Kibern. vyčisl. teh., 2015, Issue 179, pp 66-82.
Kotsur Maksim P., Assistant of the Department of Mathematical Control Problems and Cybernetics of Yuriy Fedkovych Chernivtsi National University, Ul. Kotsyubynsky, 2, Chernivtsi, 58012, Ukraine, e-mail: firstname.lastname@example.org
Nakonechniy Аleksandr G., Dr (Phys. and Math.), Prof., Academician-Secretary of the Department of Cybernetics and Systems Analysis, Head of the Department of System Analysis and Decision Theory of the Taras Shevchenko Kiev National University, Pr. Glushkov, 4 D, Kyiv, 03187, Ukraine, e-mail: email@example.com
OPTIMAL CONTROL BY TRANSIENT MODE OF STAGE THERMOELECTRIC COOLER
Introduction. A large variety of optimization problems is related to distributed parameter systems (DPSs), whose behavior is described by partial differential equations. The optimal control theory is widely employed for optimization of lumped parameter systems (LPSs), which are modeled by ordinary differential equations. Development of the theory and methods for obtaining optimal control functions for DPSs is much more difficult as compared with solving such problem for LPSs.
The purpose of the present paper is to obtain optimality conditions and to develop numerical methods for solving the optimization problem of an unsteady one-dimensional process with distributed parameters, as well as their application to optimization of transient thermoelectric cooling.
Methods. A method is proposed for solving of optimal control problem for DPS described by nonlinear partial differential equations of parabolic type with nonstandard boundary conditions. A method consists in coordinate discretization of distributed object and transition to the problem for LPS. Optimal control theory based on the Pontryagin maximum principle can be used for such system optimization.
Results. This method is applied for optimization of transient thermoelectric cooling process. Optimal dependences of current on time have been calculated for stage thermoelectric cooler power supply with the purpose of minimizing the cooling temperature within a preset time interval. Results of computer experiment for one- and two-stage coolers are presented.
Conclusion. The obtained results show that the implementation of optimal functions for cooler power supply can provide the cooling temperature in transient mode by 10 K lower than minimum temperature reached at direct current.
Keywords: distributed parameter system, optimal control, Pontryagin maximum principle, optimization, transient thermoelectric cooling.
1 Ahmed N.U. Distributed Parameter Systems. Encyclopedia of Physical Science and Technology (Third Edition), Academic PRESS, 2003, pp. 561–587. https://doi.org/10.1016/B0-12-227410-5/00183-6
2 Alessandri S.A., Gaggero M., Zoppoli R. Feedback Optimal Control of Distributed Parameter Systems by Using Finite-Dimensional Approximation Schemes. IEEE Transactions on Neural Networks and Learning Systems, 2012, vol. 23, no. 6, pp. 984–995. https://doi.org/10.1109/TNNLS.2012.2192748
3 Guangcao Ji, Clyde Martin. Optimal Boundary Control of the Heat Equation with Target Function at Terminal Time. Applied Mathematics and Computation, 2002, vol. 127, pp. 335–345. https://doi.org/10.1016/S0096-3003(01)00011-X
4 Bokhari M.A., Sadek I.S. Optimal Control of Parabolic Systems with Infinite Time Horizons. Applied Mathematics and Computation, 2008, vol. 206, pp. 678–684. https://doi.org/10.1016/j.amc.2008.05.048
5 El-Farra N.H., Armaou A., Christofides P.D. Analysis and Control of Parabolic PDE Systems with Input Constraints. Automatica, 2003, vol. 39, pp. 715–725. https://doi.org/10.1016/S0005-1098(02)00304-7
6 Kucuka I., Sadeka I., Yilmazc Y. Optimal Control of a Distributed Parameter System with Applications to Beam Vibrations Using Piezoelectric Actuators. Journal of the Franklin Institute, 2014, vol. 351, issue 2, February, pp. 656–666. https://doi.org/10.1016/j.jfranklin.2012.10.008
7 Zuyev A. Partial Asymptotic Stabilization of Nonlinear Distributed Parameter Systems. Automatica, 2005, vol. 41, pp. 1–10. https://doi.org/10.1016/S0005-1098(04)00240-7 https://doi.org/10.1016/j.automatica.2004.08.009
8 Zuazua E. Propagation, Observation and Control of Waves Approximated by Finite Difference Methods. SIAM Rev., 2005, vol. 47, no. 2, pp. 197–243. https://doi.org/10.1137/S0036144503432862
9 Krstic M., Guo B.-Z., Smyshlyaev A. Boundary Controllers and Observers for the Linearized Schrodinger Equation. SIAM J. Control Opt., 2011, vol. 49, no. 4, pp. 1479–1497. https://doi.org/10.1137/070704290
10 Subas M. An Otimal Control Problem Governed by the Potential of a Linear Schrodinger Equation. Applied Mathematics and Computation, 2002, vol. 131, pp. 95–106. https://doi.org/10.1016/S0096-3003(01)00161-8
11 Alvarez-Vazquez L.J., Fernandez F.J. Optimal Control of a Bioreactor. Applied Mathe-matics and Computation, 2010, vol. 216, pp. 559–575. https://doi.org/10.1016/j.amc.2010.03.097
12 Ryu Sang-Uk, Yagi Atsushi. Optimal Control for an Adsorbate-Induced Phase Transition Model. Applied Mathematics and Computation, 2005, vol. 171, pp. 420–432. https://doi.org/10.1016/j.amc.2005.01.044
13 Hoffman K.H., Jiang L. Optimal Control of a Phase Field Model for Solidification. Numer. Funct. Anal. and Optimiz., 1992, vol. 13, no. 1&2, pp. 11–27. https://doi.org/10.1080/01630569208816458
14 Muresan V., Abrudean M., Unguresan M., Colosi T. Cascade Control of a Residual Water Blunting System. Advances in Electrical and Computer Engineering, 2014, vol. 14, no. 2, pp. 135–144. https://doi.org/10.4316/AECE.2014.02022
15 Egorov A.I. Optimal control by thermal and diffuse processes. Moscow: Science, 1978, 463 p. (in Russian).
16 Geering Hans P. Optimal Control with Engineering Applications. NewYork, Berlin, Heidelberg: Springer-Verlag, 2007. 134 p.
17 Sethi Suresh P. Optimal Control Theory. Applications to Management Science and Economics. Springer Science+Business Media, Inc., 2000. 505 p.
18 Anatychuk L.I. Optimal Control by Properties of Thermoelectric Materials and Devices. Chernovtsy: Prut, 1992. 263 p. (in Russian).
19 Pontryagin L.S., Boltyanski V.G., Gamkrelidze R.S., Mischenko E.F. The Mathematical Theory of Optimal Processes. Moscow: Nauka, 1976. 392 p. (in Russian).
20 Butkovskii A.G. Theory of Optimal Control by Distributed Parameters Systems. Moscow: Nauka, 1965. 474 p. (in Russian).
21 Sirazetdinov T.K. Optimization of Distributed Parameters Systems. Moscow: Nauka, 1977. 479 p. (in Russian).
22 Fursikov A.V. Optimal Control by Distributed Systems. Theory and Application. Novosibirsk: Nauchnaya Kniga, 1999. 352 p. (in Russian). https://doi.org/10.1090/mmono/187
23 Shevyakov A.A., Yakovleva R.V. Control by Thermal Objects with Distributed Parameters. Moscow: Energoatomizdat, 1986. 208 p. (in Russian).
24 Anatychuk L.I. Thermoelectricity. Vol. II. Thermoelectric Energy Convectors. Kiev, Chernovtsy: Institute of Thermoelectricity, 2003. 376 p.
25 Yang Ronggui, Chen Gang, Kumar A. Ravi, Snyder G. Jeffrey, Fleurial Jean-Pierre. Transient Cooling of Thermoelectric Coolers and its Applications for Microdevices. Energy Conversion and Management, 2005, vol. 46, pp.1407–1421. https://doi.org/10.1016/j.enconman.2004.07.004
26 Kotsur M.P. Approximate Method of Optimal Control in Problems of Transient Thermoelectric Cooling. Journal of Computational and Applied Mathematics, 2013, vol. 114, no. 4, pp. 37-47. (in Ukrainian)
27 One-stage thermoelectric modules. Available at: http://www.kryothermtec.com/ru/standsard-single-stage-thermoelectric-coolers.html. Two-stage thermoelectric modules. Available at: http://www.kryothermtec.com/ru/2-stage-thermoelectric-coolers.html.