Fractional delayed control design for linear periodic systems

In this paper, the fractional Chebyshev collocation (FCC) method is proposed to design fractional delay controllers for linear systems with periodic coefficients. In our previous study, it was shown that this method can be successfully used to stabilize fractional periodic time-delay systems with the delay terms being of integer orders. In the current paper, it is shown that this method can be extended successfully to design fractional delay controllers for fractional periodic systems. For this propose, the solution of linear periodic systems with fractional delay terms is expressed in a Banach space. The short memory principle is used to show that the actual response of the system can be approximated by an approximated monodromy operator. The approximated monodromy operator yields the solution of a fixed length interval by mapping the solution of the previous interval with the same length. Usually obtaining the approximated monodromy operator is complicated or even impossible. The spectral radius of the approximated monodromy matrix indicates the asymptotic stability of the system. The efficiency of the proposed fractional delayed control is illustrated in the case of a second order system with periodic coefficients.