I built a Mindstorms robot that uses a light sensor to measure light reflected off the floor and thereby the robot's tilt. This turned out to be finicky since I had to set the zero point manually, and ambient light variations screwed things up fairly often. It worked well enough in the end though.
A full-order observer controller uses a model of the system in the control loop, which allows us to observe state information that would otherwise be hidden in the actual system. We can then use that state info in the feedback to reduce the error, which now incorporates both the system and model outputs. This can be a robust way to control high-dimensional systems while also being able to inspect the (estimated) states for useful insights.
However, we may not actually need all the state information. A minimal-order observer (aka functional observer) still uses a model, but requires fewer poles to be chosen than a full-order controller. That simplifies design and eliminates the need to calculate and compute state-space transformation matrices.
The figure shows the minimal-order observer, with the controller elements labeled as psi 0 and psi 1. In the lower diagram, psi 0 is algebraically combined with the summation block to simplify coding. As noted, each psi function is a ratio of (simple) Z-domain transfer polynomials.
|Minimal-order diagram in Simulink. In the actual system, the real robot takes the place of the "Linearized Model".|
I coded the observer controller in RobotC with the help of a couple of Matlab scripts to choose poles and calculate the coefficients of the transfer polynomials. I could have put more work into accurately modeling the robot (weighing it properly, etc.), but as you can see, it works well enough.
Code is here.