Dissipation and Controllability

Controllability and 1-Homogeneous Control Systems

4.3 Dissipation and Controllability

This section extends the controllability analysis to the case where dissipation is included in the mechanical model. The dissipation modelled is often known as Rayleigh dissipation; as a vector field it is vertical and of homogeneous order0(implying linearity as a function of the vector fiber).

˙

z=XS(z) +Y_a^lift(z)u^a= Γ(z) +D^lift(z)−Z^lift(z) +Y_a^lift(z)u^a. (4.37)

The vector fieldD^lift(z) ∈ P0 represents the additional dissipation vector field. Cort´es et al. [34] studied the effect of isotropic dissipation; more preciselyD^lift(x)was restricted to be symmetric and independent of configuration. This section attempts to generalize [34] to configuration dependent, not necessarily symmet- ric, dissipation. The dissipation vector field is seen as an interfering element in the controllability analysis, and not as a control vector field that may grant additional control authority. Although the free Lie algebras will recover additional control directions, a large set of potential Jacobi-Lie brackets will be ignored by re- stricting the analysis to the homogeneous spaces that were found to be useful in Section 4.2. Consequently, the results of this section are conservative in nature.

Free Algebras and Homogeneity

Lemma 10 IfB ∈Br₋₁(X), thenEv(φ)(B)∈ M−1. proof

If B ∈ Br₋₁(X) then B ∈ P r(X). Evaluation of the symmetric product gives a vector field in the homogeneous spaceM−k−1,−1=P−1.

The addition of dissipation, a vertical vector field of homogeneous order 0, does not alter the symmetric product calculation, which can be seen by appealing to the definition of the symetric product, Definition 37.

Consequently, the nature of good and bad brackets does not change.

Lemma 11 Letl≥2be an integer, and letB ∈Br^k(X)T

Br_−l(X)fork≥2. ThenEv(φ)(B) = 0.

proof

An elementB ∈Br_−l(X)contains brackets withlmore instances of the vector fieldsX_a,a= 1. . . m+ 1, than the vector fieldX₀. Supposing thatB ∈Br^(2k+l)(X)for anyk >0. Then, there arekinstances of the elementX0, andk+linstances of the elementsXain the products that make upB. Under the evaluation operator, Ev(φ), theX_a map to input vector fields lying in the space P−1, and the element X₀ maps to the drift vector field inM0,1. Therefore, the elementBmaps to an iterated Jacobi-Lie bracket in the space M₋k−l,−l. Forl≥2, the space contains only the zero vector field.

The remaining elements requiring analysis for configuration accessability are those in those inBr_l(X), for l≥0.

Lemma 12 Letl≥0be an integer and letB ∈Br_l(X). ThenEv_0q(φ)(B)∈ M−1,0(0_q)for allq ∈Q.

proof

An elementB∈Br_l(X)contains brackets withlmore instances of the vector fieldX₀than the vector fields X_a,a= 1. . . m+ 1. Supposing thatB ∈Br^(2k+l)(X)for anyk >0, there arekinstances of the elements Xa, and k+l instances of the element,X0. Under the evaluation operator, Ev(φ), theXa map to input vector fields lying in the spaceP₋1, and the elementX₀maps toΓ+D^lift∈ M0,1. The elementBmaps to a iterated Jacobi-Lie bracket in the spaceM−k,l=M−1,l. Of course, all vector fields of homogeneous order greater than0vanish on the zero section, therefore only the subspaceM−1,0will provide a contribution.

Previously,Ev(φ)(Br₀(X))⊂ P0, whereas nowEv(φ)(Br₀(X))⊂ M0. The vertical dissipation vector, introduces new vertical contributions to the Jacobi-Lie bracket computations. By projecting toEv(φ)(Br0(X)) to the horizontal subspace, we may recover the original (dissipation free) distribution. Equally important, the evaluation Ev(φ)(Br_k(X))is no longer the trivial space{0} fork ≥ 0. Dissipation introduces ad- ditional brackets that before had been vanishing. These brackets can provide either new control directions

if needed, or new bad brackets. The bad brackets will only occur for spaces of odd homogeneous order, Br_2k+1(X)fork ≥0.

The Form of the Accessiblity Distribution on the Zero Section for 1-Homogeneous Control Systems with Dissipation. The horizontal and vertical decomposition is no longer viable. We shall consider, in- stead, several distributions. The first consists of only those element that were previously found useful on the zero section.

C0(V) = [

k∈Z⁺

C0^(k)(V), (4.38)

where

C0^(k)(V)≡n

Ev(φ)(B)|B ∈Br^k(X)\

Br₀(X)o

. (4.39)

In the case of vanishing dissipation,C0(V)is equal toChor(V). The remaining two distributions will be used to characterize the good brackets and the bad brackets ofBr(X).

Ceven(V) = [

k∈Z⁺

Ceven^(k)(V), (4.40)

where

C^(k)even(V)≡n

Ev(φ)(B)|B ∈Br^k(X)\

Br_2l(X), l ≥0o

. (4.41)

Under the mapping Φ, we will find thate C0(V) maps intoCeven(V). Comparing this to the case of no dis- sipation, where Φ (e C0(V)) = C0(V), we see that dissipation enlarges the set of possible contributions to configuration controllability. The same can be said ofCodd(V),

Codd(V) = [

k∈Z⁺

Codd^(k)(V), (4.42)

where

C_odd^(k)(V)≡n

Ev(φ)(B)|B ∈Br^k(X)\

Br_2l−1(X), l ≥0o

. (4.43)

In the case of no dissipation,Φ (e Cver(V)) =Cver(V), whereas in the case with dissipation, Φ (e Cver(V))⊂ Codd(V). Consequently, dissipation also enlarges the space of potential bad brackets. The nice thing is that the bad brackets are in a complementary subdistribution to the good brackets.

It is critical to understand how these brackets relate to the actual system, which utilizes the setV⁰instead.

Definition 60 Define the distribution

C0(V⁰) = [

k∈Z⁺

C0^(k)(V⁰), (4.44)

which consists of

C₀^(k)(V⁰)≡n

Ev(φ)(B)e |Be=Φ(B), Be ∈Br^k(X)\

Br₀(X)o

. (4.45)

Proposition 16 C0(V⁰)⊂ Ceven(V).

proof

Take an arbitrary primitive bracketB ∈Br^(k)(X)T

Br₀(X), and calculate its replacementB⁰ = Rep(B)∈ Br^(k)(X0), whereby the following identities are obtained: I⁰ = δ(B⁰), and J⁰ = Pm

a=1δa(B⁰), where

δ₀(B⁰) = δ₀(B) +δ_m+1(B). The mapping Φ takesB⁰ and maps it into a sum of brackets from the set S(B⁰)⊂Br(X).

We shall characterize an arbitrary elementC∈S(B⁰). The element,C, will have the following degrees, I = δ₀(C), J = P_m

a=1δ_a(C) = J⁰, and K = δ_m+1(C) = I⁰ −I. Its relative degree will be δ(C) =˜ I−J−K. Whereas before, the only contributions on the zero section were those with relative degree0or

−1, contributions may be obtained for any relative degree greater than or equal to−1.

Take the element with maximal degree B ∈S(B⁰). There can only be one such element. It is the one withI =I⁰. The next highest degree terms are the terms withI = I⁰−1. Notice that, if any one of the slots withX₀in it is replaced byX_m+1, then the relative degree changes by two degrees; one degree is lost becauseX₀is removed, but one degree is also lost becauseX_m+1 is added. Since the degree drops by two degrees each time we move fromItoI−1, the relative degree of elements fromS(B⁰)have the same parity, they will be even only, or odd only. For an even parity, the only contributions can beB ∈S(B⁰)T

Br_2l(X), whereas for odd parity, the only contributions can beB ∈S(B⁰)T

Br_2l−1(X), forl≥0. Since the original parity was even,B⁰ is spanned by elements inBr_2k(X)fork≥0.

As before, elements with the Jacobi-Lie bracket[X0, Xm+1]vanish under the replacement operator.

The distributionC(V⁰)does not contain brackets of odd relative degree, so it can never contain bad brackets.

According to Lemma 12, the distributionC(V⁰)will contain vertical contributions. Given that the concern of configuration controllability is only on the horizontal subdistribution, we shall define the projection of C(V)to the horizontal subdistribution by

PC(V) = Id−A^triv

(C(V)). (4.46)

Definition 61 Define the distribution

C−1(V⁰) = [

k∈Z⁺

C₋^(k)1(V⁰), (4.47)

which consists of

C₋^(k)₁(V⁰)≡n

Ev(φ)(Be)|Be=Φ(B), Be ∈Br^k(X)\

Br₋1(X)o

. (4.48)

Proposition 17 C−1(V⁰)⊂ Codd(V) proof

Same as the proof in Proposition 16, except that now the parity is odd.

This proposition is used to characterize the space of bad brackets obtained from symmetric products. Just as the space of useful brackets was enlarged, the space of bad brackets is bigger, requiring more work to check configuration controllability. In the case of no potential,Z^lift= 0, we can improve our understanding. This is because the mappingΦe is no longer needed (it is the identity mapping since the setsXandX0map to the same objects underEv(φ)), and all of our analysis reduces to understanding the setBr(X).

Proposition 18 Assume that there does not exist a potential term in the drift vector field, i.e.,Z^lift = 0. If the space C(V⁰) is calculated on the zero section for equation (4.37) according to the algorithm, then its horizontal projection is equal toChor(V⁰), which is calculated on the zero section using equation (4.18), i.e., with the dissipation removed. Equivalently,

PC(V⁰)(0_x) =Chor(V⁰)(0_x). (4.49)

proof

In the case of no potential term,C(V⁰) =C(V). The proof in Lemma 12 reveals thatC(V) ⊂ M−1,0on the zero section. Further analysis of the proof demonstrates that theP−1terms come from the dissipation vector field only, and there is only one possible bracket that can lie inP0. This unique bracket is the same bracket that appears in theChor(V⁰)computations for the case without dissipation. Hence, the horizontal projection annihilates the new contributions due to dissipation and preserves the original contribution.

This proposition implies that the addition of dissipation does not affect the original configuration control- lability calculation in the case that there is no potential term in the drift vector field. When restricted to isotropic dissipation, this replicates the conclusion of Cort´es et al. [34].

4.3.1 Conditions for Configuration Controllability with Dissipation

Theorem 31 [85] The control system (4.18) is locally configuration accessible atx∈M ifPC(V⁰)(0_x) = H_0xE.

proof

The distributionC(V⁰)(0_x)⊂ M−1,0due to geometric homogeneity, c.f. Lemma 12. Do I need this proof?

Theorem 32 [85] Suppose thatXis such that every bad bracket inB ∈ Φ (Br_2l−1(X)), forl ≥ 0, has the property that

Ev_0x(ψ)(ψ(B)) = Xm a=1

ξ_aEv_0x(ψ)(C_a)

where C_a are good brackets in Φ (Br(X)) of lower degree thatP and ξ_a ∈ R for a = 1. . . m. Also, suppose that (4.37) is locally configuration accessible atx∈M. Then (4.37) is STLCC atx.

In the absence of a potential term,Z^lift= 0, a stronger statement holds.

Theorem 33 Suppose that the 1-homogeneous control system (4.37) has no potential term, i.e.,Z^lift = 0.

If the 1-homogeneous system with dissipation excluded, eq. (4.37), is STLCC atx ∈ M, then the original system (4.37) is STLCC atx∈E.

Again, this is a generalization of the configuration controllability results found in Cort´es et al. [34].