## Abstract

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.

Original language | English (US) |
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Pages (from-to) | 989-1002 |

Number of pages | 14 |

Journal | IEEE Transactions on Automatic Control |

Volume | 63 |

Issue number | 4 |

DOIs | |

State | Published - Apr 2018 |

## Keywords

- Optimal fractional-order controllers
- spectral methods
- stability
- time-delay systems
- time-varying systems

## ASJC Scopus subject areas

- Control and Systems Engineering
- Computer Science Applications
- Electrical and Electronic Engineering