Closed-form Solution for The Finite-horizon Linear-quadratic Control Problem of Linear Fractional-order Systems

This paper is devoted to finding an explicit solution for the finite-horizon linear-quadratic regulator optimal control problem of linear time-invariant fractional-order systems based on a spectral method. The proposed technique does not pose the burden of regular Riccati fractional differential equations that demand integrating left- and right-sided fractional differential operators. The obtained solution exhibits spectral convergence with an efficient computation time. For this purpose, the closed-form of the optimal solution and the finite horizon state-feedback control are both obtained at Chebyshev-Gauss-Lobatto points by adopting operational matrices of left- and right-sided fractional-order differentiation. The validity and effectiveness of the proposed method are examined through an illustrative example.