differential flatness
DESCRIPTION
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 PresentationTRANSCRIPT
Differential FlatnessJen Jen Chung
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Outline• Motivation
• Control Systems
• Flatness
• 2D Crane Example
• Issues
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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
Jen Jen Chung | CDMRG
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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
,,
,,,
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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
Jen Jen Chung | CDMRG
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.
Jen Jen Chung | CDMRG
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
Jen Jen Chung | CDMRG
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
Jen Jen Chung | CDMRG
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:
Jen Jen Chung | CDMRG
cos
sin
cos
sin
Rz
DRx
mgTzm
Txm
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Example: 2D Crane• Dynamic model:
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222,
,sin
RzDxzxDxgz
zzgmRT
RDx
cos
sin
cos
sin
Rz
DRx
mgTzm
Txm
12
222,
,sin
RzDxzxDxgz
zzgmRT
RDx
Example: 2D Crane
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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:
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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
Jen Jen Chung | CDMRG
Differential FlatnessJen Jen Chung