Solving state feedback control of fractional linear quadratic regulator systems using triangular functions

Abstract

The present paper is useful for two reasons. First, as far as we know, the right operational matrix of the Riemann–Liouville fractional integral for triangular functions is introduced for the first time. So, hereafter, it can be applied in approximating any fractional optimal control problem in Caputo sense, over and over. Second, the rate of convergence of the triangular functions under a new-blown norm ‖.‖p,α in its corresponding space, denoted by Lp,α, is investigated. So far, the error analysis in Lp,α space has not been studied.In this article, both left and right operational matrices of the triangular functions for arbitrary fractional order integral α > 0 in Caputo sense are applied to approximate solutions of fractional linear optimal control systems which have a quadratic performance index. The necessary and sufficient optimality conditions are stated in a fractional two-point boundary value problem. This problem is converted to a set of coupled equations involving left and right Riemann–Liouville integral operators. Using these operational matrices of the triangular functions, an extension of the functions of control, state, co-state and other engaged functions is considered. This technique is a successful approach because of reducing such systems to a system of coupled Sylvester equations. Applying the definition of Kroneker product, a linear algebraic system containing the corresponding matrices of the original problem and the operational matrices of fractional order integral can be constructed. Accordingly, fractional linear quadratic optimal control can be solved indirectly. The advantage of this methodology, in addition to simplicity, is its low computational cost and its flexible precision. The simulation results confirm the reliability and validity of this method.