Optimal Periodic Gain Fractional Delayed State Feedback Control for Linear Fractional Periodic Time Delayed Systems
This paper develops the fundamentals of optimal tuning periodic gain fractional delayed state feedback control for a class of linear fractional order periodic time delayed systems. Although there exist techniques for the state feedback control of linear periodic time delayed systems by discretization of the monodromy operator, there is no systematic method to design state feedback control for linear fractional periodic time delayed (FPTD) systems. This paper is devoted to defining and approximating the monodromy operator for a steady state solution of FPTD systems. It is shown that the monodromy operator cannot be achieved in a closed form for FPTD systems, and hence, the short memory principle along with the fractional Chebyshev collocation method is used to approximate the monodromy operator. The proposed method guarantees a near optimal solution for FPTD systems with fractional orders close to unity. The proposed technique is illustrated in examples, specifically in finding optimal linear periodic gain fractional delayed state feedback control laws for the fractional damped Mathieu equation and a double inverted pendulum subjected to a periodic retarded follower force with fractional dampers, in which it is demonstrated that the use of time periodic control gains in the fractional feedback control generally leads to a faster response.
IEEE Transactions on Automatic Control