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“Using Generalized Fibonacci Sequences for Solving the One-Dimensional LQR Problem and its Discrete-Time Riccati Equation”

Authors: Johan Byström, Lars P. Lystad and Per-Ole Nyman,
Affiliation: Luleå University of Technology and Narvik University College
Reference: 2010, Vol 31, No 1, pp. 1-18.

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Keywords: LQR, Linear quadratic control, Optimal control, Fibonacci number, Golden ratio, Binet formula

Abstract: In this article we develop a method of solving general one-dimensional Linear Quadratic Regulator (LQR) problems in optimal control theory, using a generalized form of Fibonacci numbers. We find the solution R(k) of the corresponding discrete-time Riccati equation in terms of ratios of generalized Fibonacci numbers. An explicit Binet type formula for R(k) is also found, removing the need for recursively finding the solution at a given timestep. Moreover, we show that it is also possible to express the feedback gain, the penalty functional and the controller state in terms of these ratios. A generalized golden ratio appears in the corresponding infinite horizon problem. Finally, we show the use of the method in a few examples.

PDF PDF (430 Kb)        DOI: 10.4173/mic.2010.1.1

DOI forward links to this article:
  [1] Thomas Brasch, Johan Byström and Lars Petter Lystad (2012), doi:10.1007/s10957-012-0061-2
  [2] Zhen Chen, Binglong Cong and Xiangdong Liu (2013), doi:10.1155/2013/602869
  [3] Przemys aw Ignaciuk (2016), doi:10.1016/j.jfranklin.2015.03.033

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[3] Camouzis, E. Ladas, G. (2008). Dynamics of Third Order Rational Difference Equations with Open Problems and Conjectures, Chapman and& Hall / CRC, Boca Raton.
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  title={{Using Generalized Fibonacci Sequences for Solving the One-Dimensional LQR Problem and its Discrete-Time Riccati Equation}},
  author={Byström, Johan and Lystad, Lars P. and Nyman, Per-Ole},
  journal={Modeling, Identification and Control},
  publisher={Norwegian Society of Automatic Control}


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