Controllability and 1-Homogeneous Control Systems
4.1 Geometric Homogeneity and Vector Bundles
4.1.3 The Tangent Bundle and Vector Fields
Equally important are the homogeneous properties of vectors inT E, the tangent bundle toE, or equivalently of vector fields inX(E), the space of vector fields defined onE. We focus now on the properties of the tangent bundleT E. Using the projection operator there is a natural notion of vertical forT E.
Definition 32 The vertical bundle over E, denoted byV E, is the subbundle ofT E given by the union of T π−1(0q)for allq∈Q. A vector inT Eis vertical if it lies in the kernel ofT π.
Definition 33 There is a canonical isomorphism betweenE⊕EandV E, called the vertical lift. It is given by
vlift= d dt
t=0
(ux+tvx), u, v ∈E. (4.7)
A complementary horizontal subbundle can be defined using a connection form, also called an Ehresmann connection. We rewrite below, the definition of an Ehresmann connection,
Definition 34 An Ehresmann connection,A, is a vertical valued one form on a manifoldQthat satisfies, 1. Aq :TqQ→VqQis a linear map for each pointq∈Q.
2. Ais a projection,A◦A=A.
There is no unique decomposition of the tangent bundle into horizontal and vertical, however there exists a trivial decomposition into horizontal and vertical subbundles. Its associated Ehresmann connection is called the trivial connection. On the vector bundleE, the natural projection operator,ρ2, can be used to define the trivial connection onT E,
Atriv :T E→V E, Atriv =ρ2. (4.8)
Pointwise, horizontal vectors are isomorphic to vectors on the tangent bundle of the base space using the trivial connection and the bundle projection, T π. In local trivializations for T M and T E, the injective mapping is given by
vx ∈TxM, vx =T π◦u(x,u), foru(x,u)∈H(x,u)E. (4.9) From (4.2), the horizontal bundle is in 1-1 correspondance with the Whitney sumE⊕T M.
Vector Fields
Properties that hold for tangent vectors will also hold for sections onT E, which give rise to vector fields X(E). Through the following derivations and propositions, we study how geometric homogeneity applies to vector fields.
Definition 35 The subspaceXV(E)⊂ X(E), consists of all vector fields that are vertical for all points in E.
Proposition 8 IfX, Y ∈ XV(E), then[X, Y]∈ XV(E).
The notion of homogeneity extends to vector fields via the generator,∆.
Definition 36 A vector fieldX ∈ X(E)is said to be homogeneous of orderpif, [∆, X] =pX
forp >−2.
The only vector field of homogeneous orderp ≤ −2is the zero vector field. Under the trivial splitting of vector fields into horizontal and vertical, the dilation vector field has the following properties,
Proposition 9 Let∆be the infinitesimal generator corresponding to the dilation action,δt. Given a vector fieldX ∈ X(E), the following properties hold:
1.
∆, XH
is horizontal, and 2.
∆, XV
is vertical,
for the trivial decomposition,X =XH +XV, ofX into horizontal and vertical components, respectively.
proof
Since horizontal vectors inT E are in 1-1 correspondance with vectors inT M, horizontal vector fields are in 1-1 correspondance with functions inC∞(E, T M). We may, therefore, make the identification
ρ1(X) =XH 7→Y ∈C∞(E, T M). In a local trivialization, the dynamic definition of the Jacobi-Lie bracket is
∆, XH
≡ d dt
t=0
(δt)∗XH
= d dt
t=0
(δt)∗(Y,0) = d dt
t=0
(T δ−t(Y ◦δt,0)).
Since the dilation operator leaves the base component invariant, the resulting vector field remains horizontal using the trivial decomposition. The second statement is a consequence of Proposition 8, as the dilation vector field,∆, is vertical.
Proposition 10 GivenX, Y ∈ X(E)homogeneous of order pandq, respectively, the Jacobi-Lie bracket [X, Y]is homogeneous of orderp+q.
Proposition 11 Any mappingΨ :E →Epreserving the base, i.e.,π =π◦Ψ, and homogeneous of order p, loses one degree of homogeneity when vertically lifted. That is Ψlift ∈ X(E) is homogeneous of order p−1.
proof
By Definition 33, the vertical lift of Ψ is a vector field on E. Homogeneity of Ψlift is determined by Definition 36,
h
∆,Ψlifti
= d dt
t=0
(δt)∗Ψlift= d dt
T δ−t◦Ψlift◦δ−t . In a local trivialization for the tangent bundle,
h
∆,Ψlifti
= d dt
t=0
0, e−tΨ◦δt
= d dt
t=0
0, e(p−1)tΨ
= (0,(p−1)Ψ)
= (p−1)Ψlift
Corollary 11 Given a section of the vector bundleE, its vertical lift is homogeneous of order−1.
As stated next, the converse to the corollary also holds.
Proposition 12 All vector fields of homogeneous order−1are the vertical lift of a section ofE.
proof
This is a commonly known result, c.f. Crampin [36], but will be proven regardless. Using Proposition 13, the horizontal component must be homogeneous of order−1, but by Proposition 6 the only such vector field is the zero vector field. Thus, all vector fields of homogeneous order−1must be vertical. The rest follows from Propositions 5 and 11.
Proposition 13 A vector field homogeneous of order p can be decomposed into horizontal and vertical components,X =XH +XV, that satisfy the following properties,
1. XH is in 1-1 correspondence with aY ∈C∞(E, T M)whereY is homogeneous of orderp, and 2. XV is in 1-1 correspondence with aΨlift, whereΨ∈C∞(E, E)is homogeneous of orderp+ 1.
proof
Using Proposition 9, Definition 36, and the linearity of the Jacobi-Lie bracket, the decomposition satisfies ∆, XH +XV
=
∆, XH +
∆, XV ,
That is, the horizontality and verticality of both contributions are preserved. The second statement is a consequence of Proposition 11. The first statement follows by further carrying out the proof of Proposition 9.
Proposition 13 is analogous to the discussion following Eq. (3.1) in Bullo and Lewis [25], but extended to hold for arbitrary vector bundles.
Filtrations of Homogeneous Spaces. The subbundles created by examining spaces of constant homoge- neous order form a filtration. The graded structure of a filtration means that homogeneous properties may be reduced to algebraic identities on the filtration order. Define the vector subbundle of homogeneous order kto be
Pk(E)≡ {X ∈ X(E)|Xis of homogeneous degreek.}. (4.10) Propositions 6 and 10 imply that
1. [Pi(E),Pj(E)]⊂ Pi+j(E), 2. Pk(E) ={0}, ∀k <−1.
Accordingly, we may define the following union of homogeneous spaces,
Mj,k(E) =⊕ii=j≤kPi(E), (4.11) which inherit the properties of its constitutive sets:
1.
Mi,j(E),Mi0,j0(E)
⊂ Mi+i0,j+j0(E), 2. Mi,j(E) ={0} , ∀i, j <−1.
The following notational convenience is used: when the first subscript is absent, then it is assumed to be−1, e.g.,Mi(E)≡ M−1,i(E). It can be seen that
M−1(E) =P−1(E),
meaning that this ”basic” space is not enlarged. The spacesMkform a filtration.
Corollary 12 IfX, Y ∈ M−1(E), then[X, Y] = 0.
The Jacobi identity implies a symmetry of the following bracket constructions.
Corollary 13 GivenX, Y ∈ M−1(E), then[X,[Γ, Y]] = [Y,[Γ, X]] for anyΓ∈ X(E).
Via Corollary 13, the (2,1)-tensor,[·,[Γ,·]], may be used to define a symmetric product for vector fields in M−1.
Definition 37 The symmetric product of vector fields inM−1(E)using the vector fieldΓis defined to be
hX : Y iΓ≡[X,[Γ, Y]], (4.12)
whereX, Y ∈ M−1(E).
Very often the vector fieldΓwill be specified and the relevant space of vector fields restricted such that the shortened notation symmetric product of vector fields, or even symmetric product, will make implicit sense.
In this case, we will simply writehX : Y iwithout reference to the vector fieldΓ. ByPkorMk, we will meanPk(E)orMk(E), respectively.
The symmetric product was originally derived and defined using the Riemmanian or affine connection structure of simple mechanical systems [37, 83]. However it is known, and we have shown, that is holds in the more general setting discussed here. What is critical to simple mechanical systems is not the symmetric product alone, but the homogeneous structure implied by the Lagrangian framework from which such sys- tems are derived. The homogeneous structure will imply that, for mechanical systems, it is only Jacobi-Lie brackets with a structure similar to the one above that will be important from both control-theoretic and dynamical systems perspectives.
For the systems that we will study, the homogeneous order of the vector fieldΓwill not exceed1, e.g.
Γ∈ M1(E). ThereforehX : Y i ∈ M−1, and the symmetric product of vector fields inM−1 is again in M−1. Most importantly, this implies that the symmetric product is a vertical lift, and commutes with other vertical lifts inM−1. It also implies that there exists a symmetric product defined onX(M)such that the following commutative diagram holds,
X, Y −−−−→lift Xlift, Ylift
h ·:· i
y yh ·:· i Z −−−−→lift Zlift
, (4.13)
whereX, Y, Z ∈ Γ(E) and Xlift, Ylift, Zlift ∈ M−1(E). Alternatively stated, there exists a symmetric product on sections ofEuniquely defined by
hX : Yilift≡D
Xlift : YliftE
, (4.14)
whereX, Y ∈Γ(E). Although it is not explicitly noted, the choice ofΓfor the symmetric product of vector fields plays an important role in defining the symmetric product of sections, c.f. Equation (4.12). It was
demonstrated in Lewis and Murray [85] that equation (4.14) was responsible for computational reductions in the involutive closure calculations for simple mechanical systems. The structure of vector bundles and geometric homogeneity has been used to show that it holds even whenE 6=T Q. Again, it is emphasized that the majority of canonical forms for mechanical systems discussed in the introduction satisfy the homo- geneous properties assumed above. Systems with these properties will be called 1-homogeneous systems.
We shall further refine the definition of a 1-homogeneous system in Section 4.2.