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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