Approximating Differential Equation on the Basis of Measured Values

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That looks good. The other answers don't.

Peter Nachtwey

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Velocity must be ok because y' is part of

0.1666572 y'' + 0.3333362 y' + 0 y = 1

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Yes, the velocity is OK. However, your web site is still wrong. The steady state position for the equation on your web site is

1/2,174717E-06 or 459830 which is not right. You must integrate velocity to get position so you can control position. The PID gains must be calculated for position feedback. The PID gains for velocity control and the PID gains for position control are different even if the gain and time constant are the same. For starters a first order velocity control system doesn't need a derivative gain.

Peter Nachtwey

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1 T(s) = ----------------------------- 0.1666572s^2 + 0.3333362s + 0

The level goes theoretically to infinity as any 'level' process transfer function does if e.g. more water [m^3/h] is pumped in then goes out of a tank.

In simulations you can ignore an error of 2.174717E-06

That is your hint that you need to use two differential equations like I showed in my pdf file. In this case the control output determines the steady state velocity. Then the velocity must be integrated to get position.

No, you just must use two differential equations. Not one.

See this ftp://ftp.deltacompsys.com/public/NG/Mathcad%20-%20Ode%20Opt%20First%20Order%20no%20offset.pdf

Peter Nachtwey