differential flatness jen jen chung. outline motivation control systems flatness 2d crane example...
TRANSCRIPT
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