On the Hybrid Minimum Principle On Lie Groups and the Exponential Gradient HMP Algorithm

0.39 MB·English

by  Farzin Taringoo

ON THE HYBRID MINIMUM PRINCIPLE ON LIE GROUPS AND
THE EXPONENTIAL GRADIENT HMP ALGORITHM ∗
FARZIN TARINGOO† AND PETER E. CAINES†
Abstract. This paper provides a geometrical derivation of the Hybrid Minimum Principle
(HMP) for autonomous hybrid systems whose state manifolds constitute Lie groups (G,⋆) which
are left invariant under the controlled dynamics of the system, and whose switching manifolds are
3 defined as smooth embedded time invariant submanifolds of G. The analysis is expressed in terms
1 ofextremal(i.e. optimal)trajectoriesonthecotangent bundleofthestatemanifoldG. TheHybrid
0 Maximum Principle (HMP) algorithm introduced in [26] is extended to the so-called Exponential
2 Gradientalgorithm. TheconvergenceanalysisforthealgorithmisbasedupontheLaSalleInvariance
Principleandsimulationresultsillustratetheirefficacy.
n
a Key words. HybridMinimumPrinciple,RiemannianManifolds,LieGroups.
J

7 AMS subject classifications. 34A38,49N25,34K34,49K30,93B27

]
C 1. Introduction. Lie groups have been considered as configuration manifolds
formanydynamicalsystems,see[1,2,6,9]. Manycontrolproblems,mostlymechanical
O
systems, have been addressed as control systems defined on Lie groups, see [11,21].
.
h ThisclassofcontrolsystemsalsoincludesoptimalcontrolproblemsonLiegroups,see
t [7,10,18,19]. Theproblemofhybridsystemsoptimalcontrol(HSOC)hasbeenstudied
a
m and analyzed in many papers, see e.g. [8,15,23,26,27,30,33]. In particular, [4,26,27]
present an extension of the Hybrid Maximum Principle (HMP) for hybrid systems
[
and [26] presents an iterative algorithm which is based upon the HMP necessary

1
conditions for optimality. The HMP algorithm presented in [26] is a general search
v
method applicable to both autonomous and controlled hybrid systems. A geometric
2
8 version of the Pontryagin’sMaximum Principle for a general class of state manifolds
0 is given in [2,5,27].
1 In this paper, we generalize the analysis in [26] and [29] to obtaine the HMP
.
1 statementforleftinvarianthybridsystemsdefinedonLiegroups. Theproofgivenhere
0 canalsobe appliedtorightinvarianthybridsystemswherethe correspondingadjoint
3 variable will be different. This proof can be generalized to a class of autonomous
1
hybrid systems associated with time varying switching manifolds.
:
v Inthelastpartofthepaper,specificallytheHMPalgorithmin[26]isgeneralized
Xi to the so-called exponential gradient HMP algorithm by employing the notion of ex-
ponential curves on Lie groups. The convergenceanalysis for the proposedalgorithm
r
a is based on the LaSalle Invariance Principle.
2. Hybrid systems. In the following definition the standard hybrid systems
framework(see e.g.[8,26])isgeneralizedtothe casewherethecontinuousstatespace
is a smooth manifold, where henceforth in this paper smooth means C∞.
Definition 2.1. A hybrid system with autonomous discrete transitions is a five-
tuple
(2.1) H:={H=Q×M,U,F,S,J}
where:
Q={1,2,3,...,|Q|} is a finite set of discrete (valued) states (components) and M is
∗ThisworkwassupportedbyNSERCandAFOSR.
†DepartmentofElectricalandComputerEngineeringandCentreforIntelligentMachines,McGill
University,Montreal,Canada,{taringoo,[email protected]}.
1
2 FarzinTaringoo,PeterCaines
a smooth n dimensional Riemannian continuous (valued) state (component) manifold
with associated metric g .
M
H is called the hybrid state space of H.
U ⊂ Ru is a set of admissible input control values, where U is a compact set in Ru.
The set of admissible input control functions is I := (L [t ,t ),U), the set of all
∞ 0 f
bounded measurable functions on some interval [t ,t ),t <∞, taking values in U.
0 f f
F is an indexed collection of smooth, i.e. C∞, vector fields {f } , where f :
qi qi∈Q qi
M×U →TM is a controlled vector field assigned to each discrete state; hence each
f is continuous on M×U and continuously differentiable on M for all u∈U.
qi
S := {nk : γ ∈ Q×Q,1 ≤ k ≤ K < ∞,nk ⊂ M} is a collection of embedded time
γ γ
independent pairwise disjoint switching manifolds (except in the case where γ =(p,q)
is identified with γ′ = (q,p)) such that for any ordered pair γ = (p,q),nk is an open
γ
smooth, oriented codimension 1 submanifold of M, possibly with boundary ∂nk. By
γ
abuse of notation, we describe the manifolds locally by nk ={x:nk(x)=0,x∈Rn}.
γ γ
J shall denote the family of the state jump functions on the manifold M. For an
autonomous switching event from p ∈ Q to q ∈ Q, the corresponding jump function
is given by a smooth map ζ : M → M: if x(t−) ∈ S the state trajectory jumps to
p,q
x(t)=ζ (x(t−))∈M,ζ ∈J. Thenon-jumpspecialcaseisgivenbyx(t)=x(t−).
p,q p,q
We use the term impulsive hybrid systems for those hybrid systems where the contin-
uous part of the state trajectory may have discontinuous transitions (i.e. jump) at
controlled or autonomous discrete state switching times.
We assume:
A1: The initial state h :=(x(t ),q )∈H is suchthat x =x(t )∈/ S for all q ∈Q.
0 0 0 0 0 i
A (hybrid) input function u is defined on a half open interval [t ,t ),t ≤ ∞,
0 f f
where further u ∈ I. A (hybrid) state trajectory with initial state h and (hybrid)
0
input function u is a triple (τ,q,x) consisting of a strictly increasing sequence of
times (boundary and switching times) τ = (t ,t ,t ,...), an associated sequence of
0 1 2
discrete states q = (q ,q ,q ,...), and a sequence x(·) = (x (·),x (·),x (·),...) of
0 1 2 q0 q1 q2
absolutely continuous functions x : [t ,t ) → M satisfying the continuous and
qi i i+1
discrete dynamics given by the following definition.
Definition 2.2. The continuous dynamics of a hybrid system H with initial
condition h = (x ,q ), input control function u ∈ I and hybrid state trajectory
0 0 0
(τ,q,x) are specified piecewise in time via the mappings
(2.2) (x ,u):[t ,t )→M×U, i=0,...,L, 0<L<∞,
qi i i+1
where x (.) is an integral curve of f (.,u(.)):M×[t ,t )→TM satisfying
qi qi i i+1
x˙ (t)=f (x (t),u(t)), a.e.t∈[t ,t ),
qi qi qi i i+1
where x (t ) is given recursively by
qi+1 i+1
(2.3) x (t )= lim ζ (x (t)), h =(q ,x ),t<t .
qi+1 i+1 t↑t− qi,qi+1 qi 0 0 0 f
i+1
The discrete autonomous switching dynamics are defined as follows:
For all p,q, whenever an admissible hybrid system trajectory governed by the con-
trolled vector field f meets any given switching manifold n transversally, i.e.
p p,q
fp(x(t−s ),t−s) ∈/ Tx(t−s)S, there is an autonomous switching to the controlled vector
field f , equivalently, discrete state transition p → q, p,q ∈ Q. Conversely, any
q
autonomous discrete state transition corresponds to a transversal intersection.
HybridMinimumPrincipleOnLieGroupsandExponential GradientHMPAlgorithm 3
A system trajectory is not continued after a non-transversal intersection with a
switching manifold. Given thedefinitions andassumptionsabove, standardarguments
givetheexistenceanduniquenessofahybrid statetrajectory(τ,q,x), withinitialstate
h ∈H and input function u∈I, up to T, defined to be the least of an explosion time
0
or an instant of non-transversal intersection with a switching manifold.
We adopt:
A2: (Controllability) For any q ∈ Q, all pairs of states (x ,x ) are mutually
1 2
accessible in any given time period [tˆ,t´] ⊆ [t ,t ], via the controlled vector field
0 f
x˙ (t)=f (x (t),u(t)), for some u∈I =(L [t ,t ],U).
q q q ∞ 0 f
A3: {l } ,isafamilyoflossfunctions suchthatl ∈Ck(M×U;R ),k ≥1,
qi qi∈Q qi +
and h is a terminal cost function such that h∈Ck(M;R ),k ≥1.
+
Henceforth, Hypotheses A1-A3 will be in force unless otherwise stated. Let L
be the number of switchings and u∈I then we define the hybrid cost function as
L ti+1
J(t ,t ,h ;L,u):= l (x (s),u(s))ds+h(x (t )),
0 f 0 qi qi qL f
i=0Zti
X
(2.4) t =t <T,u∈I,
L+1 f
where we observe the conditions above yield J(t ,t ,h ;L,u)<∞.
0 f 0
Definition 2.3. For a hybrid system H, given the data (t ,t ,h ;L), the Bolza
0 f 0
HybridOptimalControlProblem (BHOCP)isdefinedastheinfimizationofthehybrid
cost function J(t ,t ,h ;L,u) over the hybrid input functions u∈I, i.e.
0 f 0
(2.5) Jo(t ,t ,h ;L)=inf J(t ,t ,h ;L,u).
0 f 0 u∈I 0 f 0
Definition 2.4. AMayer HybridOptimalControlProblem (MHOCP)is defined
as the special case of the BHOCP where the cost function given in (2.4) is evaluated
only on the terminal state of the system, i.e. l =0, i=1,...,L.
qi
In general, different control inputs result in different sequences of discrete states
of different cardinality. However, in this paper, we shall restrict the infimization to
be over the class of control functions, generically denoted U ⊂I, which generates an
a priori given sequence of discrete transition events.
We adoptthe followingstandardnotationandterminology,see[12]. Thetime depen-
dentflowassociatedtoadifferentiabletimeindependentvectorfieldf isamapΦ
qi fqui
satisfying (fu(.) is used here for brevity instead of f (.,u(t)) since the calculations
qi qi
are performed with respect to a given control u):
Φ :[t ,t )×[t ,t )×M→M, (t,s,x)→Φ(t,s)(x):=Φ ((t,s),x)∈M,
fqui i i+1 i i+1 fqui fqui
(2.6)
where
(t,s) (s,s)
(2.7) Φ :M→M, Φ (x)=x,
fu fu
qi qi
4 FarzinTaringoo,PeterCaines
d
(2.8) Φ(t,s)(x)| =f Φ(t,s)(x(s)) , t,s∈[t ,t ).
dt fqui t qi fqui i i+1
(cid:0) (cid:1)
We associate TΦ(t,s)(.) to Φ(t,s) :M→M via the push-forwardof Φ(t,s).
fu fu fu
qi qi qi
(2.9) TΦ(t,s) :T M→T M.
fqui x Φ(ftqu,is)(x)
Following [12], the corresponding tangent lift of fu(.) is the time dependent vector
qi
field fT,u(.)∈TTM on TM
qi
d
(2.10) fT,u(v ):= | TΦ(t,s)(v ), v ∈T M,
qi x dt t=s fqui x x x
which is given locally as
n
∂ ∂fu,i ∂
(2.11) fT,u(x,v )= fu,i(x) +( qi vj) ,
qi x qi ∂xi ∂xj ∂vi
" #
i,j=1
and TΦ(t,s)(.) is evaluated on v ∈ T M, see [12]. The following lemma gives the
fu x x
qi
relation between the push-forward of Φ(t,s) and the tangent lift introduced in (2.11).
For simplicity and uniformity of notatiofnqi, we use f instead of fu.
qi qi
Lemma 2.5 ( [5], [29]). Consider f (x,u) as a time dependent vector field on M
q
and Φ(t,s) as the corresponding flow. The flow of fT,u, denoted by Ψ:I×I×TM→
fq q
TM, I =[t ,t ], satisfies
0 f
Ψ(t,s,(x,v))=(Φ(t,s)(x),TΦ(t,s)(v))∈TM,(x,v)∈TM.
fq fq
ForageneralRiemannianmanifoldM,theroleoftheadjointprocessλisplayedbya
trajectoryinthecotangentbundle ofM,i.e. λ(t)∈T∗ M. Similartothe definition
x(t)
ofthe tangentlift we define the cotangent lift whichcorrespondsto the variationofa
differential form α∈T∗M along x(t), see [31]:
(2.12) fT∗,u(α ):= d| T∗Φ(t,s)−1(α ), α ∈T∗M,
q x dt t=s x fqu x x x
where here x=x(t)=Φ(t,s)(x(s)) .
fu
q
Similar to (2.11) in the local coordinates (x,p) of T∗M, we have
n
(2.13) fT∗,u(x,p)= fu,i(x) ∂ −(∂ftu,ipj) ∂ .
q q ∂xi ∂xj ∂pi
" #
i,j=1
The mapping T∗ is the pull back defined on the differential forms on the cotangent
x
bundle of M. The covector α is an element of T∗M, see [31]. The following lemma
x x
givestheconnectionbetweenthecotangentliftdefinedin(2.12)anditscorresponding
flow on T∗M.
Lemma 2.6 ( [5], [29]). Consider f (x(t),u(t)) as a time dependent vector field
q
on M, then the flow Γ:I ×I×T∗M→T∗M, satisfies (I =[t ,t ])
0 f
(2.14) Γ(t,s,(x,p))=(Φ(t,s)(x),(T∗Φ(t,s))−1(p)),(x,p)∈T∗M,
fq x fq
HybridMinimumPrincipleOnLieGroupsandExponential GradientHMPAlgorithm 5
and Γ is the corresponding integral flow of fT∗,u. The mapping (T∗Φ(t,s))−1 =
q x fq
T∗Φ(−t,s)) is defined as a pull back of Φ−1 which its existence is guaranteed since
x fq
Φ : M → M is a diffeomorphism, see [5]. For a given trajectory λ(t) ∈ T∗M, its
variationwith respectto time, λ˙(t), is anelement ofTT∗M. The vectorfield defined
in(2.12)is the mappingfT∗,u :T∗M→TT∗M, fromλ(t)∈T∗M to λ˙(t)∈TT∗M.
q
Proposition 2.7 ( [5], [29]). Let f (.,u) : I ×M → M be the time dependent
q
vector field giving rise to the associated pair fT,u,fT∗,u; then along an integral curve
q q
of f (.,u) on M
q
(2.15) hΓ,Ψi:I →R,
is a constant map, where Γ is an integral curve of fT∗,u in T∗M and Ψ is an integral
q
curve of fT,u in TM.
q
Elementary Control and Tangent Perturbations.
Consider the nominal control u(.) and define the perturbed control as follows:
u t1−ǫ≤t≤t1
(2.16) u (t,ǫ):=u (t,ǫ)= 1 ,
π(t1,u1) π u(t) elsewhere
(cid:26)
where u ∈U,0≤ǫ.
1
Associated to u (.) we have the corresponding state trajectory x (t,ǫ) on M. It
π π
maybeshownthatundersuitablehypothesesofthedifferentiabilityofx withrespect
π
to ǫ at the switching times, then lim x (t,ǫ) = x(t) uniformly for t ≤ t ≤ t ,
ǫ↓0 π 0 f
see [16] and [20]. However in this paper we employ the same hypotheses of the
differentiability of x before and after switching times but must accommodate the
π
factthattheremaybe adiscontinuityof dxπ(t)| atanyswitchingtime t . Theflow
dǫ ǫ=0 i
resulting from the perturbed control is defined by:
Φ(t,s),x(ǫ):[0,τ]→M, x∈M,t,s∈[t ,t ],τ ∈R ,
π,fq 0 f +
(2.17)
whereΦ(t,s),x(.)istheflowcorrespondingtotheperturbedcontrolu (t,ǫ),i.e. Φ(t,s),x(ǫ):=
π,fq π π,fq
Φ(t,s) (x(s)). The following lemma gives the formula of the variation of Φ(t,s),x(.)
fquπ(t,ǫ) π,fq
at ǫ=0+. Recall that the point t1 ∈(t ,t ) is called Lebesgue point of u(.) if, ( [2]):
0 f
1 s
(2.18) lim |u(τ)−u(t1)|dτ =0.
s→t1 |s−t1|Zt1
For a u ∈ L ([t ,t ],U), u may be modified on a set of measure zero so that all
∞ 0 f
points are Lebesgue and the value function is unchanged (see [24], page 158).
Lemma 2.8 ( [5]). For a Lebesgue time t1, the curve Φ(t,s),x(ǫ) : [0,τ] → M is
π,fq
differentiable at ǫ=0 and the corresponding tangent vector dΦ(t1,s),x| is given by
dǫ π,fq ǫ=0
(2.19) d Φ(t1,s),x| =f (x(t1),u )−f (x(t1),u(t1))∈T M.
dǫ π,fq ǫ=0 q 1 q x(t1)
6 FarzinTaringoo,PeterCaines
Thetangentvectorf (x(t1),u )−f (x(t1),u(t1))iscalledthe elementary pertur-
q 1 q
bation vector associatedto the perturbed controlu at(x(t),t). The displacementof
π
the tangent vectors at x∈M is by the push-forwarddefined on the vector field f .
q
Proposition 2.9 ( [5]). Let Ψ:[t1,t ]→TM be the integral curve of fT,u with
f q
the initial condition Ψ(t1)=[f (x(t1),u )−f (x(t1),u(t1))]∈T M, then
q 1 q x(t1)
(2.20) d Φ(t,t1),x| =Ψ(t), t∈[t1,t ].
dǫ π,fq ǫ=0 f
By the result above and Lemma 2.5 we have
(2.21)d Φ(t,t1),x| =TΦ(t,t1)([f (x(t1),u )−f (x(t1),u(t1))])∈T M.
dǫ π,fq ǫ=0,x(t)∈M fq q 1 q x(t1)
3. Control Systems on Lie Groups. In this section we introduce control
systemsonLie groupsandthenextendthe definitionofhybridsystemsaboveto that
of hybrid systems defined on Lie groups.
3.1. Lie Groups and Lie Algebras. Definition 3.1. A group(G,⋆)is called
a Lie Group if, (see [32]):
(1): G is a smooth manifold,
(2): The group operations are smooth.
(The group operations are multiplication and inversion, ∀g ,g ,g ∈ G, g ⋆g ∈
1 2 1 2
G,g−1 ∈Gsuchthat multiplication is associative andan identity andinverses exist.).
In this paper it is assumed that the continuous part of the hybrid system evolves on
a Lie group G.
Definition 3.2. A Lie Algebra V is a real vector space endowed with a bilinear
operation [.,.]:V ×V →V such that (see [12,32]):
(1): ∀ζ,η ∈V, [ζ,η]=−[η,ζ]
(2):[ζ,[η,γ]]+[η,[γ,ζ]]+[γ,[ζ,η]]=0 ζ,η,γ ∈V
The Lie algebra L of a Lie group G is the tangent space at the identity element
e with the associatedLie bracket defined on the tangent space of G, i.e. L=T G. A
e
vector field X on G is called left invariant if
(3.1) ∀g ,g ∈G, X(g ⋆g )=TL X(g ),
1 2 1 2 g1 2
where L : G →G, L (h)= g⋆h, TL : T G→T G which immediately imply
g g g g2 g⋆g2
X(g⋆e)=X(g)=TL X(e).
g
Definition 3.3. Corresponding to a left invariant vector field X, we define the
exponential map as follows:
(3.2) exp:L→G, exp(tX(e)):=Φ(t,X),t∈R,
whereΦ(t,X)isthesolutionofg˙(t)=X(g(t))withtheboundaryconditiong(0)=
e. Thefollowingtheoremgivestheflowofaleftinvariantvectorfieldwithanarbitrary
initial state g ∈G.
HybridMinimumPrincipleOnLieGroupsandExponential GradientHMPAlgorithm 7
Theorem 3.4 ( [32]). Let G be a Lie group with the corresponding Lie algebra
L, then for a left invariant vector field X
(3.3) Φ(t,X,g)=L ◦exp(tX(e)),
g
where Φ(t,X,g) is the flow of X starting at g ∈G.
A left invariant control system defined on a given Lie group G is defined as follows:
(see [12,14,18])
(3.4) g˙(t)=f(g(t),u)=TL f(e,u), g(t)∈G,u∈Ru,
g(t)
where f(g(t),u)is a left invariantvectorfieldonG. Similar to left invariantsystems,
right invariant systems are defined. In this paper we only consider hybrid systems
where the associated vector fields are left invariant, however the analysis can also be
applied to right invariant hybrid systems.
3.2. Left Invariant Optimal Control Systems. A Bolza left invariant opti-
mal control problem is an optimal control problem where
(i): the ambient state manifold M is a Lie group G,
(ii): the correspondingvectorfieldf is a leftinvariantvectorfield definedonGsuch
q
that for any given u∈U
(3.5) f (.,u(.)):G×[t ,t ]→TG,
q 0 f
and
(iii): the cost function is defined by
tf
(3.6) J := l (g(s),u(s))ds, u∈U,
q
Zt0
wherel (g(s),u(s))isassumedtobeleftinvarianti.e. l (L g(s),u(s))=l (g(s),u(s)).
q q h q
In general, a Bolza problem can be converted to a Mayer problem using an auxiliary
state variable in the dynamics,see [26]and [5]. The following lemma gives the equiv-
alence of a Bolza problem defined on a Lie group G and its Mayer extension.
Lemma 3.5. Consider a left invariant Optimal Control Problem (OCP) defined
on a Lie group G with the following dynamics and cost function:
(3.7) g˙(t)=f(g(t),u), g(t)∈G,u∈Ru,
tf
(3.8) J = l(g(s),u(s))ds.
Zt0
Then the Mayer problem associated to the optimal control problem above is defined on
the Lie group G×R and the corresponding dynamics are left invariant.
Proof. ThestatespaceequationoftheMayerproblemcorrespondingtotheBolza
problem is as follows:
g˙ f(g(t),u(t))
(3.9) = =F(g¯(t),u),
z˙ l(g(t),u(t))
(cid:18) (cid:19) (cid:18) (cid:19)
where g¯ = (g,z), g ∈ G,z ∈ R. The group action defined on G×R is given as
follows:
(3.10) (g ,z )¯⋆(g ,z )=(g ⋆g ,z +z ),
1 1 2 2 1 2 1 2
8 FarzinTaringoo,PeterCaines
where ⋆ corresponds to the group action of G and ¯⋆ is the group action of G×R.
Since G is a Lie group it follows that (G×R,¯⋆) is also a Lie group. It remains to
show that F(g¯,u) is left invariant. The left translation on G×R is defined by
(3.11) L (h¯)=(L h,z +z ), g¯=(g,z ),h¯ =(h,z ),
g¯ g g h g h
therefore
TL F(h¯,u)=TL f(h,u)⊕l(L h,u)=f(g⋆h,u)⊕l(g⋆h,u)=F(g¯⋆¯h,u),
g¯ g g
(3.12)
whichshowsthat F(g¯,u)is left invariantsince f(g,u)andl(g,u)respectivelyareleft
invariant.
4. The Pontryagin Minimum Principle on Lie Groups. Optimal control
problems on Lie groups have been addressed in [9,10,18,19]. In this section we re-
view the Minimum Principle results presented in [19] for optimal control problems
defined ona Lie groupG. As shownin[18], the left translationgivesanisomorphism
between TGandG×L. Since Lg−1 maps g to e, TLg−1 :TgG→TeG=L is the cor-
responding isomorphism. This statement also holds between T∗G and G×L∗ where
L∗ is the dual space of the Lie algebra L. The corresponding isomorphism is given
by T∗L : T∗G → T∗G = L∗. We use the equivalence T∗G ≈ G×L∗ associated to
g g e
the isomorphism above to construct Hamiltonian functions on Lie groups.
Hamiltonian Systems on T∗M and T∗G.
By definition, for an optimal control problem defined on an n dimensional differ-
entiable manifold M, a Hamiltonian function is defined as a smooth function H :
−→
T∗M×U → R, see [2,18]. The associated Hamiltonian vector field H is defined as
follows (see [2]):
−→
(4.1) ω (.,H)=dH, λ∈T∗M,
λ
where ω is the symplectic form defined on T∗M which is locally written as follows:
λ
n
(4.2) ω = dζ ∧dx ,
λ i i
i=1
X
and (ζ,x) is the local coordinate representation of λ in T∗M.
The Hamiltonian system of the ODE corresponding to H is
−→
(4.3) λ˙ = H(λ),
where locally we have
x˙ = ∂H, i=1,...,n,
(4.4) i ∂ζi
ζ˙ =−∂H, i=1,...,n.
( i ∂xi
Similar to the case of Hamiltonian systems on smooth manifolds we can define
Hamiltonianfunctions for left invariantvectorfields onthe cotangentbundle ofa Lie
groupG. ThisisdonebyusingtheisomorphismbetweenL∗×GandT∗Gintroduced
above which is denoted by I:
(4.5) (1) I(X,g)∈T∗G, X ∈L∗,g ∈G,
HybridMinimumPrincipleOnLieGroupsandExponential GradientHMPAlgorithm 9
(4.6) (2) I(.,g):=Ig :=T∗Lg−1 :L∗ →Tg∗G is a linear isomorphism.
A Hamiltonian function for a left invariant vector field X on G is defined by
(4.7) HX(g,λ):=hλ,X(e)i=hλ,TLg−1X(g)i, λ∈L∗.
TheprecedingidentificationusesTT∗G≃T(G×L∗)=(G×L)×(L∗×L∗),therefore
the tangentvector at (g,λ)∈G×L∗ is anelement of L×L∗ denoted by T =(X,γ).
ThesymplecticformωalongagivencurveΓ(t)∈T∗Gsatisfiesthefollowingequation,
see [2,18]:
ω (T (Γ),T (Γ))=hγ (t),X (t)i−hγ (t),X (t)i−hλ(t),[X (t),X (t)]i,
Γ 1 2 2 1 1 2 1 2
(4.8)
whereT =(X ,γ ). SimilartoHamiltoniansystemsonT∗M,theHamiltonianvector
i i i
−→
field H on G×L∗ satisfies the following equation
−→
(4.9) dH =ω (.,H).
Γ(t)
The following theorem gives the Minimum Principle for optimal control problems
defined on Lie groups.
Theorem 4.1 ( [18]). For a left invariant optimal control problem defined by
(3.5) and (3.6), along the optimal state and optimal control go(t),uo(t), there exists
a nontrivial adjoint curve λo(t)∈L∗ such that the following equations hold:
(4.10) H(go(t),λo(t),uo(t))≤H(go(t),λo(t),u), ∀u∈U,
and locally
dgo ∂H
(4.11) =TL ,
dt go(t) ∂λ
(cid:0) (cid:1)
dλo
(4.12) =−(ad)∗ (λo(t)),
dt ∂∂Hλ
where H(g,λ,u):=hλ,TLg−1f(g,u)i.
For each η,ζ ∈L we have the following definition
(4.13) ad :L→L, ad η =[ζ,η].
ζ ζ
For each ζ,η ∈L,γ ∈L∗, ad∗ is defined by
(4.14) had∗(γ),ηi:=hγ,ad (η)i.
ζ ζ
For more information about the definition above see [1,32]. It should be noted that,
in general, i.e. not necessarily left invariant vector fields, for a Hamiltonian function
defined on G×L∗, the integral curve of the Hamiltonian vector field, i.e. (4.11) and
(4.12), satisfies the following equations (see [18]):
dg ∂H
(4.15) =TL ,
dt g(t) ∂λ
(cid:0) (cid:1)
dλ ∂H
(4.16) =−T∗L −(ad)∗ (λ(t)).
dt g(t) ∂g ∂∂Hλ
(cid:0) (cid:1)
Since the tangent space of T∗G is identified with L×L∗, by the definition of the
Hamiltonian H :G×L∗ →R, it is noted that ∂H ∈L∗∗ =L and ∂H ∈T∗G.
∂λ ∂g g
10 FarzinTaringoo,PeterCaines
5. Hybrid Systems on Lie Groups. The definition of hybrid systems on Lie
groupsisthespecializationofthatofhybridsystemsgiveninDefinition2.1wherethe
ambient manifold M is replaced by a Lie group G. Here we only consider a hybrid
system consisting of two different discrete states with the associated left invariant
vector fields f ,f , as follows:
q0 q1
(5.1) g˙(t)=f (g(t),u(t)), g˙(t)=f (g(t),u(t)), u(t)∈U.
q0 q1
The switching manifold S associatedto the autonomous discrete state change is con-
sidered to be a submanifold of G which is by definition a regular Lie subgroup. The
hybrid cost function is defined by
1 ti+1
(5.2) J = l (g (s),u(s))ds+h(g (t )), u∈U,
qi qi+1 qL f
i=0Zti
X
wherel ,i=0,1areleftinvariantsmoothfunctionsonG. Similartotheproofin[29],
i
we apply the needle control variation in two different steps. First, the control needle
variation is applied after the optimal switching time so there is no state propagation
alongthe statetrajectorythroughthe switchingmanifold. Second,the controlneedle
variationisappliedbeforetheoptimalswitchingtime. Inthiscasethereexistsastate
variation propagation through the switching manifold, see [26], [29].
Recalling assumption A2 in the Bolza problem and assuming the existence of
optimal controls for each pair of given switching state and switching time, let us
define a function v : G× (t ,t ) → R for a hybrid system with one autonomous
0 f
switching, i.e. L=1, as follows:
(5.3) v(g,t)=inf J(t ,t ,h ,u),
u∈U 0 f 0
where
g =Φ(t−,t0)(g )∈S ⊂G.
fq0 0
5.1. Non-Interior Optimal Switching States. In general the hybrid value
function for a Mayer type problem attains its minimum on the boundary of the at-
tainable switching states on the switching manifold and hence is not differentiable.
In this case the discontinuity ofthe adjoint processin the HMP statementis givenin
terms of a normal vector at the switching time on the switching manifold. In order
tohaveanormalvectorN ontheswitchingmanifoldweneedtodefine aRiemannian
metric on G. A left invariant Riemannian metric G on (G,⋆) satisfies the following
equation.
(5.4) G(g)(X,Y)=G(h⋆g)(TL (X),TL (Y)),
h h
where X,Y ∈T G. Let us consider I as an inner product in L where I:L×L→R.
g
the following theorem gives a Riemannian metric with respect to an inner product I
defined on L.
Lemma 5.1 ([12]). The inner product I on L determines a smooth left invariant
Riemannian metric G on G as follows:
(5.5) G(g)(X,Y):=I(TLg−1X,TLg−1Y),