Equation 16: Missing dt Term Without f(t)?

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In summary, the conversation discusses the presence of a dt term in equation 16 without any accompanying f(t) term. The participants question whether the dt term represents a small time interval or if it is part of an integral. They also mention the possibility of the integral representing all relevant history in a dynamic system. The conversation ends with a discussion of whether the f(t) term is assumed to be equal to 1 and why the dt term is included at all.
  • #1
theycallmevirgo
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TL;DR 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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  • #2
reckon ##dt## is just supposed to be some time interval, maybe smallish (can't say without seeing the book)
 
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  • #3
theycallmevirgo said:
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.
 
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  • #4
PeroK said:
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
 
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  • #5
Well, I guess if you don't need to put the range on an integral, why bother with the integral sign at all?
 
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  • #6
PeroK said:
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.
 
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  • #7
Don't we just assume ## f(t)== 1 ##? I mean, we have ##\int dt =t ##
 
  • #8
WWGD said:
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?
 
  • #9
theycallmevirgo said:
That's exactly what I thought, originally. But if so, why include it at all?
Because the result is not necessarily " neutral" when computed. You will not just ( necessarily) get a 1 multiplying . Edit: On my phone, will give you more thorough answer tmw when I get to my pc.
 

1. What is Equation 16?

Equation 16 refers to a specific mathematical equation that is commonly used in scientific research and calculations.

2. What does the "missing dt term" mean in Equation 16?

The "missing dt term" in Equation 16 refers to the absence of the variable dt, which represents a small change in time. This term is often used to calculate the rate of change in a system over a specific time interval.

3. Why is the dt term missing in Equation 16?

The dt term may be missing in Equation 16 for a variety of reasons. It could be due to simplification of the equation, assumptions made in the research, or the specific application of the equation.

4. How does the absence of the dt term affect the accuracy of Equation 16?

The absence of the dt term may affect the accuracy of Equation 16 depending on the specific context in which it is being used. In some cases, the dt term may be negligible and not significantly impact the results. However, in other cases, the absence of this term may lead to significant errors in the calculations.

5. Can the missing dt term be added back into Equation 16?

In most cases, the missing dt term can be added back into Equation 16 if necessary. However, it is important to carefully consider the implications of including this term and whether it is appropriate for the specific application of the equation.

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