June 26, 2010 Leave a comment
My last few posts have been rather abstract. I thought I’d use this one to go into some details about the actual system we’re working with.
As I mentioned before, we are looking at a cart pole in a water tunnel. A cart pole is sometimes also called an inverted pendulum. Here is a diagram from wikipedia:
The parameter we have control over is F, the force on the cart. We would like to use this to control both the position of the cart and the angle of the pendulum. If the cart is standing still, the only two possible fixed points of the system are (the bottom, or “downright”) and (the “upright”). Since is easy to get to, we will be primarily interested with getting to .
For now, I’m just going to worry about the regular cart pole system, without introducing any fluid dynamics. This is because the fluid dynamics are complicated, even with a fairly rough model (called the Quasi-steady Model), and I don’t know how to derive them anyway. Before continuing, it would be nice to have an explicit parametrization of the system. There are two position states we care about: , the cart position; and , the pendulum angle, which we will set to at the bottom with the counter-clockwise direction being positive. I realize that this is not what the picture indicates, and I apologize for any confusion. I couldn’t find any good pictures that parametrized it the way I wanted, and I’m going to screw up if I use a different parametrization than what I’ve written down.
At any rate, in addition to the two position states and , we also care about the velocity states and , so that we have four states total. For convenience, we’ll also name a variable , so that we have a control input that directly affects the acceleration of the cart. We also have system parameters (the mass of the cart), (the acceleration due to gravity), (the length of the pendulum arm), and (the inertia of the pendulum arm). With these variables, we have the following equations of motion:
You will note that the form of these equations is different from in my last post. This is because I misspoke last time. The actual form we should use for a general system is
or, if we are assuming a second-order system, then
Here we are assuming that the natural system dynamics can be arbitrarily non-linear in , but the effect of control is still linear for any fixed system state (which, as I noted last time, is a pretty safe assumption). The time when we use the form is when we are talking about a linear system — usually a linear time-invariant system, but we can also let and depend on time and get a linear time-varying system.
I won’t go into the derivation of the equations of motion of the above system, as it is a pretty basic mechanics problem and you can find the derivation on Wikipedia if you need it. Instead, I’m going to talk about some of the differences between this system and the underwater system, why this model is still important, and how we can apply the techniques from the last two posts to get a good controller for this system.
Differences from the Underwater System
In the underwater system, instead of having gravity, we have a current (the entire system is on the plane perpendicular to gravity). I believe that the effect of current is much the same as the affect of gravity (although with a different constant), but that could actually be wrong. At any rate, the current plays the role that gravity used to play in terms of defining “up” and “down” for the system (as well as creating a stable fixed point at and an unstable fixed point at ).
More importantly, there is significant drag on the pendulum, and the drag is non-linear. (There is always some amount of drag on a pendulum due to friction of the joint, but it’s usually fairly linear, or at least easily modelled.) The drag becomes the greatest when , which is also the point at which becomes useless for controlling (note the term in the affect of on ). This means that getting past is fairly difficult for the underwater system.
Another difference is that high accelerations will cause turbulence in the water, and I’m not sure what affect that will have. The model we’re currently using doesn’t account for this, and I haven’t had a chance to experiment with the general fluid model (using PDEs) yet.
Why We Care
So with all these differences, why am I bothering to give you the equations for the regular (not underwater) system? More importantly, why would I care about them for analyzing the actual system in question?
I have to admit that one of my reasons is purely pedagogical. I wanted to give you a concrete example of a system, but I didn’t want to just pull out a long string of equations from nowhere, so I chose a system that is complex enough to be interesting but that still has dynamics that are simple to derive. However, there are also better reasons for caring about this system. The qualitative behaviour of this system can still be good for giving intuition about the behaviour of the underwater system.
For instance, one thing we want to be able to do is swing-up. With limited magnitudes of acceleration and a limited space (in terms of ) to perform maneuvers in, it won’t be possible in general to perform a swing-up. However, there are various system parameters that could make it easier or harder to perform the swing-up. For instance, will increasing (the inertia of the pendulum) make it easier or harder to perform a swing-up? (You should think about this if you don’t know the answer, so I’ve provided it below the fold.)