Observability
Observability describes whether or not we have sufficient information to observe all the required states of a system. For instance, image that we wanted to know speed of a car between point A and B. Well, speed is just distance over time, so if we have a distance measurement (odometer) and a time measurement (clock), we can calculate speed. This system is observable.
Now, imagine a new system where this time we want to know the path the car took. If we still only have two instruments, the odometer and the clock, how can we possibly know what path the car took? Of course, we can't, we would need additional information, such as a GPS or overhead photos of the car along its route. Thus, with only an odometer and clock this new system isn't observable, although as I just said, with another set of measurements this system could be easily observable.
So it's important to know if the ball and plate system is observable. The answer is easy: 99% yes and 1% no.
First, let’s look at the 99% yes. It's obvious enough, though, I need hardly bother. Everyone has an instinctive understanding of where a ball will go when it starts rolling, and all we have is a photo (that our eyes take) of the ball. We “calculate” speed and acceleration based on the time and location meas urements our eyes and internal clock give us. The camera and computer has the same information, and thus the system is observable.
However... there is just one catch. Do you remember how earlier there were two solutions for every observed horizontal acceleration? That’s the 1% no. The system is only observable because we ignore the second set of solutions (when the table is tilted greater than 45 degrees) for the ball. If we wanted to consider both angles greater than and less than 45 degrees, we would need additional information to differentiate between the two solutions. We can do this in real life thanks to our stereo vision and ability to analyze size, something that my computer model cannot do. Of course, angles greater than 45 degrees aren't really of interest for us do to the physical limitations of our model.
For a more rigorous definition of observability, click here.
Controllability
Controllability, like observability, is an important property for completely controlling a system. Controllability refers to the ability to act on all the variables required to control the system. For instance, it's all fine and dandy to watch a ball roll, but if you can't touch it, or the surface it's rolling on, how can you possibly hope to alter it's trajectory?
In our case, it's again intuitively clear that the system is controllable. Try it yourself. Put a ball on a book, and tilt the book such that the ball doesn't fall off. As you can see, it's controllable, as long as you can tilt the book in any direction. Now, try to stabilize the ball by only titling the book along the axis perpendicular to your chest, thus keeping it from falling of the left or right edge. Sooner or later, it's going to fall off the front or back edge, right? Thus, this system is not controllable.
The ball and plate system has the same problem. We need a way to act in both the x- and y-directions. Fortunately, with our table we can.
For a more rigorous definition of controllability, click here.