On Nonlinear Controllability of Homogeneous Systems Linear in Control

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 1, JANUARY 2003 139 On Nonlinear Controllability of Hoogeneous Systes Linear in Control Jaes Melody, Taer Başar, and Francesco Bullo Abstract This work considers sall-tie local controllability (STLC) of single- and ultiple-input systes, _ = ( )+ where ( ) contains hoogeneous polynoials and... are constant vector fields. For single-input systes, it is shown that even-degree hoogeneity precludes STLC if the state diension is larger than one. This, along with the obvious result that for odd-degree hoogeneous systes STLC is equivalent to accessibility, provides a coplete characterization of STLC for this class of systes. In the ultiple-input case, transforations on the input space are applied to hoogeneous systes of degree two, an exaple of this type of syste being otion of a rigid-body in a plane. Such input transforations are related via consideration of a tensor on the tangent space to congruence transforation of a atrix to one with zeros on the diagonal. Conditions are given for successful neutralization of bad type (1,2) brackets via congruence transforations. Index Ters Controllability, Lie algebras, nonlinear systes. I. INTRODUCTION Various concepts of controllability for nonlinear systes were initially explored in [1] [3]. In particular, [2] is priarily concerned with the property of accessibility of the analytic control syste _x = F (x; u), naely that the set of points attainable fro a given initial point via application of feasible input is full in the sense of having a nonepty interior. Sussann and Jurdjević deonstrated in [2] that a necessary and sufficient condition for accessibility of these systes is that the Lie algebra generated by the syste have full rank, the so-called Lie algebra rank condition (LARC). In [4], Sussann explored the property of sall-tie local controllability (STLC) for affine analytic single-input systes _x = f0(x) + f1(x)u with juj 1. A syste is said to be STLC at a point x0 if that initial point is in the interior of the set of points attainable fro it in tie T for all T > 0. In this case, the Lie algebra generated by the syste, denoted by L(ff0;f1g), is the sallest involutive distribution containing ff0;f1g or, equivalently, the distribution spanned by iterated Lie brackets of f0 and f1. Sussann gave various necessary and sufficient conditions for STLC in [4]. For exaple, a necessary condition for STLC is that the tangent vector [f1; [f1;f0]] x be in the subspace spanned by all tangent vectors at x0 generated by brackets with only one occurrence of f1, which is denoted by L 1 (ff0;f1g) x. More iportantly, the conditions conjectured by Heres were proved to be sufficient conditions for STLC. These Heres local controllability conditions (HLCC) consist of (1) x0 is a (regular) equilibriu point, (2) the LARC is satisfied, and (3) S k (ff0;f1g) x S k01 (ff0;f1g) x for all even k>1where S k (ff0;f1g) x denotes the span of all tangent vectors at x0 generated by brackets with k or less occurrences of f1. Stefani [5] provided an extension of Sussann s necessary condition by deonstrating that STLC iplies (ad 2 f f0) x 2 S 201 x for all 2f1; 2;...g. For an excellent suary and tutorial of these as well as other results in the single-input case, the inquisitive reader is directed to Kawski [6]. Manuscript received January 30, 2001; revised October 17, 2002 and Septeber 13, 2002. Recoended by Associate Editor H. Wang. This work was supported in part by the U.S. Departent 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 of Illinois, Urbana, IL 61801 USA (e-ail: jelody@uiuc.edu; tbasar@control.csl.uiuc.edu; bullo@uiuc.edu) Digital Object Identifier 10.1109/TAC.2002.806667 STLC of ultiple-input affine analytic control systes was addressed by Sussann in [7], where a general sufficiency theore was proven for analytic systes of the for _x = f0(x) + f i(x)u i with the constraints ju ij 1 for all i 2 f1;...;g. In order to understand this result, it is necessary to distinguish between foral brackets and evaluated brackets. On the one hand, a foral bracket is a pairwise parenthesized word (i.e., an eleent of the free aga) with well-defined left and right factors, nuber of factors, and degree. On the other hand, an evaluated bracket is the vector field that results fro an iterated Lie bracketing of particular vector fields. When we speak of the vector field generated by a foral bracket, we ean specifically the vector field that results fro evaluating the foral bracket with respect to particular vector fields, an operation that is theoretically captured by the evaluation ap in [7]. This distinction is captured by the following notation, which we eploy both in the stateent of Sussann s general sufficiency theore and throughout the sequel: if B =(i1;...;(i k01;i k )...)represents a foral bracket of indeterinates f0;...;g, then f B = [f i ;...[f i ;f i ]...] denotes the corresponding vector field generated by the evaluation ap fro the foral bracket B with respect to a particular set of vector fields ff0;...;f g.1 Hence, for exaple, we have the foral bracket B =(1; (1; 0)) that generates the vector field resulting fro evaluating the corresponding iterated Lie bracket [f1; [f1; f0]] for a particular pair of vector fields f0 and f1. Several results were presented in [7], but in the context of this note the ost appropriate result is based on the degree of a foral bracket. For a given foral bracket B of indeterinates fig i=0, jbj i is used to denote the nuber of occurrences of i as a factor in B.For 2 [0; 1], is defined by (B) :=jbj0 + jbj i. The general theore states that systes that satisfy the LARC at x0 and have f0(x0) =0are STLC if there exists a 2 [0; 1] such that the tangent vector at x0 generated by each foral bracket B with jbj0 odd and jbj1;...; jbj all even can be expressed as a linear cobination of tangent vectors at x0 generated by soe set of brackets fb k g N k=1 with (B k ) < (B) for all k 2f1;...;Ng. It has becoe conventional to refer to the foral brackets with jbj0 odd and jbj1;...; jbj all even as bad brackets, and if these bad brackets have corresponding vector fields that are not contained 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. The obstructions are only potential because they only obstruct the known sufficient conditions. In [6], Kawski presents several exaples of systes with potential obstructions that are known to be STLC. Both Sussann in [7] and Kawski in [6] apply a generalized definition of hoogeneity to STLC. This concept of hoogeneity begins with definition of a dilation as a paraeterized ap of IR n to IR n of the for (x) =( r x1; r x2;... r x n ) where r i are nonnegative integers. A polynoial p :IR n! IR is then said to be hoogeneous of degree k with respect to the dilation, sybolically p 2 H k if p( (x)) = k p(x). Traditional hoogeneity is recovered via the dilation with r1 = 111 = r n = 1. The definition of hoogeneity is then extended to vector fields in the following anner: a vector field f is said to be hoogeneous of degree j if fp 2 H k0j whenever p 2 H k for all k 0. A related area of research that capitalizes on this generalized concept of hoogeneity is that of nilpotent and high-order approxiation of control systes presented by Heres, for exaple, in [9]. One pertinent outcoe of Heres s research is that a syste is STLC if its Taylor approxiation is STLC. However, the converse question of whether STLC can be deterined fro a finite nuber of differentiations is still open [10]. 1 Readers interested in the rich detail of foral Lie algebras ay refer to [8]. 0018-9286/03$17.00 2003 IEEE

140 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 1, JANUARY 2003 Fig. 2. Motion of a rigid body in a plane expressed in body-fixed coordinates. Fig. 1. Graphical depiction of brackets that generate nonzero vector fields of polynoial syste (1) with hoogeneity degree. In the reainder of this note, we restrict our attention to hoogeneous nonlinear systes that are linear in control, using the traditional definition of hoogeneity. We begin by addressing single-input systes. Building on Stefani s necessary condition [5] and the concepts of good and bad brackets of Sussann s general sufficiency theore [7], we deonstrate that such single-input hoogeneous systes of even degree are STLC if and only if they have a scalar state. This result, cobined with the obvious fact that for odd-degree hoogeneous single-input systes STLC is equivalent to the LARC, copletely characterizes STLC for these systes. Next, we address ultiple-input systes with the additional restriction that they be hoogeneous of degree two. In this case, we extend the applicability of Sussann s general sufficiency theore by incorporating a linear transforation on the ultidiensional input in order to neutralize potential obstructions that arise fro type (1,2) bad brackets (i.e., brackets with jbj 0 =1and jbj!0 =2). In particular, we present a foral ethod for neutralizing these type (1,2) potential obstructions wherein the proble of finding the desired linear transforation on the input space is reduced to finding a particular atrix congruence transforation. II. PROBLEM EXPOSITION In this note, we address STLC of systes of the for _x = f 0 (x) + f i u i (1) where ju i j1and x 2 IR n. f i for i 2f1;...;g are assued to be constant vector fields, i.e., f i (x) f i, and the coponents of f 0 (x) are hoogeneous polynoials of degree k 1. We use the traditional definition of hoogeneous polynoial p, naely that p(x) = k p(x). The set of such hoogeneous vector fields is denoted by H k. Our definition of degree-k hoogeneous vector fields is equivalent to that used in [6], [7], and [9] in the following anner: take : x 7! x and then H k is in the general fraework the set of vector fields hoogeneous of degree 1 0 k. With this traditional definition, we have the following eleentary facts: i) f (0) = 0 for f 2H k with k 1, and ii) [f; g] 2H j+k01 for all f 2H j and g 2H k, where H01 is interpreted as the singleton containing the zero vector field. Systes of this for are theoretically interesting because their Lie algebra at x 0 = 0has a diagonal structure, as depicted in Fig. 1. In particular, the only brackets B that generate vector fields with nonzero value at x 0 are those with jbj!0 =(k01)jbj 0 +1, where jbj!0 := jbji. This follows directly fro the eleentary facts previously given. Keeping in ind that f 0 2H k and f i 2H 0 for all i 6= 0, fro fact ii), it is clear that brackets above the diagonal have hoogeneity degree greater than zero and, hence, by fact i) have zero value at x 0. Siilarly, fro fact ii), we have that brackets below this line have hoogeneity degree less than zero and, hence, by definition are identically zero. Furtherore, systes of the for (1) coonly arise in echanics. An exaple of such a syste is the otion of a rigid body in a plane expressed in body-fixed coordinates, as depicted in Fig. 2. The equations of otion for this syste are _! =u 2 + hu 1 _v x = 0!v y _v y =!vx + u1: (2) The state consists of rotational velocity! and the two body-fixed translation velocities v x and v y. The input consists of the torque u 2 and the force u 1 applied at a oent ar of h. Hence, f 0 (x) = (0; 0x 1 x 3 ;x 1 x 2 ), f 1 = (h; 0; 1) for soe constant h, f 2 = (1; 0; 0), and the syste is of the for (1) with hoogeneity degree two. This provides a siple exaple of a syste for which Sussann s sufficient condition in [7] is not invariant with respect to input transforations. In particular, if a pure force (h = 0) and a torque are used as inputs, then the syste satisfies Sussann s sufficient condition for the ultiple-input case. 2 However, if an offset force (h 6= 0) is used, then potential obstructions appear as vector fields generated fro the type (1,2) brackets, i.e., brackets with jbj 0 =1and jbj!0 =2. In general, we eploy the phrase type (k; `) brackets to refer to all foral brackets B of indeterinates f0;...;g with jbj 0 = k and jbj!0 = `, and denote the distribution spanned by such brackets as L (k;`) (F). 3 Using this syste as a otivating exaple, we explore the neutralization via congruence transforation of potential obstructions generated by bad brackets of type (1,2) with vector fields generated by other brackets (perhaps also bad) of type (1,2). III. SINGLE-INPUT SYSTEMS In this section, we consider the single-input syste _x = f 0 (x) +f 1 u (3) where x 2 IR n, u 2 [01; 1], f 1 2 IR n, and f 0 2H k. In light of HLCC and Sussann s general sufficiency result [7], it is clear that for a syste as in (3) with odd hoogeneity degree, accessibility is equivalent to STLC, since there are no nonzero brackets with j1j 0 odd and j1j 1 even. In other words, the question of STLC reduces to the LARC. The 2 A generalized force on a rigid body consists of a pure force coponent which induces only a translational otion and a torque coponent which induces only a rotational otion. 3 The notation is used instead of to ephasize that ( ) is not necessarily a Lie subalgebra.

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 1, JANUARY 2003 141 following lea asserts that if the vector fields generated by brackets of type (1;k) do not add to the Lie algebra rank, then neither do the vector fields generated by higher-degree brackets. Lea 1: For (3) with hoogeneity degree k>0,if(ad k f f 0) x 2 spanff 1 g, then L(ff 0 ;f 1 g) x = spanff 1 g. Proof: Since the Lie algebra structure is invariant to (analytic) coordinate transforations, without loss of generality we can take f 1 to be the basis vector e 1. Let Hk 11 be the set of vector fields of hoogeneity degree k with the for (Cx k 1 + 1 (x); 2 (x);...; n (x)) where C is any scalar constant and the power of x 1 in i is less than k for all i 2f1;...;g. Then, (adf k f 0 ) x 2 spanfe 1 g iplies that f 0 2 Hk 11, since adf k corresponds to the partial derivative operator @ k =@x k 1. Furtherore, if g 2 H 11 for soe 1, then ad f g 2 H01 11 and ad f g 2H+k01. 11 Since the Lie subalgebra of (3) is spanned by vector fields of the for [f i ; [f i ;...[f i ;f i ]...]] (apply, for exaple, [11, Prop. 3.8]), all vector fields in L(ff 0;f 1g) x are a ultiple of e 1. Turning our attention to systes with even hoogeneity degree k, we see that in general L(ff 0 ;f 1 g) x does include potential obstructions generated fro bad brackets, i.e., there are bad brackets along the diagonal of Fig. 1. In particular, type (; (k 0 1) +1)brackets are (odd, even) when is odd. We ake use of the necessary condition of Stefani restated here for convenience. Theore 2 Stefani [5]: If the syste _x = f 0(x) +f 1(x)u 1 is STLC, then (adf 2k f 0 ) x 2S 2k01 (ff 0 ;f 1 g) x. When applied to even systes, Theore 2 states that (adf 2k f 0) x 2 spanff 1g is necessary for STLC. However, if (adf 2k f 0 ) x 2 spanff 1 g, then Lea 1 asserts dil(ff 0 ;f 1 g)=1, and systes with n 2 cannot be STLC (for otherwise LARC is violated). This reasoning and the fact that the syste _x = x 2k + u with x 2 IR is STLC provides the following result. Proposition 3: If the syste in (3) has odd hoogeneity degree, then it is STLC if and only if it satisfies the LARC. On the other hand, if the syste has even hoogeneity degree 2k >0, then it is STLC if and only if the state x is scalar. the unchanged equations of otion, we have the potential obstruction [f 1 ; [f 1 ;f 0 ]](0) = (0; 02h; 0) 62 spanff 1 (0);f 2 (0)g. This potential obstruction prevents the use of Sussann s sufficient condition for any h 6= 0. On the other hand, if h =0(corresponding to a force input u 1 through the center of ass) we can then apply the sufficient condition. This is so because we have the vector field [f 1 ; [f 2 ;f 0 ]] generated by the good bracket (1; (2; 0)) being (0; 01; 0), bringing the Lie algebra to full rank, and STLC is deonstrated. Furtherore, the syste for h 6= 0can be transfored into the pure-force syste (h =0)via the input transforation u 1 u 2 = 1 0 0h 1 u 1 u 2 : Since STLC is clearly invariant to full-rank input transforations, we see that a particular choice of input transforation ay provide a eans of reoving a potential obstruction, thus extending the applicability of Sussann s general theore for this class of systes. Reark 4: It is worth noting that Kawski [6] has considered techniques for neutralizing and balancing bad brackets. However, this technique is not related to ours. Kawski s technique applies in the singleinput setting, and neutralizes brackets possibly with brackets of different degree via paraeterized failies of controls carefully tailored to the syste and the brackets in question. In soe cases, these failies of controls involve switching between control liits, with the paraeter affecting the switching ties. Our technique utilizes the freedo of ultiple independent inputs to enforce a linear relation between the inputs in order to neutralize brackets of the sae type. A. Neutralization via Congruence Transfor Moving to the generic ultiple-input hoogeneous syste of degree two, suppose that there is soe bad bracket of the for (i; (i; 0)) that generates a potential obstruction, i.e., [f i; [f i;f 0]] x =2 spanff k (x 0)gk=1. We would like to find a full-rank, linear transforation T on the input space such that the transfored syste f IV. MULTIPLE-INPUT SYSTEMS We now return to consideration of the syste in (1) where f 0 2 H 2 (x). An extension of the previous results to this ultiple-input case is probleatic. In particular, the necessary condition of [5] runs into the proble of a possibility of balancing between potential obstructions generated fro bad brackets of the sae degree. This consideration, along with the otivating exaple of planar rigid-body otion lead us to investigate neutralization of potential obstructions by vector fields generated fro brackets of the sae type. Of course, for the syste in (1) the general sufficient condition of [7] can be applied to deterine STLC. Since the Lie algebra has a diagonal structure, the choice of 2 [0; 1] in the theore is iaterial. Using Sussann s concepts of good and bad brackets, the sufficient condition allows us to neutralize potential obstructions fro bad brackets with vector fields generated by good brackets of lower degree. Our goal with this section is to address the case where there are potential obstructions that cannot be neutralized in this anner, and to neutralize these potential obstructions with vector fields generated by other brackets of the sae degree via appropriate choice of linear transforation on the input space. In this endeavor, the diagonal structure of the Lie algebra will be particularly useful. Returning to the otivating exaple of planar otion in (2), it is clear that this syste is STLC. (For exaple, use the feedback transforation u 1 =!v x +u 1 and u 2 = u 2 0 hu 1 to obtain the syste described by _! = u 2, _v x = 0!v y, and _v y = u 1). However, in attepting to apply Sussann s general sufficiency result directly on _x = f 0(x)+ j=1 f it ij has the corresponding potential obstruction reoved. Of course, the full-rank input transforation will not affect spanff k gk=1. We ake the restriction ju j j1= where is the spectral radius of T in order to have the resulting u i satisfy the bounds ju ij1. 4 Suppose that there is at least one type (1,2) potential obstruction, i.e., there is soe 2 f1;...; g such that [[f ; [f ;f 0]] x =2 L (0;1) (F) x. Consider the codistribution KerL (0;1) (F) that annihilates the distribution L (0;1) (F). Then there ust exist soe differential one-for 2 KerL (0;1) (F) such that ([f ; [f ;f 0]]) x 6= 0. Let ) be the codistribution containing all covectors 2 KerL (0;1) (F) with the property ([f j ; [f j ;f 0 ]]) x 6=0 for soe j. Let us suppose that x has exactly diension one. Choosing any nonzero 2 ), we define the ap fro T IR n 2 T IR n to IR by : (f; g) 7! ([f; [g; f 0 ]]) x. This ap inherits bilinearity fro the Lie bracket and, hence, is a tensor of covariant order two at x 0. Next, we derive a atrix 9 2 IR 2 fro via (9 ) ij := (f i;f j) (4) 4 While odification of the control bound can result in difficulties in the balancing of brackets in [6], this is not a concern in our case, since the neutralization that we achieve is independent of the relative agnitudes of. u j

142 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 1, JANUARY 2003 for i; j 2f1;...;gand where f i, f j are the input vector fields of (1). By eploying the Jacobi identity and the fact that the input vector fields coute, it is clear that 9 is also syetric. Denoting by ^9 the corresponding atrix for the transfored syste, it is easy to see that ^9 = T T 9 T. In this anner, the question of whether the obstructing brackets can be neutralized is reduced to the following linear algebra question: given a syetric atrix 9 6=0, is there a full-rank, square atrix T such that the congruence transforation of 9, ^9 = T T 9 T, has all zeros along the diagonal? Supposing for a oent that such a congruence transfor exists. Since it is full rank, it ust be true that there is soe particular ~{ and ~ with ~{ 6= ~ such that ( ^9 ) ~{~ 6= 0. In siplified ters, such an input transforation not only neutralizes the potential obstructions along x, but also replaces the with vector fields generated by good brackets along x. Furtherore, the input transforation will not create type (1,2) potential obstructions that are annihilated by x. (This would be tantaount to T T 0T 6= 0). Of course, the input transforation will also affect the vector fields generated by higher degree brackets, possibly creating potential obstructions. Recalling that a syetric atrix is called indefinite if it has at least one positive eigenvalue and at least one negative eigenvalue, we have the following answer to the posed question. Lea 5: Given a atrix 9 =9 T 6= 0, there exists a full-rank atrix T such that ^9 = T T 9 T has all zeros on the diagonal if and only if 9 is indefinite. Proof: First, recall that by virtue of the syetry of the atrix 9, there exists a choice of orthonoral eigenvectors V := (v 1 ;...;v ) such that V T 9 V =diag( 1 ;...; ) where i are the real eigenvalues of 9. Expressing the coluns t i of T in ters of the orthonoral eigenvectors, we have ( ^9 ) ii = j=1 j (t T i v j ) 2. If 9 6=0is seidefinite, then without loss of generality, we can take 9 0, and hence ( ^9 ) ii = j=1 j (tt i v j ) 2 0. For necessity, we ust show that ( ^9 ) ii > 0. For any full-rank T there is soe colun t i and soe eigenvalue j such that j (ti T v j ) 2 > 0, and since j 0 for all j, wehave( ^9 ) ii > 0. Suppose 9 is indefinite, and group the eigenvalues into those which are positive f + i g, those which are negative f0 j g, and j=1 those which are zero f 0 kg k=1. The eigenvectors are siilarly grouped into fv i + g, fv0 j g, and j=1 fv0 kg k=1. We proceed by constructing the atrix T. The first 0 coluns t i of T are chosen so that t i = v 0 i, achieving ti T 9 t i = 0for i 2f1;...; 0 g. The next + coluns t j are chosen according to t j+ = v j + =(+ j )1=2 0 v 0 =(0 1 1 )1=2 for all j 2f1;...; +g. For this choice, t i?t j for all i 2f1;...; 0g and all j 2 f1;...; + g, and tj+ T 9 t j+ = 0 for all j 2 f1;...; +g. The final 0 coluns t k are chosen to be t k+ + = v + 1 =(+ 1 )1=2 + v 0 k =(0 k )1=2 for all k 2f1;...;0g. Siilarly, this final group of coluns is orthogonal to ft i g and has the property tk+ T + 9 t k+ + =0for all k 2f1;...;0g. Furtherore, ft`g `= +1 is linearly independent. This copletes the construction of T. B. Planar Vehicle Exaple Revisited Applying this line of reasoning to the planar vehicle exaple ) previously presented, the codistribution is spanned by = (0; 02h; 0). In this exaple, if we use the canonical isoorphiss T IR n T 3 IR n IR n and furtherore if we endow IR n with the natural inner product based on a particular choice of basis, we can interpret ) as the projection of the vector fields generated by the type (1,2) brackets onto the orthogonal copleent of L (0;1) (F). It represents the direction in which the vector fields generated by the type (1,2) brackets cannot be neutralized by vector fields generated by the type (0,1) brackets, i.e., the direction of potential obstruction. The tensor in coordinates is (0; 0; 2h;0; 0; 0; 2h; 0; 0). The associated atrix 9 = (4h 2 ; 2h;2h; 0) has eigenvalues 2h 2 6 2p h 4 + h 2. Of course h 6= 0 is assued, for otherwise there is no potential obstruction. It is easy to see that 9 is sign indefinite and, hence, the construction in the proof of Lea 5 provides the transforation T = 1(h)h 0 p p 2(h) 1+h 2 2(h)h 0 1(h) 1+h 2 1 (h) 2 (h) where 1 and 2 are continuous functions of h>0with i > 0 for all finite h > 0. This transforation yields ^9 = (0; 02; 02; 0), and the resulting type (1,2) brackets of the transfored syste generate [ f 1 ; [ 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 is reoved, and the Lie subalgebra generated by the syste at x 0 is spanned by vector fields corresponding to good brackets, naely ff 1 ; f 2 ; [ f 1 ; [ f 2 ;f 0 ]]g. Hence, application of Sussann s general result [7] deonstrates STLC. It is interesting to note that while T is not equal to T as previously deterined, it does transfor the syste into one with a pure force and a torque input for any h>0. C. Exaple of Balancing Two Bad Brackets Not only can this technique neutralize a potential obstruction fro a bad type (1,2) bracket with a vector field generated by a good type (1,2) bracket, but it ay also balance two potential obstructions generated by two type (1,2) brackets. Consider the two input exaple with f 0(x) = (x 2x 3;x 1x 3;x 2 1 0x 2 2), f 1 =(1; 0; 0), and f 2 =(0; 1; 0). This syste has two potential obstructions of type (1,2), naely [f 1 ; [f 1 ;f 0 ]] = (0; 0; 2) and [f 2; [f 2;f 0]] = (0; 0; 02). Furtherore, the good bracket (1; (2; 0)) evaluates as [f 1 ; [f 2 ;f 0 ]] = (0; 0; 0). For this exaple, 9 = (4; 0; 0; 04) is clearly indefinite and, hence, we have the desired transforation T = (0:5; 00:5; 0:5; 0:5). The resulting type (1,2) brackets for the transfored syste evaluate as [ f 1 ; [ f 1 ;f 0 ]] = (0; 0; 0), [ f 1 ; [ f 2 ;f 0 ]] = (0; 0; 1), and [ f 2 ; [ f 2 ;f 0 ]] = (0; 0; 0), and again STLC is achieved. An interesting variation on this exaple is obtained if we replace f 0 (x) with (x 2 x 3 ;x 1 x 3 ;x 2 1 +x 2 2 +x 1 x 2 ), where 2 IR. This syste has 9 =(4; 2;2; 4) indefinite for jj > 2. This condition has the interpretation that even when the vector fields generated by the two bad brackets have values along with the sae sign, they can still be neutralized with a vector field generated by a good bracket provided that its value along is large enough. D. Effect of Neutralization on Other Directions Next, we consider the effect of neutralization of potential obstructions along one direction within ) on the value of the brackets along another direction within ). We consider the syste that evolves on x 2 IR 4 described by f 0 (x) =(x 2 x 4 ; 0;x 2 1 +x 1 x 2 ;x 1 x 2 ), f 1 = (1; 0; 0; 0), and f 2(0; 1; 0; 0). The type (1,2) brackets for this syste are [f 1 ; [f 1 ;f 0 ]] = (0; 0; 2; 0), [f 2 ; [f 2 ;f 0 ]] = (0; 0; 0; 0), and [f 1 ; [f 2 ;f 0 ]] = (0; 0; 1; 1) and, hence, ) is spanned by the covectors (0; 0; 1; 0) and (0; 0; 0; 1). If we concentrate on neutralizing the bad bracket in the direction =(0; p 0; 2; 0), then we have 9 = (4; 2; 2; 0) with eigenvalues =262 2. The constructed transforation atrix T =(02 01=4 ; 2 01=4 ;0; 2 1=4 ) neutralizes the bad bracket along. However, the transforation produces a potential obstruction along the covector (0; 0; 0; 1), as evidenced by the resulting vector fields [ f 1 ; [ f 1 ;f 0 ]] = (0; 0; 0; 0 p 2), [ f 1 ; [ f 2 ;f 0 ]] = (0; 0; 01; 01), and [ f 2 ; [ f 2 ;f 0 ]] = (0; 0; 0; 0). In fact, this syste is known to be not STLC, as can be seen by applying the coordinate transforation y 3 := x 3 0 x 4 and y i = x i for i 6= 3.

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 1, JANUARY 2003 143 TABLE I APPLICABILITY OF NEUTRALIZATION VIA CONGRUENCE TRANSFORMATION E. Interpretation and Ipact We have developed a ethodology for neutralizing potential obstructions generated by bad brackets of type (1,2) for hoogeneous degree-two systes that requires indefiniteness of the atrix 9 defined in (4). To interpret this requireent, we recall that a atrix is positive definite if and only if all of its principal inors are positive definite. On the other hand, a atrix is negative definite if and only if all of its principal inors are negative definite when of odd diension and positive definite when of even diension. Hence, 9 will be indefinite if soe principal inor is itself indefinite. Recalling that the ijth entry of 9 is the value of the evaluated bracket [f i ; [f j ;f0]] along the covector, the iplications of the indefiniteness test becoe intuitively clear. For the oent, let us restrict our attention to cases where x has diension one, say x =spanf x g for soe. If there is a potential obstruction fro a single, type (1,2) bad bracket and a type (1,2) good bracket is not annihilated at x0 by, then the obstruction can always be reoved since the principal inor corresponding to these two brackets is always indefinite (i.e., the atrix (2a; b; b; 0) has eigenvalues a 6 p a 2 +4b 2 ). If two or ore evaluated bad brackets are not annihilated at x0 by, then they can all be siultaneously neutralized so long as a pair of the evaluated bad brackets provide opposite signs when operated on by. On the other hand, if one or ore evaluated bad brackets all have the sae sign at x0 along the covector and all good brackets are annihilated by at x0, then the technique fails. Notice that while the exaples all had just two inputs, the technique applies without odification to hoogeneous degree-two systes with ore than two inputs ( >2). When other directions are involved, the neutralization ay encounter difficulties. Supposing that x is spanned by f ig k with k 2, the question of neutralization of potential obstructions becoes one of siultaneously transforing the atrices 9 so that they all have zeros on their diagonals. If the ranges of the atrices 9 are orthogonal, then the proble can be solved with a block diagonal transforation T, where each block appropriately transfors each 9. This procedure requires a straightforward odification of the construction of T. These interpretations are suarized in Table I. Finally, notice that the hoogeneity of f0 is not essential to the developent of neutralization via congruence transfor, the construction of the atrix 9 being sufficiently general that it applies to any nonlinear syste that is linear in control. For exaple, neutralization of potential obstructions fro type (1,2) brackets for the syste f0(x) = (x2x3;x1x3; sin 2 x1 0 sin 2 x2), f1 = (1; 0; 0), and f2 = (0; 1; 0) proceeds identically to that of the previous exaple with f0(x) = (x2x3;x1x3;x1 0 x2). Clearly the proposed technique provides for neutralization of potential obstructions fro type (1,2) brackets for these ore general systes. A generalization of neutralization via congruence transforation to inhoogeneous nonlinear systes would involve incorporation of the rich differential geoetry of nilpotent and higher order approxiations and foliations described, for exaple, in [9]. V. CONCLUSION We have presented a coplete characterization of STLC for the class of single-input, hoogeneous polynoial systes linear in control, where hoogeneous is used in the traditional sense. Specifically for odd-degree systes, STLC is equivalent to the LARC, while even-degree systes are never STLC except for the degenerate case of a scalar state. For ultiple-input hoogeneous systes linear in control, we have investigated neutralization of bad brackets with brackets of the sae type. The ethodology presented in this note provides a eans of neutralizing bad brackets of type (1,2). By consideration of the tensor generated fro the bracket structure [1; [1; f0]] applied to the direction containing a potential obstruction, we have reduced the question of neutralizing an obstruction to that of finding a congruence transfor that results in a atrix with all zeros along its diagonal. It is shown that such a transforation exists if and only if the atrix in question is indefinite. When this test is translated back to type (1,2) brackets, it has intuitive iplications, which are illustrated with several siple exaples. The ethodology presented is liited in its effectiveness by the fact that it reoves a potential obstruction only along a particular direction in the tangent space, although an extension to ultiple directions appears attainable. Although the neutralization via congruence transforation result has been presented in the context of hoogeneous systes, its developent does not rely on the hoogeneity of the drift vector field and, hence, applies to neutralization of type (1,2) brackets for any nonlinear syste that is linear in control. REFERENCES [1] H. G. Heres, On the structure of attainable sets for generalized differential equations and control systes, J. Diff. Equations, vol. 9, pp. 141 154, Jan. 1971. [2] H. J. Sussann and V. Jurdjevic, Controllability of nonlinear systes, J. Diff. Equations, vol. 12, pp. 95 116, July 1972. [3] R. Herann and A. J. Krener, Nonlinear controllability and observability, IEEE Trans. Autoat. Contr., vol. AC-22, pp. 728 740, Oct. 1977. [4] H. J. Sussann, Lie brackets and local controllability: A sufficient condition for scalar-input systes, SIAM J. Control Opti., vol. 21, pp. 686 713, Sept. 1983. [5] G. Stefani, On the local controllability of a scalar input control syste, in Theory and Applications of Nonlinear Control Systes, C. I. Byrnes and A. Lindquist, Eds. New York: Elsevier, 1986. [6] M. Kawski, Higher-order sall-tie local controllability, in Nonlinear Controllability and Optial Control, H. J. Sussann, Ed. New York: Marcel Dekker, 1990, vol. 133, Pure and Applied Matheatics, ch. 14, pp. 441 477. [7] H. J. Sussann, General theore on local controllability, SIAM J. Control Opti., vol. 25, pp. 158 194, Jan. 1987. [8] J.-P. Serre, Lie Algebras and Lie Groups: Lectures Given at Harvard University, W. A. Benjain, Ed. Cabridge, MA, 1965. [9] H. G. Heres, Nilpotent and high-order approxiations of vector field systes, SIAM Rev., vol. 33, pp. 238 264, June 1991. [10] A. A. Agračhev, Is it possible to recognize controllability in a finite nuber of differentiations?, in Open Probles in Matheatical Systes and Control Theory, V. D. Blondel, Ed. New York: Springer-Verlag, 1999, Counications and Control Engineering, ch. 4, pp. 15 18. [11] H. Nijeijer and A. J. van der Schaft, Nonlinear Dynaical Control Systes. New York: Springer-Verlag, 1990.