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Differential Flatness

Differential Flatness. Jen Jen Chung. Outline. Motivation Control Systems Flatness 2D Crane Example Issues. Motivation. Easy to incorporate system constraints State and control immediately deduced from flat outputs (no integration required)

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Differential Flatness

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  1. Differential Flatness Jen Jen Chung

  2. Outline • Motivation • Control Systems • Flatness • 2D Crane Example • Issues

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

  4. Control Systems • Consider the system: • A regular dynamic compensator • A diffeomorphism such that becomes

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

  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.

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

  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 2 M. Fliess et al. A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems

  9. Example: 2D Crane

  10. Example: 2D Crane • Dynamic model:

  11. Example: 2D Crane • Dynamic model:

  12. Example: 2D Crane • 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:

  14. Example: 2D Crane

  15. Example: 2D Crane

  16. Example: 2D Crane

  17. Example: 2D Crane

  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

  19. Differential Flatness Jen Jen Chung

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