1. Nov. 22~24, 2004 1st International Conference on Research in Air Transportation, Zilina, Slovakia1 Wing Rock Dynamics and Differential Flatness W. LU F. Mora-Camino A. Acha ï bou

2. Nov. 22~24, 2004 1st International Conference on Research in Air Transportation, Zilina, Slovakia 2 Outline Introduction  Wing Rock Dynamics  Differential Flatness Flatness Based Controller Design  Flatness of Wing Rock Dynamics  Control Law Design  Simulation Results Conclusion

3. Nov. 22~24, 2004 1st International Conference on Research in Air Transportation, Zilina, Slovakia 3 Wing Rock Dynamics A rolling oscillation motion due to loss of damping at high angle-of-attack. High swept delta wing/arrow wing with slender body configurations Breakdown of asymmetrical flow/vortex (unpredictable theoretically) High nonlinearity CFD/Experiments efforts (mathematical model)

4. Nov. 22~24, 2004 1st International Conference on Research in Air Transportation, Zilina, Slovakia 4 Supersonic Transport Aircrafts

5. Nov. 22~24, 2004 1st International Conference on Research in Air Transportation, Zilina, Slovakia 5 Differential Flatness (1) Origin  Differential flatness has been first introduced by M. Fliess, J. Lévine, P. Martin, P. Rouchon. Definition  A system is said to be differentially flat if and only if the state and input can be determined by a set of variables, called flat output, which may possess physical meanings. Application  The control input of a flat system can be determined from a given trajectory of the flat output.

6. Nov. 22~24, 2004 1st International Conference on Research in Air Transportation, Zilina, Slovakia 6 Differential Flatness (2) A nonlinear system given as: is differentially flat if there exists a vector such that Numerous nonlinear systems have been proved to be differentially flat.

7. Nov. 22~24, 2004 1st International Conference on Research in Air Transportation, Zilina, Slovakia 7 Flatness of Wing Rock Dynamics (1) Mathematical Model  Proposed by Hsu and Lan

8. Nov. 22~24, 2004 1st International Conference on Research in Air Transportation, Zilina, Slovakia 8 Flatness of Wing Rock Dynamics (2) Differential Equations

9. Nov. 22~24, 2004 1st International Conference on Research in Air Transportation, Zilina, Slovakia 9 Flat output of Wing Rock Dynamics Roll angle is a flat output:

10. Nov. 22~24, 2004 1st International Conference on Research in Air Transportation, Zilina, Slovakia 10 Control Law Design (1) Trajectory based control Law  Open-loop control Performing output tracking of inherently stable systems  Close-loop control Enhance output tracking capability for stable systems Perform output tracking of inherently unstable systems

11. Nov. 22~24, 2004 1st International Conference on Research in Air Transportation, Zilina, Slovakia 11 Control Law Design (2) Linearized System

12. Nov. 22~24, 2004 1st International Conference on Research in Air Transportation, Zilina, Slovakia 12 Control Law Design (3) Close-loop control law Error dynamics Characteristic polynomial

13. Nov. 22~24, 2004 1st International Conference on Research in Air Transportation, Zilina, Slovakia 13 Control Law Design (4) Control command Example of given output trajectory

14. Nov. 22~24, 2004 1st International Conference on Research in Air Transportation, Zilina, Slovakia 14 Simulation Results (1)

15. Nov. 22~24, 2004 1st International Conference on Research in Air Transportation, Zilina, Slovakia 15 Simulation Results (2)

16. Nov. 22~24, 2004 1st International Conference on Research in Air Transportation, Zilina, Slovakia 16 Conclusions The wing rock dynamics, according to the retained model, is shown to be differentially flat. The flatness approach demonstrates a comparatively simple way to design an effective non control law for the wing rock motion. The proposed close-loop controller can stabilize the wing rock dynamics Large overshoot can emerge for small control gains. High-gain control can enhance the capability to track the desired trajectory at the expense of the saturation of actuators.

17. Nov. 22~24, 2004 1st International Conference on Research in Air Transportation, Zilina, Slovakia 17 Question...