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Contents
Chapter 6. Differentially Flat Systems
Jean LEVINE
CAS, MinesParisTech
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)
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)
2 Flatness NSC, General Case
Example of Non Holonomic Vehicle
Implicit Representation
LieBacklund 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
LieBacklund 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 highprecision resttorest
displacements.
Measurements:Motor position and velocity
z not measured.
mass
flexible beam
bumper
linear motorrail
Experiment realized with the help of
MicroControle.
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
Jean LEVINE Chapter 6. Differentially Flat Systems
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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 RepresentationLieBacklund 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
LieBacklund 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 RepresentationLieBacklund 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 RepresentationLieBacklund 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 RepresentationLieBacklund 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 RepresentationLieBacklund 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 RepresentationLieBacklund EquivalenceFlatness NSC
LieBacklund Equivalence of Implicit Systems
Consider two implicit systems (X Rn, n, F) and (Y R
n
, n , G).They are said LieBacklund 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 LieBacklund 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 RepresentationLieBacklund 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 RepresentationLieBacklund 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 RepresentationLieBacklund EquivalenceFlatness NSC
Flatness Necessary and Sufficient Conditions
(Levine (2004), Levine (2006))
Theorem
The system (X Rn, n, F), assumed firstorder 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 RepresentationLieBacklund 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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