%0 Journal Article
%T Using Generalized Fibonacci Sequences for Solving the One-Dimensional LQR Problem and its Discrete-Time Riccati Equation
%A Johan Bystr£¿m
%A Lars Petter Lystad
%A Per-Ole Nyman
%J Modeling, Identification and Control
%D 2010
%I Norwegian Society of Automatic Control
%R 10.4173/mic.2010.1.1
%X 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.
%K LQR
%K Linear quadratic control
%K Optimal control
%K Fibonacci number
%K Golden ratio
%U http://www.mic-journal.no/PDF/2010/MIC-2010-1-1.pdf