Time in Control Theory by Blas M. Vinagre

Author:Blas M. Vinagre
Language: eng
Format: epub
ISBN: 9783031540424
Publisher: Springer Nature Switzerland

Past youth we begin to perceive how memory is selective, that it is not a continuous, uniform and non-specific record of everything that happens to us, that we forget most of the things that happen to us from Monday to Friday but we remember that trip, that party or that face among a sea of faces. Our memory, as an integral, does not follow the Riemann method. The relational idea of time defended by Leibniz reappears: time exists only by virtue of events.

We get closer to the experience if we consider that from a purely empirical point of view, the idea of ‘event’–and the relationship between events–is more fundamental, and that our normal concepts of space and time are to be understood as idealized mathematical constructs of empirical data that correspond to the events, and that dynamics could be formulated in a new way in which the concepts of becoming and change are independent of any underlying temporal being [5].

Event-based sampling that accommodates Lebesgue integral sums is called level crossing sampling, but other alternatives have been studied, such as integrated level crossing sampling or the sampling by level crossing + linear prediction. In the first case, the sample is updated if the integral of the absolute value of the difference between the current value of the signal and the last sample reaches a certain threshold. In the second, it does so if the difference between the current value and the one obtained through linear prediction reaches said threshold [1]. In both cases, the event that triggers sampling is not the mere value of the function f(t) but that of a function related to it or to the independent variable that we can generalize as g(f(t)) .

We can go one step further by resorting to another type of integral, that of Stieltjes. The most common definition is

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