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## Inverted Pendulum Tutorial - Part 1

### System Model

The modelling of the inverted pendulum system comes straight from simple Newtonian physics. (Note: Modelling is covered in chapter 3 and chapter 4.) This gives us the following equations:

To make the model useful for us, we need to linearize it. Chapter 2 gives one possible procedure for this, but a more heuristic one will be presented here (which incidentally gives the same result). We first recognise that and when is small. This approximation yields:

This model is for small variations of only (i.e. a small signal model) so we assume that is small enough such that and . Then,

We next take the Laplace transform of these equations (as described in section 3.3).

Now, working with the second equation,

Note that and let , so

Recalling the definition of a, we see that

Where b2 = g / l. Using M = m = 0.5, l = 1 and g = 10 we get the following transfer function:

We can see immediately that there is one non-minimum phase zero and an unstable pole. Because of this pole, the system is open-loop unstable (i.e. if we nudge the cart a bit, the mass will fall over). Next, we try to control the inverted pendulum system based on the linear model we have just made.

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