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For example we have an inverted pendulum attached to a cart, with the cart having a speed limitation. Having derived the linearized model, how can I calculate the maximum possible deviation on the angle, and angular rate from the linearized model and the speed limitation ?

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Lets set the following variables,

1- Length of the pendulum = r

2- Mass of pendulum = m

3- Small angle from vertical = a

So the horizontal acceleration of you pendulum, "A" is

$A = \frac{mg *a}{r} $

Your control system has to be able to compensate for at least this acceleration A just to keep the tilted pendulum in check, a bit more to bring it back to stable, vertical position.

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  • $\begingroup$ If I am not mistaken, A is a force quantity ? Without the m, we would have the dynamic equation for the inverted pendulum ? So which angle offset I can compensate for can be directly read from the dynamic equations by inserting the maximum control input. $\endgroup$
    – ganto
    Jan 4 '19 at 11:22

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