Differential Flatness

Jen Jen Chung

2

Outline

• Motivation

• Control Systems

• Flatness

• 2D Crane Example

• Issues

Jen Jen Chung | CDMRG

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 | CDMRG

4

Control Systems

• Consider the system:

• A regular dynamic compensator

• A diffeomorphism

such that

becomes

Jen Jen Chung | CDMRG

mn uxuxfx RR ,,

mq vzvzxbu

vzxaz

RR

,,,

,,

qnzx R ,

GvF

vzxaz

vzxbxfx

,,

,,,

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 | CDMRG

mm vy

vy

m

111

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

mm yyyyY

YTz

x

zx TTY

11

,

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 | CDMRG

vYTbu , 11 u x

myyy ,,1

yyyBu

yyyAx

,,,

,,,

uuuxAy ,,,,

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 | CDMRG

mjyy j ,...1

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 | CDMRG

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

9

Example: 2D Crane

Jen Jen Chung | CDMRG

10

Example: 2D Crane

• Dynamic model:

Jen Jen Chung | CDMRG

cos

sin

cos

sin

Rz

DRx

mgTzm

Txm

11

Example: 2D Crane

• Dynamic model:

Jen Jen Chung | CDMRG

222,

,sin

RzDxzxDxgz

z

zgmRT

R

Dx

cos

sin

cos

sin

Rz

DRx

mgTzm

Txm

12

222,

,sin

RzDxzxDxgz

z

zgmRT

R

Dx

Example: 2D Crane

Jen Jen Chung | CDMRG

gz

zxzR

gz

zxxD

22

• 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 | CDMRG

m 01 RR

1DD 1t 02 RR

2DD 12 tt

,0,, 21 Rtztxttt

gtttt

rtzxdt

d

iRDtztx

ir

r

iiii

,, allfor

4 3, 2, 1,0,

2 1,,,

21

14

Example: 2D Crane

Jen Jen Chung | CDMRG

15

Example: 2D Crane

Jen Jen Chung | CDMRG

16

Example: 2D Crane

Jen Jen Chung | CDMRG

17

Example: 2D Crane

Jen Jen Chung | CDMRG

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 | CDMRG

Differential Flatness

Jen Jen Chung