Directory UMM :Journals:Journal_of_mathematics:OTHER:

Generic Observability of Dynamical Systems

Esfandiar Nava Yazdani

Received: January 18, 2005

Revised: July 3, 2007

Communicated by Ulf Rehmann

Abstract.

Wedealwithaertainobservationmappingdened by meansofweightedmeasurmentsonadynamialsystemandgive

ne-essaryand suientonditions,under whih thismappingis

generi-allyaninjetiveimmersion.

2000 Mathematis Subjet Classiation: 93B07, 37C20, 37C10,

94A20.

Keywords and Phrases: Observability, generi, dynamial system,

sampling.

1

Introduction

Theobservabilityproblem ofnonlineardynamial systemshasbeenan

inter-esting subjetandativeeldof researhthroughout thelastdeades. Inthe

presentworkweonsidertimeinvariantsystemsoftheform

∗

˙

x

=

f

(

x

)

y

=

h

(

x

)

.

The rst equation desribes a real dynami proess. Its state

x

(

t

)

at time

t

is assumedto beelementofasmoothseond ountable (hene paraompat)

n-dimensional manifold

M

alled state spae. The dynamis of the system

is given by the vetor eld

f

on

M

. The output funtion

h

is a mapping

from the state spae into the reals and stands for a measuring devie. The

seond equation desribes the output, whih ontains partial information

of the state. The output

y

is the only measurable quantity. There is a

very broadvariety of systems, whih anbe desribed in this way. We all the triple

(

M, f, h

)

orsimply the pair

(

f, h

)

a system. The system is alled

∗We

C

r

i

M

,

f

and

h

are

C

r

. Wedenotethesetof

C

r

vetoreldson

M

by

X

r

(

M

)

.

In many appliations it is of essential importane to know the state of the

system at any time

t

. But measurement of the entire state (e.g. all its

oordinates)is oftenimpossibleorverydiult. Forinstane beauseofhigh

ostsortehnialreasons. Inmostasesonehasonlypartialinformationfrom

measurementof theoutput

y

, mathematially beingdesribed by theseond

equation. Sotheissueistogetthestate(aswellasdistinguishdierentstates)

by using only output measurement

∗

. If this is possible, then the system is

said to be observable. Hene, the issue of the observability problem is the

following. Find riteria on the system, suh that by means of information

from theoutputtrajetory,itispossibleto distinguishdierentstatesaswell

as to reonstrut the states. There is no uniform or anonial denition of observabilityin theliterature. Theweakest andmostnaturaldenitionis the

following.

Definition 1.1

Thesystem

(

f, h

)

issaidtobeobservable(ordistinguishable) ifforeah

(

x, x

′

)

∈

M

×

M

with

x

6

=

x

′

thereexists atime

t

0

=

t

0

(

x, x

′

)

suh

that

h

(Φ

t0

(

x

))

6

=

h

(Φ

t0

(

x

′

))

.

However, the above denition is not well-suited for the treatment of the

observabilityproblem. Thereforeoneseekstoestablishastongernotionof

ob-servabilityasfollows. Consideramapping

Θ

(inthesequelalled observation

mapping), whihmaps thestatespae intosomenite dimensional Eulidian

spae, assigning states to data derived from the output trajetorieson some

observation time interval

J

. Then deide the observability of the system by

means of injetivity of

Θ

. Moreoverthefollowingnatural questionarises. Is

observabilityin thissense (,i.e. with respetto

Θ

)generi? Thelatteris the

mainissueofthepresentwork. Intheontroltheorythenoatationofobserver is generallystandingfor anothersystem havingthe output (andinput in the

ontrolledase)oftheoriginalsystemasinputandgeneratinganoutputwhih

isanasymptotiestimateoftheoriginalsystemstate.

Beside the task of reonstrution of the state, observabilityhas also

applia-tionin thetheoryofhaosand turbuleneinthefollowingsense. Suppose an

observablesystemhasaglobalattrator. Thenusing anobservationmapping

one anget a homeomorphi pitureof the attrator orat least information

about some of its harateristi properties. Examples an be found in [RT℄

and [T℄. We onsider a ertainobservation mappingintrodued in [KE℄ and

[E ℄. Wederiveneessaryandsuientonditionsforgeneriityofobservability

and loalobservability withrespet tothis mapping. Basially,there aretwo

otherwell-knwonapproahestotheobservabilitytask: samplingandhigh-gain approah. Inhislassialwork[T℄,Takensprovedgeneriityresultssimilarto

∗It

oursforthese approahes. AboutthesametimeAeyels[Ay℄ahievedsharper

results onerning the sampling approah. Further results on high-gain

ap-proahanbefoundin [GK℄,[GHK℄and[J℄. Partiularly,foraomrehensive

and intensive investigation of deterministi observation theory and

applia-tions inluding many deep resultsonerning high-gain approah we referto

[GK℄. Ourapproah hasthe disadvantagethat a suitablelinear lter hasto

beonstruted. Ontheothersideinontrasttohigh-gainapproahwedonot

needto restritourselvestosmoothsystems. Formoreonourapproahwith

appliationswereferto[N ℄.

2

Main Results

Definition 2.1

Let

0

< τ <

∞

,

(

A, b

)

be a stable and ontrollable

m

-dimensional linearlter and

(

M, f, h

)

a

C

r

-system with ow

Φ

. Weall the

mapping

M

∋

x

7→

Θ

f,h

:=

R

0

−

τ

e

−

At

bh

(Φ

t

(

x

))

dt

∈

R

m

theobservation

map-pingandthenumber

m

observationdimension. If

Θ

f,h

isinjetiveweallthe

system

(

M, f, h

) Θ

-observableorsimplyobservable.

Furthermoreweallthesystemloallyobservableif

Θ

f,h

isanimmersion.

Theindistinguishablesubsetof

M

×

M

isgivenby

Ω

f,h

:=

{

(

x, x

′

)

∈

M

×

M

\

∆

M

: Θ

f,h

(

x

) = Θ

f,h

(

x

′

)

}

.

The time interval

I

:= [

−

τ,

0]

has the physial interpretation of observation interval. We treat for simpliity of notation and in view of the physial

interpretation of the observation data ashistory of the output mapping, the

ase

I

⊂

R

≤

0

,

0

∈

I

. Forunboundedintervalsmodiationsareneeded,whih wegiveexpliitlyforthease

I

=

R

≤

0

.

The key point in theproof of the following generiity resultsis to showthat

zeroisaregularvalueoftheobservationmapping. Provingthiswethenapply

transversality density and openness theorems to get loally our statements,

whihthenwillbeglobalizedtothewholestatespae. ABaireargumentthen

yieldsthenal results. Partiularlyfor

τ <

∞

in any

C

r

neighborhoodof the

systemthereisasystemwhihis bothobservableandloally observablewith respet to

Θ

. An appropriate statement is also valid for

τ

=

∞

in the

C

1

topology.

If thestatespaeis notompat,itis moresuitableto onsidertheso alled

strongorWhitney(

C

r

-)topologyon

C

r

(

M,

R

)

and

X

r

(

M

)

. Thereasonisthat

inthistopologyonehasmoreontrolonthebehaviorofthefuntionsand

ve-tor eldsat innity. Note thatdensityin thistopologyisastrongerproperty than in the ompat-open(also alled weak)topology. For instane, roughly

speaking,asequeneofoutputfuntions

h

j

onvergesinWhitney topologyto

h

i there exists a ompatset

K

suh that

h

j

=

h

outside of

K

exeptfor

Theaseofsmoothvetoreldsissimilar. Thewellknownfatthat

C

r

(

M,

R

)

and

X

r

(

M

)

areBairespaesin theWhitney topology(proofsanbefoundin

in[H,2.4.4℄and[P ℄)isofbasiimportaneforourresults. Aresidualsubsetof

C

r

(

M,

R

)

or

X

r

(

M

)

isdense. Fromnow onthe spaes of

C

r

funtionsaswell

asvetoreldson

M

areequippedwiththeir

C

r

Whitneytopology,unless

oth-erwiseindiated. Ifthestatespae

M

isompatand

r <

∞

,then

C

r

(

M,

R

k

)

and

X

r

(

M

)

endowedwiththeompat-opentopologyareBanahspaes(while

in general Fréhet spaes), their Whitney and ompat-open topology

oin-ideandtheow

Φ

ofeahvetoreld

f

∈

X

r

(

M

)

isdenedgloballyon

M

×

R.

For some pairs in

X

r

(

M

)

×

C

r

(

M,

R

)

there is no possibility to

distin-guish or loally distinguish the states. For generiity results it is an

es-sential fat that the omplement of the set of suh pairs is residual. Let

Sing

(

f

) :=

{

x

∈

M

:

f

(

x

) = 0

x

∈

T

x

M

}

denote the set of singularities of

thevetoreld

f

(equilibria ofthe system),where

0

x

is thezeroof

T

x

M

. In

the sequelwe shalloften omit thesubsript

x

and write simply

f

(

x

) = 0

for

x

∈

Sing

(

f

)

.

Reall that asingularity

x

0

∈

Sing

(

f

)

is alled simplei the prinipialpart of thelinearizationof

f

at

x

0

, i.e. thelinearmapping

d

x0

f

:

T

x0

M

→

T

x0

M

, doesnothavezeroasaneigenvalue.

We denote the set of

C

r

vetor elds, whose singularities are all simple, by

X

r

₀

(

M

)

. Itiswellknownthatasimplesingularityisisolatedand

X

r

0

(

M

)

isan

openanddensesubsetof

X

r

(

M

)

(foraproofwereferto[PD ,3.3℄).

Definition 2.2

We all

x

0

∈

M

a

Θ

-simple singularity i there exists a otangent vetor

v

∈

T

∗

x0

M

suh that the linear system

(

T

x0

M, d

x0

f, v

T

)

is

observable,i.e.,thelinearmapping

Z

0

−

τ

e

−

tA

bve

td

x

0f

dt

:

T

x0

M

→

R

m

isinjetive. Inthisasewesaythat

v

T

isa

Θ

-oyliovetorof

d

x0

f

.

Wedenoteby

X

r

1

(

M

)

thesetof

C

r

vetoreldson

M

,whosesingularitiesare

Θ

-simple. Moreoverweset

X

r

0

,

1

(

M

) :=

X

r

0

(

M

)

∩

X

r

1

(

M

)

.

Inthelimitingase

τ

=

∞

denseorbitsaswellasnontrivialreurreneause diulties and speial onsiderationsare neessary. We investigate this ase

undertheassumptionthat

M

isompat. Appropriateresultsinthe nonom-pat ase an be similarly deriven if we further restrit (in order to ahieve

well-denedness of theobservation mapping) the systemsto be globally

Lip-shitzian. Furthermore,in order to ensuredierentiability ofthe observation

mapping,if

τ

=

∞

werestritthevetoreldstotheopenset

X

r

(

M, a

) :=

{

f

∈

X

r

(

M

) :

sup

x

∈

M

k

d

j

x

f

k

< a

forall

j

= 1

, ..., r

}

eld

f

by

C

(

f

)

andtheunionofthenegativelimitsets by

L

−

(

f

)

,weset

X

r

−

(

M

) :=

{

f

∈

X

r

(

M

) :

L

−

(

f

)

⊂

C

(

f

)

}

.

WedenotethatsomeineterestinglassesofvetoreldslikeMorse-Smaleelds

∗

areontainedin

X

r

−

(

M

)

. Partiularly,sinethesetonsistingofMorse-Smale

vetor elds is open and nonempty in

X

r

(

M

)

, the interior

int

(

X

r

−

(

M

))

of

X

r

−

(

M

)

isaBairespae(in theinduedtopology). Furthermorenotethat the

limitset ofagradienteld onsistsonlyofthe ritialpointsofthepotential

funtion. Therefore

X

r

−

(

M

)

also ontainstheset ofgradientelds. Moreover

weset

X

r

2

(

M

) :=

X

r

(

M, a

)

∩

int

(

X

r

−

(

M

))

.

It is well known that on a ompat manifold

C

1

-generially the

nonwander-ing set of a smooth vetor eld oinides with the losure of the set of its

periodi points. This statement alled general density theorem is a

onse-quene of Pugh's losing lemma, whih ensures that a nonwandering point

an be made periodi by a small

C

1

-petrubation in a neighbourhood of the

point. Seealso [Pu ℄,[AR, 7.3.6℄ andthereferenesgiventhere

∗

. Partiularly

{

f

∈

X

1

(

M

) :

L

−

(

f

)

⊂

C

(

f

)

}

isaresidualsubsetof

X

1

(

M

)

.

Wesetfor

y

∈

E

r

a,τ

:=

{

y

∈

C

r

([

−

τ,

0]

,

R

) :

R

0

−

τ

e

at

|

y

(

t

)

|

dt <

∞}

P

τ

y

:=

Z

0

−

τ

e

−

At

by

(

t

)

dt

andforxed

τ

simplywrite

P

insteadof

P

τ

.

Lemma 2.1

Let

r, τ

≤ ∞

.Furthermore let

q

0

∈

R

m

and

T, δ >

0

. Then

thefollowingshold.

a) Thereexistsafuntion

y

∈

C

r

(

R

,

R

)

beingompatlysupportedin

]

−

δ,

0[

andsatisfying

P

τ

y

=

q

0

.

b) There exists a

T

-periodi funtion

y

∈

C

r

(

R

,

R

)

with

P

τ

y

=

q

0

and

supp

(

y

|

[

−

(

k

+1)

T,

0]

)

⊂

]

−

(

k

+ 1)

δ,

0[

forall

k

∈

Z.

Proof:

Ad a) Let

ǫ

:=

min

{

δ, τ

}

. The mapping

L

1

([

−

ǫ,

0])

∋

y

7→

K

(

y

) :=

R

0

−

ǫ

e

−

At

by

(

t

)

dt

∈

R

m

is linear, ontinuousand beause of the ontrollability

of

(

A, b

)

surjetive.

C

r

ǫ

is a dense linear subspae of

L

1

([

−

ǫ,

0])

. Therefore

R

:=

K

(

C

r

ǫ

)

is a dense linear subspae of

R

m

and onsequently

R

=

R

m

.

Therefore there exists a funtion

y

0

∈

C

r

ǫ

having the property

K

(

y

0

) =

q

0

.

Thetrivialextensionof

y

0

on

R

isobviouslythedesiredfuntion.

Ad b) If

τ

≤

T

, the assertion follows diretly from part a). We prove the result for

τ > T

using sampling. Assume rst

τ <

∞

and let

N

:=

max

{

k

∈

N

:

N T

≤

τ

}

. Due to thestabilityof

A

theseries

P

∞

k

=0

e

kT A

∗

ReallthatMorse-Smalevetoreldsarestruturallystable.

∗It

isstillunknownwhetherthe

C

k

onvergesto

(

I

−

e

T A

)

−

1

and

S

N

:=

P

N

k

=0

e

kT A

= (

I

−

e

T A

)

−

1

(

I

−

e

(

N

+1)

T A

)

isinvertible. Letuswrite

ǫ

:=

min

{

τ

−

T, δ

}

. Aordingtoparta)thereexists

˜

y

0

∈

C

ǫ

r

with

P

T

y

˜

0

=

S

−

1

N

q

0

(as well as

(

I

−

e

T A

)

q

0

in ase

τ

=

∞

). Let

y

0

denote the trivial extension of

y

˜

0

on

[

−

T,

0]

. Then wehave

P

T

y

0

=

S

−

1

N

q

0

.

Let

y

denotethe

T

-periodiextensionof

y

0

on

R.

Consequently

Z

0

−

τ

e

−

At

by

(

t

)

dt

=

N

X

k

=0

Z

−

kT

−

(1+

k

)

T

e

−

At

by

0

(

t

+

kT

)

dt

=

N

X

k

=0

e

kT A

Z

0

−

T

e

−

At

by

0

(

t

)

dt

=

q

0

,

whihyieldsimmediatelythedesiredonlusion.

Let

π

denotetheanonialprojetionof thetangentbundle

T M

:

π

:

T M

→

M, π

(

v

) :=

x

for

v

∈

T

x

M.

Let

M

beendowedwithaRiemannianmetri. Denotingtheinduednormon

thetangentspaesby

|

.

|

,theunit tangentbundle

T

1

M

isgivenby

T

1

M

:=

∪

x

∈

M

{

v

∈

T

x

M

:

|

v

|

= 1

}

.

Reall that

T

1

M

is a

(2

n

−

1)

-dimensional

C

r

−

1

submanifold of

T M

. It is

ompat,if

M

isompat.

Let

K

beasubsetof

M

. Weset

T

0

K

:=

K

×

K

\

∆

K

anddenotetherestrition of

T

1

M

to

K

with

T

1

K

,i.e.,

T

1

K

:=

{

v

∈

T

1

M

:

π

(

v

)

∈

K

}

. Reallthat if

K

isan

s

-dimensionalsubmanifold,then

T

1

K

hasdimension

n

+

s

−

1

.

Definition 2.3

We dene the

τ

-history of

K

by the ow

Φ

of the ve-toreld

f

tobethelosureof

Φ(

K

;

τ

) :=

{

Φ

t

(

x

) :

−

τ < t

≤

0

, x

∈

K

}

.

Wedenote

∆Θ

f,h

(

x, x

′

) := Θ

f,h

(

x

)

−

Θ

f,h

(

x

′

)

for

x, x

′

∈

M

.

Let

V

be an open subset of

M

ontaining the

τ

-historyof

K

and

L

be the

losureof

V

. Thenwedenote

H

₀

(

L

;

K

) :=

{

h

∈

C

r

(

L,

R

) :

zeroisaregularvalueof

∆Θ

f,h

|

Λ

0

K

}

,

and

Inthesequelweset

i

= 0

,

1

aswellas

H

i

(

K

) :=

H

i

(

M

;

K

)

and

H

i

:=

H

i

(

M

)

.

If

K

′

⊂

K

and for an output funtion

h

, zero is a regular value of the

mapping

∆Θ

f,h

on

Λ

0

K

,thenitis alsoaregularvalueoftherestrited

map-ping

∆Θ

f,h

|

Λ

0K

′

,thatis,

H

0

(

K

)

⊂

H

0

(

K

′

)

. Similarly

H

1

(

K

)

⊂

H

1

(

K

′

)

holds.

In the following, if

f

∈

X

r

0

,

1

(

M

)

, then

H

0

(

L

)

stands for the set of output

funtions

h

∈

C

r

(

L,

R

)

suhthat

h

(

x

0

)

6

=

h

(

x

′

0

)

forall

(

x

0

, x

′

0

)

∈

Λ

0

(

L

∩

Sing

(

f

))

,

and

H

1

(

L

)

forthose

h

∈

C

r

+1

(

L,

R

)

suh that

d

x0

h

isa

Θ

-oyliovetorof

d

x0

f

forall

x

0

∈

L

∩

Sing

(

f

)

.

Note that if

L

isompat,then thenumberof singularitiesof

f

in

L

isnite

and for nite

r

, as an immediate appliation of the transversality openness

anddensitytheorems,itfollowsthat

H

i

(

L

)

isanopenanddensesubsetofthe

Banahspae

C

r

+

i

(

L,

R

)

.

Lemma 2.2

Assume that

i < r <

∞

and

f

∈

X

r

0

,

1

(

M

)

is omplete. Let

S

bea

C

r

submanifold of

M

suh that the

τ

-historyof

S

is ontainedin an

open subset

V

of

M

with ompat losure

L

:= ¯

V

. Consider the mappings

F

i

:

H

i

(

L

)

×

Λ

i

S

→

R

m

denedby

F

0

(

h, x, x

′

) := ∆Θ

f,h

(

x, x

′

)

and

F

1

(

h, v

) :=

d

π

(

v

)

Θ

f,h

(

v

)

.

Thenthefollowingholds.

a) Zeroisaregularvalueof

F

0

.

b)Zeroisaregularvalueof

F

1

.

Supposemoreoverthat

f

∈

X

r

2

(

M

)

. Thentheassertionsalsohold for

τ

=

∞

.

Proof:

Ad a) Let

W

0

:=

{

(

h, x, x

′

)

∈

H

0

(

L

)

×

T

0

S

:

F

0

(

h, x, x

′

) = 0

}

.

We have to show that the funtion

F

0

is submersive on

W

0

. Sine

R

m

is nite dimensional, it suies to prove that the linear mapping

d

(

h,x,x

′

)

F

0

:

T

(

h,x,x

′

)

(

H

0

(

L

)

×

T

0

S

)

→

R

m

is surjetiveforall

(

h, x, x

′

)

∈

W

0

.

Fix

(

h, x, x

′

)

∈

W

0

and

q

0

∈

R

m

. Aordingtotheondition

h

∈

H

0

(

L

)

wesee that

x

and

x

′

annotbebothequilibriumpoints. Thereforeweassumewithout

lossofgeneralitythat

x

isnotanequilibriumpoint. Sine

H

0

(

L

)

isopenand

d

ds

|

s

=0

F

0

(

h

+

sg, x, x

′

) = ∆Θ

f,g

(

x, x

′

) =

F

0

(

g, x, x

′

)

, it is suient to show

theexisteneof anoutputfuntion

g

∈

C

r

(

L,

R

)

satisfying

F

0

(

g, x, x

′

) =

q

0

.

Weusethefat thattheowthroughapointofthestatespae

M

mapseah

losednitetimeintervalonalosedsubsetof

M

anddeneasuitablemapping

Let

γ

and

γ

′

denote the

τ

-histories of the points

x

and

x

′

respetively and

Z

:=

γ

∪

γ

′

. Wedene

g

on

Z

. Wersttreatthease

τ <

∞

.

Case1: Bothorbitsareritialelements. Let

T

denotetheperiodof

x

. Inview

of lemma 2.1 there exists a

T

-periodi funtion

y

∈

C

k

(

R

,

R

)

, whih satises

theondition

P

τ

y

=

q

0

. Weset

g

(Φ

t

+

kT

(

x

)) =

y

(

t

)

for

0

≤

t

≤

T, k

∈

Z

and

g

= 0

else.

Case2: Oneof theintegralurves,say

Φ(

x

)

, isinjetiveandtheother oneis

periodi oran equilibriumpoint. AordingtoLemma 2.1there isafuntion

y

∈

C

k

([

−

τ,

0]

,

R

)

withompatsupportsuhthat

P

τ

y

=

q

0

. Wedene

g

on

γ

by

g

(Φ

t

(

x

)) =

y

(

t

)

for

−

τ

≤

t

≤

0

and

g

= 0

else. If

Φ(

x

′

)

isinjetiveand

x

is

periodi, wejust set

g

= 0

on

γ

and dene

g

on

γ

′

suhthat

P

τ

(

g

◦

Φ(

x

′

))) =

−

q

0

.

Case3: Both integralurvesareinjetive. Inthis asewedene

g

on

γ

asin

ase2andon

γ

′

by

g

= 0

.

If

τ

=

∞

,thenbeauseofeventualpreseneofdenseorbits

g

annotbesimply dened on a part of

Z

and then trivially extended. Hene the onstrution

of

g

beomes alittle moredeliate. Assuming

f

∈

X

2

ensures thenthat the reurrene is trivial and the previous proedure also works. If both orbits

are ritial elements, i.e. in ase 1, then everythingremains the sameasfor

nite observation time. Problems ould arise in ase 2 or 3 if at least one

of the integral urves is injetive and the past half of one of the orbits, say

the one through

x

′

, belongs to the negative limit set of the other orbit or

itself, i.e.,

{

Φ

t

(

x

′

) :

t

≤

₀

} ⊂

α

(

x

)

∪

α

(

x

′

)

. But this an aording to the

assumption

f

∈

X

r

2

(

M

)

onlyourif

x

′

isperiodioranequilibriumpoint. We

proeedasin ase2ofnite

τ

andndagaininviewofLemma2.1afuntion

y

∈

C

r

(

R

,

R

)

ompatly supported in an interval

[

−

ǫ,

0]

with

ǫ >

0

and set

g

(Φ

t

(

x

)) :=

y

(

t

)

forall

t

and

g

= 0

else.

Inallaseswehavedened a

C

r

funtion onthelosedsubset

Z

ofthestate

spae

M

with the property that

Z

⊂

L

and

P

τ

(

g

◦

Φ(

x

)

−

g

◦

Φ(

x

′

)) =

q

0

.

AordingtothesmoothTietzeextensiontheorem,thereexistsa

C

r

extension

of the funtion

g

to

L

. This funtion denoted again by

g

is obviously the

desiredfuntion,whihsatises

F

0

(

g, x, x

′

) =

q

0

.

Ad b) Denote

W

1

:=

{

(

h, v

)

∈

H

1

(

L

)

×

T

1

S

:

F

1

(

h, v

) = 0

}

. We x

(

h, v

)

∈

W

1

, set

x

0

:=

π

(

v

)

and show that

F

1

is submersive at

(

h, v

)

. Fix

q

0

∈

R

m

. Sine

d

ds

|

s

=0

F

1

(

h

+

sg, v

) =

F

1

(

g, v

)

for arbitrary

g

∈

C

r

(

M,

R

)

,it

sues to provethe existene of afuntion

g

∈

C

r

(

L,

R

)

with

F

1

(

g, v

) =

q

0

loally and extend it

L

. If

x

0

would be an equilibrium point, then in view

of the assumption

h

∈

H

1

(

L

)

, the linearsystem

(

T

x0

M, d

x0

f, d

x0

h

)

would be

Θ

-observableandonsequently

F

1

(

h, v

)

6

= 0

inontraditiontotheassumption

(

h, v

)

∈

W

1

. Therefore wemay assume that

x

0

is not an equilibrium point.

Hene, in viewof the straightening-out theorem there is a loal hart

(

U, ψ

)

at

x

0

suhthat

ψ

(

U

) =

U

′

×

]

−

ǫ,

−

ǫ

[

with

ǫ >

0

,

U

′

anopen subsetof

R

n

−

1

,

ψ

(

x

0

) = 0

andthevetoreld

f

hastheloalrepresentative

(

z, t

)

7→

e

n

. Here

e

n

denotesthe

n

thstandardbasevetorin

R

n

funtion on

T

1

M

at

v

by

ψ

b

. Sine

v

6

= 0

we anand do assumethat

v

has the loal representative

η

= (

η

1

, η

2

)

T

with

η

1

∈

R

n

−

1

,

η

2

∈

R,

η

6

= 0

, i.e.

v

=

∂

∂z

(

x

0

)

η

1

+

∂

∂t

(

x

0

)

η

2

. Furthermorefor

t

∈

]

−

ǫ,

−

ǫ

[

,theloalrepresentative

of

d

x0

Φ

t

readsas

Id

0

0

1

andsubsequentlythat of

d

x0

h

◦

Φ

t

v

as

∇

h

(0

, t

)

η

.

By shrinking

U

if neessary, we an (in the ase

τ

=

∞

on aount of the assumption that

f

∈

X

r

2

(

M

)

if the point

x

0

is reurrent, then it is periodi)

anddoassumethattheintersetionof

U

andthe

τ

-historyof

x

0

isonneted.

Aordingtolemma2.1thereexistsafuntion

y

ˆ

∈

C

r

+1

(

R

,

R

)

withderivative

y

being supported (on eah period, if

x

0

has period

> τ

) in

[

−

ǫ,

0]

suh

that

P

ǫ

y

(

t

)

dt

=

q

0

. Obviouslythere exists afuntion

g

∈

C

r

(

U

′

×

]

−

ǫ, ǫ

[

,

R

)

being ompatly supported, with

∇

g

(0

, t

)

η

=

y

(

t

)

. For instane dene

˜

g

(

z, t

) =

|

η

|

−

2

(

z

T

η

1

y

(

t

) +

η

2

y

ˆ

(

t

))

. Thetrivial extension of

g

˜

◦

ψ

to

L

is the

desiredfuntion

g

.

Sometimes in appliations one is interested in or limited to observation

re-stritedtoasubsetofthestatespae. Itanbeforinstanebeauseoftehnial

orphysialreasons,orifithappensthatalltheinformationneededanbe

eval-uatedfrom measurementsonaertainsubset. Thelatterasebeingperhaps

themostimportantone,oursifthesubsetunderobservationisanattrator.

Other subsets invariant under the owan also be of interest. Thereforewe

stateourgeneriityresultsforobservationsofsubsetsofthestatespaeaswell.

Lemma 2.3

Suppose that

K

is a subset of an

s

-dimensional

C

r

submani-fold of

M

denoted by

S

,

τ <

∞

and

f

∈

X

r

0

,

1

(

M

)

is omplete. Then the

followingholds.

a) Assume that

m

≥

n

+

s

−

r

and

r

≥

2

. Then

H

1

(

K

)

is residual. If

K

is losed,then

H

1

(

K

)

isalsoopen.

b) Assume that

m

≥

n

+

s

+ 1

−

r

. Then

H

0

(

K

)

is residual. If

K

is losed, then

H

0

(

K

)

ontainsanopenset.

Suppose moreover that

M

is ompat and

f

∈

X

r

2

(

M

)

. Then the assertions

holdalsoin thease

τ

=

∞

.

Proof:

Assume rst

r <

∞

,

K

is ompat,

U

is a hart domain of

S

, whih ontains

K

and hasompatlosure. Byompatnessof

U

and

nite-ness of

τ

in ase ofnite observationtime and beause of ompatnessof

M

in ase

τ

=

∞

, the

τ

-historyof

U

is ompat. By loalompatness thereis an open set

V

⊂

M

with ompat losure

L

:=

V

suh that

V

ontains the

τ

-historyof

U

.

Loal density: We prove residuality with respet to

H

i

(

L

)

. Sine the latter

is open and dense in

C

r

(

L,

R

)

, density of

H

i

(

L

;

U

)

is also then shown with

respetto

C

r

(

L,

R

)

.

map-ping

F

i

:

H

i

(

L

)

×

T

i

U

→

R

m

dened by

F

0

(

h, x, x

′

) := ∆Θ

f,h

(

x, x

′

)

and

F

1

(

h, v

) :=

d

π

(

v

)

Θ

f,h

(

v

)

.

Thereforeaordingtothetransversalitydensitytheorem(seeforinstane[AR ,

19.1℄)

H

i

(

L

;

U

)

isaresidualsubsetof

H

i

(

L

)

,andhenedensein

C

r

(

L,

R

)

.

Loal Openness: Compatnessof

K

impliesthat

T

1

K

isalsoompatin

T

1

U

.

Aordingto the transversalityopennesstheorem

H

1

(

L

;

K

)

is open (with

re-spet tothe ompat-open andby ompatnessof

L

alsowith respet to the

Whitney topology)in

C

r

(

L,

R

)

. These onlusionsdonotworkfor

H

0

(

L

;

K

)

.

Instead let

Λ

be a ompat subset of

T

0

U

. Then in view of transversality

opennesstheorem

H

′

0

(

L

; Λ) :=

{

h

∈

C

r

(

L,

R

) :

zeroisaregularvalueof

∆Θ

f,h

on

Λ

}

isopenin

C

r

(

L,

R

)

.

Sinetheassertionsareprovedfor

r

niteandsuientlylarge,theyalsohold

for

r

=

∞

. Heneweassumefromnowonthat

r

≤ ∞

.

Globalization: This part of the proof is basily standard. Therefore we

give an outline and refer to [H, 2.2℄ for details. Sine

K

⊂

U

, we have

H

1

(

L

;

U

)

⊂

H

1

(

L

;

K

)

. Hene

H

1

(

L

;

K

)

isalsodensein

C

r

(

L,

R

)

. Likewiseone

getsdensityof

H

′

0

(

L

; Λ)

in

C

r

(

L,

R

)

fromdensityof

H

0

(

L

;

U

)

,sine

Λ

⊂

T

0

U

andsubsequently

H

0

(

L

;

U

)

⊂

H

′

0

(

L

; Λ)

. Usingabump funtionwenowprove

that

H

1

(

K

)

isdensein

C

r

(

M,

R

)

. Fix

g

0

∈

C

r

(

M,

R

)

. Sine

H

1

(

L

;

K

)

isdense

in

C

r

(

L,

R

)

, thereisasequene

{

h

j

}

in

H

1

(

L

;

K

)

onvergingin the ompat-opentopologyto

g

0

|

L

. Sine

K

⊂

U

,thereisa

C

r

-funtion

ρ

:

M

→

[0

,

1]

with ompat support in

L

, suh that

ρ

= 1

onan open neighborhood of

K

. The

sequene

ρh

j

+ (1

−

ρ

)

g

0

onvergesto

g

0

withrespettotheWhitneytopology. Therefore

H

1

(

K

)

is a dense subset of

C

r

(

M,

R

)

. A similar argumentshows

that

H

′

0

(Λ)

isadensesubsetof

C

r

(

M,

R

)

.

Wenowdroptheassumptionthat

K

⊂

U

. Let

J

beaountable indexingset,

{

U

j

}

with

j

∈

J

be aoveringof

S

with hartdomains

U

j

. Furthermore let

{

K

j

}

beasubordinatefamilyofompatsetssuhthat

K

=

∪

J

K

j

and

K

j

⊂

U

j

.

Weananddoassumethatthereisaompatoveringof

M

denotedby

{

L

j

}

suhthattheinteriorof

L

j

ontainsthe

τ

-historyof

U

j

.

Opennessstatements: Supposethat

K

islosed. Then itisalso paraompat

and theoveringanbeassumedtobeloallynite. Sine

{

L

j

}

overs

M

,it holds that

H

1

(

K

) =

{

h

∈

C

r

(

M,

R

) :

h

|

L

j

∈

H

1

(

L

j

;

K

j

)

forall

j

∈

J

}

. Hene

loalnitenessoftheoveringimpliesthat

H

1

(

K

)

isopen. Simliarilyitfollows

that theset

{

h

∈

C

r

(

M,

R

) :

zeroisaregularvalueof

∆Θ

f,h

on

K

×

K

}

is open. Thelatterisontainedin

H

0

(

K

)

.

Residualitystatements: Wenowdroptheassumptionthat

K

islosed. Bythe

preedingarguments

H

1

(

K

j

)

isopenanddense. Therefore

H

1

(

K

) =

∩

j

isresidual(andinviewoftheBairepropertyof

C

r

(

M,

R

)

alsodense). Taking

a ompat overing

{

Λ

j

}

of

T

0

K

a similar argument yields the results on

H

0

(

K

)

.

The openness results in the preeding lemma an also be proved (without

applying the transversality openness theorem) as follows. For instane, by

ompatnessof

L

themapping

H

0

(

U

)

∋

h

7→

F

0

(

h, .

))

∈

C

r

(

T

0

U,

R

m

)

is

ontin-uousin theWhitneytopology. Thisfatand opennessof

{

F

∈

C

r

(

T

0

U,

R

m

) :

zeroisaregularvalueof

F

}

(this fat follows, for instane, from the lower semiontinuityofthemapping

C

r

(

L,

R

)

×

T

0

U

∋

(

h, x, x

′

)

7→

rank

(

d

(

h,x,x

′

)

F

0

)

and ompatness of

L

) in the Whitney topology implies that

H

0

(

L

;

U

)

is

open in the Whitney topology and by ompatness of

L

also open in the

ompat-opentopologyof

C

r

(

L,

R

)

. Otherpartsfollowlikewise.

In light of the preeding lemmas wean nowprove the generiity results on

outputfuntions justbyomparingdimensions.

Theorem 2.1

Suppose that

K

is ontained in an

s

-dimensional

C

r

sub-manifold of

M

and

f

∈

X

r

0

,

1

(

M

)

is omplete. Then the following assertions

hold.

a)Assumethat

r

≥

2

and

m

≥

n

+

s

. Thenfuntions

h

belongingto

C

r

(

M,

R

)

suhthat

Θ

f,h

isimmersiveateahpointof

K

,onstitutearesidualset. This set isalsoopen,if

K

islosed.

b)Assumethat

m

≥

2

s

+ 1

. Thenthesetoffuntions

h

belongingto

C

r

(

M,

R

)

suh that

Θ

f,h

is injetive (respetively an injetive immersion) on

K

, is

residual(respetively,if

r

≥

2)

. Itontainsanopenset,if

K

is losed. Supposing

m

≤

2

s

and

r >

2

s

−

m

thesameresultsholdforthesetoffuntions

h

belongingto

C

r

(

M,

R

)

suhthatthe

Θ

-unobservablepointsof

K

×

K

belong to asubmanifoldofdimension

2

s

−

m

.

Suppose moreover that

M

is ompat and

f

∈

X

r

2

(

M

)

. Then the assertions

alsoholdfor

τ

=

∞

.

Proof:

Ad a) Sine

m

≥

n

+

s

, the set of funtions

h

belonging to

C

r

(

M,

R

)

suh that

Θ

f,h

isan immersion at eah point of

K

, oinides with

H

1

(

K

)

, i.e.,

d

Θ

f,h

|

T

1

K

is transversal to

{

0

} ∈

R

m

. Therefore the assertion

followsimmediatelyfrom thepreviouslemma.

Ad b) Aording to the previous lemma

H

0

(

K

)

is residual and ontains an

open set,if

K

is losed. If

m

≥

2

s

+ 1

, thentheset of funtions

h

belonging to

C

r

(

M,

R

)

suhthattherestritionof

Θ

f,h

to

K

isinjetive(respetivelyan

injetiveimmersion), is preisely

H

0

(

K

)

(respetively

H

0

(

K

)

∩

H

1

(

K

)

). The statementontheindistinguishablesetfollowsfromthepreimagetheorem.

Ifthestatespaeisompat,wegetsharperresults. Inpartiularthefollowing

theoremisimportant,sineembeddingofthestatespaegivesinformationon limitbehaviorofthesystem.

Theorem 2.2

Suppose that

m

≥

2

n

+ 1

,

f

∈

X

r

0

,

1

(

M

)

and

M

is

om-pat. Then output funtions

h

∈

C

r

(

M,

R

)

suh that

Θ

f,h

is an embedding

onstituteanopenanddenseset.

If wefurther assumethat

f

∈

X

r

2

(

M

)

, then theassertionremainstrue in the

ase

τ

=

∞

.

Proof:

Reall that by ompatness of

M

an injetive immersion is also an embedding (ase

2

≤

r

≤ ∞

in Theorem 2.1) and an injetive mapping is also a topologial embedding (ase

r

= 1

). Hene density follows from

Theorem2.1. Opennessisaonsequeneofthefatthatbyompatnessof

M

themapping

C

r

(

M,

R

)

×

T

0

M

∋

h

7→

∆Θ

f,h

∈

C

r

(

M,

R

m

)

isontinuousand

the set of embeddings

Emb

r

(

M,

R

m

) :=

{

F

∈

C

r

(

M,

R

m

) :

F

isembedding

}

isopen.

Nextweshallproveresidualityof

X

r

0

,

1

(

M

)

byusingtheharaterizationof

sim-pliityand

Θ

-simpliityofasingularity,intermsoftransversalnonintersetion.

Lemma 2.4

Assume that

m

≥

n

. Then

X

r

0

,

1

(

M

)

is open and dense in

X

r

(

M

)

. Moreover,theassertionholdsalsofor

τ

=

∞

,ifwerestritthevetor elds to

X

r

(

M, a

)

.

Proof:

We give a proof for

τ <

∞

. The arguments for

τ

=

∞

are similar. It suies to show that

X

r

1

(

M

)

is open and dense. Let

O

resp.

U

denote the set of

n

-dimensional

Θ

-observable resp. unobservable linear

systems. Let

f

∈

X

r

1

(

M

)

,

x

0

∈

Sing

(

f

)

and

d

x0

f

denote theprinipialpartof

thelinearizationof

f

.

Note that

f

∈

X

r

1

(

M

)

if and only if there exists a

v

∈

T

x0

M

suh that the

system

(

d

x0

f, v

)

avoidsthesetof

Θ

-unobservablelinearsystemson

T

x0

M

.

We now give a loal haraterization. Let

ψ

:

U

→

R

n

be a loal hart

suh that

U

has ompat losure and

ψ

(

x

0

) = 0

. The tangent mapping

dψ

:

T U

→

R

n

×

R

n

denedby

dψ

(

v

) = (

ψ

(

x

)

, d

x

ψv

)

with

x

=

π

(

v

)

isaloal

hartfor

T M

. Let

P r

2

denotetheprojetion

R

n

×

R

n

∋

(

x, w

)

7→

w

. Consider

themapping

ξ

f

:=

P r

2

dψf

◦

ψ

−

1

on

ψ

(

U

)

. Hene

d

0

ξ

f

=

d

x0

ψd

x0

f d

0

ψ

−

1

. The

singularity

x

0

is

Θ

-simpleifand onlyif

(

d

0

ξ

f

, d

x0

v

)

∈

/

U

forsome

v

∈

T

x0

M

. Obviously

U

is losed, analyti and

6

=

End

(

R

n

)

×

R

n

, hene nite union of

losed positive odimensional real analyti submanifold of

End

(

R

n

)

×

R

n

. Givenapair

(

G, w

)

∈

O,

obviouslythere existsavetoreld

g

∈

X

r

(

M

)

suh

that

g

and

f

oinide on

M

\

U

,

x

0

∈

Sing

(

g

)

and

d

x0

ξ

g

=

G

. Animmediate appliation (details are similar to those in the proof of lemma 2.3) of the

Weannowprovethemainresultongeneriobservabilitywithrespettothe

mapping

Θ

.

Theorem 2.3

Suppose that

m

≥

2

n

+ 1

and

M

is ompat. Then pairs

(

f, h

)

in

X

r

(

M

)

×

C

r

(

M,

R

)

suh that

Θ

f,h

is an embedding onstitute an

open and dense subsetof

X

r

(

M

)

×

C

r

(

M,

R

)

. Restriing thevetorelds to

X

r

2

(

M

)

, theassertionremainsvalidfor

τ

=

∞

aswell.

Proof:

Reall that

X

r

0

,

1

(

M

)

is open and dense. Furthermore

X

r

2

(

M

)

is

open in

X

r

(

M, a

)

. The assertions follow immediately from this fats and

theorem2.2.

Similarily,generiityresultsfornonompatstatespaesanbederiven,ifwe

restritourselfto thesetof ompletevetorelds, replaeopenand denseby

residual andembedding by injetiveimmersion. We remarkalso that

onsid-eringtheobservationmapping

Θ

g,h

(

x

) :=

0

X

k

=

−

N

e

−

kA

bh

(

g

k

(

x

))

for

x

∈

M

and

(

g, h

)

∈

Dif f

r

(

M

)

×

C

r

(

M,

R

)

with

g

k

:=

g

◦

g

k

−

1

,

orrespond-ingresultsfordisretedynamialsystemsanbeprovedlikewise.

3

Concluding Remarks

Remark 3.1

Sinetheset of

m

-dimensional ontrollablelinearltersisopen anddensein

End

(

R

m

)

×

R

m

,allgeneriityresultsofthelastsetionholdalso

generiallywithrespettothelinearlters.

Thefollowingexamplesshowthattheonditionsontheobservationdimension

arealsoneessary,thus

m

≥

2

n

forgeneriloalobservabilityand

m

≥

2

n

+ 1

forgeneriobservabilityannotbeweakend.

Example 3.1

Let

M

=

S

1

and

f

(

ϕ

) = 1

, where

ϕ

denotes the standard

angular oordinate of the irle. Furthermore onsider the pair

(

λ, b

)

with

λ <

0

and

b

6

= 0

. Taking

τ

= 2

π

andtheoutputfuntion

h

(

ϕ

) =

cos ϕ

leadsto

Θ

f,h

(

ϕ

) =

1

−

e

2

λπ

1+

λ

2

(

λcos ϕ

−

sin ϕ

)

, whihis notan immersion. Moreoverthe

zeroof

d

Θ

f,h

at

ϕ

0

=

−

arctan

1

λ

istransversal. Hene thenonimmersivityof

Θ

f,h

ispreservedundersmallperturbationsoftheoutputfuntion,thevetor

eld andthelinearlter.

Example 3.2

Let

M

,

f

,

ϕ

and

τ

beasinthepreedingexample. Furthermore let

A

=

diag

(

−

1

,

−

2)

,

b

= (1

−

e

−

2

π

)

−

1

(1

,

1)

T

Thenastraightforwardomputationyields

Θ

f,h

(

ϕ

) =

cos ϕ

+

sin ϕ

+

cos

2

ϕ

+ 2

sin

2

ϕ

(

e

−

2

π

+ 1)(

2

5

(2

cos ϕ

+

sin ϕ

) +

5

4

(

cos

2

ϕ

+

sin

2

ϕ

))

.

Thefollowinggureshowstheimageof

S

1

by

Θ

f,h

.

−3

−2

−1

0

1

2

3

4

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

X

Y

Imageof

S

1

bytheontinuouslinearltermappinggivenby

ϕ

7→

(cosϕ+sinϕ+cos2ϕ+2sin2ϕ,

(e

−

2

π

+1)(

2

5

(2cosϕ+sinϕ)+

5

4

(cos2ϕ+sin2ϕ))

T

inthe

XY

-plane

Theselntersetion oftheimageis transversal. Hene thenoninjetivenessof

Θ

f,h

ispersistentundersmallperturbationsoftheoutputfuntion,thevetor

eld andthelinearlter.

Forinstane, smallperturbationsof

h

donotresultininjetivity, i.e.,thereis

an

ǫ >

0

suhthat foreahoutputfuntion

˜

h

,whihis

C

r

nearto

h

within

ǫ

,

themapping

Θ

f,

˜

h

isnotinjetive.

Notethattheonsideredsystemisalsounobservablewithrespettothe high-gainmappinggivenby

ϕ

7→

(2

cos ϕ

+ 5

cos

2

ϕ,

2

sin ϕ

−

10

sin

2

ϕ

)

T

aswellas

samplingmapping

ϕ

7→

(2

cos

(

ϕ

+

t

1

) + 5

cos

(2

ϕ

+ 2

t

1

,

2

cos

(

ϕ

+

t

2

) + 5

cos

(2

ϕ

+

2

t

2

))

T

withsamplingtimes

t

1

, t

2

. Thefollowinggureshowstheimage ofthe

−6

−4

−2

0

2

4

6

8

−6

−4

−2

0

2

4

6

8

X

Y

Imageof

S

1

bythesamplingmappinggivenby

ϕ

7→

(2cos ϕ

+ 5cos

2ϕ,

−

2sin ϕ

−

5cos

2ϕ)

T

inthe

XY

-plane

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