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Contents
Chapter 6. Differentially Flat Systems
Jean LEVINE
CAS, Mines-ParisTech
2008
Jean LEVINE Chapter 6. Differentially Flat Systems
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Contents
Contents
1 Flatness NSC, Linear Case
Introductory Example: Linear Motor with Appended Mass
General Solution (Linear Case)
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Contents
Contents
1 Flatness NSC, Linear Case
Introductory Example: Linear Motor with Appended Mass
General Solution (Linear Case)
2 Flatness NSC, General Case
Example of Non Holonomic Vehicle
Implicit Representation
Lie-Backlund Equivalence of Implicit SystemsFlatness Necessary and Sufficient Conditions
Jean LEVINE Chapter 6. Differentially Flat Systems
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Flatness NSC, Linear CaseFlatness NSC, General Case
Introductory ExampleGeneral Solution
Plan
1 Flatness NSC, Linear Case
Introductory Example: Linear Motor with Appended Mass
General Solution (Linear Case)
2 Flatness NSC, General Case
Example of Non Holonomic Vehicle
Implicit Representation
Lie-Backlund Equivalence of Implicit SystemsFlatness Necessary and Sufficient Conditions
Jean LEVINE Chapter 6. Differentially Flat Systems
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Flatness NSC, Linear CaseFlatness NSC, General Case
Introductory ExampleGeneral Solution
Introductory Example: Linear Motor with Appended Mass
Model:
Mx = F k(x z) r(x z)mz = k(x z) + r(x z)
Aim:
Fast and high-precision rest-to-rest
displacements.
Measurements:Motor position and velocity
z not measured.
mass
flexible beam
bumper
linear motorrail
Experiment realized with the help of
Micro-Controle.
Jean LEVINE Chapter 6. Differentially Flat Systems
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Flatness NSC, Linear CaseFlatness NSC, General Case
Introductory ExampleGeneral Solution
Flat Output Computation (J.L. et D.V. Nguyen, S&CL, 2003)
We want to express x, z and F as
x = a0y + a1 y + . . . = (a0 + a1ddt
+ . . .)y = Px(ddt
)y
z = b0y + b1 y + = (b0 + b1ddt
+ . . .)y = Pz(ddt
)y
F = c0y + c1 y + . . . = (c0 + c1ddt
+ . . .)y = PF(ddt
)y
Thus: Md
2
dt2+ rd
dt+ k
Pxy
rd
dt+ k
Pzy = PFy
rddt
+ kPxy + md2
dt2+ rd
dt+ kPzy = 0
Solution:
Px =1
k
m
d2
dt2+ r
d
dt+ k
, Pz =
1
k
r
d
dt+ k
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Flatness NSC, Linear CaseFlatness NSC, General Case
Introductory ExampleGeneral Solution
Flat Output:
y =r2
mk
x +1 r2
mkz
r
k
z
x = y +r
ky +
m
ky, z = y +
r
ky
F = (M+ m)
y +
r
ky(3) +
Mm
(M+ m)ky(4)
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Flatness NSC, Linear CaseFlatness NSC, General Case
Introductory ExampleGeneral Solution
General Solution (Linear Case)
Consider the linear controllable system A( ddt)x = Bu. We are lookingfor P and Q such that x = Py, u = Qy with y: flat output to bedetermined.
Let C be s.t. CTB = 0. We thus have to solveCTAP = 0, AP = BQ
General Solution
Smith Decomposition ofCTA( ddt
):U, V unimodular such that VCTAU = (|0). Thus:
P = U I
0
P0
Indeed: CTAP = CTAU
I
0
P0 = V
1 (|0)
I
0
P0 = 0
A flat output y is deduced by left inversion ofP.
Jean LEVINE Chapter 6. Differentially Flat Systems
Non Holonomic Vehicle
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Flatness NSC, Linear CaseFlatness NSC, General Case
Non Holonomic VehicleImplicit RepresentationLie-Backlund EquivalenceFlatness NSC
Plan
1 Flatness NSC, Linear Case
Introductory Example: Linear Motor with Appended Mass
General Solution (Linear Case)
2 Flatness NSC, General Case
Example of Non Holonomic Vehicle
Implicit Representation
Lie-Backlund Equivalence of Implicit SystemsFlatness Necessary and Sufficient Conditions
Jean LEVINE Chapter 6. Differentially Flat Systems
Non Holonomic Vehicle
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Flatness NSC, Linear CaseFlatness NSC, General Case
Non Holonomic VehicleImplicit RepresentationLie-Backlund EquivalenceFlatness NSC
Example of Non Holonomic Vehicle
x = u cos y = u sin
=u
ltan
After elimination of the input variables u and :
y x tan = 0Implicit representation invariant by dynamic extension.
Variational Equation
dy tan dx
x
cos2 d = 0In Polynomial Form :
tan d
dt
d
dt
x
cos2
dx
dy
d
= 0.
Jean LEVINE Chapter 6. Differentially Flat Systems
Non Holonomic Vehicle
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Flatness NSC, Linear CaseFlatness NSC, General Case
Implicit RepresentationLie-Backlund EquivalenceFlatness NSC
If(z1,z2) is a flat output, then (dz1, dz2) is a flat output of thevariational system
dx =
j0
2i=1 Px,i,jdz
(j)i = Px,1dz1 + Px,2dz2
dy = j02i=1 Py,i,jdz(j)i = Py,1dz1 + Py,2dz2d =
j0
2i=1 P,i,jdz
(j)i = P,1dz1 + P,2dz2
and we must have
tan
d
dt
d
dt x
cos2 Px,1 Px,2
Py,1 Py,2P,1 P,2
= 0
Jean LEVINE Chapter 6. Differentially Flat Systems
Non Holonomic Vehicle
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Flatness NSC, Linear CaseFlatness NSC, General Case
Implicit RepresentationLie-Backlund EquivalenceFlatness NSC
Smith Decomposition
tan
d
dt
d
dt
x
cos2
0 0 10 1 0 cos
2 x
cos2 x
ddt
sin cos xddt
=
1 0 0
One can verify that P is given by
P =
0 11 0
cos2 x
ddt
sin cos xddt
P0, (P0 arbitrary).
and thus dxdy
d
=
0 1
1 0
cos2
x
d
dt
sin cos
x
d
dt
P0
dz1dz2
or dx = dz2, dy = dz
1. Therefore: x = z
2, y = z
1.
Jean LEVINE Chapter 6. Differentially Flat Systems
Non Holonomic Vehicle
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Flatness NSC, Linear CaseFlatness NSC, General Case
Implicit RepresentationLie-Backlund EquivalenceFlatness NSC
Implicit Representation of Nonlinear Systems
Consider the explicit system x = f(x, u) with rank f
u
= m locallyon X Rm.A representation invariant by endogenous dynamic extension is
obtained by elimination of the input u = (x, x), yielding the(n m)-dimensional implicit system:
F(x, x) = 0
with rankFx
= n m.
We introduce the global coordinates
x = (x, x,x, . . .)
on the manifold X Rn endowed with the trivial Cartan field n.The implicit representation is thus given by the triple
(XR
n
, n, F).Jean LEVINE Chapter 6. Differentially Flat Systems
Non Holonomic Vehicle
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Flatness NSC, Linear CaseFlatness NSC, General Case
Implicit RepresentationLie-Backlund EquivalenceFlatness NSC
Lie-Backlund Equivalence of Implicit Systems
Consider two implicit systems (X Rn, n, F) and (Y R
n
, n , G).They are said Lie-Backlund equivalent iff there exists a locally C
mapping : Y Rn
X Rn, with locally C
inverse s.t.
(i) n = n and n = n ;
(ii) for every y s.t. LknG(y) = 0 for all k 0, then
x = (y) satisfies Lkn F(x) = 0 for all k 0 andconversely.
The system (XRn, n, F) is flat iff it is Lie-Backlund equivalent to(Rm, m, 0).
Variational Property
The system (XRn, n, F) is flat iff there exists a locally C and
invertible mapping : X Rn Rm such that
dF = 0.
Jean LEVINE Chapter 6. Differentially Flat Systems
Fl NSC Li CNon Holonomic VehicleI li i R i
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Flatness NSC, Linear CaseFlatness NSC, General Case
Implicit RepresentationLie-Backlund EquivalenceFlatness NSC
We have:
dF =F
xdx +
F
xdx =
F
x+
F
x
d
dt
dx
= P(F)dx
and
dF = P(F) P(0)dy
with
P(0) =j0
0
y(j)dj
dtj
We thus have to find a polynomial matrix P(0) solution to
P(F) P(0) = 0.
This solution is deduced from the Smith decomposition ofP(F).
Jean LEVINE Chapter 6. Differentially Flat Systems
Fl t NSC Li CNon Holonomic VehicleI li it R t ti
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Flatness NSC, Linear CaseFlatness NSC, General Case
Implicit RepresentationLie-Backlund EquivalenceFlatness NSC
Notations:
K: field of meromorphic functions from XRn
to R and K[d
dt]principal ideal ring of polynomials of d
dt= Ln with coefficients
in K.
Mp,q[ddt
]: module of the p q matrices over K[ ddt
], with p and qarbitrary integers.
Up[ddt
]: group ofunimodular matrices ofMp,p[ddt
].
Smith Decomposition: IfP(F) Mnm,n[ddt
], there exist
V Unm[ddt
] and U Un[ddt
] such that
VP(F)U = (, 0nm,m) .
We note V L Smith (P(F)) and U R Smith (P(F)).Moreover, if the linear tangent system around an arbitrary trajectory is
controllable, one can prove that = Inm.
Jean LEVINE Chapter 6. Differentially Flat Systems
Flatness NSC Linear CaseNon Holonomic VehicleImplicit Representation
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Flatness NSC, Linear CaseFlatness NSC, General Case
Implicit RepresentationLie-Backlund EquivalenceFlatness NSC
Flatness Necessary and Sufficient Conditions
(Levine (2004), Levine (2006))
Theorem
The system (X Rn, n, F), assumed first-order controllable, is flat
iff for all U R Smith (P(F)) andQ L SmithU, withU = U
0nm,m
Im
, there exists an m m matrix such that for all
p(X)m and all p N, p+1(X), and a matrix M Um[ddt
]such that, if we denote by
= (Im, 0m,nm) Q(x)dx ,we have:
d = , d () = 2, d (M) = Mwith d extension of the exterior derivative d to polynomial matrices
with coefficients in (X).
Jean LEVINE Chapter 6. Differentially Flat Systems
Flatness NSC Linear CaseNon Holonomic VehicleImplicit Representation
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Flatness NSC, Linear CaseFlatness NSC, General Case
Implicit RepresentationLie-Backlund EquivalenceFlatness NSC
Comments
The last conditions generalize the moving frame structure
equations of Cartan to manifolds of jets of infinite order.
They are equivalent to the existence of M s.t. d(M) = 0, orequivalently to the existence ofy s.t. dy = M (flat output).
Their validity may be checked by computer algebra.
Jean LEVINE Chapter 6. Differentially Flat Systems
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