# dt without f(t)?

Summary:
How can an equation contain a time derivative without any f(t)?
In equation 16 they seem to have a dt term without f(t). Am I missing something?

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ergospherical
Gold Member
reckon ##dt## is just supposed to be some time interval, maybe smallish (can't say without seeing the book)

theycallmevirgo and PeroK
PeroK
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2020 Award
Summary:: How can an equation contain a time derivative without any f(t)?

In equation 16 they seem to have a dt term without f(t). Am I missing something?
Context is everything here. It looks more like there's an integral with respect to time in there, but it's highly contextual notation.

theycallmevirgo and ergospherical
ergospherical
Gold Member
Context is everything here. It looks more like there's an integral with respect to time in there, but it's highly contextual notation.
fwiw I'm assuming the formula in the picture is the same one as (or a variation of) this here:
https://en.wikipedia.org/wiki/PID_controller#Controller_theory

the bit in the brackets in the picture corresponding to ##\int e(\tau) d\tau## on the wiki version

theycallmevirgo, DaveE and PeroK
PeroK
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Well, I guess if you don't need to put the range on an integral, why bother with the integral sign at all?

theycallmevirgo
DaveE
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Well, I guess if you don't need to put the range on an integral, why bother with the integral sign at all?
Yes, I agree their notation sucks.
In dynamic systems (control systems) it is common to leave the range out, with an assumption it's "all relevant history". This is because most of the interest is in the behavior (stability, etc.), not the actual operating points. One of the cheats you get from linear systems, the integral can be treated like an operator; it might not matter what the actual value is.

theycallmevirgo and PeroK
WWGD
Gold Member
Don't we just assume ## f(t)== 1 ##? I mean, we have ##\int dt =t ##

Don't we just assume ## f(t)== 1 ##? I mean, we have ##\int dt =t ##
That's exactly what I thought, originally. But if so, why include it at all?

WWGD