Differential Flatness

Differential FlatnessJen Jen Chung

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Outline• Motivation

• Control Systems

• Flatness

• 2D Crane Example

• Issues

Jen Jen Chung | CDMRG

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Motivation

• Easy to incorporate system constraints

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

• Useful for trajectory generation and implementation


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Control Systems• Consider the system:

• A regular dynamic compensator

• A diffeomorphism

such that

becomes


mn uxuxfx RR ,,

mq vzvzxbu

vzxaz

RR

,,,

,,

qnzx R ,

GvF

vzxaz

vzxbxfx

,,

,,,

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Control Systems• In Brunovsky canonical form

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

• Thus


mm vy

vy

m

111

m ,,1 1111 ,,,,,, 1 m

mm yyyyY

YTzx

zx TTY

11

,

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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.


vYTbu , 11 u x

myyy ,,1

yyyBu

yyyAx

,,,

,,,

uuuxAy ,,,,

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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


mjyy j ,...1

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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

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Example: 2D Crane


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Example: 2D Crane• Dynamic model:


cos

sin

cos

sin

Rz

DRx

mgTzm

Txm

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Example: 2D Crane• Dynamic model:


222,

,sin

RzDxzxDxgz

zzgmRT

RDx

cos

sin

cos

sin

Rz

DRx

mgTzm

Txm

12

222,

,sin

RzDxzxDxgz

zzgmRT

RDx

Example: 2D Crane


gzzxzR

gzzxxD

22

• Flat outputs:

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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:


m 01 RR1DD 1t 02 RR

2DD 12 tt

,0,, 21 Rtztxttt

gtttt

rtzxdtd

iRDtztx

ir

r

iiii

,, allfor

4 3, 2, 1,0,

2 1,,,

21

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Example: 2D Crane


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Example: 2D Crane


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Example: 2D Crane


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Example: 2D Crane


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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


Differential FlatnessJen Jen Chung

Differential Flatness

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