State estimation and control of the Fibonacci system

In this paper the second-order unstable Fibonacci system is defined by adding a control input to the recursion relation that generates the Fibonacci numbers, and the characteristics of its Kalman state estimator and its control using both the LQR and deadbeat control approaches are investigated. It is found that the elements of the Kalman estimator's steady state gain and error covariance matrices are functions of the golden ratio, and, under certain assumptions on the noise variance and the initial-state error covariance, that the recursion relations for the elements of the a priori error covariance matrix involve functions of the Fibonacci numbers. On the control side, as the control weight increases relative to the weights on the states in the LQR design, the elements of the feedback gain matrix and the closed loop pole locations approach functions of the golden ratio. Finally, the deadbeat design is capable of bringing the unstable Fibonacci system from any initial state to an “idle” state of (1, 1) in two control iterations, of maintaining it there with the simple control sequence {−1, −1, −1, …}, and then of regenerating the Fibonacci sequence at any later time by simply turning off the control.