5 states is at
Philippe MARTIN y
Pierre ROUCHON z
Systems and Control Letters. Vol: 25, pp:167{173, 1995
Abstract
A very simple classication of PfaÆan systems of dimension 2 in 5 variables is
given. It is used to show that any controllable driftlesssystem with 3 inputs and 5
states is 0-at and can be put into multi-input chained form by dynamic feedback
and coordinate change.
Key words: nonlinear control, atness, dynamic feedback, driftless systems, chained
form,PfaÆan systems, absolute equivalence,nonholonomicsystems.
1 Introduction
In this paper, driftless systems with 3 inputs and 5states
_ x=u
1
f
1
(x)+u 2
f
2
(x)+u 3
f
3
(x); x2X R 5
;
are investigated. Driftless systems oftenappear inengineering askinematicsof mechanical
systems subjected to nonholonomicconstraints,and for that reason have recently received
considerableattentionincontroltheory(see e.g.[14,10]andthe bibliographytherein). We
show here that driftless systems with 3 inputs and 5 states are particularly simple: they
are at [6,11, 7],and even 0-at, as soonas they are controllable, which means that their
dierential behavior is summarized in a map depending only on the state x. Moreover,
aroundaregularpoint, they canbeconverted by dynamicfeedback and coordinatechange
intothe "multi-inputchainedform" [14, 19]
_ x 1
=u 1
; x_ 2
=x 3
u 1
; x_ 3
=u 2
; x_ 4
=x 5
u 1
; x_ 5
=x 6
u 1
; x_ 6
=u 3
:
This work was supported in partby INRIA, NSF grantECS-9203491,G.R. \Automatique"(CNRS)
andDRED(Ministeredel'
EducationNationale).
y
Center for Control Engineeringand Computation, Engr. I, University of California, Santa Barbara,
CA93106,USA.Email: martin@xanadu.ece.ucsb.edu
z
CentreAutomatiqueetSystemes,
EcoledesMinesdeParis,60BdSaint-Michel,75272ParisCedex06,
FRANCE.Tel: 33(1)40519115. Email: rouchon@cas.ensmp.fr
ingly popular in control theory (see e.g. [8, 17, 15, 13, 12, 18]). More precisely we show,
and then interpret in control terms, that any totally nonholonomic PfaÆan system of di-
mension 2 in 5 variables is absolutely equivalent [4] toa very simple one. It is interesting
tocompare this simple result with the much ner (and complicated) classication up to a
coordinatechange obtained by Cartanin hisfamous 1910 paper[3].
The paperisorganized asfollows: section 2shortly discusses atnessand its linkswith
driftless, PfaÆan and chained systems. The heart of the paperis section3, where PfaÆan
systemsof dimension2in5variablesare showntohaveavery simplestructure. Theresult
isthen rephrased fordriftless systems insection 4.
2 Flatness and driftless, chained and PfaÆan systems
Let be a (smooth) control system, x_ = f(x;u), dened on an open subset X U of
R n
R m
. The prolongationof a (smooth) map, y:=h(x;u;u;_ u; :::;u
(k)
),depending onx
and nitely many derivativesof u is the map
_ y:=@
x
h(x;u;u;_ u; :::;u
(k)
):f(x;u)+ k
X
i=0
@
u
(i)
h(x;u;u;_ u; :::;u
(k) ):u
(i+1) :
As we willdeal onlywith a nite number of maps and prolongations,we consider them as
maps ofXUU
1
:::U
r
for some\large" integer r, wherethe U
i
's are open subsets
of R m
. We use the short-hand notations u := (u;u;_ u; :::;u
(r)
), U := U U
1
:::U
r ,
etc. We can now dene the concept of atness [6,11, 7]:
Denition1. Wesay isat at(x
0
;u
0
) ifthereexist (smooth) mapsy:=h(x;u) dened
around (x
0
;u
0
) and '; dened around y
0
:= h(x
0
;u
0
) such that x = '(y) and u = (y)
hold around (x
0
;u
0
). We say is at if it is at at every point of a dense open subset
of XU.
In other words every choice of a map y(t) leads to an allowable trajectory (x(t);u(t))
onlyvia dierentiation. No integration is needed.
The map y := h(x;u) in the denition is called a at (or linearizing) output. We say
the system is k-at if it has a at output depending on derivatives of u of order at most
k 1. Hence a 0-at system has a at output depending onlyon the state x.
Flatnessisanimportantstructuralproperty: forinstance,theknowledgeofaatoutput
gives a simple answer to the motion planning problem [16] and allows to transform the
system into a linearone by (dynamic) feedback [11]. Moreover, many engineering systems
(or atleast \idealized"systems) happen to be at.
Checking that a system is at is a challenging and widely open problem. Checking k-
atnessforagivenkamountstostudyingtheintegrabilityofasystemofpartialdierentials
equations, which is in theory doable though extremely tedious. Yet, the main conceptual
problemis that itis not known whether k may be a prioribounded.
Things look a little simpler for driftless systems x_ = P
m
i=1 u
i
f
i
(x); which frequently
arise in nonholonomic mechanics. The problem is solved for 2-input systems [12], and
inputsystems (andinparticular atnessdoesnot imply0-atness). Areason forthe lesser
complexityisthat timeplaysnoroleinadriftlesssystem: solutionsareinvariantby atime
reparametrization. A driftlesssystem isnothingbut a(time-independent)PfaÆan system;
the machinery of exterior dierentialsystems directly applies, whereas in the generalcase
time must becarefullyhandled [8,17].
Of course, a at driftless system is never at at an equilibrium point (x
0
;0), which is
often a point of interest, and thus cannot be feedback linearized around that point (the
linear tangent approximation is not controllable [5]). Hence, a further question is to nd
a simple local model in which the system could be converted by feedback and coordinate
change, provided the point of interest is \regular" in some sense to be dened. For the
2-inputcase, a naturalcandidate is the so-called \chainedform"[14]
_ x 1
=u 1
; x_ 2
=x 3
u
1
:::; x_ n 1
=x n
u 1
; x_ n
=u 2
:
Combining results of [12, 13], a at 2-input driftless system can be converted around a
\regular" pointintochained formby static feedback and coordinatechange. Generalizing
to driftless systems with more than 2 inputs raises new problems, and conversion into an
adequate simple local modelmay require not only static, but also dynamic feedback [19],
which is not completelysurprising.
We nally recall some facts about PfaÆan systems (see [2, chapter 2] for a modern
introduction). Let X be an open subset of R n
and C 1
(X) the ring of smooth functions
on X. A PfaÆan system on X is a C 1
(X)-module 1
of smooth dierential 1-forms on X.
Some importantproperties of a PfaÆan system I can be characterized by its derived ag,
i.e. the descending chain of PfaÆan systems I 0
:=I I 1
:::dened by
I k+1
:=f2I k
; d 0modI k
g:
We say I istotally nonholonomicif I k
=0 fork large enough. Wethen denea point-wise
notionof regularityby considering the real vector spaces 2
~
I 0
x :=(I
0
)
x and
~
I k+1
x
:=f (x); 2I k
; d (x)0 mod(I k
)
x g:
Clearly (I k
)
x
~
I k
x
. We say x
0
2X is a weakly regular point if
~
I k
x
has constant dimension
in a neighborhood of x
0
for k 0, in which case dim(I k
)
x
= dim
~
I k
x
= dimI k
in that
neighborhood. Weakly regular points forma dense open subset of X.
To adriftless controlsystem D: x_ = P
m
i=1 u
i
f
i
(x), with f
1
;:::;f
m
independent vector
eldsonX,isnaturallyassociatedaPfaÆansystemI :=ff
1
;:::;f
m g
?
ofdimensionn m.
I representsthe kinematicconstraintsobtained by eliminatingthe controls inD. Notice D
iscontrollable if and onlyif I istotallynonholonomic. We say x
0
is aweakly regular point
of D if itis a weakly regularpointof I.
Notice that weak regularity is equivalent for PfaÆan systems of dimension 2 to the
stronger property considered in[13], but may ingeneral hide subtlersingularities [9].
1
To discardirrelevant non-local problems, we always assumethat aC 1
(X)-module has thesame di-
mensionasitsrestrictionstoanyopensubsetofX.
2
WedenoteS
x
:=f(x); 2Sgtherealvectorspaceinducedatapointx2X byaC 1
(X)-moduleS.
We now turn to the study of driftless systems with 3 inputs and 5 states, or equivalently
PfaÆan systems of dimension 2 in 5 variables. If I is such a PfaÆan system, the rank
structure of its derived ag splits into fourcases:
1. dimI 1
=2
2. dimI 1
=1; dimI 2
=1
3. dimI 1
=1; dimI 2
=0
4. dimI 1
=0.
The rst three cases are not very diÆcult, and we refer the reader to the beginning of
Cartan'spaper[3] (seealso[9]foramorecareful treatmentregarding regularityproblems).
The rst two cases correspond to non controllable, hence non-at control systems [7]. In
suitablelocalcoordinates,wehaverespectivelyI =fdx 1
; dx 2
gandI =fdx 1
; dx 2
x 3
dx 4
g.
The third case is more interesting:
Theorem 1. Let I be a PfaÆan system of dimension 2 and x
0
be a weakly regular point.
Assume dimI 1
= 1 and dimI 2
= 0. Then I = fdx 1
x 2
dx 4
; dx 2
x 3
dx 4
g in suitable
coordinates dened around x
0 .
The ideaofthe proof istoshowthatthe systemdepends infactononly4variablesand
toapply Engel's theorem [2, chapter 2].
The fourth case (which is the generic situation)is the heart of Cartan's paper, and its
nestudyup toacoordinatechangeisquiteintricate. Wewillshowthatitcannonetheless
be reduced to a unique normal form when considering a larger class of transformations.
Before statingthe result,we prove auseful lemma:
Lemma 1. Let I be a totally nonholonomic PfaÆan system of dimension 2 in 5 variables
and x
0
be a weakly regular point. Then I contains a form ! such that, around x
0 ,
(i)d!^d!^! =0 and (ii) d!^! 6=0:
Aform! satisfying(i)and (ii)issaid tohaverank1atx
0
, andaclassicalresultasserts
that, insuitable coordinates dened aroundx
0
,! is colinearto dx 1
+x 2
dx 3
[2,chapter2].
Proof. IfdimI 1
=1and dimI 2
=0,the assertion stems fromtheorem 1. Wethus assume
dimI 1
=0;now (ii)is satisedfor any forminI,otherwise dim
~
I 1
x0
>0. Let! 1
;! 2
span I
aroundx
0
;wewanttondafunction denedaroundx
0
suchthat !:=! 1
+! 2
satises
(i). Notice that (d!) 2
^! is colineartodx 1
^dx 2
^dx 3
^dx 4
^dx 5
(itis aforma degree5
ina space of dimension 5)and that
(d!) 2
^! = (d!
1
) 2
^! 1
+(2d!
1
^d!
2
^! 1
+(d!
1
) 2
^! 2
)
+ 2
(2d!
1
^d!
2
^! 2
+(d!
2
) 2
^! 1
)+ 3
(d!
2
) 2
^! 2
+2d^(d!
1
^! 2
^! 1
+d!
2
^! 2
^! 1
):
Hence (d!) ^! =0 reduces tothe rst-order quasilinear partialdierentialequation
d(x):(f(x)+(x)g(x)) =a(x;(x)); (1)
wherethe twovectorselds f;g and the function aare known expressions. Moreover since
dim
~
I 1
x
= 0 around x
0 , d!
1
^! 2
^! 1
and d!
2
^! 2
^! 1
, thus f and g, are independent
at x
0
. Hence f(x
0
)+(x
0 )g(x
0
) 6= 0 whatever (x
0
), and it is well-known [1, chapter 2]
that equation (1)then admitssolutions dened around x
0 .
Theorem2. LetI be atotallynonholonomicPfaÆansystem of dimension2in 5variables
and x
0
be a weakly regular point. Then, in suitable coordinates dened around x
0
, I takes
one of the two following normal forms:
I =fdx 1
x 2
dx 4
; dx 2
x 3
dx 4
g if dimI 1
=1 and dimI 2
=0
I =fdx 1
a(x)dx 3
x 5
dx 4
; dx 2
x 3
dx 4
g if dimI 1
=0,
where a is somefunction. Moreover I can be prolonged around x
0 into
J :=fdz 1
z 5
dz 4
; dz 2
z 3
dz 4
; dz 3
z 6
dz 4
g:
Using Cartan's terminology [4], this result asserts that any totally nonholonomic sys-
tems of dimension 2 in 5 variables is absolutely equivalent to a \contact" system whose
generalsolution depends ontwo arbitrary functionsof one argument.
Proof. We rst prove the second claim. If I is in the rst normal form, prolong it into
J := I +fdx 5
x 6
dx 4
g and perform the coordinate change z := (x 5
;x 1
;x 2
;x 4
;x 6
;x 3
). If
I is in the second normal form, prolong it into J := I +fdx 3
x 6
dx 4
g and perform the
coordinate change z := (x 1
;x 2
;x 3
;x 4
;x 5
+x 6
a(x);x 6
), which is well-dened as soon as x 6
issmallenough. In the new coordinates, J is obviously inthe requested form.
Wenowturn tothe rstclaim. Bylemma1,I contains aformofrank1aroundx
0 and,
up to acoordinate change, isspanned by the forms
! 1
:= dx 1
+dx 3
+dx 4
+Ædx 5
and ! 2
:=dx 2
x 3
dx 4
;
forsomefunctions ;;;Æ. Sincedim
~
I 0
x
0
=2,oneofthesefourfunctionsmustbedierent
from0 atx
0 .
If (x
0
)=Æ(x
0
)=0,wemayassume(x
0
)6=0andwithoutlossofgenerality(x)= 1
aroundx
0
(if(x
0
)=0but(x
0
)6=0,weexchangetherolesof andwiththecoordinate
change z :=(x 1
;x 2
x 3
x 4
; x 4
;x 3
;x 5
)). Hence
d!
2
=dx 4
^dx 3
=dx 4
^( dx 1
+dx 4
+Ædx 5
! 1
) dx 4
^dx 1
+Ædx 4
^dx 5
modI;
which implies dim
~
I 1
x0
1. But dim
~
I 1
x
must have non-zero constant dimension around x
0 ,
thus and Æ equal 0 in a neighborhood of x
0
. We conclude that, around x
0 , dim
~
I 1
x has
constant rank1 and dim
~
I 2
x
constant rank 0,and we may use theorem 1.
Otherwise (x
0
)6=0(up toa renumbering of x),and the system is spanned by
! 1
:=dx 1
dx 3
dx 4
Ædx 5
and ! 2
:=dx 2
x 3
dx 4
:
Perform now the coordinatechange z :=g(x), z :=x, i=2;:::;5,where g isa function
suchthat g
1 (x
0
)6=0 and is yet tobedetermined 3
. It follows
dz 1
= g
1 (!
1
+dx 3
+dx 4
+Ædx 5
)+g
2 (!
2
+x 3
dx 4
)+g
3 dx
3
+g
4 dx
4
+g
5 dx
5
= (g
1 +g
3 )dx
3
+(g
1 +x
3
g
2 +g
4 )dx
4
+(Æg
1 +g
5 )dx
5
+g
1
! 1
+g
2
! 2
:
Ifg is asolutionof the rst-orderlinear partialdierentialequationÆg
1 +g
5
=0such that
g
1 (x
0
)6=0(such asolution always exists [1, chapter 2]), I is clearly spanned by
1
:=dz 1
a(z)dz 3
b(z)dz 4
and 2
:=dz 2
z 3
dz 4
;
where a;b followfromthe expression of dz 1
. Computing the exterior derivative,we get
d 1
cdz 3
^dz 4
+a
5 dz
3
^dz 5
+b
5 dz
4
^dz 5
modI; d 2
= dz 3
^dz 4
;
where cis some functioninvolvinga;b and their partial derivatives.
If a
5 (z
0 ) = b
5 (z
0
) = 0, then dim
~
I 1
x
0
= 1. But dim
~
I 1
x
must have constant dimension 1
aroundz
0
, thusa
5 and b
5
equal0in aneighborhoodof z
0
,i.e. the system doesnot depend
onz
5
. Once again we conclude by theorem 1.
Otherwise wemayassumeb
5 (z
0
)6=0andwithoutlossofgenerality b(z)=z 5
aroundz
0
(ifb
5 (z
0
)=0buta
5 (z
0
)6=0,wecan exchangetheroles ofaand bby thecoordinatechange
x := (z 1
;z 2
z 3
z 4
; z 4
;z 3
;z 5
)). This shows I is spanned by dz 1
a(z)dz 3
z 5
dz 4
and
dz 2
z 3
dz 4
.
4 Application to driftless systems
We now interprettheorem 2in terms of driftless systems:
Corollary 1. Let D: x_ =u 1
f
1
(x)+u 2
f
2
(x)+u 3
f
3
(x) be a driftless system with 3 inputs
and 5 states. The following three statements are equivalent:
(i) D is 0-at; (ii) D is at ; (iii) D is controllable:
If D satises these conditions then, around any weakly regular point, it can be put by
dynamic feedback and coordinate change into the multi-input chained form
_ z 1
=v 1
; z_ 2
=z 3
v 1
; z_ 3
=v 2
; z_ 4
=z 5
v 1
; z_ 5
=z 6
v 1
; z_ 6
=v 3
:
Proof. Clearly(i))(ii))(iii),andweonlyhavetoprove(iii))(i). SupposeDiscontrollable
and x
0
is a weakly regular point. By the previous theorem, the PfaÆan system I :=
ff
1
;f
2
;f
2 g
?
can beassumed innormal formaround x
0 .
If dimI 1
=0then, up toinvertible static feedback and coordinatechange, D reads
_ x 1
=a(x)u 1
+x 5
u 2
; x_ 2
=x 3
u 2
; x_ 3
=u 1
; x_ 4
=u 2
; x_ 5
=u 3
:
3
Ifhissomefunctionofx,wedenotebyh
i
thepartialderivative
@h
@x i
,i=1;:::;5.
We claim y:=(x ;x ;x ) is aat output. Indeed, simple computationsgive
x 1
=y 1
; x 2
=y 2
; x 3
= _ y 2
_ y 3
; x 4
=y 3
; u 1
=
y 2
_ y 3
y 3
_ y 2
(y_ 3
) 2
; u 2
=y_ 3
and x 5
is obtained by solving the equation
_ y 1
=a(y 1
;y 2
; _ y 2
_ y 3
;y 2
;x 5
)
y 2
_ y 3
y 3
_ y 2
(y_ 3
) 2
+x 5
_ y 3
;
the expression for u 3
follows by dierentiation. The system is thus at at every point
(x
0
;u
0
) such that x
0
is weakly regular, u 2
0
6= 0 and a
5 (x
0 )u
1
0 +u
2
0
6= 0 (where as usual
a
i :=
@a
@x i
). These points clearly forma dense open subset of XU.
Apply nowthe dynamicfeedback
_ x 6
=v 3
; u 1
=x 6
v 1
; u 2
=v 1
;
u 3
= v
2
[(x 5
+x 6
a(x))a
1
(x)+x 3
a
2
(x)+x 6
a
3
(x)+a
4 (x)]x
6
v 1
a(x)v 3
1+x 6
a
5 (x)
and the coordinate change z := (x 4
;x 1
;x 5
+x 6
a(x);x 2
;x 3
;x 6
), which are well-dened as
soonasx 6
issmallenough, toput the systemintomulti-inputchainedform. The dynamic
feedback is the counterpart of the prolongationin theorem 2.
If dimI 1
=1and dimI 2
=0,the system reads
_ x 1
=x 2
u 2
; x_ 2
=x 3
u 2
; x_ 3
=u 1
; x_ 4
=u 2
; x_ 5
=u 3
;
and y:=(x 1
;x 4
;x 5
) is aat output. Detailsare leftto the reader.
The dynamic feedback in the proof respects the driftless structure of the system. It is
alsoendogenous[11], i.e. built onlyfrom x and derivativesof u.
Performing a dynamic feedback (or prolonging the PfaÆan system) when dimI 1
= 1
and dimI 2
= 0 has little practical interest, since the system is already in a very simple
form.
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