1. Differential Flatness Jen Jen Chung

2. Outline Motivation Control Systems Flatness 2D Crane Example Issues Jen Jen Chung | CDMRG2

3. Motivation Easy to incorporate system constraints State and control immediately deduced from flat outputs (no integration required) Useful for trajectory generation and implementation Jen Jen Chung | CDMRG3

4. Control Systems Consider the system: A regular dynamic compensator A diffeomorphism such that becomes Jen Jen Chung | CDMRG4

5. Control Systems In Brunovsky canonical form Where are controllability indices and ______________________ is another basis vector spanned by the components of. Thus Jen Jen Chung | CDMRG5

6. Control Systems Therefore, and both and can be expressed as real-analytic functions of the components of and of a finite number of its derivatives: The dynamic feedback is endogenous iff the converse holds, i.e. Jen Jen Chung | CDMRG6

7. Flatness A dynamics which is linearisable via such an endogenous feedback is (differentially) flat The set is called a flat or linearising output of the system State and input can be completely recovered from the flat output without integrating the system differential equations Jen Jen Chung | CDMRG7

8. Flatness Flat outputs: “…since flat outputs contain all the required dynamical informations to run the system, they may often be found by inspection among the key physical variables.” 2 Jen Jen Chung | CDMRG8 2 M. Fliess et al. A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems

9. Example: 2D Crane Jen Jen Chung | CDMRG9

10. Example: 2D Crane Dynamic model: Jen Jen Chung | CDMRG10

11. Example: 2D Crane Dynamic model: Jen Jen Chung | CDMRG11

12. Example: 2D Crane Jen Jen Chung | CDMRG12 Flat outputs:

13. Example: 2D Crane How to carry a load m from the steady- state R = R1 and D = D1 at time t1, to the steady-state R = R2 > 0 and D = D2 at time ? Consider the smooth curve: Constraints: Jen Jen Chung | CDMRG13

14. Example: 2D Crane Jen Jen Chung | CDMRG14

15. Example: 2D Crane Jen Jen Chung | CDMRG15

16. Example: 2D Crane Jen Jen Chung | CDMRG16

17. Example: 2D Crane Jen Jen Chung | CDMRG17

18. Issues No general computable test for flatness currently exists “There are no systematic methods for constructing flat outputs.” 1 Does not handle uncertainties/noise/disturbances Jen Jen Chung | CDMRG18

19. Differential Flatness Jen Jen Chung