Golden section, Fibonacci sequence and the time invariant Kalman and Lainiotis filters

Abstract We consider the discrete time Kalman and Lainiotis filters for multidimensional stochastic dynamic systems and investigate the relation between the golden section, the Fibonacci sequence and the parameters of the filters. Necessary and sufficient conditions for the existence of this relation are obtained through the associated Riccati equations. A conditional relation between the golden section and the steady state Kalman and Lainiotis filters is derived. A Finite Impulse Response (FIR) implementation of the steady state filters is proposed, where the coefficients of the steady state filter are related to the golden section. Finally, the relation between the Fibonacci numbers and the discrete time Lainiotis filter for multidimensional models is investigated.

[1]  Petros G. Voulgaris,et al.  On optimal ℓ∞ to ℓ∞ filtering , 1995, Autom..

[2]  Nicholas J. Higham,et al.  Matrix Powers in Finite Precision Arithmetic , 1995, SIAM J. Matrix Anal. Appl..

[3]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[4]  Alexey Stakhov,et al.  The Generalized Principle of the Golden Section and its applications in mathematics, science, and engineering , 2005 .

[5]  Ke Chen,et al.  Applied Mathematics and Computation , 2022 .

[6]  Douglas J. Klein,et al.  A recursion formula for resistance distances and its applications , 2013, Discret. Appl. Math..

[7]  John F. Donoghue,et al.  The Kalman Filter for Complex Fibonacci Systems , 2012 .

[8]  Fevzi Büyükkiliç,et al.  Cumulative growth with fibonacci approach, golden section and physics , 2009 .

[9]  John Donoghue.  State estimation and control of the Fibonacci system , 2011, Signal Process..

[10]  Alfonso Farina,et al.  Expressing stochastic filters via number sequences , 2010, Signal Process..

[11]  N. Assimakis,et al.  Discrete Time Kalman and Lainiotis 1 Filters Comparison , 2007 .

[12]  Demetrios G. Lainiotis,et al.  Partitioned linear estimation algorithms: Discrete case , 1975 .

[13]  T. Başar,et al.  A New Approach to Linear Filtering and Prediction Problems , 2001 .

[14]  N. J. A. Sloane,et al.  The On-Line Encyclopedia of Integer Sequences , 2003, Electron. J. Comb..

[15]  Alexey Stakhov,et al.  The “golden” matrices and a new kind of cryptography , 2007 .

[16]  Maria Adam,et al.  Lainiotis filter, golden section and Fibonacci sequence , 2013, Signal Process..

[17]  Maria Adam,et al.  FIR implementation of the Steady-State Kalman Filter , 2008 .

[18]  Maria Adam,et al.  -Step Sum and -Step Gap Fibonacci Sequence , 2014 .

[19]  Luigi Chisci,et al.  Fibonacci sequence, golden section, Kalman filter and optimal control , 2009, Signal Process..

[20]  Norman Morrison,et al.  The Fibonacci Sequence , 2017 .

[21]  Iwona Wloch,et al.  On a new type of distance Fibonacci numbers , 2013, Discret. Appl. Math..

[22]  Demetrios G. Lainiotis,et al.  Steady State Kalman Filter: A New Approach , 2003, Neural Parallel Sci. Comput..

[23]  Νικόλαος Ασημάκης,et al.  Optimal decentralized Kalman filter and Lainiotis filter , 2015 .

[24]  Alexey Stakhov,et al.  The golden section in the measurement theory , 1989 .