What causes poor performance in Linear Observer based dynamic controllers?

Question

I have a simple linear system(time invariant) which is unstable. I have implemented linear state feedback system to stabilize it. In addition I have implemented a state estimator to estimate the system states(Luenberger observer). Both function fine on their own but when I combine them together with the feedback system using the observers output I get terrible startup performance. Like huge orders of magnitude wrong.

What could be causing this? Is this normal/expected behavior? How do you normally fix it?

Answer 1

The problem with linear state feedback with an observer is, that you might lose all the desirable properties of the linear quadratic regulator or the pole placement if you are not able to use full state feedback. There is also a famous very short paper by John C. Doyle that is giving an example for which the combination of state feedback and observer (in this case the optimal Kalman filter) fails. Doyle also mentions that introducing noise might help to solve this problem. A more reliable solution is to use $\mathcal{H}_{\infty}$ methods or other robust control strategies like $l_1$ adaptive control and sliding mode control.

Answer 2

You didnt give any information about your system, but as a rule of thumb this problems get worse with the order of the system. Try to imagine this as the 'depth' of the system which has to be reconstructed. If you are implementing this in a discrete manner, this effect becomes worse.

It can be said, that your observer poles should be at least about 5 times faster than the fastest closed loop controller poles. Please try to verify whether you can recover the closed loop performance when pushing the observer poles near infinity. In a continuous simulation this shouldnt be a problem.