REMARKS ON THE TIME-OPTIMAL CONTROL OF A CLASS OF
HAMILTONIAN SYSTEMS∗
Eduardo D.Sontag
SYCON-Rutgers Center for Systems and Control
Department of Mathematics,Rutgers University,New Brunswick,NJ08903
ABSTRACT
This paper introduces a subclass of Hamiltonian control systems motivated by me-chanical models.Some problems of time-optimal control are studied,and results on singular trajectories are obtained.
1Introduction
We deal here with time-optimal control and some other questions for multivariable sys-tems for which a certain Hamiltonian structure is prent.The main results characterize regions of the state space where
用格外造句singular trajectories cannot exist,and provide high-order conditions for optimality.This work was motivated by previous studies([5],[6]) on robotic manipulators.Pursuing further the ideas expresd in[6],we identify a class of Hamiltonian systems which includes many mechanical models,particularly tho in which the energy is at most quadratic in the momenta.
This work is only preliminary,in that more questions are left open than are answered. We believe that the further study of the class of systems identified here,–or of some variant,–for instance in the direction of realization theory and algebraic properties, would be very fruitful and might be the“right”level of generality for many results on nonlinear mechanical systems(as oppod to dealing,for example,with the class of all Hamiltonian systems).
2A Class of Hamiltonian Systems
The Appendix reviews some of the basic terminology about symplectic manifolds and Hamiltonian vectorfields.We start here with the definition of Hamiltonian control system as in[4],but not introducing outputs.The systems that we consider are of the
type
˙x(t)=f(x(t))+
m
i=1
u i(t)g i(x(t))(Σ)
where f,g1,...,g m are smooth vectorfields on the n−dimensional manifold M.A control u(·)is a measurable locally esntially bounded function u:[0,T]→I R m.We will only ∗Proc.IEEE Conf.Decision and Control,Tampa,Dec.1989,IEEE Publications,1989,pp.217-221.
be interested in Hamiltonian systems.The system(Σ)is called Hamiltonian if M is a symplectic manifold and there exist smooth functions
H0,H1,···,H m:M→I R
so that the vectorfields ofΣare tho associated to the functions:
f=X H
0,g i=X H
i
,
It may be of interest to weaken this definition and allow f to be only locally Hamilto-nian as done in[4],or to consider what are sometimes called“Poisson systems”(
[3]).But we wish here to instead restrict this class further by imposingfive conditions, thefirst four of which follow and the last of which will be given later.We assume from now on:
(A1)2m=n
(A2){H i,H j}=0∀i,j≥1
(A3)dH1∧dH2∧···∧dH m=0everywhere
(A4){H i,{H j,{H k,H0}}}=0∀i,j,k≥1
All the axioms are satisfied,for instance,when dealing with robotic manipulators for which each link can be parately controlled.Condition(3)can be dropped for a few of the results,but is esntial othe
rwi;it corresponds to the requirement that each control act freely on one of the degrees of freedom of the system.It would be of great interest to study how far one can go without this assumption,which would allow studying failure modes.
2.1A natural gradation
We introduce the following ts of smooth functions on M.Let F k:=0for k<0,and for k≥0:
F k:={F:M→I R|ad k+1H F=0}
where H is the t of functions{H i,},and where ad l H F is the linear span of the t of functions of the form
ad a
辣椒育苗1···ad a
文明上网黑板报
l
F
over all possible quences of elements a1,···,a l in H.
Note that,directly from the definition of the F k’s,it follows that{F i,F j}⊆F i+j for all i,j.This is becau for each smooth functions F∈F i and G∈F j,and for each quence a1,···,a i+j+1of elements of H,repeated applications of the Jacobi identity give that
ad a
1···ad a
i+j+1
{F,G}
is a linear combination of terms of the form
{ad b1···ad b r F,ad c1···ad c s G}
with the b1,···,b r,c1,···,c s in H and r+s=i+j+1.This last equality implies that either r>i or s>j,which in turn implies that each such term must vanish.Under the assumptions(A1)to(A4),one can in fact prove an inclusion in F i+j−1:
Lemma2.1The following properties hold:
1.0=···=F−1⊆F0⊆F1⊆···
2.H i∈F0,
3.H0∈F2
4.{F i,F j}⊆F i+j−1for all i,j.
Proof.Thefirst of this properties is immediate from the definition of the F k’s,while the cond and third follow from(A2)and(A4)respectively.To prove the last property, note that by the Darboux/Lie Theorem and using property(A4),locally we may assume that each H i=q i(in some t of canonical coordinates),which implies that F k consists of tho functions that can be expresd as polynomials of degree at most k on the p i’s (and arbitrary on the other coordinates).From this,property(4)follows.
Note that using canonical coordinates as in the above proof,property(A4)says that H0is at most quadratic on the“generalized momenta”.
We now introduce
E:=Poisson algebra generated by H i,
The algebra E completely characterizes the control theoretic properties of the system, since the associated vectorfields do.If we now let
E i:=
F i∩E
then E is graded in the same manner as the t of all functions,in the n that all the properties in the above Lemma hold with E l replaced for F l.
2.2Strong Accesibility Condition
Locally,each F∈F0is a smooth function of H1,···,H m,
F(x)=h(H1(x),···,H m(x)),
again by the characterization in terms of degree with respect to momenta.Thus,by (4)in the Lemma,also for each i,j≥1it holds that{H i,{H0,H j}}is a function of H1,···,H m.
We let
A ij:={H i,{H0,H j}}
and let A be the m by m matrix{A ij}.The remaining axiom is:
(A5)rank A=m everywhere
Note that,for mechanical systems,A is typically the inver of the inertia matrix.Under this condition(which we assume from now on):
Lemma2.2The following holds everywhere on M:
dH1∧···∧dH m∧d{H0,H1}∧···∧d{H0,H m}=0
Proof.In local canonical coordinates,we can write
{H0,H j}=b j(q)+
浪妞
m
i=1
A ij(q)p i
(for some functions b j)so when computing the expression in the coordinates there results
(det A)dp1∧···∧dp m∧dq1∧···∧dq m
and the determinant is nonzero by hypothesis.
Equivalently,the t of vectorfields
{g1,···,g m,[f,g1],···,[f,g m]}
forms afield of n-frames on M,which implies that the original system is strongly acces-sible.
3Time Optimal Control
We now restrict controls to satisfy|u i(t)|≤1for all i,t.Other constraints ts could be ud,in which ca the material to follow would have to be modified accordingly.
The time-optimal problem is:given states x1,x2,find a controlled trajectory that steers x1to x2in as small time as possible.A pair(u(·),x(·))of functions defined on some interval I=[0,T]and satisfying the
equations ofΣon that interval,will be called an optimal trajectory if u is a control with|u i(t)|≤1for all i,t and for each other solution (u (·),x (·))on any interval[0,T ]for which x(0)=x (0)and x(T)=x (T )necessarily T ≥T.
Among the questions one wishes to understand for such problems are tho dealing with the existence of bang-bang and/or singular optimal trajectories,the possibility of Fuller-like phenomena,and the feedback synthesis of controls.Lie-theoretic techniques have proved very uful,but mostly for single-input systems.Our goal here is to show how some conclusions can be drawn for multiple input systems if(A1)-(A5)are assumed.
3.1Pontryagin’s Maximum Principle
In order to elegantly state the PMP,we introducefirst the Hamiltonian extension(or lift)Σ∗of the systemΣ([4]for more details).
The state space ofΣ∗is T∗M en as symplectic manifold.Note that in our applica-tions,M is itlf a symplectic manifold,but for now we do not u that fact.
With each vectorfield X on M one associates a Hamiltonian
H X:T∗M→I R
as follows:
谜语游戏大全
H X(λ,x):= λ,X(x) ,x∈M,λ∈T∗x M
and the notation(λ X)(x)is ud instead of H X(λ,x).
When M happens to be itlf a symplectic manifold and X,Y are Hamiltonian vector fields,X=X H,Y=X K,and H,K:M→I R are smooth,one has the formula
{λ X H,λ X K}=λ X{H,K}(1) where the brackets on the left indicate Poisson product on T∗M but in the right they indicate the Poisson bracket on M.In particular,given the vectorfields f,g1,···,g m definingΣ,we introduce the Hamiltonians
ϕ:=λ f,γi:=λ g i,
From(A2)and(1),it follows that
{γi,γj}=0(2) for all i,j.
Now the Hamiltonian extension ofΣis defined as the control system
˙ξ=f∗(ξ)+m
i=1u i g∗
i
(ξ)(Σ∗)
预算管控where f∗(respectively,g∗i)is the Hamiltonian vectorfield associated toϕ(respectively γi)under the symplectic structure of T∗M.
In local coordinates(x,λ)for T∗M,the equations of the Hamiltonian extension are the usual ones
˙x=f(x)+
m
i=1
u i g i(x)
˙λ=−(f+m
i=1
u i g i) xλ.
Pontryagin’s Maximum Principle(PMP)states that if(u(t),x(t)),t∈I is an optimal trajectory then it must be an here exists an absolutely continuous curve ξ:I→T∗M so that for each t:
ξ(t)=(x(t),p(t))
with p(t)∈T∗
入党积极分子总结x(t)M,the pairξ,u satisfies the equations ofΣ∗,and the following properties
hold:
1.p(t)=0for all t
2.for almost all t,
max
|v i|≤1
m
i=1
v iγi(t)=
m
i=1
u i(t)γi(t)
3.ϕ+ m
i=1
u iγi is constant and≥0along the trajectory.
4Singular Trajectories
Note that for each t and i for whichγi(t)=0,property(2)in the PMP implies that
u i(t)=signγi(t).
If the t of zeroes ofγi is discrete in I,it follows that u i is piecewi constant,equal to ±1,withfinitely many switchings.We say in that ca that the control is u i-bang-bang. The worst ca in this regard is whenγi≡0on I;we say then that the trajectory (x(·),u(·))is u i-singular.We now want to say something about singularity.
If H is any Hamiltonian on M,we may consider the valueα(t)of the Hamiltonian λ X H evaluated along any trajectory ofΣ∗.A calculation shows that its derivative
becomes
˙α(t)={α(t),ϕ(t)}+
归来饱饭黄昏后m
i=1
u(t){α(t),γi(t)}(3)
(bracket here is in T∗M),or equivalently,using equation(1),
˙α(t)=λ X{H,H
0}+
m
i=1
u(t)λ X{H,H
m}
(4)
where the right hand side functions are also evaluated alongξ(·).In particular,taking H to be any of the H i’s,i≥1,and using(A2),there results
˙γi=λ X{H
i,H0}
={γi,ϕ}(5) along any trajectory.
Now obrve that along an extremal trajectory and corresponding lift there cannot be anyτwhere
γ1(τ)=···=γm(τ)={γ1,ϕ}(τ)=···{γm,ϕ}(τ)=0
becau the existence of a nonzeroλ∈T∗
x(τ)
M so that
λ g1(x)=···=λ g m(x)=λ [f,g1](x)=···=λ [f,g m](x)=0
would contradict Lemma(2.2).
Proposition4.1There cannot be any optimal trajectory which is u i-singular for all i. Moreover,if for some i an extremal is u j-singular for all j=i,then u i is bang-bang.
