Fractional Order Linear Quadratic Regulator

In this paper, we formulate the fractional linear quadratic regulator (LQR) problem. The analytical solution of fractional optimal control near the origin and infinity are derived. It is shown that the optimal control to the linear fractional system can be computed through the corresponding fractional Euler-Lagrange equations. Moreover, the analytical analysis of right-sided fractional equation is discussed, which relates to the construction and solution of fractional LQR problems. The matrix Mittag-Leffler function is used here to serve as the fundamental solution of high-dimension linear fractional equations. More detailed discussions about fractional systems and right-sided fractional operators are provided and summarized. Finally, two illustrated simulation results are provided as a proof of concept.