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Tn is simply integration time constant in PI standard form, usually designated Ti. "Follow-up time" sounds strange, no commonly understood term. "Follow-up controller" is however a usual term describing the specific controller form that uses manipulated value Yn as integrator state variable.

Means nothing, it's just the specific math how P- and I-term are combined in this controller form, but they can be analyzed separately. I-term must not necessarily derived from Xn-1, more often it uses Xn, see other PI examples in quoted manual.