Electronics > Beginners
Control theory - integrator?
IanB:
--- Quote from: nForce on December 10, 2018, 07:35:38 pm ---Why is at the end difference between Y(n) and Y(n-1)? So we get a derivative?
--- End quote ---
It's giving a first order backward difference approximation to the first derivative dY/dt.
nForce:
What about first order term?
rstofer said in another topic, that this is a filter, where we have one pole.
First order term is a first order differential equation? But we got this here, which kim.dd pointed out.
T3sl4co1l:
It's called first order, because it's the first (simplest) finite difference. This is a method to approximate the derivative:
https://en.wikipedia.org/wiki/Numerical_differentiation#Higher-order_methods
Again, the notation seems to be wrong. "x[n] - x[n-1]" typically means a discrete-time (e.g., digital or switched-capacitor filter) application. It's impossible to know what "Y(n)" is supposed to mean -- n usually represents an integer, yet Y(s) is usually representing a continuous function. There would be no point in taking the value of the function at integer values, except for discrete-time purposes.
In the context of numerical analysis, it still seems to be wrong, because the step size must be adjustable in order to form a better and better approximation, to a function that may be changing arbitrarily fast (a step size of 1 is arbitrary, and will miss any details smaller than it).
Tim
nForce:
Ok, I will use your (standard) notation in the future.
rstofer:
--- Quote from: nForce on December 13, 2018, 03:58:25 pm ---What about first order term?
rstofer said in another topic, that this is a filter, where we have one pole.
First order term is a first order differential equation? But we got this here, which kim.dd pointed out.
--- End quote ---
Here is an explanation of RC filters including their Laplace Transform. Note from equation 1 the 1/s factor. Yes, that looks like integration because an RC filter IS an integrator. Think about what happens if you hit it with a step function. Just after equation 10 (that is a LOT of math...) there is a graph of the step response and it winds up as an exponential. Vo = Vi (1-e- t/RC), just what we expect.
But that's all in the time domain. After equation 23, there is a graph of the frequency response and the end result is a roll-off of 20 dB per decade for a single pole filter.
https://coertvonk.com/hw/filters/low-pass-rc-filter-14273
Notice the Octave code further down, well, Octave is free and I have it so I might as well play with the equations. Or I can convert the code to MATLAB (the two are amazingly similar) which I prefer.
I have bookmarked this page - it is the most coherent explanation I have ever read.
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