Transfer function models for distributed-parameter systems with impedance boundary conditions. (English) Zbl 1405.93073
Summary: A transfer function description is derived for a general class of linear distributed parameter systems dependent on time and one spatial variable. Suitable functional transformations are the Laplace transformation for the time variable and the Sturm-Liouville transformation for the space variable. A practical problem is the determination of the eigenfunctions of the Sturm-Liouville transformation since these depend on the type and the parameters of the boundary conditions. This contribution shows that the design of a transfer function model can be separated from the correct treatment of the boundary conditions. The presented approach exhibits strong parallels to state feedback techniques from control theory. Examples for an electrical transmission line demonstrate how terminations with arbitrary complex impedances can be considered without redesigning the transmission line model.
MSC:
| 93B17 | Transformations |
| 93B52 | Feedback control |
| 93C05 | Linear systems in control theory |
| 93C15 | Control/observation systems governed by ordinary differential equations |
| 34B24 | Sturm-Liouville theory |
Keywords:
transfer functions; boundary conditions; Sturm-Liouville transformation; biorthogonal basis functions; state feedback; closed-loop systemReferences:
| [1] | Antonini, G., A Dyadic Green’s function based method for the transient analysis of lossy and dispersive multiconductor transmission lines, IEEE Transactions on Microwave Theory and Techniques,, 56, 4, 880-895 (2008) |
| [2] | Antonini, G., Spectral models of lossy nonuniform multiconductor transmission lines, IEEE Transactions on Electromagnetic Compatibility,, 54, 2, 474-481 (2012) |
| [3] | Avanzini, F.; Marogna, R., A modular physically based approach to the sound synthesis of membrane percussion instruments, IEEE Transactions on Audio, Speech, and Language Processing,, 18, 4, 891-902 (2010) |
| [4] | Berners, D. P., The discontinuous conical bore as a Sturm- Liouville problem, The Journal of the Acoustical Society of America,, 100, 4, 2812-2812 (1996) |
| [5] | Butkovskiy, A., Structural theory of distributed systems (1983), Chichester, England: Ellis Horwood Ltd, Chichester, England |
| [6] | Chou, C. Y.; Hong, B. S.; Chiang, P. J.; Wang, W. T.; Chen, L. K.; Lee, C. Y., Distributed control of heat conduction in thermal inductive materials with 2D geometrical isomorphism, Entropy,, 16, 9, 4937-4959 (2014) |
| [7] | Churchill, R. V., Operational mathematics (1972), Boston, MA: McGraw Hill, Boston, MA · Zbl 0227.44001 |
| [8] | Curtain, R.; Zwart, H., An introduction to infinite-dimensional systems theory (1995), New York, NY: Springer-Verlag, New York, NY · Zbl 0839.93001 |
| [9] | Deutscher, J., Zustandsregelung verteilt-parametrischer Systeme (2012), Heidelberg: Springer, Heidelberg |
| [10] | Deutscher, J.; Harkort, C., Parametric state feedback design for second-order Riesz-spectral systems, 282-287 (2009), IEEE |
| [11] | Deutscher, J.; Harkort, C., Parametric state feedback design of linear distributed-parameter systems, International Journal of Control,, 82, 6, 1060-1069 (2009) · Zbl 1168.93011 |
| [12] | Eringen, C., The finite Sturm- Liouville-transform, The Quarterly Journal of Mathematics, Oxford Second Series,, 5, 120-129 (1954) · Zbl 0059.10301 |
| [13] | Hong, B. S., Realization of inhomogeneous boundary conditions as virtual sources in parabolic and hyperbolic dynamics, Applied and Computational Mathematics,, 3, 5, 197-204 (2014) |
| [14] | Kato, T., Perturbation theory for linear operators (1976), Berlin, Germany: Springer-Verlag, Berlin, Germany · Zbl 0342.47009 |
| [15] | Luther, U.; Rost, K., Matrix exponentials and inversion of confluent vandermonde matrices, Electronic Transactions on Numerical Analysis,, 18, 91-100 (2004) · Zbl 1065.34001 |
| [16] | Pazy, A., Semigroups of linear operators and applications to partial differential equations, (1983), New York, NY: Springer, New York, NY · Zbl 0516.47023 |
| [17] | Petrausch, S.; Rabenstein, R.; Astola, J.; Egiazarian, K.; Saramäki, T., A simplified design of multidimensional transfer function models, 35-40 (2004), Tampere International Center for Signal Processing |
| [18] | Pryce, J. D., Numerical solution of Sturm- Liouville problems (1993), Oxford: Clarendon Press, Oxford · Zbl 0795.65053 |
| [19] | Rabenstein, R.; Trautmann, L., Digital sound synthesis of string instruments with the functional transformation method, Signal Processing,, 83, 8, 1673-1688 (2003) · Zbl 1144.94356 |
| [20] | Schäfer, M.; Frens̆Tátský, P.; Rabenstein, R.; Rajmic, P.; Rund, F.; Schimmel, J., A physical string model withadjustable boundary conditions (2016), Tampere International Center for Signal Processing |
| [21] | Schäfer, M.; Rabenstein, R.; Paszke, W.; Galkowski, K.; Rogers, E., Calculation of the transformation kernels for the functional transformation method, 1-6 (2017) |
| [22] | Trautmann, L.; Rabenstein, R., Digital sound synthesis by physical modeling using the functional transformation method (2003), New York, NY: Kluwer Academic Publishers, New York, NY · Zbl 1041.94002 |
| [23] | Willems, J. C., The behavioral approach to open and interconnected systems, IEEE Control Systems,, 27, 6, 46-99 (2007) · Zbl 1395.93111 |
| [24] | Woittennek, F.; Mounier, H., Controllability of networks of spatially one-dimensional second order PDEs-An algebraic approach, SIAM Journal on Control and Optimization,, 48, 6, 3882-3902 (2010) · Zbl 1217.93042 |
| [25] | Zwart, H.; De Moor, B.; Motmans, B., Transfer functions for infinite-dimensional systems. Proceedings Sixteenth International Symposium on Mathematical Theory of Networks and Systems, 1-5 (2004), Leuven, Belgium: Katholieke Universiteit Leuven, Leuven, Belgium |
| [26] | Zwart, H., Transfer functions for infinite-dimensional systems, Systems & Control Letters,, 52, 3, 247-255 (2004) · Zbl 1157.93420 |
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