IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 1, JANUARY 2003 139

On Nonlinear Controllability of HomogeneousSystems Linear in Control

James Melody, Tamer Basar, and Francesco Bullo

AbstractThis work considers small-time local controllability (STLC)of single- and multiple-input systems, _ = ( ) + where( ) contains homogeneous polynomials and . . . are constant

vector fields. For single-input systems, it is shown that even-degree homo-geneity precludes STLC if the state dimension is larger than one. This,along with the obvious result that for odd-degree homogeneous systemsSTLC is equivalent to accessibility, provides a complete characterizationof STLC for this class of systems. In the multiple-input case, transforma-tions on the input space are applied to homogeneous systems of degree two,an example of this type of system being motion of a rigid-body in a plane.Such input transformations are related via consideration of a tensor on thetangent space to congruence transformation of a matrix to one with zeroson the diagonal. Conditions are given for successful neutralization of badtype (1,2) brackets via congruence transformations.

Index TermsControllability, Lie algebras, nonlinear systems.

I. INTRODUCTION

Various concepts of controllability for nonlinear systems wereinitially explored in [1][3]. In particular, [2] is primarily concernedwith the property of accessibility of the analytic control system_x = F (x; u), namely that the set of points attainable from a giveninitial point via application of feasible input is full in the sense ofhaving a nonempty interior. Sussmann and Jurdjevic demonstrated in[2] that a necessary and sufficient condition for accessibility of thesesystems is that the Lie algebra generated by the system have full rank,the so-called Lie algebra rank condition (LARC).

In [4], Sussmann explored the property of small-time local control-lability (STLC) for affine analytic single-input systems _x = f0

(x) +

f

1

(x)u with juj 1. A system is said to be STLC at a point x0

if thatinitial point is in the interior of the set of points attainable from it intime T for all T > 0. In this case, the Lie algebra generated by thesystem, denoted by L(ff0

; f

1

g), is the smallest involutive distributioncontaining ff0

; f

1

g or, equivalently, the distribution spanned by iter-ated Lie brackets of f0

and f1

. Sussmann gave various necessary andsufficient conditions for STLC in [4]. For example, a necessary condi-tion for STLC is that the tangent vector [f1

; [f

1

; f

0

]]

x

be in the sub-space spanned by all tangent vectors at x0

generated by brackets withonly one occurrence of f1

, which is denoted by L1(ff0

; f

1

g)

x

. Moreimportantly, the conditions conjectured by Hermes were proved to besufficient conditions for STLC. These Hermes local controllability con-ditions (HLCC) consist of (1) x0

is a (regular) equilibrium point, (2)the LARC is satisfied, and (3) Sk(ff0

; f

1

g)

x

S

k1

(ff

0

; f

1

g)

x

for all even k > 1 where Sk(ff0

; f

1

g)

x

denotes the span of all tan-gent vectors at x0

generated by brackets with k or less occurrences off

1

. Stefani [5] provided an extension of Sussmanns necessary condi-tion by demonstrating that STLC implies (ad2mf

f

0

)

x

2 S

2m1

x

forallm 2 f1; 2; . . .g. For an excellent summary and tutorial of these aswell as other results in the single-input case, the inquisitive reader isdirected to Kawski [6].

Manuscript received January 30, 2001; revised October 17, 2002 andSeptember 13, 2002. Recommended by Associate Editor H. Wang. This workwas supported in part by the U.S. Department of Energy and in part by the U.S.National Science Foundation under Grant CMS-0100162.

The authors are with the Coordinated Science Laboratory, University ofIllinois, Urbana, IL 61801 USA (e-mail: jmelody@uiuc.edu; tbasar@con-trol.csl.uiuc.edu; bullo@uiuc.edu)

Digital Object Identifier 10.1109/TAC.2002.806667

STLC of multiple-input affine analytic control systems was ad-dressed by Sussmann in [7], where a general sufficiency theorem wasproven for analytic systems of the form _x = f0

(x) +

m

i=1

f

i

(x)u

i

with the constraints jui

j 1 for all i 2 f1; . . . ; mg. In order tounderstand this result, it is necessary to distinguish between formalbrackets and evaluated brackets. On the one hand, a formal bracketis a pairwise parenthesized word (i.e., an element of the free magma)with well-defined left and right factors, number of factors, and degree.On the other hand, an evaluated bracket is the vector field that resultsfrom an iterated Lie bracketing of particular vector fields. When wespeak of the vector field generated by a formal bracket, we meanspecifically the vector field that results from evaluating the formalbracket with respect to particular vector fields, an operation that istheoretically captured by the evaluation map in [7]. This distinctionis captured by the following notation, which we employ both in thestatement of Sussmanns general sufficiency theorem and throughoutthe sequel: if B = (i1

; . . . ; (i

k1

; i

k

) . . .) represents a formal bracketof indeterminates f0; . . . ; mg, then fB = [fi

; . . . [f

i

; f

i

] . . .]

denotes the corresponding vector field generated by the evaluationmap from the formal bracket B with respect to a particular set ofvector fields ff0

; . . . ; f

m

g.1 Hence, for example, we have the formal

bracket B = (1; (1; 0)) that generates the vector field resulting fromevaluating the corresponding iterated Lie bracket [f1

; [f

1

; f

0

]] for aparticular pair of vector fields f0

and f1

.

Several results were presented in [7], but in the context of this notethe most appropriate result is based on the

degree of a formal bracket.For a given formal bracketB of indeterminates figmi=0

, jBji

is used todenote the number of occurrences of i as a factor in B. For 2 [0; 1],

is defined by

(B) := jBj

0

+

m

i=1

jBj

i

. The general theoremstates that systems that satisfy the LARC atx0

and have f0

(x

0

) = 0 are

STLC if there exists a 2 [0; 1] such that the tangent vector at x0

gen-erated by each formal bracketB with jBj0

odd and jBj1

; . . . ; jBj

m

alleven can be expressed as a linear combination of tangent vectors at x0

generated by some set of brackets fBk

g

N

k=1

with

(B

k

) <

(B) forall k 2 f1; . . . ; Ng. It has become conventional to refer to the formalbrackets with jBj0

odd and jBj1

; . . . ; jBj

m

all even as bad brackets, andif these bad brackets have corresponding vector fields that are not con-tained in the span of vector fields generated by good brackets of lower

degree at x0

then they are referred to as potential obstructions to STLC.Theobstructionsareonlypotentialbecause theyonlyobstruct theknownsufficient conditions. In [6], Kawski presents several examples of sys-tems with potential obstructions that are known to be STLC.

Both Sussmann in [7] and Kawski in [6] apply a generalized defi-nition of homogeneity to STLC. This concept of homogeneity beginswith definition of a dilation

as a parameterized map of IRn to IRnof the form

(x) = (

r

x

1

;

r

x

2

; . . .

r

x

n

) where ri

are nonnega-tive integers. A polynomial p : IRn ! IR is then said to be homoge-neous of degree k with respect to the dilation, symbolically p 2 Hk

if p(

(x)) =

k

p(x). Traditional homogeneity is recovered via thedilation with r1

= = r

n

= 1. The definition of homogeneity isthen extended to vector fields in the following manner: a vector fieldf is said to be homogeneous of degree j if fp 2 Hkj

wheneverp 2 H

k

for all k 0. A related area of research that capitalizes on thisgeneralized concept of homogeneity is that of nilpotent and high-orderapproximation of control systems presented by Hermes, for example,in [9]. One pertinent outcome of Hermess research is that a systemis STLC if its Taylor approximation is STLC. However, the conversequestion of whether STLC can be determined from a finite number ofdifferentiations is still open [10].

1Readers interested in the rich detail of formal Lie algebras may refer to [8].

0018-9286/03$17.00 2003 IEEE

140 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 1, JANUARY 2003

Fig. 1. Graphical depiction of brackets that generate nonzero vector fields ofpolynomial system (1) with homogeneity degree .

In the remainder of this note, we restrict our attention to homoge-neous nonlinear systems that are linear in control, using the traditionaldefinition of homogeneity. We begin by addressing single-input sys-tems. Building on Stefanis necessary condition [5] and the conceptsof good and bad brackets of Sussmanns general sufficiency theorem[7], we demonstrate that such single-input homogeneous systems ofeven degree are STLC if and only if they have a scalar state. Thisresult, combined with the obvious fact that for odd-degree homoge-neous single-input systems STLC is equivalent to the LARC, com-pletely characterizes STLC for these systems. Next, we address mul-tiple-input systems with the additional restriction that they be homoge-neous of degree two. In this case, we extend the applicability of Suss-manns general sufficiency theorem by incorporating a linear transfor-mation on the multidimensional input in order to neutralize potentialobstructions that arise from type (1,2) bad brackets (i.e., brackets withjBj

0

= 1 and jBj!0

= 2). In particular, we present a formal method forneutralizing these type (1,2) potential obstructions wherein the problemof finding the desired linear transformation on the input space is re-duced to finding a particular matrix congruence transformation.

II. PROBLEM EXPOSITION

In this note, we address STLC of systems of the form

_x = f

0

(x) +

m

i=1

f

i

u

i

(1)

where jui

j 1 and x 2 IRn. fi

for i 2 f1; . . . ;mg are assumed to beconstant vector fields, i.e., fi

(x) f

i

, and the components of f0

(x)

are homogeneous polynomials of degree k 1. We use the traditionaldefinition of homogeneous polynomial p, namely that p(x) = kp(x).The set of such homogeneous vector fields is denoted byHk

. Our defi-nition of degree-k homogeneous vector fields is equivalent to that usedin [6], [7], and [9] in the following manner: take

: x 7! x andthen Hk

is in the general framework the set of vector fields homoge-neous of degree 1 k. With this traditional definition, we have thefollowing elementary facts: i) f(0) = 0 for f 2 Hk

with k 1, andii) [f; g] 2 Hj+k1

for all f 2 Hj

and g 2 Hk

, whereH1

is inter-preted as the singleton containing the zero vector field.

Systems of this form are theoretically interesting because their Liealgebra at x0

= 0 has a diagonal structure, as depicted in Fig. 1. Inparticular, the only bracketsB that generate vector fields with nonzerovalue at x0

are those with jBj!0

= (k 1)jBj

0

+ 1, where jBj!0

:=

m

i=1

jBj

i

. This follows directly from the elementary facts previouslygiven. Keeping in mind that f0

2 H

k

and fi

2 H

0

for all i 6= 0,

Fig. 2. Motion of a rigid body in a plane expressed in body-fixed coordinates.

from fact ii), it is clear that brackets above the diagonal have homo-geneity degree greater than zero and, hence, by fact i) have zero valueat x0

. Similarly, from fact ii), we have that brackets below this line havehomogeneity degree less than zero and, hence, by definition are iden-tically zero.

Furthermore, systems of the form (1) commonly arise in mechanics.An example of such a system is the motion of a rigid body in a planeexpressed in body-fixed coordinates, as depicted in Fig. 2. The equa-tions of motion for this system are

_! =u

2

+ hu

1

_v

x

= !v

y

_v

y

=!v

x

+ u

1

: (2)The state consists of rotational velocity ! and the two body-fixedtranslation velocities vx

and vy

. The input consists of the torqueu

2

and the force u1

applied at a moment arm of h. Hence,f

0

(x) = (0;x

1

x

3

; x

1

x

2

), f1

= (h; 0; 1) for some constant h,f

2

= (1; 0; 0), and the system is of the form (1) with homogeneitydegree two. This provides a simple example of a system for whichSussmanns sufficient condition in [7] is not invariant with respectto input transformations. In particular, if a pure force (h = 0) anda torque are used as inputs, then the system satisfies Sussmannssufficient condition for the multiple-input case.2 However, if anoffset force (h 6= 0) is used, then potential obstructions appear asvector fields generated from the type (1,2) brackets, i.e., bracketswith jBj0

= 1 and jBj!0

= 2. In general, we employ the phrase type(k; `) brackets to refer to all formal brackets B of indeterminatesf0; . . . ; mg with jBj0

= k and jBj!0

= `, and denote the distributionspanned by such brackets as L(k;`)(F).3 Using this system as amotivating example, we explore the neutralization via congruencetransformation of potential obstructions generated by bad brackets oftype (1,2) with vector fields generated by other brackets (perhaps alsobad) of type (1,2).

III. SINGLE-INPUT SYSTEMS

In this section, we consider the single-input system

_x = f

0

(x) + f

1

u (3)where x 2 IRn, u 2 [1; 1], f1

2 IR

n

, and f0

2 H

k

. In light ofHLCC and Sussmanns general sufficiency result [7], it is clear that fora system as in (3) with odd homogeneity degree, accessibility is equiva-lent to STLC, since there are no nonzero brackets with jj0

odd and jj1

even. In other words, the question of STLC reduces to the LARC. The

2A generalized force on a rigid body consists of a pure force component whichinduces only a translational motion and a torque component which induces onlya rotational motion.

3The notation is used instead of to emphasize that ( ) is notnecessarily a Lie subalgebra.

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 1, JANUARY 2003 141

following lemma asserts that if the vector fields generated by bracketsof type (1; k) do not add to the Lie algebra rank, then neither do thevector fields generated by higher-degree brackets.

Lemma 1: For (3) with homogeneity degree k > 0, if (adkf

f

0

)

x

2

spanff

1

g, then L(ff0

; f

1

g)

x

= spanff

1

g.

Proof: Since the Lie algebra structure is invariant to (analytic) co-ordinate transformations, without loss of generality we can take f1

to bethe basis vector e1

. LetH11k

be the set of vector fields of homogeneitydegree k with the form (Cxk1

+

1

(x);

2

(x); . . . ;

n

(x)) where Cis any scalar constant and the power of x1

in i

is less than k for alli 2 f1; . . . ;mg. Then, (adkf

f

0

)

x

2 spanfe

1

g implies that f0

2

H

11

k

, since adkf

corresponds to the partial derivative operator @k=@xk1

.

Furthermore, if g 2 H11m

for some m 1, then adf

g 2 H

11

m1

and adf

g 2 H

11

m+k1

. Since the Lie subalgebra of (3) is spanned byvector fields of the form [fi

; [f

i

; . . . [f

i

; f

i

] . . .]] (apply, for ex-ample, [11, Prop. 3.8]), all vector fields inL(ff0

; f

1

g)

x

are a multipleof e1

.

Turning our attention to systems with even homogeneity degree k,we see that in general L(ff0

; f

1

g)

x

does include potential obstruc-tions generated from bad brackets, i.e., there are bad brackets along thediagonal of Fig. 1. In particular, type (m; (k 1)m+ 1) brackets are(odd, even) whenm is odd. We make use of the necessary condition ofStefani restated here for convenience.

Theorem 2Stefani [5]: If the system _x = f0

(x) + f

1

(x)u

1

isSTLC, then (ad2kf

f

0

)

x

2 S

2k1

(ff

0

; f

1

g)

x

.

When applied to even systems, Theorem 2 states that(ad

2k

f

f

0

)

x

2 spanff

1

g is necessary for STLC. However, if(ad

2k

f

f

0

)

x

2 spanff

1

g, then Lemma 1 asserts dimL(ff0

; f

1

g) = 1,

and systems with n 2 cannot be STLC (for otherwise LARC isviolated). This reasoning and the fact that the system _x = x2k + uwith x 2 IR is STLC provides the following result.

Proposition 3: If the system in (3) has odd homogeneity degree,then it is STLC if and only if it satisfies the LARC. On the other hand,if the system has even homogeneity degree 2k > 0, then it is STLC ifand only if the state x is scalar.

IV. MULTIPLE-INPUT SYSTEMS

We now return to consideration of the system in (1) where f0

2

H

2

(x). An extension of the previous results to this multiple-input caseis problematic. In particular, the necessary condition of [5] runs into theproblem of a possibility of balancing between potential obstructionsgenerated from bad brackets of the same degree. This consideration,along with the motivating example of planar rigid-body motion lead usto investigate neutralization of potential obstructions by vector fieldsgenerated from brackets of the same type.

Of course, for the system in (1) the general sufficient condition of [7]can be applied to determine STLC. Since the Lie algebra has a diagonalstructure, the choice of 2 [0; 1] in the theorem is immaterial. UsingSussmanns concepts of good and bad brackets, the sufficient condi-tion allows us to neutralize potential obstructions from bad bracketswith vector fields generated by good brackets of lower degree. Our goalwith this section is to address the case where there are potential obstruc-tions that cannot be neutralized in this manner, and to neutralize thesepotential obstructions with vector fields generated by other brackets ofthe same degree via appropriate choice of linear transformation on theinput space. In this endeavor, the diagonal structure of the Lie algebrawill be particularly useful.

Returning to the motivating example of planar motion in (2), it isclear that this system is STLC. (For example, use the feedback trans-formation u1

= !v

x

+ u

1

and u2

= u

2

hu

1

to obtain the systemdescribed by _! = u2

, _vx

= !v

y

, and _vy

= u

1

). However, in at-tempting to apply Sussmanns general sufficiency result directly on

the unchanged equations of motion, we have the potential obstruction[f

1

; [f

1

; f

0

]](0) = (0;2h; 0) 62 spanff

1

(0); f

2

(0)g. This potentialobstruction prevents the use of Sussmanns sufficient condition for anyh 6= 0. On the other hand, if h = 0 (corresponding to a force input u1

through the center of mass) we can then apply the sufficient condition.This is so because we have the vector field [f1

; [f

2

; f

0

]] generated bythe good bracket (1; (2; 0)) being (0;1; 0), bringing the Lie algebrato full rank, and STLC is demonstrated. Furthermore, the system forh 6= 0 can be transformed into the pure-force system (h = 0) via theinput transformationu

1

u

2

=

1 0

h 1

u

1

u

2

:

Since STLC is clearly invariant to full-rank input transformations,we see that a particular choice of input transformation may providea means of removing a potential obstruction, thus extending theapplicability of Sussmanns general theorem for this class of systems.

Remark 4: It is worth noting that Kawski [6] has considered tech-niques for neutralizing and balancing bad brackets. However, this tech-nique is not related to ours. Kawskis technique applies in the single-input setting, and neutralizes brackets possibly with brackets of dif-ferent degree via parameterized families of controls carefully tailoredto the system and the brackets in question. In some cases, these familiesof controls involve switching between control limits, with the param-eter affecting the switching times. Our technique utilizes the freedomof multiple independent inputs to enforce a linear relation between theinputs in order to neutralize brackets of the same type.

A. Neutralization via Congruence TransformMoving to the generic multiple-input homogeneous system

of degree two, suppose that there is some bad bracket ofthe form (i; (i; 0)) that generates a potential obstruction, i.e.,[f

i

; [f

i

; f

0

]]

x

=2 spanff

k

(x

0

)g

m

k=1

. We would like to find a full-rank,linear transformation T on the input space such that the transformedsystem

_x = f

0

(x) +

m

j=1

f

m

i=1

f

i

T

ij

u

j

has the corresponding potential obstruction removed. Of course, thefull-rank input transformation will not affect spanffk

g

m

k=1

. We makethe restriction juj

j 1=

where is the spectral radius of T in orderto have the resulting ui

satisfy the bounds jui

j 1.4

Suppose that there is at least one type (1,2) potential ob-struction, i.e., there is some | 2 f1; . . . ;mg such that[[f

|

; [f

|

; f

0

]]

x

=2 L

(0;1)

(F)

x

. Consider the codistributionKerL

(0;1)

(F) that annihilates the distribution L(0;1)(F). Then theremust exist some differential one-form 2 KerL(0;1)(F) such that([f

|

; [f

|

; f

0

]])

x

6= 0. Let ( ; ) be the codistribution containing allcovectors 2 KerL(0;1)(F)with the property ([fj

; [f

j

; f

0

]])

x

6= 0

for some j. Let us suppose that (1;2)x

has exactly dimension one.Choosing any nonzero 2 ( ; ), we define the map

fromT IR

n

T IR

n to IR by

: (f; g) 7! ([f; [g; f

0

]])

x

. This mapinherits bilinearity from the Lie bracket and, hence, is a tensor ofcovariant order two at x0

. Next, we derive a matrix

2 IR

mm

from

via

(

)

ij

:=

(f

i

; f

j

) (4)4While modification of the control bound can result in difficulties in the bal-

ancing of brackets in [6], this is not a concern in our case, since the neutralizationthat we achieve is independent of the relative magnitudes of .

142 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 1, JANUARY 2003

for i; j 2 f1; . . . ;mg and where fi

, fj

are the input vector fields of (1).By employing the Jacobi identity and the fact that the input vector fieldscommute, it is clear that

is also symmetric. Denoting by ^

thecorresponding matrix for the transformed system, it is easy to see that^

= T

T

T . In this manner, the question of whether the obstructingbrackets can be neutralized is reduced to the following linear algebraquestion:

given a symmetric matrix

6= 0, is there a full-rank, squarematrix T such that the congruence transformation of

,^

=

T

T

T , has all zeros along the diagonal?Supposing for a moment that such a congruence transform exists.

Since it is full rank, it must be true that there is some particular ~{ and~| with ~{ 6= ~| such that (^

)

~{~|

6= 0. In simplified terms, such an inputtransformation not only neutralizes the potential obstructions along(1;2)

x

, but also replaces them with vector fields generated by goodbrackets along (1;2)x

. Furthermore, the input transformation will notcreate type (1,2) potential obstructions that are annihilated by (1;2)x

.

(This would be tantamount to T T 0T 6= 0). Of course, the input trans-formation will also affect the vector fields generated by higher degreebrackets, possibly creating potential obstructions.

Recalling that a symmetric matrix is called indefinite if it has at leastone positive eigenvalue and at least one negative eigenvalue, we havethe following answer to the posed question.

Lemma 5: Given a matrix

=

T

6= 0, there exists a full-rankmatrix T such that ^

= T

T

T has all zeros on the diagonal if andonly if

is indefinite.Proof: First, recall that by virtue of the symmetry of the

matrix

, there exists a choice of orthonormal eigenvectorsV := (v

1

; . . . ; v

m

) such that V T

V = diag(

1

; . . . ;

m

)where i

are the real eigenvalues of

. Expressing the columns ti

of T in termsof the orthonormal eigenvectors, we have (^

)

ii

=

m

j=1

j

(t

T

i

v

j

)

2

.

If

6= 0 is semidefinite, then without loss of generality, we can take

0, and hence (^

)

ii

=

m

j=1

j

(t

T

i

v

j

)

2

0. For necessity,we must show that (^

)

ii

> 0. For any full-rank T there is somecolumn ti

and some eigenvalue j

such that j

(t

T

i

v

j

)

2

> 0, andsince j

0 for all j, we have (^

)

ii

> 0.

Suppose

is indefinite, and group the eigenvalues into thosewhich are positive f+i

g

m

i=1

, those which are negative fj

g

m

j=1

, andthose which are zero f0k

g

m

k=1

. The eigenvectors are similarly groupedinto fv+i

g

m

i=1

, fv

j

g

m

j=1

, and fv0k

g

m

k=1

. We proceed by constructingthe matrix T . The firstm0

columns ti

of T are chosen so that ti

= v

0

i

,

achieving tTi

t

i

= 0 for i 2 f1; . . . ; m0

g. The next m+

columnst

j

are chosen according to tj+m

= v

+

j

=(

+

j

)

1=2

v

1

=(

1

)

1=2 forall j 2 f1; . . . ; m+

g. For this choice, ti

?t

j

for all i 2 f1; . . . ;m0

g

and all j 2 f1; . . . ;m+

g, and tTj+m

t

j+m

= 0 for allj 2 f1; . . . ;m

+

g. The final m

columns tk

are chosen to bet

k+m +m

= v

+

1

=(

+

1

)

1=2

+ v

k

=(

k

)

1=2 for all k 2 f1; . . . ;m

g.

Similarly, this final group of columns is orthogonal to fti

g

m

i=1

and hasthe property tTk+m +m

t

k+m +m

= 0 for all k 2 f1; . . . ;m

g.

Furthermore, ft`

g

m

`=m +1

is linearly independent. This completes theconstruction of T .

B. Planar Vehicle Example Revisited

Applying this line of reasoning to the planar vehicle examplepreviously presented, the codistribution ( ; ) is spanned by = (0;2h; 0). In this example, if we use the canonical isomor-phisms T IRn T IRn IRn and furthermore if we endow IRn withthe natural inner product based on a particular choice of basis, we caninterpret ( ; ) as the projection of the vector fields generated by thetype (1,2) brackets onto the orthogonal complement of L(0;1)(F). Itrepresents the direction in which the vector fields generated by the

type (1,2) brackets cannot be neutralized by vector fields generated bythe type (0,1) brackets, i.e., the direction of potential obstruction. Thetensor

in coordinates is (0; 0; 2h; 0; 0; 0; 2h; 0; 0). The associatedmatrix

= (4h

2

; 2h; 2h; 0) has eigenvalues 2h2 2p

h

4

+ h

2

.

Of course h 6= 0 is assumed, for otherwise there is no potentialobstruction. It is easy to see that

is sign indefinite and, hence, theconstruction in the proof of Lemma 5 provides the transformation

T

=

1

(h)h

2

(h)

p

1 + h

2

2

(h)h

1

(h)

p

1 + h

2

1

(h)

2

(h)

where 1

and 2

are continuous functions of h > 0 with i

> 0 forall finite h > 0. This transformation yields ^

= (0;2;2; 0),

and the resulting type (1,2) brackets of the transformed systemgenerate [ f1

; [

f

1

; f

0

]] = (0; 0; 0), [

f

1

; [

f

2

; f

0

]] = (0; 1=h; 0), and[

f

2

; [

f

2

; f

0

]] = (0; 0; 0). Thus the potential obstruction to STLC isremoved, and the Lie subalgebra generated by the system at x0

isspanned by vector fields corresponding to good brackets, namelyf

f

1

;

f

2

; [

f

1

; [

f

2

; f

0

]]g. Hence, application of Sussmanns generalresult [7] demonstrates STLC. It is interesting to note that while T

isnot equal to T as previously determined, it does transform the systeminto one with a pure force and a torque input for any h > 0.

C. Example of Balancing Two Bad BracketsNot only can this technique neutralize a potential obstruction from a

bad type (1,2) bracket with a vector field generated by a good type (1,2)bracket, but it may also balance two potential obstructions generated bytwo type (1,2) brackets. Consider the two input example with f0

(x) =

(x

2

x

3

; x

1

x

3

; x

2

1

x

2

2

), f1

= (1; 0; 0), and f2

= (0; 1; 0). This systemhas two potential obstructions of type (1,2), namely [f1

; [f

1

; f

0

]] =

(0; 0; 2) and [f2

; [f

2

; f

0

]] = (0; 0;2). Furthermore, the good bracket(1; (2; 0)) evaluates as [f1

; [f

2

; f

0

]] = (0; 0; 0). For this example,

= (4; 0; 0;4) is clearly indefinite and, hence, we have the de-sired transformation T = (0:5;0:5; 0:5; 0:5). The resulting type(1,2) brackets for the transformed system evaluate as [ f1

; [

f

1

; f

0

]] =

(0; 0; 0), [

f

1

; [

f

2

; f

0

]] = (0; 0; 1), and [ f2

; [

f

2

; f

0

]] = (0; 0; 0), andagain STLC is achieved. An interesting variation on this example is ob-tained if we replace f0

(x)with (x2

x

3

; x

1

x

3

; x

2

1

+x

2

2

+x

1

x

2

), where 2 IR. This system has

= (4; 2; 2; 4) indefinite for jj > 2.This condition has the interpretation that even when the vector fieldsgenerated by the two bad brackets have values along with the samesign, they can still be neutralized with a vector field generated by agood bracket provided that its value along is large enough.

D. Effect of Neutralization on Other DirectionsNext, we consider the effect of neutralization of potential obstruc-

tions along one direction within ( ; ) on the value of the bracketsalong another direction within ( ; ). We consider the system thatevolves on x 2 IR4 described by f0

(x) = (x

2

x

4

; 0; x

2

1

+x

1

x

2

; x

1

x

2

),

f

1

= (1; 0; 0; 0), and f2

(0; 1; 0; 0). The type (1,2) brackets for thissystem are [f1

; [f

1

; f

0

]] = (0; 0; 2; 0), [f2

; [f

2

; f

0

]] = (0; 0; 0; 0),

and [f1

; [f

2

; f

0

]] = (0; 0; 1; 1) and, hence, ( ; ) is spanned by thecovectors (0; 0; 1; 0) and (0; 0; 0; 1). If we concentrate on neutralizingthe bad bracket in the direction = (0; 0; 2; 0), then we have

=

(4; 2; 2; 0)with eigenvalues = 22p

2. The constructed transforma-tion matrix T = (21=4;21=4; 0; 21=4) neutralizes the bad bracketalong . However, the transformation produces a potential obstruc-tion along the covector (0; 0; 0; 1), as evidenced by the resulting vectorfields [ f1

; [

f

1

; f

0

]] = (0; 0; 0;

p

2), [

f

1

; [

f

2

; f

0

]] = (0; 0;1;1),

and [ f2

; [

f

2

; f

0

]] = (0; 0; 0; 0). In fact, this system is known to benot STLC, as can be seen by applying the coordinate transformationy

3

:= x

3

x

4

and yi

= x

i

for i 6= 3.

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 1, JANUARY 2003 143

TABLE IAPPLICABILITY OF NEUTRALIZATION VIA CONGRUENCE TRANSFORMATION

E. Interpretation and ImpactWe have developed a methodology for neutralizing potential ob-

structions generated by bad brackets of type (1,2) for homogeneous de-gree-two systems that requires indefiniteness of the matrix

definedin (4). To interpret this requirement, we recall that a matrix is positivedefinite if and only if all of its principal minors are positive definite.On the other hand, a matrix is negative definite if and only if all of itsprincipal minors are negative definite when of odd dimension and pos-itive definite when of even dimension. Hence,

will be indefinite ifsome principal minor is itself indefinite. Recalling that the ijth entry of

is the value of the evaluated bracket [fi

; [f

j

; f

0

]] along the covector, the implications of the indefiniteness test become intuitively clear.For the moment, let us restrict our attention to cases where (1;2)x

hasdimension one, say (1;2)x

= spanf

x

g for some . If there is a po-tential obstruction from a single, type (1,2) bad bracket and a type (1,2)good bracket is not annihilated at x0

by , then the obstruction can al-ways be removed since the principal minor corresponding to these twobrackets is always indefinite (i.e., the matrix (2a; b; b; 0) has eigen-values ap

a

2

+ 4b

2). If two or more evaluated bad brackets are notannihilated at x0

by , then they can all be simultaneously neutralizedso long as a pair of the evaluated bad brackets provide opposite signswhen operated on by . On the other hand, if one or more evaluated badbrackets all have the same sign at x0

along the covector and all goodbrackets are annihilated by at x0

, then the technique fails. Noticethat while the examples all had just two inputs, the technique applieswithout modification to homogeneous degree-two systems with morethan two inputs (m > 2).

When other directions are involved, the neutralization may en-counter difficulties. Supposing that (1;2)x

is spanned by fi

g

k

i=1

with k 2, the question of neutralization of potential obstructionsbecomes one of simultaneously transforming the matrices

so thatthey all have zeros on their diagonals. If the ranges of the matrices

are orthogonal, then the problem can be solved with a block diagonaltransformation T , where each block appropriately transforms each

. This procedure requires a straightforward modification of theconstruction of T . These interpretations are summarized in Table I.

Finally, notice that the homogeneity of f0

is not essential to the de-velopment of neutralization via congruence transform, the constructionof the matrix

being sufficiently general that it applies to any non-linear system that is linear in control. For example, neutralization ofpotential obstructions from type (1,2) brackets for the system f0

(x) =

(x

2

x

3

; x

1

x

3

; sin

2

x

1

sin

2

x

2

), f1

= (1; 0; 0), and f2

= (0; 1; 0)

proceeds identically to that of the previous example with f0

(x) =

(x

2

x

3

; x

1

x

3

; x

1

x

2

). Clearly the proposed technique provides forneutralization of potential obstructions from type (1,2) brackets forthese more general systems. A generalization of neutralization via con-gruence transformation to inhomogeneous nonlinear systems would in-volve incorporation of the rich differential geometry of nilpotent andhigher order approximations and foliations described, for example, in[9].

V. CONCLUSION

We have presented a complete characterization of STLC for the classof single-input, homogeneous polynomial systems linear in control,where homogeneous is used in the traditional sense. Specifically forodd-degree systems, STLC is equivalent to the LARC, while even-de-gree systems are never STLC except for the degenerate case of a scalarstate. For multiple-input homogeneous systems linear in control, wehave investigated neutralization of bad brackets with brackets of thesame type. The methodology presented in this note provides a means ofneutralizing bad brackets of type (1,2). By consideration of the tensorgenerated from the bracket structure [; [; f0

]] applied to the directioncontaining a potential obstruction, we have reduced the question of neu-tralizing an obstruction to that of finding a congruence transform thatresults in a matrix with all zeros along its diagonal. It is shown that sucha transformation exists if and only if the matrix in question is indefinite.When this test is translated back to type (1,2) brackets, it has intuitiveimplications, which are illustrated with several simple examples. Themethodology presented is limited in its effectiveness by the fact thatit removes a potential obstruction only along a particular direction inthe tangent space, although an extension to multiple directions appearsattainable. Although the neutralization via congruence transformationresult has been presented in the context of homogeneous systems, itsdevelopment does not rely on the homogeneity of the drift vector fieldand, hence, applies to neutralization of type (1,2) brackets for any non-linear system that is linear in control.

REFERENCES[1] H. G. Hermes, On the structure of attainable sets for generalized dif-

ferential equations and control systems, J. Diff. Equations, vol. 9, pp.141154, Jan. 1971.

[2] H. J. Sussmann and V. Jurdjevic, Controllability of nonlinear systems,J. Diff. Equations, vol. 12, pp. 95116, July 1972.

[3] R. Hermann and A. J. Krener, Nonlinear controllability and observ-ability, IEEE Trans. Automat. Contr., vol. AC-22, pp. 728740, Oct.1977.

[4] H. J. Sussmann, Lie brackets and local controllability: A sufficient con-dition for scalar-input systems, SIAM J. Control Optim., vol. 21, pp.686713, Sept. 1983.

[5] G. Stefani, On the local controllability of a scalar input control system,in Theory and Applications of Nonlinear Control Systems, C. I. Byrnesand A. Lindquist, Eds. New York: Elsevier, 1986.

[6] M. Kawski, Higher-order small-time local controllability, in Non-linear Controllability and Optimal Control, H. J. Sussmann, Ed. NewYork: Marcel Dekker, 1990, vol. 133, Pure and Applied Mathematics,ch. 14, pp. 441477.

[7] H. J. Sussmann, General theorem on local controllability, SIAM J.Control Optim., vol. 25, pp. 158194, Jan. 1987.

[8] J.-P. Serre, Lie Algebras and Lie Groups: Lectures Given at HarvardUniversity, W. A. Benjamin, Ed. Cambridge, MA, 1965.

[9] H. G. Hermes, Nilpotent and high-order approximations of vector fieldsystems, SIAM Rev., vol. 33, pp. 238264, June 1991.

[10] A. A. Agrachev, Is it possible to recognize controllability in afinite number of differentiations?, in Open Problems in Mathemat-ical Systems and Control Theory, V. D. Blondel, Ed. New York:Springer-Verlag, 1999, Communications and Control Engineering, ch.4, pp. 1518.

[11] H. Nijmeijer and A. J. van der Schaft, Nonlinear Dynamical ControlSystems. New York: Springer-Verlag, 1990.

Index:

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index:

INDEX:

ind: