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Massachusetts InstituteofTechnology

DepartmentofElectricalEngineeringandComputerScience

6.243j(Fall2003): DYNAMICSOFNONLINEARSYSTEMS

byA.Megretski

Lecture1: Input/OutputandState-SpaceModels

1

Thislecturepresentssomebasicdefinitionsandsimpleexamplesonnonlineardynamicalsystemsmodeling.

1.1 BehavioralModels.

Themostgeneral(thoughrarelythemostconvenient)waytodefineasystemisbyusing

a

behavioral

input/output

model.

1.1.1 What isasignal?

Intheselectures,asignalisalocallyintegrablefunctionz: R+ Rk,whereR+ denotes

thesetofallnon-negativerealnumbers. Thenotionoflocal integrabilitycomes fromthe Lebesque measure theory, and means simply that the function can be safely andmeaningfully integrated over finite intervals. Generalized functions, such as the deltafunction(t),arenotallowed. TheargumenttR+ ofasignalfunctionwillbereferredtoastime(whichitusuallyis).

Example1.1Functionz=z()definedby

t0.9sgn(cos(1/t)) fort>0,z(t)=

0 fort=0

1VersionofSeptember3,2003

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2

isavalidsignal,while

1/t

for

t

>

0,

z(t)

= 0 fort=0

andz(t)=(t)arenot.

Thedefinitionaboveformallycoverstheso-calledcontinuous time(CT)signals. Discretetime(DT)signalscanberepresentedwithinthis frameworkasspecialCTsignals.More precisely, a signal z : R+ R

k is called aDT signal if it is constant on everyinterval[k, k+1)wherek=0, 1, 2, . . . .

1.1.2 What isasystem?

Systemsareobjectsproducingsignals(calledoutputsignals),usuallydependingonothersignals

(inputs)

and

some

other

parameters

(initial

conditions).

In

most

applications,

mathematicalmodelsofsystemsaredefined(usuallyimplicitly)bybehaviorsets. Foranautonomoussystem(i.e. forasystemwithnoinputs),abehaviorsetisjustasetB={z}consistingofsomesignalsz: R+ R

k (kmustbethesameforallsignalsfromB). Forasystemwithinputvandoutputw,thebehaviorsetconsistsofallpossibleinput/outputpairsz = (v(), w()). There is no real difference between the two definitions, since thepairofsignalsz=(v(), w())canbeinterpretedasasinglevectorsignalz(t)= [v(t);w(t)]containingbothinputandoutputstackedoneovertheother.

Note that in this definition a fixed input v() may occur in many or in no pairs(v, w)B,whichmeansthatthebehaviorsetdoesnotnecessarilydefinesystemoutput

as

a

function

of

an

arbitrary

system

input.

Typically,

in

addition

to

knowing

the

input,onehastohavesomeotherinformation(initialconditionsand/oruncertainparameters)

todeterminetheoutputinauniqueway.

Example 1.2 The familiar ideal integrator system (the one with the transfer functionG(s) = 1/s) can be defined by its behavioral set of all input/output scalar signal pairs(v, w)satisfying

t2

w(t2)w(t1)= v()d, t1, t2

[0,).t1

Inthisexample,todeterminetheoutputuniquelyitissufficienttoknowvandw(0).

InExample1.1.2asystemischaracterisedbyanintegralequation. Thereisavariety

of

other

ways

to

define

the

same

system

(by

specifying

a

transfer

function,

by

writing

a

differentialequation,etc.)

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3

1.1.3 What isa linear/nonlinearsystem?

A

system

is

called

linear

if

its

behavior

set

satisfies

linear

superposition

laws,

i.e.

when

foreveryz1, z2 BandcRwehavez1 +z2 Bandcz1 B.Excludingsomeabsurdexamples2,linearsystemsarethosedefinedbyequationswhich

arelinearwithrespecttovandw. Inparticular,theidealintegratorsystemfromExample1.1.2islinear.

Anonlinearsystemissimplyasystemwhichisnotlinear.

1.2 SystemState.

It is important to realize that system state can be defined for an arbitrary behavioral

model

B

=

{z(}.

1.2.1 Twosignalsdefiningsamestateattime t.

System state atagiven time instance t0 is supposed to contain all informationrelatingpast(tt0)behavior. Thisleadsustothefollowingdefinitions.

Definition LetBbeabehaviorset. Signalsz1, z2 Baresaidtocommuteattimet0 ifthesignals

z1(t) fortt0,z12(t)= z2(t) fort>t0

and z2(t)

for

t

t0,

z21(t)= z1(t) fort>t0

alsobelongtothebehaviorset.

Definition LetB beabehaviorset. Signalsz1, z2 B aresaidtodefine same stateofB

at time t0 ifthesetofz B commutingwithz1 att0 isthesameasthesetofz Bcommutingwithz2 att0.

Definition Let B be a behavior set. LetX be any set. A functionx : R B Xis called a state of system B if z1 and z2 define same state of B at time t wheneverx(t, z1())=x(t, z2()).

Example

1.3

Consider

a

system

in

which

both

input

v

and

output

w

are

binary

signals,

i.e. DT signals taking values from the set {0, 1}. Define the input/output relation bythe following rules: w(t) = 1 only if v(t) = 1, and for every t1, t2 Z+ such that

2Suchasthe(linear)systemdefinedbythenonlinearequation(v(t)w(t))2=0t

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w(t1)=w(t2)=1andw(t)=0forallt(t1, t2) Z,thereareexactlytwointegerstin

the

interval

(t1, t2)

such

that

v(t)

=

1.

In other words, the system counts the 1s in the input and, every time the countreachesthree,thesystemresets itscountertozero,andoutputs1(otherwiseproducing0s).

It

is

easy

to

see

that

two

input/output

pairsz1 =(v1, w1)andz2 =(v2, w2)commute

ata(discrete)timet0

ifandonlyifN(t0, z1)=N(t0, z2),whereN(t0, z)forz=(v, w) Bis

the

number

of

1s

inv(t)fort(t0, t1) Z,wheret1 meansthenext(aftert0)integer

time t when w(t) = 1. Hence the state of the system can be defined by a functionx: R+ B {0, 1, 2},x(t, z)=N(t, z).

Inthisexample,knowingasystemstateallowsonetowritedownstatespaceequationsforthesystem:

x(t+

1)

=

f(x(t), v(t)),

w(t)

=

g(x(t), v(t)),

(1.1)

wheref(x, v)=(x+v)mod3,

andg(x, v)=1ifandonlyifx=2andv=1.

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Massachusetts

Institute

of

Technology

Department

of

Electrical

Engineering

and

Computer

Science


byA.Megretski

Lecture2: DifferentialEquationsAsSystemModels1

Ordinary differential requations (ODE) are the most frequently used tool for modelingcontinuous-time nonlinear dynamical systems. This section presens results on existenceofsolutionsforODEmodels,which,inasystemscontext,translateintowaysofprovingwell-posednessofinterconnections.

2.1 ODEmodelsandtheirsolutions

Ordinary

differential

equations

are

used

to

describe

responses

of

a

dynamical

system

to

allpossibleinputsandinitialconditions. Equationswhichdonothaveasolutionforsomevalidinputsandinitialconditionsdonotdefinesystemsbehaviorcompletely,and,hence,areinappropriateforuseinanalysisanddesign. This isthereasonaspecialattention ispaid

in

this

lecture

to

the

general

question

of

existence

of

solution

of

differential

equation.

2.1.1

ODE

and

their

solutions

An ordinary differential equation on a subset Z Rn R is defined by a functiona : Z Rn. Let T be a non-empty convex subset ofR (i.e. T can be a single pointset,oranopen,closed,orsemi-open interval inR). A functionx: T Rn iscalleda

solution

of

the

ODE x(t)=a(x(t), t) (2.1)

if(x(t), t)Z foralltT,and

t2

x(t2)x(t1)= a(x(t), t)dt t1, t2

T. (2.2)t1


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2

Thevariabletisusuallyreferredtoasthetime.

Note

the

use

of

an

integral

form

in

the

formal

definition

(2.2):

it

assumes

that

the

functionta(x(t), t)isintegrableonT,butdoesnotrequirex=x(t)tobedifferentiableatanyparticularpoint,whichturnsouttobeconvenientforworkingwithdiscontinuousinputsignals,suchassteps,rectangularimpulses,etc.

Example

2.1Letsgndenotethesignfunctionsgn: R{0, 1, 1}definedby

1, y>0,sgn(y)= 0, y=0,

1, y

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2.1.3

Well-posedness

of

standard

ODE

system

models

As

it

was

mentioned

before,

not

all

ODE

models

are

adequate

for

design

and

analysis

purposes. The notion of well-posedness introduces some typical constraints aimed atinsuring

their

applicability.

Definition A standard ODE model ODE(f, g) is called well posed if for every signal

v(t)V and foreverysolutionx1 : [0, t1] X of(2.4)withx1(0)X0 thereexistsasolutionx: R+ X of(2.4)suchthatx(t)=x1(t)forallt[0, t1].

The ODE from Example 2.1.1 can be used to define a standard autonomous ODEsystemmodel

x(t)=sgn(x(t)), w(t)=x(t),

whereV =X =X

0

=

R,f(x,v,t)=sgn(x)andg(x,v,t)=x. Itcanbeverifiedthat

this

autonomous

system

is

well-posed.

However,

introducing

an

input

into

the

model

destroyswell-posedness,asshowninthefollowingexample.

Example

2.2ConsiderthestandardODEmodel

x(t)=sgn(x(t))+v(t), w(t)=x(t), (2.6)

wherev(t)isanunconstrainedscalarinput. Here

V =X=X0 =R, f(x,v,t)=sgn(x)+v, g(x,v,t)=x.

Whilethismodelappearstodescribeaphysicallyplausiblesituation(velocitydynamicssubject

to

dry

friction

and

external

force

input

v),

the

model

is

not

well-posed.

Toprovethis, considerthe inputv(t) = 0.5=const. It issufficienttoshowthatnosolutionoftheODE

x(t)=0.5sgn(x(t))

satisfyingx(0)=0existsonatimeinterval[0, tf]fortf >0. Indeed,letx=x(t)besuchsolution. Asan integralofa bounded function, x=x(t) witllbe acontinuous functionoftime. Acontinuousfunctionoveracompactintervalalwaysachievesamaximum. Lettm [0, tf]beanargumentofthemaximumovert[0, tf].

Ifx(tm)>0thentm

>0and,bycontinuity,x(t)>0inaneighborhoodoftm,hencethereexists>0suchthatx(t)>0forallt[tm , tm]. Accordingtothedifferentialequation,

this

means

that

x(tm

)

=

x(tm)+ 0.5

>

x(tm),

which

contradicts

the

selection

oftm asanargumentofmaximum. Hencemaxx(t)=0. Similarly,minx(t)=0. Hencex(t)=0forallt. Buttheconstantzerofunctiondoesnotsatisfythedifferentlialequation.Hence,nosolutionexists.

It can be shown that the absense of solutions in Example 2.1.3 is caused by lack ofcontinuityoffunctionf =f(x,v,t)withrespecttox(discontinuitywithrespecttovandtwouldnotcauseasmuchtrouble).

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4

2.2 Existenceofsolutions forcontinuousODE

This

section

contains

fundamental

results

establishing

existence

of

solutions

of

differential

equationswithacontinuousrightside.

2.2.1

Local

existence

of

solutions

for

continuous

ODE

Inthissubsectionwestudysolutionsx: [t0, tf

]Rn ofthestandardODE

x(t)=a(x(t), t) (2.7)

(sameas(2.1)),subjecttoagiveninitialcondition

x(t0)

=

x0.

(2.8)

Herea: Z Rn isagivencontinuous function,definedonZ Rn R. Itturnsoutthatasolutionx=x(t)of(2.7)withinitialcondition(2.8)exists,atleastonasufficientlyshorttimeinterval,wheneverthepointz0 =(x0, t0)lies,inacertainsense,intheinteriorofZ.

Theorem

2.1 Assume thatforsomer>0

Dr(x0, t0)={( x, t)Rn

R: |x x0| r, t[t0, t0 +r]}

is

a

subset

of

Z.

Let

x, t)|: (M =max{|a( x, t)Dr(x0, t0)}.

Then,fortf =min{t0 +r/M,t0 +r},

there exists a solution x : [t0, tf] Rn of (2.7) satisfying (2.8). Moreover, any such

solutionalsosatisfies |x(t)x0|rforallt[t0, tf

].

Example

2.3TheODE

x(t)=c0 +c1cos(t)+x(t)2,

where

c0, c1 are given constants, belongs to the class ofRiccati equations, which play aprominent role in the linear system theory. According to Theorem 2.1, for any initialconditionx(0)=x0 thereexistsasolutionoftheRiccatiequation,definedonsometimeinterval [0, tf

] of positive length. This does not mean, however, that the corresponding

autonomous

system

model

(producing

output w(t) =x(t)) is well-posed, since such

solutionsarenotnecessarilyextendabletothecompletetimehalf-line[0, ).

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2.2.2

Maximal

solutions

If

x1 : [t0, t1]

Rn and

x2 : [t1, t2]

Rn are

both

solutions

of

(2.7),

and

x1(t1)

=

x2(t1),

thenthefunctionx: [t0, t2]Rn,definedby

x1(t), t[t0, t1],x(t)=x2(t), t[t1, t2],

(i.e.

the

result

of

concatenating x1 and x2) isalso a solutionof (2.7). This means that

somesolutionsof(2.7)canbeextendedtoalargertimeinterval.A solution x : T Rn of (2.7) is calledmaximal if there exists no other solution

x : T Rn for which T is a proper subset of T, and x(t) = x(t) for all t T. Inparticular,well-posednessofstandardODEsystemmodelscontainstherequirementthat

all

maximal

solutions

must

be

defined

on

the

whole

time-line

t

[0, ).

Thefollowingtheoremgivesausefulcharacterization ofmaximalsolutions.

Theorem

2.2 Let X beanopen subsetofRn. Let a : XR Rn bea continuous

function. Thenallmaximalsolutionsof(2.7)aredefinedonopenintervalsand,wheneversuch solution x : (t0, t1) X has afinite interval end t = t0 R or t = t1 R (as

opposed to t0 = or t1 = ), there exists no sequence tk (t0, t1) such that tkconverges to t whilex(tk)converges toa limit inX.

Inotherwords,intheabsenseofa-prioriconstraintsonthetimevariable,asolutionisnotextendableonlyifx(t)convergestotheboundaryofthesetonwhichaisdefined. Inthe

most

typical

situation,

the

domain

Z

of

f

in

(2.4)

is

Rn

R+,whichmeansnoa-prioriconstraintsoneitherxort. Inthiscase,accordingtoTheorem2.2,asolutionx=x(t)notextendableoverafinitetimeinterval[0, tf),tf

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discontinuous for a fixedfinite set t1

0such thatt0+r

|a(x1(t), t)a(x2(t), t)|dt0isintegrableovereveryfiniteinterval,andtheinequality

t1

t1

|t1/3x1(t)t1/3

x2(t)|dt t1/3dt max |x1(t)x2(t)|

0 0 t[0,t1]

holds.Onthecontrary,thedifferentialequation

t1x(t),

t

>

0x(t)= x(0)=x00, t=0,

doesnothaveasolutionon[0, )foreveryx0 =0. Indeed,ifx: [0, t1]Risasolutionforsomet1 >0then

d x(t)=0

dt t

forallt=0. Hencex(t)=ctforsomeconstantc,andx(0)=0.

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2.2.4

Differential

inclusions

Let

X

be

a

subset

of

Rn,

and

let

:

X

2Rn

be

a

function

which

maps

every

point

of

X toasubsetofRn. Suchafunctiondefinesadifferential inclusion

x(t)(x(t)). (2.9)

Byasolutionof (2.1)onaconvexsubsetT ofRwemeana functionx : T X suchthat

t2

x(t2)x(t1)= u(t)dt t1, t2 Tt1

for some integrable function u : T Rn satisfying the inclusion u(t) (x(t)) for

all

t

T.

It

turns

out

that

differential

inclusions

are

a

convenient,

though

not

always

adequate,wayofre-definingdiscontinuousODEtoguaranteeexistenceofsolutions.Itturnsoutthatdifferentialinclusion(2.9)subjecttofixedinitialconditionx(t0)=x0

hasasolutiononasufficientlysmallintervalT = [t0, t1]whenevertheset-valuedfunction iscompactconvexset-valuedandsemicontinuouswithrespecttoitsargument(plus,asusually,x0

mustbeaninteriorpointofX).

Theorem

2.4 Assume thatforsomer>0

(a) thesetBr(x0)={xR

n : |x x0|r}

is

a

subset

of

X;

xBr(x0) theset((b)forevery x)isconvex;

(c)for every sequenceof xk Br(x0) converging toa limit x Br(x0)andfor everysequenceuk

( xk

) thereexistsasubsequencek=k(q)asqsuch thatthesubsequence x).uk(q)

hasa limit in(

Thenthesupremum

M =sup{| x), xDr(x0, t0)}u|: u(

is

finite,

and,

for

tf =min{t0 +r/M,t0 +r},

there exists a solution x : [t0, tf] Rn

of (2.9) satisfying x(t0) = x0. Moreover, anysuchsolutionalsosatisfies |x(t)x0| rforallt[t0, tf

].

Thediscontinuousdifferentialequation

x(t)=sgn(x(t))+c,

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wherec isafixedconstant,canbere-definedasacontinuousdifferential inclusion(2.9)

by

introducing

{c 1}, y >0,(y)= [c 1, c+1], y=0,

{c+1}, y

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Massachusetts

Institute

of

Technology

Department

of

Electrical

Engineering

and

Computer

Science


byA.Megretski

Lecture3: ContinuousDependenceOnParameters1

Arguments based on continuity of functions are common in dynamical system analysis.They rarely apply to quantitative statements, instead being used mostly for proofs ofexistenceofcertainobjects (equilibria, open or closed invariantset, etc.) Alternatively,continuityargumentscanbeusedtoshowthatcertainqualitativeconditionscannotbesatisfiedforaclassofsystems.

3.1 UniquenessOfSolutions

InthissectionourmainobjectiveistoestablishsufficientconditionsunderwhichsolutionsofODEwithgiveninitialconditionsareunique.

3.1.1

A

counterexample

Continuityofthefunctiona: Rn Rn ontherightsideofODE

x(t)=a(x(t)), x(t0)= x0 (3.1)

doesnotguaranteeuniquenessofsolutions.

Example

3.1

The

ODE

x(t)=3|x(t)|2/3, x(0)=0

hassolutionsx(t)0andx(t)t3 (actually,thereare infinitelymanysolutions inthiscase).


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3.1.2

A

general

uniqueness

theorem

The

key

issue

for

uniqueness

of

solutions

turns

out

to

be

the

maximal

slope

of

a

=

a(x):

to guaranteeuniquenessontime intervalT = [t0, tf], it issufficienttorequireexistenceofaconstantM suchthat

|a( x2)| M|x1 x1)a( x2|

x1,forall x2 fromaneigborhoodofasolutionx: [t0, tf]Rn of(3.1). Theproofofboth

existence and uniqueness is so simple in this case that we will formulate the statementforamuchmoregeneralclassofintegralequations.

Theorem

3.1 LetX beasubsetofRn containingaball

x0)

=

{ x

Br( x

Rn :

| x0|

r}

ofradiusr>0,and let t1 >t0 berealnumbers. Assume thatfunctiona: X[t0, t1][t0, t1]R

n issuch that thereexistconstantsM, K satisfying

|a(x1, , t)a(x2, , t)|K|x1 x2| x1,x2 Br(x0), t0 tt1, (3.2)

and|a(x , ,t)|M xBr(x0), t0 tt1. (3.3)

Then,for a sufficiently small tf

> t0, there exists uniquefunction x : [t0, tf

] Xsatisfying t

x(t)

=

x0 +

a(x(), , t)d

t

[t0, tf].

(3.4)

t0

Aproofofthetheorem isgiven inthenextsection. Whenadoesnotdependonthethirdargument,wehavethestandardODEcase

x(t)=a(x(t), t).

Ingeneral,Theorem3.1coversavarietyofnonlinearsystemswithaninfinitedimensionalstate space, suchas feedback interconnectionsof convolution operators andmemorylessnonlinear

transformations.

For

example,

to

prove

well-posedness

of

a

feedback

system

in

whichtheforwardloopisanLTIsystemwithinputv,outputw,andtransferfunction

es 1G(s)= ,

s

andthefeedbackloopisdefinedbyv(t)=sin(w(t)),onecanapplyTheorem3.1with

sin(x)+h(t), t 1 t,a(x ,,t)=

h(t), otherwise,

whereh=h(t)isagivencontinuousfunctiondependingontheinitialconditions.

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3

3.1.3

Proof

of

Theorem

3.1.

First

prove

existence.

Choose

tf >

t1 such

that

tf

t0

r/M

and

tf

t0

1/(2K).

Definefunctionsxk

: [t0, tf

]X by t

x0, xk+1(t)= x0(t) x0 + a(xk(), , t)d.t0

By(3.3)andbytf t0 r/M wehavexk(t)Br(x0)forallt[t0, tf]. Henceby(3.2)andbytf t0 1/(2K)wehave

t

|xk+1(t)xk(t)| |a(xk(), , t)a(xk1(), , t)|dt0

t

K|xk()

xk1()|d

t0

0.5 max {|xk(t)xk1(t)|}.t[t0,tf

]

Thereforeonecanconcludethat

max {|xk+1(t)xk

(t)|}0.5 max {|xk

(t)xk1(t)|}.t[t0,tf

] t[t0,tf

]

Hence xk(t) converges exponentially to a limit x(t) which, due to continuity of a withrespoecttothefirstargument,isthedesiredsolutionof(3.4).

Now letusproveuniqueness. Notethat,duetotf t0 r/M,allsolutionsof(3.4)

must

satisfy

x(t)

Dr(x0)

for

t

[t0, tf].

If

xa and

xb are

two

such

solutions

then

t

|xa(t)xb(t)| |a(xa(), , t)a(xb(), , t)|dt0

t

K|xa()xb()|dt0

0.5 max {|xa(t)xb(t)|},

t[t0,tf

]

whichimmediatelyimplies

max {|xa(t)xb(t)|}=0.t[t0,t

f

]

The proof is complete now. Note that the same proof applies when (3.2),(3.3) arereplacedbytheweakerconditions

x1, , t)a( x1 x1, x0), t0 tt1,|a( x2, , t)|K()| x2| x2 Br(

andx,,t)|m(t) x0), t0

tt1,|a( xBr

(

wherethefunctionsK()andM()areintegrableover[t0, t1].

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3.2 ContinuousDependenceOnParameters

In

this

section

our

main

objective

is

to

establish

sufficient

conditions

under

which

solutions

ofODEdependcontinuouslyoninitialconditionsandotherparameters.Consider

the

parameterized

integral

equation

t

x(t, q)= x0(q)+ a(x(, q), , t , q )d, t[t0, t1], (3.5)t0

where q R is a parameter. For every fixed value of q integral equation (3.5) has theformof(3.4).

Theorem

3.2 Letx0 : [t0, tf] R

n beasolutionof(3.5)withq=q0. Forsomed>0

let

Xd ={ xRn : t[t0, tf] : |x x0(t)|0thereexists>0such that

|x0(q1) x0(q2)| q1, q2 (q0 d, q0 +d): |q1 q2|

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3.3 Implicationsofcontinuousdependenceonparameters

This

section

contains

some

examples

showing

how

the

general

continuous

dependence

of solutions on parameters allows one to derive qualitative statements about nonlinearsystems.

3.3.1

Differential

flow

Consideratime-invariantautonomousODE

x(t)=a(x(t)), (3.8)

wherea:Rn Rm issatisfiestheLipschitzconstraint

|a( x2)| M|x1 x1)a( x2| (3.9)

on every bounded subset ofRn. According to Theorem 3.1, this implies existence anduniqueness of a maximal solution x : (t, t+) R

n of (3.8) subject to given initial

conditionsx(t0) = x0 (bythisdefinition, t 0suchthat |x(t, x0|

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In

other

words,

all

solutions

starting

sufficiently

close

to

an

asymptotically

stableequilibrium x0

convergeto itast,andnoneofsuchsolutionscanescape farawaybeforefinallyconvergingtox0.

Theorem

3.3 Let x0 R

n be an asymptotically stable equilibrium of (3.8). The set

x0)ofall x) A=A( xRn such thatx(t, x0

as t isanopensubsetofRn,anditsboundary is invariantunderthetransformations x).xx(t,

Theproofofthetheoremfollowseasilyfromthecontinuityofx(,).

3.3.3

Limit

points

of

a

trajectory

x0 Rn

,

the

set

of

all

possible

limits x(tk, as k , whereFor a fixed x0) x

the sequence {tk

} also converges to infinity, is called the limit set of the trajectorytx(t,x0).

Theorem

3.4 The limit set of a given trajectory is always closed and invariant under

the transformations x).xx(t,

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Lecture4: AnalysisBasedOnContinuity 1

This lecturepresentsseveraltechniquesofqualitativesystemsanalysisbasedonwhat isfrequently called topological arguments, i.e. on the arguments relying on continuity offunctionsinvolved.

4.1 Analysisusinggeneraltopologyarguments

Thissectioncoversresultswhichdonotrelyspecificallyontheshapeofthestatespace,

and

thus

remain

valid

for

very

general

classes

of

systems.

We

will

start

by

proving

generalizationsoftheoremsfromtheprevious lecturetothecaseofdiscrete-timeautonomoussystems.

4.1.1

Attractor

of

an

asymptotically

stable

equilibrium

Consideranautonomoustimeinvariantdiscretetimesystemgovernedbyequation

x(t+1)=f(x(t)), x(t)X, t=0, 1, 2, . . . , (4.1)

where X is a given subset ofRn, f : X X is a given function. Remember that fis called continuous if f(xk) f(x) as k whenever xk, x X are such thatxk x ask). Inparticular,thismeansthateveryfunctiondefinedonafinitesetX iscontinuous.

Oneimportantsourceofdiscretetimemodelsisdiscretizationofdifferentialequations.RnAssumethatfunctiona: Rn issuchthatsolutionsoftheODE

x(t)=a(x(t)), (4.2)


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with x(0)=x existandare unique onthetime interval t [0, 1] forall x Rn. Then

discrete

time

system

(4.1)

with

f()

=

x(1,)

describes

the

evolution

of

continuous

time

x xsystem(4.2)atdiscretetimesamples. Inparticular,ifaiscontinuousthensoisf.LetuscallapointintheclosureofX locallyattractiveforsystem(4.1)ifthereexists

d >0suchthatx(t) x0

ast foreveryx=x(t)satisfying(4.1)with|x(0)x0|< d.Note

that

locally

attractive

points

are

not

necessarily

equilibria,

and,

even

if

they

are,

theyarenotnecessarilyasymptoticallystableequilibria.x0 R

n the

setA=A( x X in(4.1)whichdefineaFor x0)ofall initialconditions

solutionx(t)convergingto xx0

ast iscalledtheattractorof 0.

Theorem

4.1 Iff iscontinuousand x0 is locallyattractivefor(4.1) then theattractor

A=A(x0)isa(relatively)opensubsetofX,anditsboundaryd(A)(inX)isf-invariant,

i.e.

f()

d(A)

whenever

x x d(A).

RememberthatasubsetY XRn iscalledrelativelyopeninX ifforeveryy Ythereexistsr >0suchthatallx X satisfying|x y|< rbelongtoY. AboundaryofasubsetY XRn inXishesetofallx Xsuchthatforeveryr >0thereexisty YandzX/Y suchthat |y x|< r andz x|< r. Forexample,thehalf-open intervalY =(0, 1]isarelativelyclosedsubsetofX=(0, 2),and itsboundary inX consistsofasinglepointx=1.

Example

4.1 Assume system (4.1), defined on X = Rn by a continuous function

f : Rn Rn, is such that all solutions with |x(0)|

100

converge

to

infinity

as

t

.

Then,

according

to

Theorem

4.1,

the

boundary

of

the

attractor

A=

A(0)

is

a

non-empty

f-invariant

set.

By

assumptions, 1 |x| 100 for all x A(0). Hence we can conclude that there existsolutionsof(4.1)whichsatisfytheconstraints1 |x(t)| 100forallt.

Example

4.2Forsystem(4.1),definedonX=Rn byacontinuousfunctionf : Rn

Rn,itispossibletohaveeverytrajectorytoconvergetooneoftwoequilibria. However,itisnotpossibleforbothequilibriatobelocallyattractive. Otherwise,accordingtoTheorem4.1,Rn wouldberepresentedasaunionoftwodisjointopensets,whichcontradictsthenotionofconnectednessofRn.

4.1.2

Proof

of

Theorem

4.1

Accordingtothedefinitionoflocalattractiveness,thereexistsd >0suchthatx(t)x0as t for every x = x(t) satisfying (4.1) with |x(0)x0| 0such

x xthat |x(t1) x1(t1)|< d/2whenever |x(0)1|< . Sincethis implies |x(t1)0|< d,

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wehave x x X suchthat |x 1|< ,whichprovesthatA=A(0)x A(0)forevery x x

is

open.

Toshowthatd(A) isf-invariant,notefirstthatA is itselff-invariant. Nowtakeanarbitrary x d(A). By the definition of the boundary, there exists a sequence xk A

x xk)convergestof(convergingto . Hence,bythecontinuityoff,thesequencef( x). Iff()A,thisimpliesf(x x)d(A). Letusshowthattheoppositeisimpossible. Indeed,if f() A then, since A is proven open, there exists > 0 such that z A for everyxz X such that |zf()| 0 such thatx|f(y)f( xx)|

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(c) the limit set is a union of trajectories of maximal solutions x : (t1, t2) R2 of

(4.2),

each

of

which

has

a

limit

(possibly

infinite)

as

t

t1 or

t

t2.

TheproofofTheorem4.3 isbasedonthemorespecifictopologicalarguments,tobediscussedinthenextsection.

4.2 Map index insystemanalysis

The

notion

of index of a continuous function is a remarkably powerful tool for proving

existenceofmathematicalobjectswithcertainproperties,and,assuch, isveryusefulinqualitativesystemanalysis.

4.2.1

Definition

and

fundamental

properties

of

index

Forn=1, 2, . . . letSn ={xRn+1 : |x|=1}

denote the unit sphere inRn+1. Note the use of n, not n+1, in the S-notation: itindicatesthat locallythesphereinRn+1 lookslikeRn. Thereexistsawaytodefinetheindex ind(F) of every continuous map F : Sn Sn in such a way that the followingconditionswillbesatisfied:

(a)

ifH : Sn [0, 1]Sn iscontinuousthen

ind(H(, 0))=ind(H(, 1))

(such

maps

H

is

called

a

homotopy

between

H(, 0)

and

H(, 1));

(b) ifthemapF : Rn+1 Rn+1 definedby

F(z)=|z|F(z/|z|)

iscontinuouslydifferentiableinaneigborhoodofSn then

ind(F)= det(Jx(F))dm(x),xSn

whereJx(F)istheJacobianofF atx,andm(x)isthenormalizedLebesquemeasureonSn (i.e. m is invariantwithrespecttounitarycoordinatetransformations,and

the

total

measure

of

Sn

equals

1).

Once it is proven that the integral in (b) is always an integer (uses standard vol-ume/surface integrationrelations), it iseasytoseethatconditions(a),(b)define ind(F)correctly anduniquelly. For n=1, the index of acontinuous map F : S1 S1 turnsouttobesimplythewindingnumberofF, i.e. thenumberofrotationsaroundzerothetrajectoryofF makes.

Itisalsoeasytoseethatind(FI)=1fortheidentitymapFI(x)=x,andind(Fc)=0foreveryconstantmapFc(x)=x0 =const.

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4.2.2

The

Browers

fixed

point

theorem

One

of

the

classical

mathematical

results

that

follow

from

the

very

existence

of

the

index

functionisthefamousBrowersfixedpointtheorem,whichstatesthatforeverycontinuousfunction

G: Bn Bn,where

Bn ={xRn+1 : |x|1},

equationF(x)=xhasatleastonesolution.The statement is obvious (though still very useful) when n=1. Let us prove it for

n > 1, starting with assume the contrary. Then the map G : Bn Bn which mapsxBn to the point of Sn1 which isthe (unique) intersection of theopenray startingfromG(x)andpassingthroughxwithSn1. ThenH : Sn1 [0, 1]Sn1 definedby

H(x, t)=G(tx)

is a homotopy between the identity map H(, 1) and the constant map H(, 0). Due toexistenceoftheindexfunction,suchahomotopydoesnotexist,whichprovesthetheorem.

4.2.3

Existence

of

periodic

solutions

Let a : Rn R Rn be locally Lipschitz and T-periodic with respect to the secondargument,i.e.

x, t+T)=a(a( x, t) x, t

where

T

>

0

is

a

given

number.

Assume

that

solutions

of

the

ODE

x(t)=a(x(t), t) (4.3)

withinitialconditionsx(0)Bn remaininBn foralltimes. Then(4.3)hasaT-periodicsolutionx=x(t)=x(t+T)foralltR.

x x(T, 0,Indeed,themap x) isacontinuousfunctionG: Bn Bn. Thesolutionx=G(of x)definestheinitialconditionsfortheperiodictrajectory.

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Lecture5: LyapunovFunctionsandStorageFunctions1

This lecture gives an introduction into system analysis using Lyapunov functions andtheirgeneralizations.

5.1 RecognizingLyapunov functions

ThereexistsanumberofslightlydifferentwaysofdefiningwhatconstitutesaLyapunov

function

for

a

given

system.

Depending

on

the

strength

of

the

assumptions,

a

variety

of

conclusionsaboutasystemsbehaviorcanbedrawn.

5.1.1

Abstract

Lyapunov

and

storage

functions

Ingeneral,Lyapunovfunctionsarereal-valuedfunctionsofsystemsstatewhicharemonotonically non-increasing on every signal from the systems behavior set. More generally,stotagefunctionsarereal-valuedfunctionsofsystemsstateforwhichexplicitupperboundsofincrementsareavailable.

Let B = {z} be a behavior set of a system (i.e. elements of B are are vector signals, which represent all possible outputs for autonomous systems, and all possible in-put/output

pairs

for

systems

with

an

input).

Remember

that

by

a

state

of

a

system

we

mean

a

function

x

:

B [0, )

X

such

that

two

signals

z1, z2 B define samestateofBattimetwheneverx(z1(), t)=x(z2(), t)(seeLecture1notesfordetailsandexamples). HereX isasetwhichcanbecalledthestatespaceofB. Notethat,giventhebehaviorsetB,statespaceX isnotuniquellydefined.


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Definition A real-valued function V : X R defined on state space X of a system

with

behavior

set

B

and

state

x

:

B

[0, )

X

is

called

a

Lyapunov

function

if

tV(t)=V(x(t))=V(x(z(), t))isanon-increasingfunctionoftimeforeveryzB.

According to this definition, Lyapunov functions provide limited but very explicitinformationaboutsystembehavior. Forexample, if X =Rn and V(x(t))= |x(t)|2 isaLyapunov function then we now that system state x(t) remains bounded for all times,thoughwemayhavenoideaofwhattheexactvalueofx(t)is.

Forconservativesystems inphysics,thetotalenergy isalwaysaLyapunov function.Even fornon-conservativesystems, it is frequently importantto look forenergy-likeexpressionsasLyapunovfunctioncandidates.

OnecansaythatLyapunovfunctionshaveanexplicitupperbound(zero)imposedontheirincrementsalongsystemtrajectories:

V(x(z(), t1))V(x(z(), t0))0 t1 t0 0, zB.

Ausefulgeneralizationofthisisgivenbystoragefunctions.

Definition Let B be a set of n-dimensional vector signals z : [0, ) Rn. Let

: Rn Rbeagivenfunctionsuchthat(z(t))islocallyintegrableforallz()B. A

real-valuedfunctionV : X RdefinedonstatespaceX ofasystemwithbehaviorsetBandstatex: B[0, )X iscalledastoragefunctionwithsupplyrate if

t1

V(x(z(), t1))V(x(z(), t0)) (z(t))dt t1 t0 0, zB. (5.1)

t0

Inmanyapplications isafunctioncomparingtheinstantaneousvaluesofinputandoutput. Forexample, ifB ={z(t) = [v(t);w(t)]} is thesetofall possible input/outputpairs of a given system, existence of a non-negative storage function with supply rate(z(t))=|v(t)|2 |w(t)|2 provesthatpoweroftheoutput,definedas

1 t

2w()p = lim sup |w()|2d,

TtT t 0

neverexceedpoweroftheinput.

Example

5.1

Let

behavior

set

B

=

{(i(t), v(t))}

descrive

the

(dynamcal)

voltage-current

relation of a passive single port electronic circuit. Then the total energy E = E(t)accumulatedinthecircuitcanserveasastoragefunctionwithsupplyrate

(i(t), v(t))=i(t)v(t).

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5.1.2

Lyapunov

functions

for

ODE

models

It

is

important

to

have

tools

for

verifying

that

a

given

function

of

a

systems

state

is

monotonicallynon-increasingalongsystemtrajectories,withoutexplicitlycalculatingsolutions of system equations. For systems defined by ODE models, this can usually bedone.

ConsideranautonomoussystemdefinedbyODEmodel

x(t)=a(x(t)), (5.2)

wherea: X Rn isafunctiondefinedonasubsetofRn. AfunctionalV : X R isaLyapunov functionforsystem(5.2) ift V(x(t)) ismonotonicallynon-increasing foreverysolutionof(5.2). Rememberthatx: [t0,t1]X iscalledasolutionof(5.2)ifthe

composition

a

x

is

absolutely

integrable

on

[t0,

t1]

and

equality

t

x(t)=x(t0)+ a(x())dt0

holdsforallt[t0,t1].TocheckthatagivenfunctionV isaLyapunovfunctionforsystem(5.2),oneusually

attemptstodifferentiateV(x(t))withrespecttot. IfX isanopenset,andbothV andxaredifferentiable(notethatthedifferentiabilityofxisassuredbythecontinuityofa),thecompositiontV(x(t))isalsodifferentiable,andthemonotonicityconditionisgivenby

V

( x)

0

x)a( x

X,

(5.3)

whereV(x)denotesthegradientofV atx.Insomeapplicationsonemaybeforcedtoworkwithsystemsthathavenon-differentiable

solutions (for example, because of ajump in an external input signal). The convenientLyapunov functioncandidatesV mayalsobenon-differentiableatsomepoints. Insuchsituations, it istemptingtoconsider,forevery xX inxX,thesubgradientofV at thedirectiona(x). Onemayexpectthatnon-positivityofsuchsubgradients,whichcanbeexpressedas

V( x))V(x+ta( x)

lim sup 0 xX, (5.4)0,>0 0

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familyT ={T}ofopendisjointintervalsT [0, 1]oftotallength1. Indeed,forafixed

Kantor

function

k

define

V( x)+k(1floor(x)=floor( x)),

x) denotes the largest integer not larger than x x) be zero on everywhere floor( . Let a(interval(m +t1, m+t2),wheremisanintegerand(t1, t2)T,anda(x)=0otherwise.Thenx(t)tisasolutionofODE(5.2),butV(x(t))isstrictlymonotonicallyincreasing,

x+ta(despite the fact that t V( x)) is constant in a neigborhood of t = 0 for every

x

R.

However,ifV andallsolutionsof(5.2)aresmoothenough,condition(5.4)issufficient

for

V

to

be

a

Lyapunov

function.

Theorem

5.1 IfX isanopensetinRn,V : X RislocallyLipschitz,a: X Rn is

continuous,andcondition(5.4) issatisfied thenV(x(t)) ismonotonicallynon-increasingforallsolutionsx: [t0, t1]X of(5.2).

Proof Wewillusethefollowingstatement: ifh: [t0, t1]Riscontinuousandsatisfies

h(t +)h(t)lim sup 0 t[t0, t1), (5.5)

d0,d>0(0,d)

then h ismonotonically non-increasing. Indeed, for every r > 0 let hr(t) = h(t)rt.

If

hr is

monotonically

non-increasing

for

all

r

>

0

then

so

is

h.

Otherwise,

assume

thathr(t3)>hr(t2) forsomet0 t2 0. Lett4 bethemaximalsolutionof

equationhr(t)=hr(t2)witht[t2, t3]. Thenhr(t)>hr(t4)forallt(t4, t3],andhence(5.5)isviolatedatt=t4.

Now let M be the Lipschitz constant for V in a neigborhood of the trajectory of x.Sinceaiscontinuous,

x(t +)x(t) lim a(x(t))=0 t.

0,>0

Hencethemaximum(overt[t0, t1 ])of

V(x(t +))V(x(t)) V(x(t)+a(x(t)))V(x(t)) V(x(t +))V(x(t)+a(x(t)))= +

V(x(t)+a(x(t)))V(x(t)) x(t +)x(t)a(x(t))

+

M

converges

to

a

non-positive

limit

as

0.

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Atime-varyingODEmodel

x1(t)

=

a1(x1(t),

t)

(5.6)

canbeconvertedto(5.2)byintroducing

x(t)

= [x1(t);t], a([barx;])=[a1(x,);1],

inwhichcasetheLyapunovfunctionV =V(x(t))=V(x1(t),t)cannaturallydependontime.

5.1.3

Storage

functions

for

ODE

models

ConsidertheODEmodel

x(t)

=

f(x(t),

u(t))

(5.7)

with statevector x(t) X Rn, input u(t) U Rm, where f : XU Rn is agiven function. Let : XU R be agiven functional. A function V : X R iscalledastoragefunctionwithsupplyrateforsystem(5.7)

t1

V(x(t1))V(x(t0)) (x(t),u(t))dt

t0

for every pair of integrable functions x : [t0,t1] X, u : [t0,t1] U such that thecompositiontf(x(t),u(t))satisfiestheidentity

tx(t)

=

x(t0)

+

f(x(t),

u(t))dt

t0

forallt[t0,t1].WhenX isanopenset,f andarecontinuous,andV iscontinuouslydifferentiable,

verifyingthatagivenf isavalidstoragefunctionwithsupplyrate isstraightforward:itissufficienttocheckthat

V f( u)( u) uU.x, x, xX,

WhenV islocallyLipschitz,thefollowinggeneralizationofTheorem5.1isavailable.

Theorem

5.2 IfX isanopensetinRn,V : X RislocallyLipschitz,f,: XU

Rn are

continuous,

and

condition

V( x, x)x+tf( u))V(

x, xX,lim sup ( u) uU (5.8)

0,>0 0

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Lecture

6:

Storage

Functions

And

Stability

Analysis1

This lecture presents results describing the relation between existence of Lyapunov orstoragefunctionsandstabilityofdynamicalsystems.

6.1 Stabilityofanequilibria

InthissectionweconsiderODEmodels

x(t)=a(x(t)), (6.1)

wherea: X Rn isacontinuousfunctiondefinedonanopensubsetX ofRn. Remem-berthatapoint x0)=0,i.e. ifx(t)x0 Xisanequilibriumof(6.1)ifa( x0 isasolutionof (6.1). Depending on the behavior of other solutions of (6.1) (they may stay close tox0,orconvergeto x0 ast,orsatisfysomeotherspecifications)theequilibriummaybecalledstable,asymptoticallystable,etc. Varioustypesofstabilityofequilibriacanbederived using storage functions. On the other hand, in many cases existence of storagefunctionswithcertainpropertiesisimpledbystabilityofequilibria.

6.1.1

Locally

stable

equilibria

Remember

that

a

point

x0

X

is

called

a

(locally)

stable

equilibrium

of

ODE

(6.1)

if

forevery >0thereexists >0suchthatallmaximalsolutionsx=x(t)of(6.1)withx0| aredeinfedforallt0,andsatisfy|x(t)0|< forallt0.|x(0) x

Thestatementbelowusesthenotionofalowersemicontinuity: afunctionf : Y R,defined

on

a

subsetY ofRn,iscalledlowersemicontinuousif

lim inf f()f( xx x) Y.r0,r>0xY: | x

|

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Theorem

6.1 x0 X is a locally stable equilibrium of (6.1) if and only if there exist

c

>

0

and

a

lower

semicontinuous

function

V

:

Bc(x0)

R,

defined

on

x0)={ x00suchthat

x) x V(min{,c/2})>V( x: | x0|

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Forthecaseofa linearsystem,however, localstabilityofequilibrium x0 =0 implies

existence

of

a

Lyapunov

function

which

is

a

positive

definite

quadratic

form.

Theorem

6.2 Ifa: Rn Rn isdefinedby

a()=Axx

whereAisagivenn- by-nmatrix,thenequilibriumx0 =0of(6.1)islocallystableifandonly if thereexistsamatrixQ=Q >0such thatV(x(t))=x(t)Qx(t) ismonotonically

non-increasingalongthesolutionsof(6.1).

Theproofofthistheorem,whichcanbebasedonconsideringaJordanformofA, is

usually

a

part

of

a

standard

linear

systems

class.

6.1.2

Locally

asymptotically

stable

equilibria

A point x0 is called a (locally) asymptotically stable equilibrium of (6.1) if it is a stableequilibria, and, in addition, there exists e0 > 0 such that every solution of (6.1) with

x0|< 0 convergesto |x(0) x0 ast.

Theorem

6.3 IfV : X R isacontinuousfunctionsuchthat

V( x) xx0)< V( xX/{0},

and

V(x(t))

is

strictly

monotonically

decreasing

for

every

solution

of

(6.1)

except

x(t)

x0 then x0 isa locallyasymptoticallystableequilibriumof(6.1).

Proof From Theorem 6.1, x0 is a locally stable equilibrium. It is sufficient to show

that every solution x = x(t) of (6.1) starting sufficiently close to x0 will converge tox0 as t . Assume the contrary. Then x(t) is bounded, and hence will have at

x which is not xleast one limit point 0. In addition, the limit V of V(x(t)) will exist.Consider a solution x = x(t) starting from that point. By continuous dependence on

initial conditions we conclude that V(x(t)) = V is constant along this solution, whichcontradictstheassumptions.

A

similar

theorem

deriving

existence

of

a

smooth

Lyapunov

function

is

also

valid.

Theorem

6.4 If0 isanasymptoticallystableequilibriumofsystem(6.1)wherea: X x

Rn isacontinuouslydifferentiablefunctiondefinedonanopensubsetX ofRn thentherex x xexistsacontinuouslydifferentiablefunctionV : B(0)RsuchthatV(0)< V()for

all = x x0 andV( x)

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Proof DefineV by

V

(x(0))

=

(|x(t)|2

)dt,

0

where : [0,) [0,) is positive for positive arguments and continuously differen-tiable. IfV iscorrectlydefinedanddifferentiable,differentiationofV(x(t))withrespecttotatt=0yields

V(x(0))a(x(0))=(|x(0)|2),

which proves the theorem. To make the integral convergent and continuously differen-tiable,itissufficienttomake(y)convergingtozeroquicklyenoughasy0.

Forthecaseofalinearsystem,aclassicalLyapunovtheoremshowsthatlocalstabilityofequilibrium x0 =0 impliesexistenceofastrictLyapunovfunctionwhich isapositive

definite

quadratic

form.

Theorem

6.5 Ifa: Rn Rn isdefinedby

a( xx)=A

whereAisagivenn- by-nmatrix,thenequilibriumx0 =0of(6.1)islocallyasymptoticallystable ifandonly if thereexistsamatrixQ=Q >0such that,forV( x x,x)= Q

V( x=|x)A x|2.

6.1.3

Globally

asymptotically

stable

equilibria

Hereweconsiderthecasewhena: Rn Rn indefinedforallvectors. Anequilibriumx0 of(6.1) iscalledgloballyasymptotically stable if it is locallystableandeverysolutionof(6.1)convergestox0 ast.

Theorem6.6 IffunctionV : Rn Rhas a unique minimumatx0, isstrictlymonotonicallydecreasingalongeverytrajectoryof(6.1)exceptx(t)x0,andhasboundedlevelsets then x0 isagloballyasymptoticallystableequilibriumof(6.1).

TheproofofthetheoremfollowsthelinesoftheproofofTheorem6.4. Notethatthe

assumption

that

the

level

sets

of

V

are

bounded

is

critically

important:

without

it,

somesolutions

of

(6.1)

may

converge

to

infinity

instead

ofx0.

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Massachusetts

Institute

of

Technology

Department

of

Electrical

Engineering

and

Computer

Science


byA.Megretski

Lecture7: FindingLyapunovFunctions1

ThislecturegivesanintroductionintobasicmethodsforfindingLyapunovfunctionsandstoragefunctionsforgivendynamicalsystems.

7.1 Convexsearch forstorage functions

The set of all real-valued functions of system state which do not increase along systemtrajectoriesisconvex,i.e. closedundertheoperationsofadditionandmultiplicationbya

positive

constant.

This

serves

as

a

basis

for

a

general

procedure

of

searching

for

Lyapunov

functionsorstoragefunctions.

7.1.1

Linearly

parameterized

storage

function

candidates

Considerasystemmodelgivenbydiscretetimestatespaceequations

x(t+1)=f(x(t),w(t)), y(t)=g(x(t),w(t)), (7.1)

wherex(t)XRn isthesystemstate,w(t)W Rm issysteminput,y(t)Y Rk

issystemoutput,andf : XW X,g: XW Y aregivenfunctions. AfunctionalV : X Risastoragefunctionforsystem(7.1)withsupplyrate: Y W Rif

V(x(t+1))V(x(t))(y(t)) (7.2)

foreverysolutionof(7.1),i.e. if

x, x)(g( w), xX, wW. (7.3)V(f( w))V( x, w)

1Version

of

September

26,

2003

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Inparticular,when0,thisyieldsthedefinitionofaLyapunovfunction.

Finding,

for

a

given

supply

rate,

a

valid

storage

function

(or

at

least

proving

that

one

exists)isamajorchallengeinconstructiveanalysisofnonlinearsystems. Themostcommonapproachisbasedonconsideringa linearlyparameterizedsubsetofstoragefunctioncandidatesV definedby

N

x)= qVq(V={V( x), (7.4)q=1

where{Vq}isafixedsetofbasisfunctions,andk areparameterstobedetermined. HereeveryelementofV isconsideredasastoragefunctioncandidate,andonewantstosetupanefficientsearchforthevaluesofk whichyieldafunctionV satisfying(7.3).

Example

7.1 Consider the finite state automata definedby equations (7.1) with value

sets

X={1,2,3}, W ={0,1}, Y ={0,1},

andwithdynamicsdefinedby

f(1,1)=2, f(2,1)=3, f(3,1)=1, f(1,0)=1, f(2,0)=2, f(3,0)=2,

g(1,1)=1, g( w)=0( w)x, x, =(1,1).

Inordertoshowthattheamountof1sintheoutputisnevermuchlargerthanonethirdofthe amountof 1s inthe input, onecan trytofinda storage function V withsupplyrate

( w)

=

w

3y,

y.

TakingthreebasisfunctionsV1,V2,V3 definedby

1, x=k,Vk(x)= 0, x=k,

theconditionsimposedon1,2,3 canbewrittenasthesetofsixaffineinequalities(7.3),twoofwhich(with( w)=(1,0)and( w)=(2,0))willbesatisfiedautomatically,whilex, x,theotherfourare

x,2

3

1 at( w)=(3,0),

x,

2 1 2

at

( w)

=

(1,

1),

x,3 2 1 at( w)=(2,1),

x,1 3 1 at( w)=(3,1).

Solutionsofthislinearprogramaregivenby

1 =c, 2 =c 2, 3 =c 1,

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wherec R isarbitrary. It is customary tonormalizestorageandLyapunov functions

so

that

their

minimum

equals

zero,

which

yields

c

=

2

and

1

=2, 2

=0, 3

=1.

Now,summingtheinequalities(7.2)fromt=0tot=T yields

T1 T1

3 y(t)V(x(0)) V(x(T))+ w(t),t=0 t=0

which is impliesthedesiredrelationbetweenthenumbersof1s inthe inputand intheoutput,sinceV(x(0)) V(x(T))cannotbelargerthan2.

7.1.2

Storage

functions

via

cutting

plane

algorithms

Thepossibilitytoreducethesearch foravalidstorage functiontoconvexoptimization,asdemonstratedbytheexampleabove,isageneraltrend. Onegeneralsituationinwhichanefficientsearchforastoragefunctioncanbeperformed iswhenacheapprocedureofcheckingcondition(7.3)(anoracle)isavailable.

AssumethatforeverygivenelementV Vitispossibletofindoutwhethercondition(7.3)issatisfied,and,inthecasewhentheanswerisnegative,toproduceapairofvectorsxX,wW forwhichtheinequality in(7.3)doesnothold. Selectasufficientlylarge setT0

(apolytopeoranellipsoid) inthespaceofparametervector =(q)qN

=1 (thisset

will

limit

the

search

for

a

valid

storage

function).

Let

be

the

center

of

T0.

Define

V by the , and apply the verification oracle to it. If V is a valid storage function,the search for storage function ends successfully. Otherwise, the invalidity certificatex,( w) produced by the oracle yields a hyperplane separating and the (unknown) set

of definingvalid storage functions, thus cutting a substantial portion from thesearchset T0, reducing it to a smaller set T1. Now re-define

as the center of T1 and repeattheprocessbyconstructingasequenceofmonotonicallydecreasingsearchsetsTk,untileitheravalidstoragefunctionisfound,orTk shrinkstonothing.

With an appropriate selection of a class of search sets Tk (ellipsoids or polytopesare most frequently used) and with an adequate definition of a center (the so-called

analytical

center

is

used

for

polytopes),

the

volume

of

Tk can

be

made

exponentially

decreasing,

which

constitutes

fast

convergence

of

the

search

algorithm.

7.1.3

Completion

of

squares

The success of the search procedure described in the previous section depends heavilyonthechoiceofthebasis functionsVk. Amajordifficultytoovercome isverificationof(7.3) for a given V. It turns out that the only known large linear space of functionals

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4

F : Rn Rwhichadmitsefficientcheckofnon-negativityof itselements isthesetof

quadratic

forms

x x F(x)= Q , (Q=Q)

1 1

forwhichnonnegativityisequivalenttopositivesemidefinitenessofthecoefficientmatrixQ.

Thisobservationisexploitedinthelinear-quadraticcase,whenf,garelinearfunctions

f( w)=A w, g( w)=C w,x, x+B x, x+D

and isaquadraticform

x x ( w)=

.x,

w w

Thenitisnaturaltoconsiderquadraticstoragefunctioncandidates

V( x xx)= P

only,and(7.3)transformsintothe(symmetric)matrixinequality

PA+AP PB

. (7.5)BP 0

Since this inequality is linear with respect to its parameters P and , it can be solvedrelatively

efficiently

even

when

additional

linear

constraints

are

imposed

onP and.

Notethataquadraticfunctionalisnon-negativeifandonlyifitcanberepresentedasa

sum

of

squares

of

linear

functionals.

The

idea

of

checking

non-negativity

of

a

functional

bytryingtorepresent itasasumofsquaresof functions fromagiven linearsetcanbeused in searching forstorage functions of general nonlinear systemsaswell. Indeed, letH : RnRm RM andV : Rn RN bearbitraryvector-valuedfunctions. Forevery RN,condition(7.3)with

x)=V(V( x)

isimpliedbytheidentity

x, x)+ x, H( w)=( w) V(f( w)) V( H( w)S x, x, xX, wW, (7.6)

as long as S =S 0 is a positive semidefinite symmetric matrix. Note that both thestorage

function

candidate

parameter

and

the

sum

of

squares

parameter

S

=

S

0

enterconstraint(7.6) linearly. This,thesearchforavalidstoragefunction isreducedtosemidefiniteprogram.

Inpractice,thescalarcomponentsofvectorHshouldincludeenoughelementssothatidentity(7.6)canbeachievedforevery RN bychoosinganappropriateS=S (notnecessarily positivie semidefinite). For example, if f,g, are polynomials, it may be agoodideatouseapolynomialV andtodefineHasthevectorofmonomialsuptoagivendegree.

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7.2 Storage functionswithquadraticsupplyrates

As

described

in

the

previous

section,

one

can

search

for

storage

functions

by

considering

linearly parameterized sets of storage function candidates. It turns out that storagefunctions

derived

for

subsystems

of

a

given

system

can

serve

as

convenient

building

blocks

(i.e. thecomponentsVq ofV). Indeed, assumethatVq =Vq(x(t))arestorage functionswithsupplyratesq =q(z(t)). Typically,z(t) includesx(t)as itscomponent,andhassome additional elements, such as inputs, outputs, and othe nonlinear combinations ofsystem states and inputs. If the objective is to find a storage function V with a givensupplyrate,onecansearchforV intheform

N

V(x(t))= Vq(x(t)), q 0, (7.7)q=1

whereq arethesearchparameters. Notethatinthiscaseitisknowna-priorithateveryV in(7.7)isastoragefunctionwithsupplyrate

N

(z(t))= qq(z(t)). (7.8)q=1

Therefore,inordertofindastoragefunctionwithsupplyrate

=(z(t)),itissufficienttofindq 0suchthat

N1q( z)

z)

( z.

(7.9)

q=1

When,q aregenericfunctions,eventhissimplifiedtaskcanbedifficult. However,intheimportantspecialcasewhen andq arequadraticfunctionals,thesearchforq in(7.9)becomesasemidefiniteprogram.

Inthissection,theuseofstoragefunctionswithquadraticsupplyratesisdiscussed.

7.2.1

Storage

functions

for

LTI

systems

x)= xisastoragefunctionforLTIsystemAquadraticformV( xP

x =

Ax

+

Bw

(7.10)

withquadraticsupplyrate

x x ( w)=

x,

w w

ifandonlyifmatrixinequality(7.5)issatisfied.Thewell-knownKalman-Popov-YakubovichLemma,orpositivereallemmagivesuseful

frequencydomainconditionforexistenceofsuchP =P forgivenA,B,.

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6

Theorem

7.1 Assume that thepair(A,B) iscontrollable. AsymmetricmatrixP =P

satisfying

(7.5)

exists

if

and

only

if

xw

xw

0

whenever jx=Ax+Bw forsomeR. (7.11)

Moreover, if thereexistsamatrixK such thatA+BK isaHurwitzmatrix,and

I I 0,

K K

thenallsuchmatricesP =P arepositivesemidefinite.

Example

7.2LetG(s) =C(sI A)1B+D beastabletransfer function(i.e. matrix

A

is

a

Huewitz

matrix)

with

a

controllable

pair

(A,

B).

Then

|G(j)| 1

for

all

R

ifandonlyifthereexistsP =P 0suchthat

w|2 w|2x x+B x+D 2 P(A w) | |C xRn, wRm.

ThiscanbeprovenbyapplyingTheorem7.1with

( w)=| x+Dx, w|2 |C w|2

andK=0.

7.2.2

Storage

functions

for

sector

nonlinearities

Whenever two components v = v(t) and w = w(t) of the system trajectory z = z(t)arerelated insuchawaythatthepair(v(t),w(t))liesintheconebetweenthetwolinesw=k1vandv=k2v,V 0isastoragefunctionfor

(z(t))=(w(t) k1v(t))(k2v(t) w(t)).

For example, if w(t) = v(t)3 then (z(t)) = v(t)w(t). If w(t) = sin(t)sin(v(t)) then2(z(t))=|v(t)|2 |w(t)| .

7.2.3

Storage

for

scalar

memoryless

nonlinearity

Whenever

two

components

v

=

v(t)

and

w

=

w(t)

of

the

system

trajectory

z

=

z(t)

are

related by w(t) = (v(t)), where : R R is an integrable function, and v(t) is acomponentofsystemstate,V(x(t))=(v(t))isastoragefunctionwithsupplyrate

(z(t))=v(t)w(t),

where y

(y)= ()d.0

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7.3 Implicitstorage functions

A

number

of

important

results

in

nonlinear

system

analysis

rely

on

storage

functions

for

whichnoexplicitformulaisknown. Itisfrequentlysufficienttoprovidealowerboundforthe

storage

function

(for

example,

to

know

that

it

takes

only

non-negative

values),

and

tohaveananalyticalexpressionforthesupplyratefunction.Inordertoworkwithsuchimplicitstoragefunctions,itishelpfultohavetheorems

whichguaranteeexistenceofnon-negativestoragefunctionsforagivensupplyrate. Inthisregard,Theorem7.1canbeconsideredasanexampleofsuchresult,statingexistenceofastoragefunctionforalinearandtimeinvariantsystemasanimplicationofafrequency-dependent matrix inequality. In this section we present a number of such statementswhichcanbeappliedtononlinearsystems.

7.3.1

Implicit

storage

functions

for

abstract

systems

ConsiderasystemdefinedbybehavioralsetB={z}of functionsz : [0, ) Rq. Asusually,thesystemcanbeautonomous,inwhichcasez(t)istheoutputattimet,orwithan input, inwhichcasez(t) = [v(t);w(t)]combinesvector inputv(t)andvectoroutputw(t).

Theorem

7.2 Let: Rq Rbeafunctionand letBbeabehavioralset,consistingof

somefunctionsz: [0, ) Rq. Assume that thecomposition(z(t)) is integrableovereverybounded interval(t0, t1) inR+forallzB. Fort0, tR+ define

t

I(z, t0, t)= (z())d.t0

Thefollowingconditionsareequivalent:

(a)forevery z0 B and t0 R+ the setofvaluesI(z, t0, t), takenforall tt0 andforallzBdefiningsamestateasz0

at timet0, isboundedfrombelow;

(b) thereexistsanon-negativestoragefunctionV : B R+ R+ (suchthatV(z1, t)=V(z2, t)wheneverz1 andz2 definesamestateofBat timet)withsupplyrate.

Moreover,whencondition(a)issatisfied,astoragefunctionV from(b)canbedefinedby

V(z0(), t0)= infI(z, t0, t), (7.12)

wheretheinfimumistakenoveralltt0 andoverallzBdefiningsamestateasz0 attimet0.

Proof Implication (b)(a) follows directly from the definition of a storage function,

whichrequiresV(z0, t1)V(z0, t0)I(z, t0, t1) (7.13)

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fort1

t0,z0

B. CombiningthiswithV 0yields

I(z, t0, t1)V(z, t0)=V(z0, t0)

forallz, z0

definingsamestateofBattimet0.Now let us assume that (a) is valid. Then a finite infimum in (7.12) exists (as an

infimumoveranon-emptysetboundedfrombelow)andisnotpositive(sinceI(z0, t0, t0)=0). HenceV iscorrectlydefinedandnotnegative. Tofinishtheproof, letusshowthat(7.13)holds. Indeed,ifz1

definessamestateasz0

attimet1

then

z0(t), tt1,z01(t)= z1(t), t>t1

defines

same

state

as

z0 at

time

t0 0wW, x0) wW xB(

(7.15)willbesatisfied. However,using(7.15)requiresalotofcautioninmostcases,since,evenforverysmoothf,,theresultingstoragefunctionV doesnothavetobedifferentiable.

7.3.3

Zames-Falb

quadratic

supply

rate

Anon-trivialandpowerfulcaseofanimplicitlydefinedstoragefunctionwithaquadraticsupplyratewasintroducedinlate60-sbyG.ZamesandP.Falb.

Theorem

7.3 LetA,B,C bematricessuch thatA isaHurwitzmatrix,and

|CeAtB|dt

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10

Theorem

7.4 Assume thatmatrices Ap,Bp,Cp are such that Ap is aHurwitzmatrix,

and

there

exists

>

0

such

that

Re(1G(j))(1H(j)) R,

whereH isaFourier transformofafunctionwithL1normnotexceeding1,and

G(s)=Cp(sIAp)1Bp.

Thensystemx(t)=Apx(t)+Bp(Cx(t)+v(t))

hasfiniteL2gain, in thesense that thereexists>0such that

|x(t)|2dt

(|x(0)|2 +

|v(t)|2dt

0 0

forallsolutions.

7.4 Examplewithcubicnonlinearityanddelay

Considerthefollowingsystemofdifferentialequations2 withanuncertainconstantdelayparameter:

x1(t) = x1(t)3 x2(t)

3 (7.16)

x2(t) =

x1(t)

x2(t)

(7.17)

Analysis of this system is easy when = 0, and becomes more difficult when is anarbitraryconstant inthe interval [0,0]. The system isnotexponentially stable for anyvalueof. Ourobjectiveistoshowthat,despitetheabsenceofexponentialstability,themethodofstoragefunctionswithquadraticsupplyratesworks.

The

case =0

For =0,webeginwithdescribing(7.16),(7.17)bythebehaviorset

Z={z= [x1;x2;w1;w2]},

where3 3w1 =x1, w2 =x2, x1 =w1 w2, x2 =x1 x2.

Quadraticsupplyratesforwhichfollowfromthe linearequationsofZ aregivenby

x1 w1 w2LTI(z)=2 P x1 x2,

x2

2Suggested

by

Petar

Kokotovich

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11

whereP =P isanarbitrarysymmetric2-by-2matrixdefiningstoragefunction

VLT I(z(), t)

=

x(t)P x(t).

Amongthenon-trivialquadraticsupplyratesvalidforZ,thesimplestaredefinedby

N L(z)=d1x1w1 +d2x2w2 +q1w1(w1 w2)+q2w2(x1 x2),

withthestoragefunction

VN L(z(), t)=0.25(q1x1(t)4 +q2x2(t)

4),

wheredk 0. Itturnsout(and iseasytoverify)thattheonlyconvexcombinationsofthese supply rates which yield 0 are the ones that make = LT I +N L = 0, forexample

0.5 0P = , d1 =d2 =q2 =1, q1 =0.0 0

The absence of strictly negative definite supply rates corresponds to the fact that thesystem

is

not

exponentially

stable.

Nevertheless,

a

Lyapunov

function

candidate

can

be

constructedfromthegivensolution:

4 4

2

4V(x)=xP x +0.25(q1x1 +q2x2)=0.5x1 +0.25x2.

ThisLyapunovfunctioncanbeusedalongthestandardlinestoproveglobalasymptotic

stability

of

the

equilibrium

x

=

0

in

system

(7.16),(7.17).

7.4.1

The

general

case

Now consider the case when [0, 0.2] is an uncertain parameter. To show that thedelayedsystem(7.16),(7.17)remainsstablewhen 0.2,(7.16),(7.17)canberepresentedbyamoreelaboratebehaviorsetZ={z()}with

z= [x1;x2;w1;w2;w3;w4;w5;w6]R8,

satisfyingLTIrelations

x1 =

w1

w2 +

w3, x2 =

x1

x2

andthenonlinear/infinitedimensionalrelations

3 3 3w1(t)=x1, w2 =x2, w3 =x2 (x2 +w4)3,

3w4(t)=x2(t )x2(t), w5 =w4

, w6 =(x1 x2)3

.

Some additional supply rates/storage functions are needed to bound the new variables.These will be selected using the perspective of a small gain argument. Note that the

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12

perturbationw4

caneasilybeboundedintermsofx2

=x1

x2. Infact,theLTIsystem

with

transfer

function

(exp( s)

1)/s

has

a

small

gain

(in

almost

any

sense)

when

is

small. Henceasmallgainargumentwouldbeapplicableprovidedthatthegainfromw4

tox2couldbeboundedaswell.It turns out that the L2-induced gain from w4 to x2 is unbounded. Instead, we can

usetheL4 norms. Indeed,thelasttwocomponentsw5, w6 ofwwereintroducedinordertohandleL4 normswithintheframeworkofquadraticsupplyrates. Morespecifically,inadditiontotheusualsupplyrate

x1

w1

w2

+w3LTI(z)=2 P x1 x2,

x2

thesetZ hassupplyrates

(z)

=d1x1w1 +d2x2w2 +q1w1(w1 w2 +w3)+q2w2(x1 x2)

+d3[0.99(x1w1 +x2w2)x1w3 +2.54

w4w5 0.54(x1 x2)w6]

+q3[0.24(x1 x2)w6 w4w5],

di 0. Herethesupplyrateswithcoefficientsd1, d2, q1, q2 aresameasbefore. Thetermwithd3,basedonazerostoragefunction,followsfromtheinequality 4

4

x1

x24 4 30.99(x1 +x2)x1(x2 (x2 +w4)3)+

5w4

0

2 2

(whichissatisfiedforallrealnumbersx1, x2, w4,andcanbecheckednumerically).

The

term

with

q3 follows

from

a

gain

bound

on

the

transfer

function

G(s)

=

(exp( s)1)/s from x1 x2 to w4. It is easy to verify that the L1 norm of its impulse response

equals,andhencetheL4 inducedgainofthecausalLTIsystemwithtransferfunctionG willnotexceed1. Considerthefunction 4 t

Vd(v(), T)=inf 0.24|v1(t)|

4 v1(r)dr dt, (7.18)T t

where the infimum is taken over all functions v1 which are square integrable on (0, )andsuchthatv1(t)=v(t)fortT. BecauseoftheL4 gainboundofG with [0, 0.2]doesnotexceed0.2,theinfimumin(7.18)isbounded. Sincewecanalwaysusev1(t)=0

for

t

>

T,

the

infimum

is

non-positive,

and

hence

Vd is

non-negative.

The

IQC

defined

bytheq3termholdswithV =q3Vd(x1 x2, t).Let

4 40(z)=0.01(x1w1

+x2w2)=0.01(x1 +x2),

whichreflectsourintentiontoshowthatx1, x2

willbeintegrablewithfourthpowerover(0, ). Using

0.5 0P = , d1 =d2 =0.01, d3 =q2 =1, q1 =0, q3 =2.5

4

0 0

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yieldsaLyapunovfunction

V(xe(t))=0.5x1(t)2

+

0.25x2(t)4

+

2.54Vd(x1 x2, t),

where xe is the total state of the system (in this case, xe(T) = [x(T);vT()], wherevT()L2(0, )denotesthesignalv(t)=x1(T +t) x2(T +t)restrictedtotheintervalt(0, )). Itfollowsthat

dV(xe(t)) 0.01(x1(t)

4 +x2(t)4).

dt

Ontheotherhand,wesawpreviouslythatV(xe(t))0 isboundedfrombelow. Therefore, x1(), x2() 4 (fourth powersof x1, x2 are integrable over (0, )) as long as the

initial

conditions

are

bounded.

Thus,

the

equilibrium

x

=

0

in

system

(7.16),(7.17)

is

stablefor0 0.2.

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Massachusetts

Institute

of

Technology

Department

of

Electrical

Engineering

and

Computer

Science


byA.Megretski

Lecture8: LocalBehavioratEqilibria1

ThislecturepresentsresultswhichdescribelocalbehaviorofautonomoussystemsintermsofTaylorseriesexpansionsofsystemequationsinaneigborhoodofanequilibrium.

8.1 Firstorderconditions

ThissectiondescribestherelationbetweeneigenvaluesofaJacobiana(x0)andbehaviorofODE

x(t)

=

a(x(t))

(8.1)

oradifferenceequationx(t+1)=a(x(t)) (8.2)

in

a

neigborhood

of

equilibriumx0.

In the statements below, it is assumed that a : X Rn is a continuous functiondefinedonanopensubsetX Rn. It isfurtherassumedthat x0 X,andthereexistsann-by-nmatrixAsuchthat

|a( x0)A|x0 +)a(0 as ||0. (8.3)

||

Ifderivativesdak/dxi ofeachcomponentak ofawithrespecttoeachcpomponentxiofxexistatx0,Aisthematrixwithcoefficientsdak/dxi,i.e. theJacobianofthesystem.However,differentiabilityatasinglepointx0 doesnotguaranteethat(8.3)holds. Ontheother

hand,

(8.3)

follows

from

continuous

differentiability

ofainaneigborhoodofx0.

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Example

8.1Functiona: R2 R2,definedby

22

(2 x2)2

x1 x1x2 x1 2

x1

a =22 x2 x2 x1x2 +(

2

2)2 x2x1

for x x1 and = 0, and by a(0) = 0, is differentiable with respect to x2 at every pointx R2, and its Jacobian a(0) = A equals minus identity matrix. However, condition(8.3)isnotsatisfied(notethata( x=0).x)isnotcontinuousat

8.1.1

The

continuous

time

case

Letuscallanequilibriumx0 of(8.1)exponentiallystableifthereexistpositiverealnum

bers

,r,C

such

that

every

solution

x

:

[0, T]

X

with

|x(0)

x0

|

0 there exists > 0 such that the nonlinearcomponent

w(t)

satisfies

the

sector

constraint

2NL(x(t),w(t))=|x(t)|2 |w(t)| 0,

aslongas|x(t)|

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4

Toprove(b),takearealnumberd(0, r/2)suchthatnotwoeigenvaluesofAsum

up

to

2d.

Then

P

=

P

be

the

unique

solution

of

the

Lyapunov

equation

P(A+dI)+(A +dI)P =I.

NotethatP isnon-singular: otherwise,ifP v=0forsomev=0,itfollowsthat

(A|v|2 =v(P(A+dI)+(A +dI)P)v=(P v)(A+dI)v+v +dI)(P v)=0.

In addition, P = P is not positive semidefinite: since, by assumption, A+dI has aneigenvectoru=0whichcorrespondstoaneigenvaluewithapositiverealpart,wehave

|u|2 =2Re()uP u,

henceuP u0besmallenoughsothat

2 2 xP w0.5|x| for |w| |x|.

Byassumption,thereexists>0suchthat

x) A |a( x| |x| for |x| .

Then

d(e2dt 2dtx(t)P x(t))=e (2dx(t)P x(t)+2x(t)P Ax(t)+2x(t)P(a(x(t)) Ax(t)))

dt

2 0.5e2dt|x(t)|

as long as x(t) is a solution of (8.1) and |x(t)| . In particular, this means that ifx(0)P x(0) R2d.

Theproofof(c)issimilartothatof(a).

8.1.3

The

discrete

time

case

Theresults forthediscretetimecasearesimilartoTheorem8.1, withtherealpartsoftheeigenvaluesbeingreplacedbythedifferencebetweentheirabsolutevaluesand1.

Let us call an equilibrium x0 of (8.2)exponentially stable if there exist positive realnumbers,r,C suchthateverysolutionx:[0, T]X with|x(0)x0|

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5

(a)

ifA=a(0)isaSchurmatrix(i.e. ifalleigenvaluesofA haveabsolutevalue lessx

than

one)

then

0 is

a

(locally)

exponentially

stable

equilibrium

of

(8.2);

x(b)

ifA=a(0)hasaneigenvaluewithabsolutevaluegreaterthan1then 0 isnotanx x

exponentially

stable

equilibrium

of

(8.2);

(c)

ifA=a(0)hasaneigenvaluewithabsolutevaluestrictly larger than1 then x x0 is

notastableequilibriumof(8.2).

8.2 Higherorderconditions

When

the

JacobianA=a( x0 hasnoeigenvaluesx0)of(8.1)evaluatedattheequilibrium

with positive real part, but has some eigenvalues on the imaginary axis, local stability

analysis

becomes

much

more

complicated.

Based

on

the

proof

of

Theorem

8.1,

it

is

natural

to expect that system states corresponding to strictly stable eigenvalues will behave ina predictably stable fashion, and hence the behavior of system states corresponding tothe eigenvalues on the imaginary axis will determine local stability or instability of theequilibrium.

8.2.1

A

Center

Manifold

Theorem

Inthissubsectionweassumeforsimplicitythatx0 =0isthestudiedequilibriumof(8.1),i.e. a(0)=0. Assumealsothataisk timescontinuouslydifferentiableinaneigborhoodofx0 =0,wherek1,andthatA=a

(0)hasnoeigenvalueswithpositiverealpart,buthas

eigenvalues

on

the

imaginary

axis,

as

well

as

in

the

open

left

half

plane

Re(s)

< 0.

ThenalinearchangeofcoordinatesbringsAintoablock-diagonalform

Ac

0A=

0 As,

whereAs isaHurwitzmatrix,andalleigenvaluesofAc havezerorealpart.

Theorem

8.3

Leta: Rn Rn bek2timescontinuouslydifferentiableinaneigbor

hood

of x0 =0. Assume thata(0)=0and

Ac 0

a(0)

=

A=

0

As ,

where As is a Hurwitzp-by-p matrix, and all eigenvalues of the q-by-qmatrix Ac have

zero

real

part.

Then

(a)

there

exists > 0 and afunction h : Rq Rp, k1 times continuously differ

entiable

in

a

neigborhood

of

the

origin,

such

that h(0) = 0, h(0) = 0, and every

solution x(t) = [xc(t);xs(t)] of (8.1) with xs(0) = h(xc(0)) and with |xc(0)| <

satisfiesxs(t)=h(x0(t))foras longas |xc(t)|< ;

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(b)

for

every

function hfrom (a), the equilibrium x0 = 0 of (8.1) is locally stable

(asymptotically

stable)

[unstable]

if

and

only

if

the

equilibrium

xc =

0

of

the

ODE

dotxc(t)=a([xc(t);h(xc(t))]) (8.4)

is

locally

stable

(asymptotically

stable)

[unstable];

(c)

if

the

equilibrium xc = 0 of (8.4) is stable then there exist constants r > 0, > 0

such

that

for

every

solutionx=x(t)of(8.1)with |x(0)|< r thereexistsasolution

xc =xc(t)of(8.4)suchthat

limet|x(t)[xc(t);h(xc(t))]|=0.t

Thesetofpointsx= [xc;h( xMc ={ xc)]: |c|< },

where >0 issmallenough, iscalledthecentralmanifoldof(8.1). Theorem8.3,calledfrequentlythecentermanifoldtheorem,allowsonetoreducethedimensionofthesystemtobeanalyzedfromntoq,aslongasthefunctionhdefiningthecentralmanifoldcanbecalculatedexactlyortoasufficientdegreeofaccuracytojudgelocalstabilityof(8.4).

Example

8.3ThisexampleistakenfromSastry,p. 312. Considersystem

x1(t) = x1(t)+kx2(t)2,

x2(t) =

x1(t)x2(t),

where k is a real parameter. In this case n = 2,p = q = 1, Ac = 0, As = 1, and kcanbearbitrarily large. AccordingtoTheorem8.3,thereexistsak timesdifferentiablefunctionh: RRsuchthatx1 =h(x2)isaninvariantmanifoldoftheODE(atleast,inaneigborhoodoftheorigin). Hence

ky2 =h(y)+h(y)h(y)y

forallsufficientlysmally. Forthe4thorderTaylorseriesexpansion

3 4 4h(y)=h2y2 +h3y +h4y

4 +o(y ), h(y)=2h2y+3h3y2 +4h4y +o(y

3),

comparing the coefficients on both sides of the ODE for h yields h2 =k, h3 = 0, h4 =2k2. HencethecentermanifolsODEhastheform

xc(t)=kxc(t)3 +o(xc(t)

3),

whichmeansstabilityfork 0.

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Massachusetts

Institute

of

Technology

Department

of

Electrical

Engineering

and

Computer

Science


byA.Megretski

Lecture9: LocalBehaviorNearTrajectories1

This lecturepresentsresultswhichdescribe localbehaviorofODEmodels inaneigborhoodofagiventrajectory,withmainattentionpaidtolocalstabilityofperiodicsolutions.

9.1 SmoothDependenceonParameters

InthissectionweconsideranODEmodel

x(t)=a(x(t), t , ), x(t0)= x0(), (9.1)

whereisaparameter. Whenaandx0

aredifferentiablewithrespectto,thesolutionx(t)=x(t, )isdifferentiablewithrespecttoaswell. Moreover,thederivativeofx(t, )withrespecttocanbefoundbysolvinglinearODEwithtime-varyingcoefficients.

Theorem

9.1 Let a : Rn RRk Rn be a continuousfunction, 0 R

k. Letx0 : [t0, t1] R

n be a solution of (9.1)with = 0. Assume that a is continuouslydifferentiable with respect to itsfirst and third arguments on an open set X such that(x0(t), t , 0)Xforallt[t0, t1].Thenforallinaneigborhoodof0 theODEin(9.1)hasauniquesolutionx(t)=x(t, ).Thissolutionisacontinuouslydifferentiablefunctionof,anditsderivativewithrespecttoat=0 equals(t),where: [t0, t1]R

n,k

is

the

n-by-k

matrix-valued

solution

of

the

ODE

(t)=A(t)(t)+B(t), (t0)=0, (9.2)

xa( xat whereA(t)isthederivativeofthemap x,t,0)withrespectto x=x0(t),B(t)is thederivativeof themap a(x0(t), t , )at =0,and 0

is thederivativeof themapx0()at=0.

1Version

of

October

10,

2003

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2

Proof Existenceanduniquenessofx(t, )and(t)followfromTheorem3.1. Hence,in

order

to

prove

differentiability

and

the

formula

for

the

derivative,

it

is

sufficient

to

show

thatthereexistafunctionC : R+

R+

suchthatC(r)/r 0asr 0and>0suchthat

|x(t, ) (t)( 0) x0(t)| C(| 0|)

whenever | 0| . Indeed,duetocontinuousdifferentiabilityofa,thereexistC1, 0suchthat

|a( x x0(t)) B(t)( 0)| C1(|x x,t,) a(x0(t), t , 0) A(t)( x0(t)| +| 0|)

and|x0() x0(0) 0( 0)| C1(| 0|)

whenever|x x0(t)| +| 0| , t [t0, t1].

Hence,for(t)=x(t, ) x0(t) (t)( 0)

wehave|(t)| C2|(t)| +C3(| 0|),

aslongas|(t)| 1 and| 0| 1,where1 >0issufficientlysmall. Togetherwith

|(t0)| C4(| 0|),

this

implies

the

desired

bound.

Example

9.1Considerthedifferentialequation

y(t)=1+ sin(y(t)), y(0)=0,

whereisasmallparameter. For=0,theequationcanbesolvedexplicitly: y0(t)=t.Differentiatingy(t)withrespecttoat=0yields(t)satisfying

(t)=sin(t), (0)=0,

i.e.

(t)

=

1

cos(t).

Hence

y(t)=t+(1 cos(t))+O(2)

forsmall.

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3

9.2 Stabilityofperiodicsolutions

In

the

previous

lecture,

we

were

studying

stability

of

equilibrium

solutions

of

differential

equations. Inthissection,stabilityofperiodicsolutionsofnonlineardifferentialequationsis

considered.

Our

main

objective

is

to

derive

an

analog

of

the

Lyapunovs

first

method,

statingthat aperiodicsolution is asymptoticallystable ifsystems linearization aroundthesolutionisstableinacertainsense.

9.2.1

Periodic

solutions

of

time-varying

ODE

Considersystemequationsgivenintheform

x(t)=f(x(t), t), (9.3)

wheref : RnR Rn iscontinuous. Assumethatais(, x)-periodic,inthesensethatthereexist >0andx Rn suchthat

f(t+, r)=f(t, r), f(t, r+x)=f(t, r) t R, r Rn. (9.4)

Notethatwhilethefirstequation in(9.4)meansthatf isperiodic intwithaperiod,it ispossible that x=0, in whichcase the second equation in (9.4)does not bringanyadditionalinformation.

Definition A solution x0 : R R

n of a (, x)x)-periodic system (9.3) is called (,periodicif

x0(t+

)

=

x0(t)

+

x

t

R.

(9.5)

Example

9.2

According

to

the

definition,

the

solutiony(t)=tofthe forcedpendulum

equationy(t)+y(t)+sin(y(t))=1+sin(t) (9.6)

as a periodic one (use = x = 2). This is reasonable, since y(t) in the pendulumequation represents an angle, so that shifting y by 2 does not change anything in thesystemequations.

Definition Asolutionx0 : [t0, ) R

n of(9.3)iscalledstableifforevery>0there

exists

>

0

such

that

|x(t) x0(t)| t 0, (9.7)

whenever x() is a solution of (9.3) such that |x(0) x0(0)|0suchthat

x(t) x0(t) Cexp(t)|x(0) x0(0)| t 0 (9.8)

whenever|x(0) x0(0)| issmallenough.

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4

To derive a stability criterion for periodic solutions x0

: R Rn of (9.3), assume

continuous

differentiability

of

function

f

=

f( x, t)

with

respect

to

the

first

argument

x

for |xx0(t)| , where > 0 is small, and differentiate the solution as a function ofinitialconditionsx(0)x0(0).

Theorem

9.2 Let f : Rn R Rn be a continuous (,x)-periodicfunction. Let

x0 : R Rn bea (,x)-periodic solutionof(9.3). Assume that thereexists >0 such

that f iscontinuouslydifferentiablewith respect to itsfirstargumentfor |xx0(t)|0. Anon-constant(,x)-periodicsolutionx0 : R R

n ofsystem(9.11)iscalledastable limitcycleif

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(a) forevery>0thereexists>0suchthatdist(x(t), x0()) 0 such that dist(x(t), x0()) 0 as t for every solution of(9.11)suchthatdist(x(0), x0())0 suchthata iscontinuouslydifferentiableon theset

X={ xRn : |x x0(t)|

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Massachusetts

Institute

of

Technology

Department

of

Electrical

Engineering

and

Computer

Science


byA.Megretski

Lecture

10:

Singular

Perturbations

and

Averaging1

Thislecturepresentsresultswhichdescribelocalbehaviorofparameter-dependentODEmodelsincaseswhendependenceonaparameterisnotcontinuousintheusualsense.

10.1 SingularlyperturbedODE

Inthissectionweconsiderparameter-dependentsystemsofequations

x(t) = f(x(t), y(t), t),(10.1)

y =

g(x(t), y(t), t),

where [0, 0] is a small positive parameter. When > 0, (10.1) is an ODE model.For = 0, (10.1) is a combination of algebraic and differential equations. Models suchas (10.1), where y represents a set of less relevant, fast changing parameters, are frequently

studied

in

physics

and

mechanics.

One

can

say

that

singular

perturbations

is

the

classicalapproachtodealingwithuncertainty,complexity,andnonlinearity.

10.1.1

The

Tikhonovs

Theorem

A typical question asked about the singularly perturbed system (10.1) is whether its

solutions

with

>

0

converge

to

the

solutions

of

(10.1)

with

=

0

as

0.

A

sufficientconditionforsuchconvergence isthattheJacobianofg withrespectto itssecondargumentshouldbeaHurwitzmatrixintheregionofinterest.

Theorem

10.1 Let x0 : [t0, t1] R

n, y0 : [t0, t1] Rm be continuousfunctions

satisfyingequations

x0(t)=f(x0(t), y0(t), t), 0=g(x0(t), y0(t), t),

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of

October

15,

2003

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2

wheref : Rn Rm RRn andg: Rn Rm RRm arecontinuousfunctions.

Assume

that

f, g

are

continuously

differentiable

with

respect

to

their

first

two

arguments

inaneigborhoodof the trajectoryx0(t), y0(t),and that thederivative

A(t)=g2

(x0(t), y0(t), t)

isaHurwitzmatrixforall t[t0, t1]. Thenforevery t2 (t0, t1) thereexistsd>0andC>0such that inequalities |x0(t)x(t)| Cforallt[t0, t1]and |y0(t)y(t)| Cforall t[t2, t1]forallsolutionsof(10.1)with |x(t0)x0(t0)| , |y(t0)y0(t0)| d,and(0, d).

The theorem was originally proven by A. Tikhonov in 1930-s. It expresses a simple

principle,

which

suggests

that,

for

small

>

0,

x

=

x(t)

can

be

considered

a

constant

when

predicting

the

behavior

ofy. Fromthisviewpoint, foragiven t (t0, t1),onecan

expectthaty(t +)y1(),

wherey1 : [0, )isthesolutionofthefastmotionODE

y1()=g(x0(t), y1()), y1(0)=y(t).

Since y0(t) is an equilibrium of the ODE, and the standard linearization around thise

6243 spring 2008

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