Application of the KF

1. Sensor fusion: So given $p$ measurement of the same state $x(t)$, where each sensor has it’s own model $C_p$, we can still apply the KF to find the estimate $\hat{x}$, this time with more information, so it’s better.

2. Control In adaptive control we can also change how the controller controls the system, this can be due to the lack of a mathematical model of the plant.

3. Fault detection:

4. Recursive system identification: So we can only identify the system once “all” the output is given, or we can still use it at it is at each time $t$, but is not optimized. For simplicity let’s define two function $R(t)$ and $f(t)$ such that: So: We can then write a recursive equation: We have now obtained a recursive algorithm, to find the parameter vector $\theta$ : Still the algorithm could be upgraded: Finally we can summarize and define the RLS (Recursive Least Squares) algorithm: