜  󰂐
󰂐󰇟󰇛󰇜󰇛󰇜󰇛󰇜󰇠(11)
where k is the thermal conductivity.
For the investigation of the dynamics of temperatures , the following equations are used:
󰇛󰇜  󰂐
󰂐(12)
󰇛󰇜 󰂐
󰂐󰇟󰇛󰇜󰇛󰇜󰇛󰇜󰇠= 󰂐
󰂐[1/󰇛󰇜] (13)
The transfer function of the direct problem linking [3] is:
󰇟
󰇛󰇜󰇠=sech(KL) (14)
The transfer function for the inverse problem [1] to [3]is:
󰇛󰇜 (15)
The open loop control block diagrams shown in Fig. 1.

Fig. 1: Open loop block diagram.
MATLABTM and SimulinkTM are used for simulating the above system.
In this formulation, the hyperbolic functionsG1 and G2 contain square root of s:
x=
(16)
Taylor series expansion provides equations in integer powers of s. For G1 = sech(x) Taylor series
expansion is:
󰇛󰇜 󰇛
󰇜
 for | x| <
(17)
where Euler numbers En are zero for odd-indexed numbers, while even indexed numbers are:
E0=1
E2=-1
E4=5
E6=-61
E8=1385
E10=-50521
E12=2702765
E14=-199360981 etc.
For G2 = cosh(x), also an even function results:
󰇛󰇜 󰇛
󰇛󰇜󰇜
 for -∞< x< ∞ (18)
The above Taylor series expansions of 1/G1=cosh(x) and G2=cosh(x) contain only even-indexed terms,
and give integer number powers polynomials in s for simulation. This polynomial approximation, easy to
compute, is particularly useful in real-time control of the plate temperature.
For the Simulink simulation, Taylor expansion of the direct problem transfer function,G1, will be limited
to N terms, and inverse problem transfer function,G2, is limited to M terms. For the transfer function
G1*G2,N and M are chosen such that NM. In the simulated case, here a thin Aluminum plate, has the
thickness L = 0.03 [m] and thermal diffusivity α= 9.715e-5 [m2/sec], such that:
x=
=
(19)
For the simulations was chosen M=4and M=8, i.e. N >M:
󰇛󰇜 󰇛󰇜  (20)
󰇛󰇜
󰇛󰇜󰇛 
󰇜 (21)
3. Results and Discussion
Simulations were carried out for open loop control for different values of input frequency and for the
desired sinusoidal temperature amplitude of 200above the original temperature, i.e., for 20 sin 󰂐.
Simulations were carried for the direct problem G1 for M=8 while for inverse problem G2 for M= 4. The
input was
= 20sin(󰂐t).Simulation results for󰂐=0.1, 1, 5, and 10 are shown in Fig. 2-6
(a) (b)
Fig. 2.The outputs (a) of inverse problem and of (b) open loop control for󰂐= 0.1 rad/sec.
(a) (b)
Fig. 3.The outputs (a) of inverse problem and (b) of open loop control for󰂐 = 1 rad/sec.
(a) (b)
Fig. 4.The outputs (a) of inverse problem and (b) of open loop control for󰂐 = 5 rad/sec.
(a) (b)
Fig. 5.The outputs (a) of inverse problem and (b) of open loop control for󰂐 = 10 rad/sec.
(a) (b)
Fig. 6.The output of open loop control with N=8, M=4 for (a), 󰂐= 12and (b) 15 rad/sec.
Fig. 7 shows the Bode diagram of open loop control transfer function G1*G2 for N=8 and M=4.
Fig. 7.Bode diagram of open loop control transfer function G1*G2 for N=8 and M=4
The simulation results in Fig. 2-5 represent (a) the outputs of the inverse problem and (b) of open loop
control with N=8, M=4 for󰂐 = 0.1, 1, 5 and 10 rad/sec. The output temperature results in Fig 2 and
3,for lower frequencies of 0.1 and 1 rad/sec, compared to desired one,
󰂐,and of the
command temperature , are very close. The results in Fig 4 and 5, for output temperature , for higher
frequencies of 5 and 10 rad/sec, compared to desired one,
󰂐, and of the command
temperature, are significantly different. This can be explained by the very high amplitudes of the output
of the inverse problem in fig. 4a and 5a, which lead eventually to an ill-posed inverse problem,
particularly with regard to parameters L and α uncertainty. For 󰂐 = 12, Fig. 6a shows an amplitude lower
than in Fig 5b and for 󰂐 15, shows an amplitude much lower than in Fig 5b. Bode diagram of open loop
control transfer function in Fig. 7 explains this by indicating significantly lower magnitudes for 󰂐>11
rad/sec.In open loop control, since there is no feedback from the output, parameter uncertainty and
disturbance effects cannot be reduced. Further study will focus on closed loop temperature control.
4. Conclusions
The temperature on the one face of a plate can be open loop controlled in real-time to a desired value from
the other face using the proposed polynomial approach. The resulting hyperbolic functions are
approximated by Taylor expansions. The proposed approach is different from previous regularization
methods of ill-posed problems due to using truncated polynomials and is particularly suitable for real-time
open loop temperature control. Simulation results indicate the advantages and the limitations of the
proposed approach.
References
[1] D. Maillet, S. Andre, J. Batsale, A. Degiovanni and C. Moyne, Thermal Quadrupoles,” Solving the
Heat Equation through integral Transform. England: Wiley& Sons, 2000.
[2] J. Beck, B. Blackwell and C. Clair, Inverse Heat Conduction, John Wiley& Sons, 1985.
[3] D. Necsulescu,Advanced Mechatronics,”Monitoring and Control of Spatially Distributed Systems ,
World Scientific publishing, 2009.
[4] Z. Feng, J. Chen, Y .Zhang and S. Montgomery-smith, “Temperature and heat flux estimation from
sampled transient sensor measurements,” Int. Journal of Thermal Sciences, pp. 2385-2390, 2010.
[5] Z. Feng, J. Chen, Y. Zhang, “Real-time solution of heat conduction in a finite slab for inverse
analysis,” International Journal of Thermal Science, vol. 49, pp. 762-768, 2010.
[6] Z. Feng, J. Chen, Y. Zhang and J. Griggs, “Estimation of front surface temperature and heat flux of a
locally heated plate from distributed sensor data on the back surface,” International Journal of Heat and
Mass Transfer, vol. 54, pp. 3431-3439, 2011.
[7] C. Fan, F. Sun and L. Yang, A simple method for inverse estimation of surface temperature
distribution on a flat plate,” Inverse Problems in Science and Engineering, Vol. 17, No. 7, pp. 885-
899,October 2009.
[8] M. Monde, “Analytical method in inverse heat transfer problem using Laplace transform technique,”
International Journal of Heat and Mass Transfer, vol. 43, pp. 3965-3975, 2000.
... It can be seen that these functions appear in the scope of integrals, linear differential equations, and Laplace's equation. Among the applications of hyperbolic functions, one can mention the description of wave movement in elastic bodies, the shape of electric power transmission lines, temperature distribution in metal blades, tracking curves and the geometry of general relativity theory and other cases in various sciences(see [1], [10], [11] and [15]). ...
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... 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. ...
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  • D Maillet
  • S Andre
  • J Batsale
  • A Degiovanni
  • C Moyne
D. Maillet, S. Andre, J. Batsale, A. Degiovanni, and C. Moyne, " Thermal Quadrupoles, " in Solving the Heat Equation through integral Transform. England: Wiley& Sons, 2000.