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Open-loop unstable feedback systems with double-sided inputs : an explicit demonstration of self-consistency

Leithead, W.E. and Ragnoli, E. and O'Reilly, J. (2005) Open-loop unstable feedback systems with double-sided inputs : an explicit demonstration of self-consistency. In: Proceedings of the 44th IEEE Conference on Decision and Control, 2005 and 2005 European Control Conference. IEEE, pp. 5198-5203. ISBN 0-7803-9567-0

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The standard formulation of linear shift-invariant feedback systems in the doubly infinite time axis setting lacks self-consistency with seemingly irreconcilable difficulties having been identified when the system is open-loop unstable. The available options for circumventing these difficulties for discrete-time SISO systems are highlighted and the manner in which they are exploited to obtain a self-consistent framework by reformulating the feedback systems in the space of distributions is clarified. In addition, it is explicitly demonstrated that causality and stability of a standard example are implied by the causality and stability of the equivalent system when reformulated in the self-consistent framework.