You are here : Control System Design - Index | Simulations | Inverted Pendulum | Part 1 ## Inverted Pendulum Tutorial - Part 1## System ModelThe 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
Where
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. |