A consistent Mathematical Framework for linear feedback systems using distributions.
Ragnoli, Emanuele (2006) A consistent Mathematical Framework for linear feedback systems using distributions. PhD thesis, National University of Ireland Maynooth.
In this thesis, when the signals are either discrete time or continuous time signals, three different Mathematical Formalisms for feedback systems are investigated. The first one, the Standard Formalism, uses mathematical elements that are adopted from the conventional analysis for feedback systems (conventional analysis as in ). It is shown how consistency can be regained, but with the effect of severely restricting the class of signals. Moreover, it is shown under which conditions the Standard Formalism becomes a consistent Framework. The second one, the Generalised Formalism, extends the class of systems but lacks of a transform domain analysis. The third one, a Framework using Distributions, is shown to be consistent. Moreover, the class of signals does not need any restriction and a transform domain analysis can be performed. The class of signals is the space of distributions, or generalised functions. Being those an extension of the concept of ”classical” function, the traditional class of signals is largely increased. The class of systems on the distributions are, in time domain the convolutes on the distributions, and in transform domain the multipliers on the Fourier transforms of distributions. Convolutes and multipliers are a broader class of systems than the traditional class of convolutions and algebraic functions, in time and transform domain, respectively. Since this is a consistent Framework, paradoxes and inconsistencies, such as the Georgiou Smith paradox, do not occur. Hence, it is proved that the Framework using Distributions is suitable for the analysis and design of feedback systems.
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