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Differential Flatness Jen Jen Chung
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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) - PowerPoint PPT Presentation

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Differential FlatnessJen Jen Chung1OutlineMotivation

Control Systems

Flatness

2D Crane Example

IssuesJen Jen Chung | CDMRG22MotivationEasy to incorporate system constraints

State and control immediately deduced from flat outputs (no integration required)

Useful for trajectory generation and implementationJen Jen Chung | CDMRG33Control SystemsConsider the system:

A regular dynamic compensator

A diffeomorphism

such that

becomesJen Jen Chung | CDMRG4

Regularity implies the invertibility of the system with input v and output uDiffeomorphism: smooth and invertible mapping between mainfolds4Control SystemsIn Brunovsky canonical form

Where are controllability indices and ______________________ is another basis vector spanned by the components of .

Thus Jen Jen Chung | CDMRG5

5Control SystemsTherefore, 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

6FlatnessA 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 equationsJen Jen Chung | CDMRG7

7FlatnessFlat 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.2Jen Jen Chung | CDMRG82 M. Fliess et al. A Lie-Bcklund approach to equivalence and flatness of nonlinear systemsUltimately, want flat outputs that allow for intuitive trajectory representation.8Example: 2D CraneJen Jen Chung | CDMRG9

9Example: 2D CraneDynamic model:Jen Jen Chung | CDMRG10

10Example: 2D CraneDynamic model:

Jen Jen Chung | CDMRG11

Rearrange (3) to get sin(theta) equationSubstitute (4) into (2) and rearrange to get T equationSubstitute sin(theta) into (1) to get xdd*z equationSquare (4) and sin(theta), add to get R^2 equation11

Example: 2D CraneJen Jen Chung | CDMRG12

Flat outputs:

12Example: 2D CraneHow 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

13Example: 2D CraneJen Jen Chung | CDMRG14

Crane given 7 seconds to move from SS at (0,10) to SS at (20,10)14Example: 2D CraneJen Jen Chung | CDMRG15

15Example: 2D CraneJen Jen Chung | CDMRG16

Required to reach a vertical position of 5 at 3.5 seconds16Example: 2D CraneJen Jen Chung | CDMRG17

17IssuesNo 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 | CDMRG1818Differential FlatnessJen Jen Chung19