Abstract
This note exposes the mathematical setting of initial value problems for causal time-invariant linear systems, given by ordinary differential equations within the framework of generalized functions. We show the structure of the unique solutions for such equations, and apply it to problems with causal or persistent inputs using time-domain methods and generalized Laplace and Fourier transforms. In particular, we correct a widespread inconsistency in the use of the Laplace transform.
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References
Mäkilä PM (2006) A note on the Laplace transform method for initial value problems. Int J Control 79(1): 36–41
Lundberg KH, Miller HR, Trumper DL (2007) Initial conditions, generalized functions, and the Laplace transform. IEEE Control Syst Mag 27(1): 22–35
Kailath T (1980) Linear systems. Prentice Hall, Englewood Cliffs
Siebert WM (1986) Circuits, signals and systems. MIT Press, Cambridge
Dorf RC, Bishop RH (1992) Modern control systems. Prentice Hall, Upper Saddle River
Nise NS (1992) Control systems engineering. Wiley, New York
Oppenheim AV, Willsky AS (1997) Signals and systems. Prentice Hall, Upper Saddle River
Zemanian AH (1965) Distribution theory and transform analysis. McGraw Hill, New York
Schwartz L (1966) Mathematics for the physical sciences. Hermann, Paris
Brigola R (1996) Fourieranalysis, Distributionen und Anwendungen. Vieweg, Wiesbaden
Schwartz L (1957) Théorie des distributions. Hermann, Paris
Triebel H (1992) Höhere Analysis. VEB Dt. Verlag der Wissenschaften, Berlin
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Brigola, R., Singer, P. On initial conditions, generalized functions and the Laplace transform. Electr Eng 91, 9–13 (2009). https://doi.org/10.1007/s00202-009-0110-5
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DOI: https://doi.org/10.1007/s00202-009-0110-5