bavo.pdf

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


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


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






Car q = (x,y,), U IR2, u = (v, ) f =

v cos v sin

7/28/2019 bavo.pdf

11/62

GEOMETRIC CONTROL 7






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


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


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



view

q0

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



view


What can I say about controls that optimise a given cost ?

7/28/2019 bavo.pdf

21/62



view


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


A Differential Geometric Framework: families of vector fields

What are the essential ingredients:

7/28/2019 bavo.pdf

23/62




every control u(t) determines a time dependent vector field on the state space

7/28/2019 bavo.pdf

24/62





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





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


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


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



Lq(F) = {t



dimension of accessible set is lower then leaf dimension non-smooth boundary

7/28/2019 bavo.pdf

29/62



Lq(F) = {t



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


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


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


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





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





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


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


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


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


ControllabilitySufficient conditions for Accq(F) = M or controllability

7/28/2019 bavo.pdf

39/62



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




If N(F) = X(M),

7/28/2019 bavo.pdf

41/62




If N(F) = X(M),

then Accq(N(F)) = M,

7/28/2019 bavo.pdf

42/62




If N(F) = X(M),


then cl(Accq(F)) = M,

7/28/2019 bavo.pdf

43/62




If N(F) = X(M),


then cl(Accq(F)) = M,

then Accq(F) = M.

7/28/2019 bavo.pdf

44/62



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




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




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


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


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


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



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



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


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


Part II: Optimisation

7/28/2019 bavo.pdf

54/62


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


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


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


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


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


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


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


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


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.

bavo.pdf

Documents