### Abstract

his paper develops the fundamentals of optimaltuning periodic-gain fractional delayed state feedback control for a class of linear fractional order periodic time-delayed systems.his 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.T Although there exist techniques for the state feedback control of linear periodic timedelayed 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 steady state solution of FPTD systems. It is shown that the monodromy operator cannot be achieved in a close form for FPTD systems, and hence the short memory principle along with the fractional Chebyshev collocation (FCC) method are 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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Journal | IEEE Transactions on Automatic Control |

DOIs | |

State | Accepted/In press - Jul 23 2017 |

### Keywords

- Chebyshev approximation
- Delay effects
- Differential equations
- Feedback control
- Mathematical model
- Stability analysis
- State feedback

### ASJC Scopus subject areas

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