## “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. |

**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 (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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**BibTeX:**

@article{MIC-2010-1-1,

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},

volume={31},

number={1},

pages={1--18},

year={2010},

doi={10.4173/mic.2010.1.1},

publisher={Norwegian Society of Automatic Control}

};