# How can I calculate the maximum possible deviation from equilibrium, that can still be compensated, given a limitation on the control input?

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 ?

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.

• 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. Jan 4 '19 at 11:22