I have tried to find out about this from quite a few sources but it still remains unclear to me. I know that the Luenberger observer is applicable for a deterministic system with known control inputs while the Kalman filter is used for a stochastic linear dynamic system. There are versions of Kalman filters that are implementable for nonlinear systems with process and measurement noise, which is what we have in practice. Although there is an extended version of the Luenberger observer as well which could theoretically be applied to nonlinear systems, I have not found much about its practical applicability in the real world. So my question is - Is Luenberger observer used in practice? If yes, please provide a few examples to support your answer.
I don't understand the question here. It seems like the question boils down to "Kalman filters can be adapted to nonlinear systems, so why use Luenberger observers?" There are plenty of linear systems, and plenty of nonlinear systems that can be linearized.
Anywhere that you could use a standard Kalman filter you could also use a Luenberger observer. Personally I think the Luenberger observer is easier to understand and (importantly) easier to tune. The thumbrule I was taught was to have observer poles 5 to 10 times faster than the controller poles, to ensure your controller is acting on plant dynamics and not observer dynamics.
I don't know of any equivalent guidance for Kalman filters. Kalman filters require some values for process and measurement noise. The measurement noise is generally straightforward, but I have yet to read anything compelling on how to characterize process noise. Usually what winds up happening is the creation of a "tuning knob" that creates some ratio of process-to-measurement noise, and then you adjust that until you get some output that looks okay.
My issue with random adjustments is that I don't get a good feel for filter dynamics with that approach, which means then that it's hard to ensure that my controller is controlling the system and not just getting excited by filter dynamics.
In short, if you know you've got a good system model, I think you can get better control with the Luenberger observers. If you don't have a great system model, or the entirety of your model is essentially noise effects, then you can go with some basic motion model and use a Kalman filter. I've used Kalman filters to estimate wind effects, for example.
This is just my personal experience doing controls development.
State observers don't need to be linear, either. You just need to be more thoughtful about the correction tuning--those gains can be scheduled, too.
Another advantage of the Luenberger form is that the correction can often be interpreted as an estimate of the disturbance/load, and can be used in control.