Recently, I’ve become interested in multi-modal observers, e.g. particle filters. I like the idea of tracking different hypotheses simultaneously. In this viewpoint, we consider our estimation of the system’s state as pdf with arbitrary shape. In classical observers (e.g. Kalman filters), there is only a single unimodal hypothesis about the state. But in these particle filter-like methods, we consider different state estimations with different probabilities.
Well! This kind of viewpoint is not what was emphasized in my control theory courses. In those courses, everything were about proving the stability of the system (observer in this case) in a deterministic setting. We made no explicit attention to the probabilistic interpretation of the state estimation except that we assume that the measurment and state transition model are contaminated with a normal noise (so unimodal), and we evaluate the covariance of the state over time.
Now, I wonder what would happen to the control problem: Knowing the pdf of probable current states, what is the best control action? Suppose that one hypothesis says that your state is (10,0) and the other says that it is in (-10,0). If your controller is linear in states, what would you select? -K1*10 or -K1*(-10)?!
I must read more about these things. I am completely new in this field!