We Will Control the Vertical

[Title from The Outer Limits intro.]

Recently, my research has involved working with the control systems of the LISA spacecraft, which allow them to remain properly oriented. One of the main tools we use for this is a State Space Model. This type of structure models a physical system as a set of inputs, outputs, and states, which change in time. The system is defined by 4 matrices:

where u is a vector of inputs, y is a vector of outputs, and x is a vector of states. The dot indicates a time derivative, which tells us how the states change in time. Because this is a linear model, it will only work if we stay close to some equilibrium, but that's exactly what we hope to achieve. If we choose our inputs in relation to the outputs, we can try to stabilize the system.

I decided to play around with this model a bit by using the classic example of an inverted pendulum. This type of system may be familiar to you if you've ever tried to balance a pole on the palm of your hand. We want to keep the pole straight by moving its base. Gravity makes it tip in one direction or the other, and we react by moving our hand in the same direction. The trouble is how to avoid overshooting. For my model, I used a sliding cart in place of a hand. If we don't apply any external force to the cart, the rod will quickly fall over:

We want to apply a force to keep the cart underneath the pole, but I found that if I applied a force exactly opposite to the one making the pole tip, the pole would stay at a fixed angle while the cart zoomed off the screen. I played around a bit with different forces based on the pole's angle and speed, and found a configuration that looks pretty good:

It been interesting to learn some of the techniques that are more engineering-focused. I'm just glad I have more experienced people to help me out – I don't want to be responsible for another Mars Climate Orbiter!