Approximate state-space and transfer function models for 2x2 linear hyperbolic systems with collocated boundary inputs

• Autorzy: Bartecki Krzysztof

Identyfikatory

DOI

10.34768/amcs-2020-0035

• Języki publikacji: EN

Abstrakty

EN

Two approximate representations are proposed for distributed parameter systems described by two linear hyperbolic PDEs with two time- and space-dependent state variables and two collocated boundary inputs. Using the method of lines with the backward difference scheme, the original PDEs are transformed into a set of ODEs and expressed in the form of a finite number of dynamical subsystems (sections). Each section of the approximation model is described by state-space equations with matrix-valued state, input and output operators, or, equivalently, by a rational transfer function matrix. The cascade interconnection of a number of sections results in the overall approximation model expressed in finite-dimensional state-space or rational transfer function domains, respectively. The discussion is illustrated with a practical example of a parallel-flow double-pipe heat exchanger. Its steady-state, frequency and impulse responses obtained from the original infinite-dimensional representation are compared with those resulting from its approximate models of different orders. The results show better approximation quality for the “crossover” input–output channels where the in-domain effects prevail as compared with the “straightforward” channels, where the time-delay phenomena are dominating.

Słowa kluczowe

EN

distributed parameter system; hyperbolic equations; approximation model; state space; transfer function

PL

układ o parametrach rozłożonych; równanie hiperboliczne; model aproksymacyjny; przestrzeń stanu; funkcja przesyłowa

• Wydawca: Oficyna Wydawnicza Uniwersytetu Zielonogórskiego
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2020
• Tom: Vol. 30, no. 3
• Strony: 475--491
• Opis fizyczny: Bibliogr. 47 poz., rys., tab., wykr.

Twórcy

autor

Bartecki Krzysztof

• Institute of Control Engineering, Opole University of Technology, ul. Prószkowska 76, 45-758 Opole, Poland

Bibliografia

• [1] Ahmad, I. and Berzins, M. (2001). MOL solvers for hyperbolic PDEs with source terms, Mathematics and Computers in Simulation 56(2): 115–125.
• [2] Albertos, P. and Sala, A. (2004). Multivariable Control Systems: An Engineering Approach, Springer, London.
• [3] Anfinsen, H. and Aamo, O.M. (2018). Adaptive control of linear 2×2 hyperbolic systems, Automatica 87: 69–82.
• [4] Anfinsen, H. and Aamo, O.M. (2019). Adaptive Control of Hyperbolic PDEs, Springer, Cham.
• [5] Anfinsen, H., Diagne, M., Aamo, O. and Krstić, M. (2017). Estimation of boundary parameters in general heterodirectional linear hyperbolic systems, Automatica 79: 185–197.
• [6] Arov, D.Z., Kurula, M. and Staffans, O.J. (2012). Boundary control state/signal systems and boundary triplets, in A. Hassi et al. (Eds), Operator Methods for Boundary Value Problems, Cambridge University Press, Cambridge, pp. 73–86.
• [7] Bartecki, K. (2013a). Computation of transfer function matrices for 2×2 strongly coupled hyperbolic systems of balance laws, Proceedings of the 2nd Conference on Control and Fault-Tolerant Systems, Nice, France, pp. 578–583.
• [8] Bartecki, K. (2013b). A general transfer function representation for a class of hyperbolic distributed parameter systems, International Journal of Applied Mathematics and Computer Science 23(2): 291–307, DOI: 10.2478/amcs-2013-0022.
• [9] Bartecki, K. (2015a). Abstract state-space models for a class of linear hyperbolic systems of balance laws, Reports on Mathematical Physics 76(3): 339–358.
• [10] Bartecki, K. (2015b). Transfer function-based analysis of the frequency-domain properties of a double pipe heat exchanger, Heat and Mass Transfer 51(2): 277–287.
• [11] Bartecki, K. (2016). Modeling and Analysis of Linear Hyperbolic Systems of Balance Laws, Springer, Cham.
• [12] Bartecki, K. (2019). Approximation state-space model for 2×2 hyperbolic systems with collocated boundary inputs, 24th International Conference on Methods and Models in Automation and Robotics (MMAR), Międzyzdroje, Poland, pp. 513–518.
• [13] Bastin, G. and Coron, J.-M. (2016). Stability and Boundary Stabilization of 1-D Hyperbolic Systems, Birkhäuser, Basel.
• [14] Callier, F.M. and Winkin, J. (1993). Infinite dimensional system transfer functions, in R.F. Curtain et al. (Eds), Analysis and Optimization of Systems: State and Frequency Domain Approaches for Infinite-Dimensional Systems, Springer, Berlin/Heidelberg, pp. 75–101.
• [15] Cockburn, B., Johnson, C., Shu, C. and Tadmor, E. (1998). Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Springer, Heidelberg.
• [16] Coron, J., Li, T. and Li, Y. (2019). One-Dimensional Hyperbolic Conservation Laws and Their Applications, World Scientific, Singapore.
• [17] Curtain, R.F. and Zwart, H. (1995). An Introduction to Infinite-Dimensional Linear Systems Theory, Springer, New York, NY.
• [18] Curtain, R. and Morris, K. (2009). Transfer functions of distributed parameters systems: A tutorial, Automatica 45(5): 1101–1116.
• [19] Deutscher, J. (2017). Finite-time output regulation for linear 2×2 hyperbolic systems using backstepping, Automatica 75: 54–62.
• [20] Doyle, J., Francis, B. and Tannenbaum, A. (1992). Feedback Control Theory, Macmillan, New York, NY.
• [21] Emirsajłow, Z. and Townley, S. (2000). From PDEs with boundary control to the abstract state equation with an unbounded input operator: A tutorial, European Journal of Control 6(1): 27–49.
• [22] Engel, K.-J. and Nagel, R. (2000). One-Parameter Semigroups for Linear Evolution Equations, Springer, New York, NY.
• [23] Godlewski, E. and Raviart, P.-A. (1996). Numerical Approximation of Hyperbolic Systems of Conservation Laws, Springer, New York, NY.
• [24] Grabowski, P. and Callier, F.M. (2001). Circle criterion and boundary control systems in factor form: Input–output approach, International Journal of Applied Mathematics and Computer Science 11(6): 1387–1403.
• [25] Gugat, M., Herty, M. and Yu, H. (2018). On the relaxation approximation for 2×2 hyperbolic balance laws, in C. Klingenberg and M. Westdickenberg (Eds), Theory, Numerics and Applications of Hyperbolic Problems I, Springer, Cham, pp. 651–663.
• [26] Hu, L., Di Meglio, F., Vazquez, R. and Krstić, M. (2016). Control of homodirectional and general heterodirectional linear coupled hyperbolic PDEs, IEEE Transactions on Automatic Control 61(11): 3301–3314.
• [27] Jones, B.L. and Kerrigan, E.C. (2010). When is the discretization of a spatially distributed system good enough for control?, Automatica 46(9): 1462–1468.
• [28] Kitsos, C., Besançon, G. and Prieur, C. (2019). A high-gain observer for a class of 2×2 hyperbolic systems with C1 exponential convergence, IFAC-PapersOnLine 52(2): 174–179.
• [29] Koto, T. (2004). Method of lines approximations of delay differential equations, Computers & Mathematics with Applications 48(1–2): 45–59.
• [30] Lalot, S. and Desmet, B. (2019). The harmonic response of counter-flow heat exchangers—Analytical approach and comparison with experiments, International Journal of Thermal Sciences 135: 163–172.
• [31] LeVeque, R. (2002). Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge.
• [32] Levine, W.S. (Ed.) (2011). The Control Systems Handbook: Control System Advanced Methods, Electrical Engineering Handbook, CRC Press, Boca Raton, FL.
• [33] Li, H.-X. and Qi, C. (2010). Modeling of distributed parameter systems for applications—A synthesized review from time-space separation, Journal of Process Control 20(8): 891–901.
• [34] Litrico, X. and Fromion, V. (2009a). Boundary control of hyperbolic conservation laws using a frequency domain approach, Automatica 45(3): 647–656.
• [35] Litrico, X. and Fromion, V. (2009b). Modeling and Control of Hydrosystems, Springer, London.
• [36] Maidi, A., Diaf, M. and Corriou, J.-P. (2010). Boundary control of a parallel-flow heat exchanger by input–output linearization, Journal of Process Control 20(10): 1161–1174.
• [37] Mattheij, R.M.M., Rienstra, S.W. and ten Thije Boonkkamp, J.H.M. (2005). Partial Differential Equations: Modeling, Analysis, Computation, SIAM, Philadelphia, PA.
• [38] Partington, J.R. (2004). Some frequency-domain approaches to the model reduction of delay systems, Annual Reviews in Control 28(1): 65–73.
• [39] Polyanin, A.D. and Manzhirov, A.V. (1998). Handbook of Integral Equations, CRC Press, Boca Raton, FL.
• [40] Rauh, A., Senkel, L., Aschemann, H., Saurin, V.V. and Kostin, G.V. (2016). An integrodifferential approach to modeling, control, state estimation and optimization for heat transfer systems, International Journal of Applied Mathematics and Computer Science 26(1): 15–30, DOI: 10.1515/amcs-2016-0002.
• [41] Ray, W.H. (1981). Advanced Process Control, McGraw-Hill New York, NY.
• [42] Russell, D.L. (1978). Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Review 20(4): 639–739.
• [43] Schiesser, W.E. and Griffiths, G.W. (2009). A Compendium of Partial Differential Equation Models: Method of Lines Analysis with Matlab, Cambridge University Press, New York, NY.
• [44] Shakeri, F. and Dehghan, M. (2008). The method of lines for solution of the one-dimensional wave equation subject to an integral conservation condition, Computers & Mathematics with Applications 56(9): 2175–2188.
• [45] Tucsnak, M. and Weiss, G. (2009). Observation and Control for Operator Semigroups, Birkhäuser, Basel.
• [46] Zavala-Río, A., Astorga-Zaragoza, C.M. and Hernández-González, O. (2009). Bounded positive control for double-pipe heat exchangers, Control Engineering Practice 17(1): 136–145.
• [47] Zwart, H. (2004). Transfer functions for infinite-dimensional systems, Systems and Control Letters 52(3–4): 247–255.

Uwagi

PL

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).