Floquet Theory and Periodic Orbits Applied to Airborne Wind Energy Systems

Dr. G. Sánchez-Arriaga

Bioengineering and Aerospace Engineering Department, Universidad Carlos III de Madrid (UC3M)

Friday, August 17, 2018, 23:00

"Room 02-012, Georges-Köhler Allee 102, Freiburg 79110, Germany"

Airborne Wind Energy (AWE) systems are normally designed to operate periodically, like for instance by doing reel-in and reel-out manoeuvres combined with figure-of-eight trajectories or circles. Moreover, it has been shown that periodic orbits are ubiquitous in tethered flying objects even without control [1,2]. For these reasons, the determination of periodic orbits and their stability by using flight simulators is a topic of high interest for AWE community. This talk summarizes the main elements of the Floquet theory and its application to the numerical determination of periodic orbits and their stability. The most important bifurcation scenarios, both local and global, for periodic orbits are also discussed. The applications of these theoretical concepts to three practical AWE systems are presented. First, it is shown how the equilibrium state of a 1-line kite constrained in a vertical plane can become unstable through a supercritical Hopf bifurcations, thus yielding a branch of stable periodic orbits [1]. Second, for a two-line kite, it is shown that stable figure-of-eight trajectories can be achieved with and without controlling the tether lengths if the stability derivative of the kite is designed properly [2]. Finally, some periodic figure-of-eight orbits for a 1-line AWE system controlled by a time-dependent bridle are also presented [3].

References

[1] G. Sánchez-Arriaga, Dynamics and control of single-line kites, The Aeronautical Journal, 110, 615, 2006.

[2] G. Sánchez-Arriaga, M. García-Villalba and R. Schmehl, Modeling and dynamics of a two-line kite, Applied Mathematical Modelling 47, 473, 2017.

[3] G. Sánchez-Arriaga, A. Pastor-Rodríguez, M. Sanjurjo-Rivo, and R. Schemhl, A Lagrangian Flight Simulator for Airborne Wind Energy System, submitted to Applied Mathematical Modelling.