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

  • Context: Undergrad 
  • Thread starter Thread starter theycallmevirgo
  • Start date Start date
Click For Summary
SUMMARY

The discussion centers on the presence of a dt term in equation 16 without an accompanying f(t), raising questions about its notation and meaning. Participants suggest that the dt term likely represents a time interval and may imply an integral with respect to time, similar to the notation found in PID controller theory. The conversation highlights the common practice in dynamic systems to omit the range of integrals, assuming relevance to "all relevant history" rather than specific operating points. The notation's ambiguity is acknowledged, with participants debating the necessity of including f(t) in the equation.

PREREQUISITES
  • Understanding of integral calculus and its notation
  • Familiarity with PID controller theory
  • Knowledge of dynamic systems and control theory
  • Basic grasp of time derivatives in mathematical equations
NEXT STEPS
  • Research the mathematical foundations of PID controllers and their applications
  • Explore the concept of integrals in dynamic systems and their implications
  • Study the role of time derivatives in control theory
  • Examine various notational conventions used in mathematical equations
USEFUL FOR

Students and professionals in engineering, particularly those focused on control systems, dynamic systems analysis, and mathematical modeling.

theycallmevirgo
Messages
108
Reaction score
25
TL;DR
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?
 

Attachments

  • Screenshot_20211013-131215_ReadEra.jpg
    Screenshot_20211013-131215_ReadEra.jpg
    28.7 KB · Views: 139
Physics news on Phys.org
reckon ##dt## is just supposed to be some time interval, maybe smallish (can't say without seeing the book)
 
  • Like
Likes   Reactions: theycallmevirgo and PeroK
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.
 
  • Like
Likes   Reactions: theycallmevirgo and ergospherical
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
 
  • Like
Likes   Reactions: theycallmevirgo, DaveE and PeroK
Well, I guess if you don't need to put the range on an integral, why bother with the integral sign at all?
 
  • Like
Likes   Reactions: theycallmevirgo
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.
 
  • Informative
  • Like
Likes   Reactions: theycallmevirgo and PeroK
Don't we just assume ## f(t)== 1 ##? I mean, we have ##\int dt =t ##
 
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?
 
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.
 

Similar threads

Undergrad What is the position of this moving particle over time?
  • · Replies 8 ·
Replies
8
Views
2K
High School How does this derivative work? And why the speed of light remains?
  • · Replies 7 ·
Replies
7
Views
430
Graduate Non solvable integral? (dx/dt)^2 dt
  • · Replies 3 ·
Replies
3
Views
3K
MHB How to Integrate $\int(e^{t^2}+16)te^{t^2} dt$ Using Substitution?
  • · Replies 2 ·
Replies
2
Views
1K
Undergrad What does dv/dt mean, how is it used, and when?
  • · Replies 9 ·
Replies
9
Views
3K
Undergrad On convolution theorem of Laplace transform: Schiff
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Is there a way to simplify this integral involving an exponential function?
  • · Replies 6 ·
Replies
6
Views
1K
Undergrad Possibility of an integral system
  • · Replies 3 ·
Replies
3
Views
1K
Undergrad Difference between d/dt and d(theta)/dt? Why is it dr or ds/dt?
  • · Replies 3 ·
Replies
3
Views
3K
MHB Proving Equation (1): Let r(t) be a Vector in $\mathbb{R^3}$
  • · Replies 2 ·
Replies
2
Views
1K