Fibonacci sequence, golden section, Kalman filter and optimal control

A connection between the Kalman filter and the Fibonacci sequence is developed. More precisely it is shown that, for a scalar random walk system in which the two noise sources (process and measurement noise) have equal variance, the Kalman filter's estimate turns out to be a convex linear combination of the a priori   estimate and of the measurements with coefficients suitably related to the Fibonacci numbers. It is also shown how, in this case, the steady-state Kalman gain as well as the predicted and filtered covariances are related to the golden ratio φ=(5+1)/2. Furthermore, it is shown that, for a generic scalar system, there exist values of its key parameters (i.e. system dynamics and ratio of process-to-measurement noise variances) for which the previous connection is preserved. Finally, by exploiting the duality principle between control and estimation, similar connections with the linear quadratic control problem are outlined.