A general transfer function representation for a class of hyperbolic distributed parameter systems

Krzysztof Bartecki

International Journal of Applied Mathematics and Computer Science, Tome 23 (2013), p. 291-307 / Harvested from The Polish Digital Mathematics Library

Résumé

Results of transfer function analysis for a class of distributed parameter systems described by dissipative hyperbolic partial differential equations defined on a one-dimensional spatial domain are presented. For the case of two boundary inputs, the closed-form expressions for the individual elements of the 2×2 transfer function matrix are derived both in the exponential and in the hyperbolic form, based on the decoupled canonical representation of the system. Some important properties of the transfer functions considered are pointed out based on the existing results of semigroup theory. The influence of the location of the boundary inputs on the transfer function representation is demonstrated. The pole-zero as well as frequency response analyses are also performed. The discussion is illustrated with a practical example of a shell and tube heat exchanger operating in parallel- and countercurrent-flow modes.

Détail

Publié le : 2013-01-01

Zbl 1282.93182

EUDML-ID : urn:eudml:doc:257120

Bibliographie

[000] Ancona, F. and Coclite, G.M. (2005). On the boundary controllability of first-order hyperbolic systems, Nonlinear Analysis: Theory, Methods & Applications 63(5-7): e1955-e1966. | Zbl 1224.93010

[001] Arbaoui, M.A., Vernieres-Hassimi, L., Seguin, D. and Abdelghani-Idrissi, M.A. (2007). Counter-current tubular heat exchanger: Modeling and adaptive predictive functional control, Applied Thermal Engineering 27(13): 2332-2338.

[002] Bartecki, K. (2007). Comparison of frequency responses of parallel- and counter-flow type of heat exchanger, Proceedings of the 13th IEEE IFAC International Conference on Methods and Models in Automation and Robotics, Szczecin, Poland, pp. 411-416.

[003] Bartecki, K. (2009). Frequency- and time-domain analysis of a simple pipeline system, Proceedings of the 14th IEEE IFAC International Conference on Methods and Models in Automation and Robotics, Międzyzdroje, Poland, pp. 366-371.

[004] Bartecki, K. (2010). On some peculiarities of neural network approximation applied to the inverse kinematics problem, Proceedings of the Conference on Control and FaultTolerant Systems, Nice, France, pp. 317-322.

[005] Bartecki, K. (2011). Approximation of a class of distributed parameter systems using proper orthogonal decomposition, Proceedings of the 16th IEEE International Conference on Methods and Models in Automation and Robotics, Międzyzdroje, Poland, pp. 351-356.

[006] Bartecki, K. (2012a). Neural network-based PCA: An application to approximation of a distributed parameter system, in L., Rutkowski, M., Korytkowski, R., Scherer, R., Tadeusiewicz, L. Zadeh and J. Zurada (Eds.), Artificial Intelligence and Soft Computing, Lecture Notes in Computer Science, Vol. 7267, Springer, Berlin/Heidelberg, pp. 3-11.

[007] Bartecki, K. (2012b). PCA-based approximation of a class of distributed parameter systems: Classical vs. neural network approach, Bulletin of the Polish Academy of Sciences: Technical Sciences 60(3): 651-660.

[008] Bartecki, K. and Rojek, R. (2005). Instantaneous linearization of neural network model in adaptive control of heat exchange process, Proceedings of the 11th IEEE International Conference on Methods and Models in Automation and Robotics, Międzyzdroje, Poland, pp. 967-972.

[009] Bounit, H. (2003). The stability of an irrigation canal system, International Journal of Applied Mathematics and Computer Science 13(4): 453-468. | Zbl 1115.93333

[010] Callier, F.M. and Winkin, J. (1993). Infinite dimensional system transfer functions, in R.F, Curtain, A. Bensoussan and J.L. Lions (Eds.), Analysis and Optimization of Systems: State and Frequency Domain Approaches for Infinite-Dimensional Systems, Lecture Notes in Control and Information Sciences, Vol. 185, Springer, Berlin/Heidelberg, pp. 75-101. | Zbl 0793.93040

[011] Cheng, A. and Morris, K. (2003). Well-posedness of boundary control systems, SIAM Journal on Control and Optimization 42(4): 1244-1265. | Zbl 1049.35041

[012] Chentouf, B. and Wang, J.M. (2009). Boundary feedback stabilization and Riesz basis property of a 1-D first order hyperbolic linear system with $L^∞$-coefficients, Journal of Differential Equations 246(3): 1119-1138. | Zbl 1160.35045

[013] Christofides, P.D. and Daoutidis, P. (1997). Finite-dimensional control of parabolic PDE systems using approximate inertial manifolds, Journal of Mathematical Analysis and Applications 216(2): 398-420. | Zbl 0890.93051

[014] Christofides, P.D. and Daoutidis, P. (1998a). Distributed output feedback control of two-time-scale hyperbolic PDE systems, Applied Mathematics and Computer Science 8(4): 713-732. | Zbl 0981.93045

[015] Christofides, P.D. and Daoutidis, P. (1998b). Robust control of hyperbolic PDE systems, Chemical Engineering Science 53(1): 85-105. | Zbl 0981.93045

[016] Contou-Carrere, M.N. and Daoutidis, P. (2008). Model reduction and control of multi-scale reaction-convection processes, Chemical Engineering Science 63(15): 4012-4025.

[017] Coron, J., d'Andrea Novel, B. and Bastin, G. (2007). A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws, IEEE Transactions on Automatic Control 52(1): 2-11.

[018] Curtain, R.F. (1988). Equivalence of input-output stability and exponential stability for infinite-dimensional systems, Mathematical Systems Theory 21(1): 19-48. | Zbl 0657.93050

[019] Curtain, R.F., Logemann, H., Townley, S. and Zwart, H. (1992). Well-posedness, stabilizability, and admissibility for Pritchard-Salamon systems, Journal of Mathematical Systems, Estimation, and Control 4(4): 1-38. | Zbl 0815.93046

[020] Curtain, R.F. and Sasane, A.J. (2001). Compactness and nuclearity of the Hankel operator and internal stability of infinite-dimensional state linear systems, International Journal of Control 74(12): 1260-1270. | Zbl 1011.93024

[021] Curtain, R.F. and Weiss, G. (2006). Exponential stabilization of well-posed systems by colocated feedback, SIAM Journal on Control and Optimization 45(1): 273-297. | Zbl 1139.93026

[022] Curtain, R.F. and Zwart, H. (1995). An Introduction to InfiniteDimensional Linear Systems Theory, Springer-Verlag, New York, NY. | Zbl 0839.93001

[023] Curtain, R. and Morris, K. (2009). Transfer functions of distributed parameters systems: A tutorial, Automatica 45(5): 1101-1116. | Zbl 1162.93300

[024] Delnero, C.C., Dreisigmeyer, D., Hittle, D.C., Young, P.M., Anderson, C.W. and Anderson, M.L. (2004). Exact solution to the governing PDE of a hot water-to-air finned tube cross-flow heat exchanger, HVAC&R Research 10(1): 21-31.

[025] Diagne, A., Bastin, G. and Coron, J.-M. (2012). Lyapunov exponential stability of 1-D linear hyperbolic systems of balance laws, Automatica 48(1): 109-114. | Zbl 1244.93143

[026] Ding, L., Johansson, A. and Gustafsson, T. (2009). Application of reduced models for robust control and state estimation of a distributed parameter system, Journal of Process Control 19(3): 539-549.

[027] Dooge, J.C.I. and Napiorkowski, J.J. (1987). The effect of the downstream boundary conditions in the linearized St Venant equations, The Quarterly Journal of Mechanics and Applied Mathematics 40(2): 245-256. | Zbl 0611.76049

[028] Dos Santos, V., Bastin, G., Coron, J.-M. and d'Andréa Novel, B. (2008). Boundary control with integral action for hyperbolic systems of conservation laws: Stability and experiments, Automatica 44(5): 1310-1318. | Zbl 1283.93211

[029] Evans, L.C. (1998). Partial Differential Equations, American Mathematical Society, Providence, RI. | Zbl 0902.35002

[030] Filbet, F. and Shu, C.-W. (2005). Approximation of hyperbolic models for chemosensitive movement, SIAM Journal on Scientific Computing 27(3): 850-872. | Zbl 1141.35396

[031] Friedly, J.C. (1972). Dynamic Behaviour of Processes, Prentice Hall, New York, NY.

[032] Górecki, H., Fuksa, S., Grabowski, P. and Korytowski, A. (1989). Analysis and Synthesis of Time Delay Systems, Wiley, New York, NY. | Zbl 0695.93002

[033] Grabowski, P. and Callier, F.M. (2001a). Boundary control systems in factor form: Transfer functions and input-output maps, Integral Equations and Operator Theory 41(1): 1-37. | Zbl 1009.93041

[034] Grabowski, P. and Callier, F.M. (2001b). Circle criterion and boundary control systems in factor form: Input-output approach, International Journal of Applied Mathematics and Computer Science 11(6): 1387-1403. | Zbl 0999.93061

[035] Gvozdenac, D.D. (1990). Transient response of the parallel flow heat exchanger with finite wall capacitance, Archive of Applied Mechanics 60(7): 481-490.

[036] Jacob, B. and Zwart, H.J. (2012). Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces, Operator Theory: Advances and Applications, Vol. 223, Springer-Verlag, Basel. | Zbl 1254.93002

[037] 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. | Zbl 1201.93027

[038] Jovanović, M.R. and Bamieh, B. (2006). Computation of the frequency responses for distributed systems with one spatial variable, Systems Control Letters 55(1): 27-37. | Zbl 1129.93421

[039] Kowalewski, A. (2009). Time-optimal control of infinite order hyperbolic systems with time delays, International Journal of Applied Mathematics and Computer Science 19(4): 597-608, DOI: 10.2478/v10006-009-0047-x. | Zbl 1300.49007

[040] 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.

[041] Litrico, X. and Fromion, V. (2009a). Boundary control of hyperbolic conservation laws using a frequency domain approach, Automatica 45(3): 647-656. | Zbl 1168.93015

[042] Litrico, X. and Fromion, V. (2009b). Modeling and Control of Hydrosystems, Springer, London. | Zbl 1168.93015

[043] Litrico, X., Fromion, V. and Scorletti, G. (2007). Robust feedforward boundary control of hyperbolic conservation laws, Networks and Heterogeneous Media 2(4): 717-731. | Zbl 1165.35321

[044] Lumer, G. and Phillips, R.S. (1961). Dissipative operators in a Banach space, Pacific Journal of Mathematics 11(2): 679-698. | Zbl 0101.09503

[045] 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.

[046] Mattheij, R.M.M., Rienstra, S.W. and ten Thije Boonkkamp, J.H.M. (2005). Partial Differential Equations: Modeling, Analysis, Computation, Society for Industrial and Applied Mathematics, Philadelphia, PA. | Zbl 1090.35001

[047] Miano, G. and Maffucci, A. (2001). Transmission Lines and Lumped Circuits, Academic Press, San Diego, CA. | Zbl 1157.78317

[048] Park, H.M. and Cho, D.H. (1996). The use of the Karhunen-Loève decomposition for the modeling of distributed parameter systems, Chemical Engineering Science 51(1): 81-98.

[049] Patan, M. (2012). Distributed scheduling of sensor networks for identification of spatio-temporal processes, International Journal of Applied Mathematics and Computer Science 22(2): 299-311, DOI: 10.2478/v10006-012-0022-9. | Zbl 1283.93298

[050] Phillips, R.S. (1957). Dissipative hyperbolic systems, Transactions of the American Mathematical Society 86(1): 109-173. | Zbl 0081.31102

[051] Pritchard, A.J. and Salamon, D. (1984). The linear quadratic optimal control problem for infinite dimensional systems with unbounded input and output operators, in F. Kappel and W. Schappacher (Eds.), Infinite-Dimensional Systems, Lecture Notes in Mathematics, Vol. 1076, Springer, Berlin/Heidelberg, pp. 187-202. | Zbl 0545.93042

[052] Rabenstein, R. (1999). Transfer function models for multidimensional systems with bounded spatial domains, Mathematical and Computer Modelling of Dynamical Systems 5(3): 259-278. | Zbl 0935.93038

[053] Rauch, J. and Taylor, M. (1974). Exponential decay of solutions to hyperbolic equations in bounded domains, Indiana University Mathematical Journal 24(1): 79-86. | Zbl 0281.35012

[054] Salamon, D. (1987). Infinite dimensional linear systems with unbounded control and observation: A functional analytic approach, Transactions of the American Mathematical Society 300(2): 383-431. | Zbl 0623.93040

[055] Sasane, A. (2002). Hankel Norm Approximation for Infinitedimensional Systems, Springer-Verlag, Berlin. | Zbl 1009.93501

[056] Staffans, O.J. and Weiss, G. (2000). Transfer functions of regular linear systems, Part II: The system operator and the Lax-Phillips semigroup, Transactions of the American Mathematical Society 354(8): 3229-3262. | Zbl 0996.93012

[057] Strikwerda, J. (2004). Finite Difference Schemes and Partial Differential Equations, Society for Industrial and Applied Mathematics, Philadelphia, PA. | Zbl 1071.65118

[058] Sutherland, J.C. and Kennedy, C.A. (2003). Improved boundary conditions for viscous, reacting, compressible flows, Journal of Computational Physics 191(2): 502-524. | Zbl 1134.76736

[059] Tucsnak, M. and Weiss, G. (2006). Passive and Conservative Linear Systems, Nancy University, Nancy.

[060] Tucsnak, M. and Weiss, G. (2009). Observation and Control for Operator Semigroups, Birkhäuser, Basel. | Zbl 1188.93002

[061] Uciński, D. (2012). Sensor network scheduling for identification of spatially distributed processes, International Journal of Applied Mathematics and Computer Science 22(1): 25-40, DOI: 10.2478/v10006-012-0002-0. | Zbl 1273.93168

[062] Weiss, G. (1994). Transfer functions of regular linear systems, Part I: Characterizations of regularity, Transactions of the American Mathematical Society 342(2): 827-854. | Zbl 0798.93036

[063] Wu, W. and Liou, C.-T. (2001). Output regulation of two-time-scale hyperbolic PDE systems, Journal of Process Control 11(6): 637-647.

[064] Xu, C.-Z. and Sallet, G. (2002). Exponential stability and transfer functions of processes governed by symmetric hyperbolic systems, ESAIM: Control, Optimisation and Calculus of Variations 7: 421-442. | Zbl 1040.93031

[065] 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.

[066] Zwart, H. (2004). Transfer functions for infinite-dimensional systems, Systems and Control Letters 52(3-4): 247-255. | Zbl 1157.93420

[067] Zwart, H., Gorrec, Y.L., Maschke, B. and Villegas, J. (2010). Well-posedness and regularity of hyperbolic boundary control systems on a one-dimensional spatial domain, ESAIM: Control, Optimisation and Calculus of Variations 16(04): 1077-1093. | Zbl 1202.93064