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Control theory - integrator?

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nForce:
Hello,

I have a derivation, which I think it's for integrator. Is this from control theory?

Why is at the end difference between Y(n) and Y(n-1)? So we get a derivative?

kim.dd:
The Laplace or continuous transfer function can be transformed to a discrete transfer function using more than one transformation rule.
Not all transformation will lead to a stable or implementable solution. e.g. a stable continuous pole could end up outside the unit circle for your discrete implementation.

The three basic transformations:



For the first transform given you would get:
Y(z)*(1 + (-1)*z^-1) = X(z)*(0 + (ts/Ti)*z^-1)
Y[n] - Y[n-1] = (ts/Ti)*X[n-1]
Y[n] = Y[n-1] + (ts/Ti)*X[n-1]

For the second one:
Y(z)*(1 + (-1)*z^-1) = X(z)*((ts/Ti) + 0*z^-1)
Y[n] - Y[n-1] = (ts/Ti)*X[n]
Y[n] = Y[n-1] + (ts/Ti)*X[n]

For the last one:
Y(z)*(1 + (-1)*z^-1) = X(z)*((0.5*ts/Ti) + (0.5*ts/Ti)*z^-1)
Y[n] - Y[n-1] = 0.5*(ts/Ti)*(X[n]+X[n-1])
Y[n] = Y[n-1] + 0.5*(ts/Ti)*(X[n]+X[n-1])

rstofer:
If you look at the last 2 equations, you will see that they are identical when delta t in the last equation approaches 0 - the very definition of a derivative which is the form of the next-to-last equation.  With a little rearranging, of course.

The last equation represents delta Y over delta t  which, when delta t approaches 0 is, in fact, dY/dt.

kim.dd:
I agree all lines are equivalent, he just converted the integral equation to a differential equation. They all say the same thing :)

nForce:
So this is the integrator term in control theory?

And for a differentiator is the same only reverse?

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