The Kalman filter was developed by Rudolf Kalman in the early 1960's. Its viability was proven when it was used in the navigational computers for the Apollo missions that brought American astronauts to the moon. Today, you will find Kalman filters in many electronic systems such as radar, robotics, automation, and vehicle navigation. The Kalman filter has the advantage of being able to use a minimal amount of computations and memory compared to other observers. This was important back then, when electronic computers were far more primitive. It can still be important today, if we want to use small, portable microcontrollers to control machines as opposed to big, heavy desktops. The way a Kalman filter works is that it treats the white noise as an error. The optimum observer equation will correct any error in its state estimation based on the discrepancy between the measured and calculated output. So we transform the state equations,
into the optimum observer equation.
So let's split this matrix equation into its 3 rows.
We can easily get a small microprocessor to run these three equations in an iterative loop. The difference between the a priori and a posteriori values is
What we now need to do is to figure out what the Kalman gain coefficients are.