Transcript
7/28/2019 bavo.pdf
1/62
Geometric Control theoryBavo Langerock
Department of Mathematical Physics and AstronomyGhent University
21st INTERNATIONAL WORKSHOP onDIFFERENTIAL GEOMETRIC METHODS IN THEORETICAL MECHANICS
Madrid, 2006
7/28/2019 bavo.pdf
2/62
GEOMETRIC CONTROL 1
Plan of the talk
Examples of control systems
Regularity assumptions
What are interesting problems ?
Families of vector fields: accessible sets: topology, optimality, smoothness,controllability
Lie determined systems
The maximum principle
7/28/2019 bavo.pdf
3/62
GEOMETRIC CONTROL 2
Control Theory: examples of open systems
Open systems: influenceable x = f(x, u)
future is not determined by current state
dynamics determined by steering devices
the system
state space M
u1
u2
u3
7/28/2019 bavo.pdf
4/62
GEOMETRIC CONTROL 3
A (simplified) car
state space is (x,y,) SE(2)
suppose that we can(A) steer the car such that we can move forward and backward with velocityv IR
(B) rotate the car with angular velocity IR...
7/28/2019 bavo.pdf
5/62
GEOMETRIC CONTROL 3
A (simplified) car
state space is (x,y,) SE(2)
suppose that we can(A) steer the car such that we can move forward and backward with velocityv IR
(B) rotate the car with angular velocity IR...
x = v cos y = v sin
=
Question: Can we reach any point in the plane ?
How to concatenate motions to reach these points ?
7/28/2019 bavo.pdf
6/62
GEOMETRIC CONTROL 4
A hovercraft
state space is (x,y,) SE(2)
A fan exerts a steering force F IR+, S1 on the boat
x = Fcos( + )
y = F sin( + )I = F h sin
F
7/28/2019 bavo.pdf
7/62
GEOMETRIC CONTROL 5
The upward pendulum
state space is (x, ) IR S1
the system is such that any force can be exerted at point x on the pendulum
L = 12(m
2 + I)2 + m sin x + 12mx2 mg sin
Q = F dx
x F
Can we invent F(x,, ) such that the solutions tends to = /2 ?
7/28/2019 bavo.pdf
8/62
GEOMETRIC CONTROL 6
Damping a vibrating spring
state space x IR
an exterior force is acting on the mass m:
mx = mg k(x ) + F
where the force is restricted to |F| 1.
m
F
mg
Find the force law such that it damps the oscillation in the least amount of time.
7/28/2019 bavo.pdf
9/62
GEOMETRIC CONTROL 7
The input-state model
The state space IRn q = (q1, . . . , q n)
The inputs u = (u1, . . . , uk) U IRk
U is called the control domain, is arbitrary, often with boundary !
Dynamical steering law q = f(q, u) , with f : IRn IRk Rn.
7/28/2019 bavo.pdf
10/62
GEOMETRIC CONTROL 7
The input-state model
The state space IRn q = (q1, . . . , q n)
The inputs u = (u1, . . . , uk) U IRk
U is called the control domain, is arbitrary, often with boundary !
Dynamical steering law q = f(q, u) , with f : IRn IRk Rn.
Car q = (x,y,), U IR2, u = (v, ) f =
v cos v sin
7/28/2019 bavo.pdf
11/62
GEOMETRIC CONTROL 7
The input-state model
The state space IRn q = (q1, . . . , q n)
The inputs u = (u1, . . . , uk) U IRk
U is called the control domain, is arbitrary, often with boundary !
Dynamical steering law q = f(q, u) , with f : IRn IRk Rn.
Car q = (x,y,), U IR2, u = (v, ) f =
v cos v sin
Damping q = (x, v), U = [1, 1], u = damping force F,
f =
0 1
k 0
xv
+ (mg + kl + u)
01
7/28/2019 bavo.pdf
12/62
GEOMETRIC CONTROL 8
Control systems are encountered in population dynamics, economics,electric circuits, quantum mechanics and
mechanical systems (control forces book by Bullo and Lewis)
7/28/2019 bavo.pdf
13/62
GEOMETRIC CONTROL 8
Control systems are encountered in population dynamics, economics,electric circuits, quantum mechanics and
mechanical systems (control forces book by Bullo and Lewis)
Can be used to model non-holonomic constraints
Sub-Riemannian geometry
Lie algebroids
7/28/2019 bavo.pdf
14/62
GEOMETRIC CONTROL 9
What with regularity assumptions ?
controls are smooth with a finite number of discontinuous jumpsReflects the control nature of u.
t
U
u1
u2
u3
7/28/2019 bavo.pdf
15/62
GEOMETRIC CONTROL 9
What with regularity assumptions ?
controls are smooth with a finite number of discontinuous jumpsReflects the control nature of u.
q = f(q, u(t)) still has continuous unique solutions, q(t) are piecewise
smooth curves
u1 u2 u3
q0
q1
M = IRn
t
U
u1
u2
u3
7/28/2019 bavo.pdf
16/62
GEOMETRIC CONTROL 10
let q(t) be the solution to the time dependent system
q = f(q, u(t)) through a fixed initial state q0.
The solution is called a controlled curve.
7/28/2019 bavo.pdf
17/62
GEOMETRIC CONTROL 11
More general point of view:
controls are measurable and bounded curves
guarantees that q = f(q, u) has absolutely continuous solutions.
Many results depend on considering tangent maps to f(q, u): how to define
it within this general framework. . .
Regularity assumptions are important: results depend on it !
We do not wish to consider this generality: it is already difficult enough tostudy families of smooth vector fields.
7/28/2019 bavo.pdf
18/62
GEOMETRIC CONTROL 12
Control Theory: main problems from a geometers point of
view
q0
q(t)
What can I say about the set points that I can reach following a controlled q(t) ?
7/28/2019 bavo.pdf
19/62
GEOMETRIC CONTROL 12
Control Theory: main problems from a geometers point of
view
q0
q(t)
What can I say about the set points that I can reach following a controlled q(t) ?
Topology, Boundary, interior, . . .
Smoothness:manifold with boundary, corners, stratified space . . .
sufficient conditions for the accessible sets to be the entireconfiguration space preferable from an engineering pointof view !
7/28/2019 bavo.pdf
20/62
GEOMETRIC CONTROL 13
Control Theory: main problems from a geometers point of
view
What can I say about the set points that I can reach following a controlled q(t) ?
What can I say about controls that optimise a given cost ?
7/28/2019 bavo.pdf
21/62
GEOMETRIC CONTROL 13
Control Theory: main problems from a geometers point of
view
What can I say about the set points that I can reach following a controlled q(t) ?
What can I say about controls that optimise a given cost ?
Cost is a certain function that one wants to minimise.
Typical question: What control minimises time needed to take me from q0 toq1
7/28/2019 bavo.pdf
22/62
GEOMETRIC CONTROL 14
A Differential Geometric Framework: families of vector fields
What are the essential ingredients:
7/28/2019 bavo.pdf
23/62
GEOMETRIC CONTROL 14
A Differential Geometric Framework: families of vector fields
What are the essential ingredients:
every control u(t) determines a time dependent vector field on the state space
7/28/2019 bavo.pdf
24/62
GEOMETRIC CONTROL 14
A Differential Geometric Framework: families of vector fields
What are the essential ingredients:
every control u(t) determines a time dependent vector field on the state space
arbitrary family F of vector fields on configuration manifold M, (forget thestructure given by f(q, u))
controlled curves are concatenations of integral curves
7/28/2019 bavo.pdf
25/62
GEOMETRIC CONTROL 14
A Differential Geometric Framework: families of vector fields
What are the essential ingredients:
every control u(t) determines a time dependent vector field on the state space
arbitrary family F of vector fields on configuration manifold M, (forget thestructure given by f(q, u))
controlled curves are concatenations of integral curves
Other possibility:
control domain translates to bundle over configuration manifoldU M, typical fibre of U is control domain: manifold with boundary,manifold with corners,. . .
dynamical law translates to bundle map U T M, fibred over identity f is linear in control variables (distribution approach).
7/28/2019 bavo.pdf
26/62
GEOMETRIC CONTROL 15
Accessible sets: general definition and observations
Often family of vector fields picture used.
M configuration manifold, equipped with an arbitrary family of smooth vectorfields F
Accq(F) = {t
1t1(q) | ti IR+, {i} flow of Xi F}
Acc(q,T)(F){t
1t1(q) | ti IR+,
i=1 ti T , {i} flow of Xi F}
Study the structure of the set Accq(F)
7/28/2019 bavo.pdf
27/62
GEOMETRIC CONTROL 16
Accq(F) and Accq,T(F) are subsets of the leaf Lq(F) through q of thesmallest integrable distribution containing FDefinition (Sussmann):
Lq(F) = {t
1t1(q) | ti IR , {i} flow of Xi F}
What is the structure of Accq(F) as a subset of Lq(F) ?
dimension of accessible set is lower then leaf dimension
7/28/2019 bavo.pdf
28/62
GEOMETRIC CONTROL 16
Accq(F) and Accq,T(F) are subsets of the leaf Lq(F) through q of thesmallest integrable distribution containing FDefinition (Sussmann):
Lq(F) = {t
1t1(q) | ti IR , {i} flow of Xi F}
What is the structure of Accq(F) as a subset of Lq(F) ?
dimension of accessible set is lower then leaf dimension non-smooth boundary
7/28/2019 bavo.pdf
29/62
GEOMETRIC CONTROL 16
Accq(F) and Accq,T(F) are subsets of the leaf Lq(F) through q of thesmallest integrable distribution containing FDefinition (Sussmann):
Lq(F) = {t
1t1(q) | ti IR , {i} flow of Xi F}
What is the structure of Accq(F) as a subset of Lq(F) ?
dimension of accessible set is lower then leaf dimension non-smooth boundary dimension of accessible set is not constant
(x, y)
7/28/2019 bavo.pdf
30/62
GEOMETRIC CONTROL 17
Accessible sets: some results on topology
Definition: The topology induced by F is the finest topology for which allmaps IR
+ M given by t
1t1(q) are continuous, with ti 0 and{i} flow of a member of F.
7/28/2019 bavo.pdf
31/62
GEOMETRIC CONTROL 17
Accessible sets: some results on topology
Definition: The topology induced by F is the finest topology for which allmaps IR
+ M given by t
1t1(q) are continuous, with ti 0 and{i} flow of a member of F.
This topology on Accq(F) is in general finer than the subset topology w.r.tthe topology of Lq(F).
X0 = x, defined on IR2;
X1 = y, defined on ]0, [IR;X2 = x on IR]0, [X3 = yy on IR
2
xx
7/28/2019 bavo.pdf
32/62
GEOMETRIC CONTROL 18
Full-rank systems or Lie-determined systems
A diff. geom. condition guaranteeing that locally F will determine the
structure of Accq(F) (cf. books of Jurdjevic, Agrachev).
7/28/2019 bavo.pdf
33/62
GEOMETRIC CONTROL 18
Full-rank systems or Lie-determined systems
A diff. geom. condition guaranteeing that locally F will determine the
structure of Accq(F) (cf. books of Jurdjevic, Agrachev).
Definition: Lie determined or Full-rank systems are families F for whichT(Lq(F)) = Lie(F) with Lie(F) the distribution generated by all iterated Liebrackets of elements of F.
7/28/2019 bavo.pdf
34/62
GEOMETRIC CONTROL 18
Full-rank systems or Lie-determined systems
A diff. geom. condition guaranteeing that locally F will determine the
structure of Accq(F) (cf. books of Jurdjevic, Agrachev).
Definition: Lie determined or Full-rank systems are families F for whichT(Lq(F)) = Lie(F) with Lie(F) the distribution generated by all iterated Liebrackets of elements of F.
Acc(q,T)(F) has non-empty interior interior points are normally accessible (full-rank paths) If Full-rank then F-generated topology and subset topology coincide (at
least on interior points) int(cl(Accq(F))) = int(Accq(F)) (no separation of interior by boundary) Acc(q,T)(F) cl(int(Acc(q,T)(F))) (no isolated boundary points)
No dense accessible sets in an orbit occur
7/28/2019 bavo.pdf
35/62
GEOMETRIC CONTROL 19
Families with coincident accessible sets (up to boundary)
Closure of accessible sets should be regarded as the analogue of leaf of
foliation . . .
7/28/2019 bavo.pdf
36/62
GEOMETRIC CONTROL 19
Families with coincident accessible sets (up to boundary)
Closure of accessible sets should be regarded as the analogue of leaf of
foliation . . .
Acc(q,T)(convex(F)) cl(Acc(q,T)(F))
Acc(q,T)(cl(F)) cl(Acc(q,T)(F))
Accq(cone(F)) cl(Accq(F))
A map Diff(M) is a normalizer if
(Acc1(q)(F)) cl(Accq(F)).
Then, if N(F) is the family of the pull-back vector fields of F undernormalizers, Accq(N(F)) cl(Accq(F)).
7/28/2019 bavo.pdf
37/62
GEOMETRIC CONTROL 20
If cl(Accq(F)) = M then Accq(F) = M. Important for controllability. Closureis essential !
The Lie saturate S(F) is the largest subset of Lie(F) such thatcl(Accq(S(F))) = cl(Accq(F)).
Lie saturate is invariant under normalizer
If Lie saturate = T M then any point can be accessed from an arbitrarypoint Accq(F) = M (:= controllability)
Not very applicable.
7/28/2019 bavo.pdf
38/62
GEOMETRIC CONTROL 21
ControllabilitySufficient conditions for Accq(F) = M or controllability
7/28/2019 bavo.pdf
39/62
GEOMETRIC CONTROL 21
ControllabilitySufficient conditions for Accq(F) = M or controllability
Trick: Extend family F to cl(F), cone(F) and N(F). Results say thatAccq(N(F)) cl(Accq(F)).
7/28/2019 bavo.pdf
40/62
GEOMETRIC CONTROL 21
ControllabilitySufficient conditions for Accq(F) = M or controllability
Trick: Extend family F to cl(F), cone(F) and N(F). Results say thatAccq(N(F)) cl(Accq(F)).
If N(F) = X(M),
7/28/2019 bavo.pdf
41/62
GEOMETRIC CONTROL 21
ControllabilitySufficient conditions for Accq(F) = M or controllability
Trick: Extend family F to cl(F), cone(F) and N(F). Results say thatAccq(N(F)) cl(Accq(F)).
If N(F) = X(M),
then Accq(N(F)) = M,
7/28/2019 bavo.pdf
42/62
GEOMETRIC CONTROL 21
ControllabilitySufficient conditions for Accq(F) = M or controllability
Trick: Extend family F to cl(F), cone(F) and N(F). Results say thatAccq(N(F)) cl(Accq(F)).
If N(F) = X(M),
then Accq(N(F)) = M,
then cl(Accq(F)) = M,
7/28/2019 bavo.pdf
43/62
GEOMETRIC CONTROL 21
ControllabilitySufficient conditions for Accq(F) = M or controllability
Trick: Extend family F to cl(F), cone(F) and N(F). Results say thatAccq(N(F)) cl(Accq(F)).
If N(F) = X(M),
then Accq(N(F)) = M,
then cl(Accq(F)) = M,
then Accq(F) = M.
7/28/2019 bavo.pdf
44/62
GEOMETRIC CONTROL 21
ControllabilitySufficient conditions for Accq(F) = M or controllability
Example: M = IRn, U = IR, f(x, u) = Ax + ub with A IRnn, b IRn.F = {Ax + ub | u IR}
7/28/2019 bavo.pdf
45/62
GEOMETRIC CONTROL 21
ControllabilitySufficient conditions for Accq(F) = M or controllability
Example: M = IRn, U = IR, f(x, u) = Ax + ub with A IRnn, b IRn.F = {Ax + ub | u IR}
Full rank ? b,Ax sp(F), [Ax + ub,Ax + ub] = (u u)Ab,
[Ab, Ax + u
b] = u
A2
b, . . .
if sp{b,Ab,A2b , . . . , An1b} = IRn then Full rank.
7/28/2019 bavo.pdf
46/62
GEOMETRIC CONTROL 21
ControllabilitySufficient conditions for Accq(F) = M or controllability
Example: M = IRn, U = IR, f(x, u) = Ax + ub with A IRnn, b IRn.F = {Ax + ub | u IR}
Full rank ? b,Ax sp(F), [Ax + ub,Ax + ub] = (u u)Ab,
[Ab, Ax + u
b] = u
A2
b, . . .
sp{b} cl(cone(F)), thus F1 := F sp{b}.
b
0
Ax
Ax+ b
7/28/2019 bavo.pdf
47/62
GEOMETRIC CONTROL 22
the map u(x) = x + ub is a normalizer for F1. (u is flow of (x b) F1)uF
1 N(F1).
7/28/2019 bavo.pdf
48/62
GEOMETRIC CONTROL 22
the map u(x) = x + ub is a normalizer for F1. (u is flow of (x b) F1)uF
1 N(F1).
Ab cl(cone(N(F1))) from u(x Ax) = x Ax + uAb, thus extendF2 := F1 sp{Ab} and so on
if sp{b,Ab,A2b , . . . , An1b} = IRn then Accx(F) = IRn.
7/28/2019 bavo.pdf
49/62
GEOMETRIC CONTROL 23
Accessible sets: smooth structure
Open problem : provide sufficient conditions to show that Accq(F) is a
smooth manifold with singularities.
7/28/2019 bavo.pdf
50/62
GEOMETRIC CONTROL 23
Accessible sets: smooth structure
Open problem : provide sufficient conditions to show that Accq(F) is a
smooth manifold with singularities.
Manifold with boundary ?
Manifold with corners
charts are homeomorphisms from open subsets of Accq(F) to opensubsets of IR
n
+, such that compatibility holds.
Stratified SpacesPartitioned topological spaces such that a.o. strata are smooth manifolds....
7/28/2019 bavo.pdf
51/62
GEOMETRIC CONTROL 24
Accessible sets: smooth structure
Example: Given a family of n independent globally defined vector fields on an
n-dimensional manifold M, then if
[Xi, Xj] C(Xi, Xj) and
any point is accessible by a maximal path, then
Accx(F) is a manifold with corners.
7/28/2019 bavo.pdf
52/62
GEOMETRIC CONTROL 25
Example: Consider the family of vector fields{X1 = /x,X2 = /x + ex/y,X3 = /x + ex/z}. It is easily seen that
[X1, X2] = ex/y C(X1, X2),
[X1, X3] = ex/z C(X1, X3),
[X2, X3] = ex(/z /y) C(X2, X3).
x
y
z
7/28/2019 bavo.pdf
53/62
GEOMETRIC CONTROL 26
Part II: Optimisation
7/28/2019 bavo.pdf
54/62
GEOMETRIC CONTROL 27
Optimisation: variational problems in control theory
Problem: Given a control system q = f(q, u); an initial state q0, initial time t0,
a final state q1 and a final time t1. Among all controlled curves(q, u) : [t0, t1] M U, which one minimises a given cost function L(q, u).
Maximum principle gives necessary conditions for a controlled curve (q, u) tobe optimal:
Define H(q,u,p,) = pifi
(q, u) + L(q, u);
Define a multiplier for (q(t), u(t)) to be a pair (p(t), ) satisfying:
qi(t) = fi(q(t), u(t)) =H
pii = 1, . . . , n
pi(t) = Hqi
= pj(t)fj
qi L
qii = 1, . . . , n
Fix t, then u H(q(t), u , p(t), ) is maximal for u = u(t) .
7/28/2019 bavo.pdf
55/62
GEOMETRIC CONTROL 28
The Maximum Principle If a controlled curve (q, u) is optimal then there is anonzero multiplier (p, ), with either 0, 1.
Sufficient condition originates from following sequence of arguments:
Add a coordinate to the configuration space: (q, J) M IR.
Consider an extended control system: ( q, J) = (f(q, u), L(q, u)).
A optimal controlled curve (q, u) has to belong to the boundary of theaccessible set of the extended controlled system !
Study of the boundary of accessible sets is important ! A more detailedinvestigation of the structure of the boundary may lead to stronger MP !
7/28/2019 bavo.pdf
56/62
GEOMETRIC CONTROL 29
The Maximum Principle for time optimality
Find (q, u) such that it minimises time among all controlled curves starting at
q0 and ending at qf (where the time interval may vary). L = 1
The Maximum Principle If a controlled curve (q, u) is optimal then there is anonzero multiplier (p, ), with either 0, 1 and such that H(q,u,p,) = 0.
7/28/2019 bavo.pdf
57/62
GEOMETRIC CONTROL 30
Maximum Principle: some examplesBang Bang: the rocket car.
Control Law:
ddt
xv
=
0 10 0
xv
+
0u
.
Hamiltonian: H = pxv +pvu + .
Equations: px = 0 and pv = px, or px = c0 and pv = c0t + c1.
Maximality condition: c0v (c0t c1)u attains maximal value ifu = sgn((c0tf + c1)) = 1.
7/28/2019 bavo.pdf
58/62
GEOMETRIC CONTROL 31
sgn((c0tf + c1)) changes sign at most once at most one discontinuousjump in u.
x(t) = t2
2 + v0t + x0
7/28/2019 bavo.pdf
59/62
GEOMETRIC CONTROL 32
Optimal Damping of an oscillation.
Dynamics: x = x + u, with |u| 1. Control law
d
dt x
v
= 0 1
1 0 x
v
+ 0
u
.
Hamiltonian: H = pxv +pv(x + u) + .
Equations: px = pv and pv = px,
px = A cos(t + )pv = A sin(t + )
.
7/28/2019 bavo.pdf
60/62
GEOMETRIC CONTROL 33
Maximality condition: pxv +pv(x + u) attains maximal value ifu = sgn(sin(t )) = 1.
sgn(sin(t )) changes sign with period .
x(t) = B sin(t + ) 1 circles in phase space with centre in (1, 0) or(1, 0).
7/28/2019 bavo.pdf
61/62
GEOMETRIC CONTROL 34
Lagrangian Equations
Trivial control system: q = f(q, u) = u (controls are velocities, control domain
= IRn)
Hamiltonian: H = piui + L.
Equations: p = Lqi
Maximality condition: Hui
= pi + Lqi
= 0 ( = 0 gives contradiction, so only
= 1)
Reduces to d/dt(L/qi) L/qi = 0.
7/28/2019 bavo.pdf
62/62
GEOMETRIC CONTROL 35
References
[1] A. Agrachev and Y. Sachkov. Control Theory from the GeometricViewpoint, volume 87 of Encyclopaedia of Mathematical Sciences.Springer-Verlag, 2004.
[2] V. Jurdjevic. Geometric Control Theory, volume 51 of Cambridge Studiesin Advanced Mathematics. Cambridge University Press, 1997.
[3] L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamklelidze, and E.F. Mishchenko.The Mathematical Theory of Optimal Processes. Wiley, Interscience,1962.
[4] L.C. Evans. An Introduction to Mathematical Optimal Control Theory.Prepint.