### NONLINEAR CONTROL OF TEMPERATURE PROFILE OF UNSTABLE HEAT CONDUCTION SYSTEMS: A PORT HAMILTONIAN APPROACH

#### Abstract

This paper focuses on boundary control of distributed parameter systems (also called infinite dimensional systems). More precisely, a passivity based approach for the stabilization of temperature profile inside a well-insulated bar with heat conduction in a one-dimensional described by parabolic partial differential equations (PDEs) is developed. This approach is motivated by an appropriate model reduction schema using the finite difference approximation method. On this basis, it allows to discretize and then, write the original parabolic PDEs into a Port Hamiltonian (PH) representation. From this, the boundary control input is therefore synthesized using passive tools to stabilize the temperature at a desired reference profile asymptotically. The infinite dimensional nature of the original distributed parameter system in the PH framework is also discussed. Numerical simulations illustrate the application of the developments.

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[[1] Groot, S.R. de and Mazur, P. (1962). Non-equilibrium thermodynamics. Dover Pub. Inc., Amsterdam, 1st edition.

[[2] Sandler, S.I. (1999). Chemical and engineering thermodynamics. John Wiley & Sons Inc, New York, 3rd edition.

[[3] Luyben, W.L. (1990). Process modeling, simulation, and control for chemical engineers. McGraw-Hill, Singapore.

[[4] Christofides, P.D. (2001). Control of nonlinear distributed process systems: Recent developments and challenges. AIChE Journal. 47(3):514–518.

[[5] Ray, W.H. (1978). Some recent applications of distributed parameter systems theory - A survey. Automatica. 14:281-287.

[[6] Curtain, R. and Zwart, H. (1995). An introduction to infinite-dimensional linear systems theory. New York: Springer-Verlag.

[[7] Ruszkowski, M., Garcia-Osorio, V. and Ydstie, B.E. (2005). Passivity based control of transport reaction systems. AIChE Journal. 51:3147-3166.

[[8] Alonso, A.A. and Ydstie, B.E. (2001). Stabilization of distributed systems using irreversible thermodynamics. Automatica. 37:1739-1755.

[[9] Antelo, L.T., Otero-Muras, I., Banga, J.R. and Alonso, A.A. (2007). A systematic approach to plant wide control based on thermodynamics. Computers & Chemical Engineering. 31:677-691.

[[10] Burden, R. and Faires, J.D. (2010). Numerical analysis. Brooks, Cole, USA, 9th edition.

[[11] Moulla, R., Lefèvre, L. and Maschke, B. (2011). Geometric pseudospectral method for spatial integration of dynamical systems. Mathematical and Computer Modelling of Dynamical Systems. 17(1):85-104.

[[12] Peng, H., Baaiu, A., Couenne, F. and Le Gorrec, Y. (2007). Energy based discretization of a class of distributed parameters processes comparison with classical approach. Proc. the 46th IEEE Conference on Decision and Control. 3513–3518.

[[13] Khalil, H.K. (2002). Nonlinear systems. Prentice Hall, 3rd edition.

[[14] Van Der Schaft, A. (2000). L2-gain and passivity techniques in nonlinear control. Springer-Verlag, London, 2nd edition.

[[15] Brogliato, B., Lozano, R., Maschke, B. and Egeland, O. (2007). Dissipative systems analysis and control. Springer, London, 2nd edition.

[[16] Dubljevic, S., Mhaskar, P., El-Farra, N.H. and Christofides P.D. (2005). Predictive control of transport-reaction processes. Computers & Chemical Engineering. 29:2335-2345.

[[17] Christofides, P.D., Robust control of parabolic PDE systems. Chemical Engineering Science. 53(16):pp. 2949-2965, 1998.

[[18] Vilsa, C., Garcia, M.R., Banga, J.R. and Alonso, A.A., Robust feedback control of distributed chemical reaction systems. Chemical Engineering Science. 62, pp. 2941-2957, 2007.

[[19] El-Farra, N., Armaou, A. and Christofides P.D., Analysis and control of parabolic PDE systems with input constraints. Automatica. 39, pp. 715-725, 2003.

[[20] Hamroun, B., Lefèvre, L. and Mendes, E., Passivity based control of a reduced Port-controlled Hamiltonian model for the shallow water equations. Proc. the 47th IEEE Conference on Decision and Control. pp. 3917-3922, 2008.

[[21] Li, M. and Christofides, P.D., Optimal control of diffusion-convection-reaction processes using reduced order models. Computers & Chemical Engineering. 32, pp.2123-2135, 2008.

[[22] Qi, C. and Li, H.X., A time/space separation based hammerstein modeling approach for nonlinear distributed parameter processes. Computers & Chemical Engineering. 33(7), pp.1247-1260, 2009.

[[23] Godasi, S., Karakas, A. and Palazoglu, A., Control of nonlinear distributed parameter processes using symmetry groups and invariance conditions. Computers & Chemical Engineering. 26, pp. 1023-1036, 2002.

[[24] Ramírez, H., Le Gorrec, Y., Macchelli, A. and Zwart H., Exponential stabilization of boundary controlled port-Hamiltonian systems with dynamic feedback. IEEE Transactions on Automatic Control. 59(10), pp. 2849-2855, 2014.

[[25] Hoang, N.H. and Dochain, D., A thermodynamic approach to the passive boundary control of tubular reactors. Proc. the 9th IFAC-Symposium on Nonlinear Control Systems. Toulouse, France, pp. 383-388, 2013.

[[26] Van Der Schaft, A., Port-controlled Hamiltonian systems: towards a theory for control and design of nonlinear physical systems. SICE Journal. 39(2), pp. 91-98, 2000.

[[27] Maschke, B., Ortega, R. and Van Der Schaft, A., Energy based Lyapunov functions for forced Hamiltonian systems with dissipation. IEEE Transactions on Automatic Control. 45(8), pp. 1498-1502, 2000.

[[28] Ortega, R., Van Der Schaft, A., Maschke, B. and Escobar, G., Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian Systems. Automatica. 38, pp. 585-596, 2002.

[[29] Hoang, H., Couenne F., Jallut, C. and Le Gorrec Y., The Port Hamiltonian approach to modeling and control of Continuous Stirred Tank Reactors. Journal of Process Control. 21, pp. 1449-1458, 2011.

[[30] Žecová, M., Terpák, J. and Dorčák L, Usage of the heat conduction model for the experimental determination of thermal diffusivity. Proc. the 14th International Carpathian Control Conference. Rytro, Poland. pp. 436-441, 2013.

[[31] Nguyen, T.S. and Hoang, N.H, Optimization and simulation in chemical engineering: Application to the production of cyclopentenol from cyclopentadiene. VNU Journal of Science. 30(6S-A), pp.32-43, 2014.

DOI: https://doi.org/10.15625/1813-9663/32/1/6401

*Journal of Computer Science and Cybernetics *ISSN: 1813-9663**Published by Vietnam Academy of Science and Technology**