3-5 Julho Ciência 2017 , Lisbon

Nonlinear Systems: An Introduction to Lyapunov Stability Theory

Antonio Pascoal

Modelação e Simulação, 2017-2018

Linear versus Nonlinear Control

Nonlinear Plant

u y

Linearbasedcontrollaws

- Lack of global stability and performance results

+ Good engineering intuition for linear designs (local stability and performance)

- Poor physical intuition

Nonlinearcontrollaws

+ Powerful robust stability analysis tools

+ Possible deep physical insight

- Need for stronger theoretical background

- Limited tools for performance analysis

Linear versus Non-Linear N

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T

T −βv − fv 2 =mTdvdt;mT =m +ma

vAUVspeedcontrol

Dynamics

Nonlinear Plant

T v

)(tvrObjective:generateT(t)sothat )(tv tracksthereferencespeed

Trackingerror vve r −=

mTdedt

=mTdvrdt

−mTdvdtErrorDynamics

Nonlinear control: key ideas N

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mTdedt

=mTdvrdt

−mTdvdt

ErrorDynamics

2)( fvvTdtdvm

dttdem r

TT ++−= β

22)( fvvfvvkedtdvm

dtdvm

dtdem r

Tr

TT ++⎥⎦⎤

⎢⎣⎡ +++−= ββ

0=+ kedtde 00 ≥−= tktete );exp()()(

NonlinearControlLaw

2)( fvvKedtdvmT r

T +++= β

Non

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Nonlinear control: key ideas

00 ≥−= tktete );exp()()(

Trackingerrortendstozeroexponentiallyfast.

Simpleandelegant!

Catch:thenonlineardynamicsareknownEXACTLY.

Keyidea:i)use“simple”concepts,ii)dealwithrobustnessagainstparameteruncertainty.

2)( fvvKedtdvmT r

T +++= β

Newtoolsareneeded:LYAPUNOVtheory

Nonlinear control: key ideas

0=+ fvdtdvm

(freemass,subjectedtoasimplemotionresistingforce)v fv

vmf

dtdv −=

)()( 0

)0(tvetv

ttmf −−

=

v

m/f

0 v

t

v=0 is an equilibrium point; dv/dt=0 when v=0!

v=0 is attractive (trajectories converge to 0)

SIMPLEEXAMPLE N

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Lyapunov theory of stability: a soft Intro

SIMPLEEXAMPLE

0 v

How can one prove that the trajectories go to the equilibrium point WITHOUT SOLVING the differential equation?

2

21)( mvvV =

(energyfunction)

0,0;0,0)(

==≠

vVvvV ≻

0

)(.))((

2

)(|

≺fvdtdvmv

dtdV

dttdv

vV

dttvdV

tv

−==

→∂∂=

V positive and bounded below by zero; dV/dt negative implies convergence of V to 0!

Lyapunov theory of stability: a soft Intro N

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)(

);(

2212

1121

xkxdtdx

xkxdtdx

−−=

−=

)(21)( 2

221 xxxV +=

⎥⎦

⎤⎢⎣

⎡=

2

1

xx

x

State vector

0;21)( ≻IQQxxxV T ==

Q-positive definite

)(xfdtdx =

2-D case

0,0;0,0)(

==≠

vVvvV ≻

Lyapunov theory of stability: a soft Intro N

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2-D case

)(

);(

2212

1121

xkxdtdx

xkxdtdx

−−=

−=⎥⎦

⎤⎢⎣

⎡=

2

1

xx

x )(xfdtdx =

ttxtxV ⇐⇐ )())((

)(21)( 2

221 xxxV +=

RtRtxRtxV ∈⇐∈⇐∈ 2)())((

dtdx

xV

dtxdV T

∂∂=)(

1x2 2x1 1x1 0)()()(2221211121 ≺xkxxxxkxxx

dtxdV −−−=

[ ] ⎥⎦

⎤⎢⎣

⎡−−

−=

)()(

,)(

221

11221 xkx

xkxxx

dtxdV

V positive and bounded below by zero; dV/dt negative implies convergence of V to 0! x tends do 0!

x1 k (x1 ) > 0, x1 > 0; x2 k (x2 ) > 0, x2 > 0

Non

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Shifting IstheoriginalwaystheTRUEorigin?

2

2

)()(dtydmmg

dtdyfyk =+−−

mg

)(yk

y

)(dtdyf

y-measured from spring at rest

Examine if yeq is “attractive”!

ζ+= eqxx

dtd

dtd

dtdx

dtdx eq ζζ =+=

)()( ζζζ GxFdtdx

dtd

eq =+==

Equilibrium point yeq: dx/dt=0 mgyk eq =)(

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡= = 0

; eqeq

yx

dtdyy

x 0)();( == eqxFxFdtdx

0)0()0( =+= eqxFGExamine the ZERO eq. Point!

Non

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Lyapunov theory of stability: a soft Intro

Shifting IstheoriginalwaystheTRUEorigin?

Examine if xref(t) is “attractive”!

ζ+= refxx

dtdxF

dtd

dtdx

dtdx

ref

ref

ζ

ζ

+

=+=

)(

),()()()( tGxFxFxFdtdx

dtd

refrefref ζζζ =−+=−=

0))(()0)((),0( =−+= txFtxFtG refref

))(()(

));(()( txFdttdx

txFdttdx

refref ==

xref(t) is a solution

Examine the ZERO eq. Point!

Lyapunov theory of stability: a soft Intro N

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ControlAction

f (0,0) = 0

0)0();( == hyhu

Nonlinear plant

y u

Static control law

dxdt

= f (x ,h (g (x ))); f (0,0) = 0

0)0();( == FxFdtdx

Investigate if 0 is attractive!

0)0();();,( === gxgyuxfdtdx

Lyapunov theory of stability: a soft Intro N

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Stability of the zero solution

0)0(;)( == fxfdtdx

0 x-space

The zero solution is STABLE if

0);0()()0()(:0)(,0 ttBtxBtx o ≥∈⇒∈>=∃>∀ εδεδδε

δ

ε Non

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Lyapunov Stability Theory

0)0(;)( == fxfdtdx

0 x-space

The zero solution is locally ATTRACTIVE if

0)(lim)0()(:0 =⇒∈>∃ →∞to txBtx αα

Attractiveness of the zero solution

α

Lyapunov Stability Theory N

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0)0(;)( == fxfdtdx

The zero solution is locally ASYMPTOTICALLY STABLE if it is STABLE and ATTRACTIVE

(thetwoconditionsarerequiredforAsymptoticStability!)

OnemayhaveattractivenessbutNOTStability!

ε

δ

Non

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Lyapunov Stability Theory

)1()(xfdtdx =

Non

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Lyapunov Theory: a formal approach

(thetwoconditionsarerequiredforAsymptoticStability!)ε

δ

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Lyapunov Theory: a formal approach

Thereareatleastthreewaysofassessingthestability(ofanequilibriumpointofa)system:

• Solvethedifferentialequation(brute-force)

• Linearizethedynamicsandexaminethebehaviouroftheresultinglinearsystem(localresultsforhyperboliceq.pointsonly)

• UseLypaunov´sdirectmethod(elegantandpowerful,mayyieldglobalresults)

Stability Analysis N

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Lyapunov Theory: a formal approach N

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If

∞→∞→< x as )(;0)( xVdtxdV

thentheoriginisgloballyasymptoticallystable

Non

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Lyapunov Theory: a formal approach

Whathappenswhen ?0)( ≤dtxdV

Isthesituationhopeless? No!

⎭⎬⎫

⎩⎨⎧ ==Ω

≤

0)(::

;0)(

dtxdVx

definedtxdVLet

SupposetheonlytrajectoryofthesystementirelycontainedinW isthenulltrajectory.Then,theoriginisasymptoticallystable

(LetMbethelargestinvariantsetcontainedinW.ThenallsolutionsconvergetoM.IfMistheorigin,theresultsfollows)

Krazovskii-LaSalle

Lyapunov Theory: a formal approach N

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2

2

)()(dtydm

dtdyfyk =−−

)(yk

y )(dtdyf

⎥⎦

⎤⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡= = 0

0; eqx

dtdyy

x

0)0();( == FxFdtdx

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

)(1)(121

2

2

1

xfm

xkm

x

dtdxdtdx

EnergyPotentialEnergyKineticxV +=)(

V (x) = 12mx2

2 + k(ς )dς0

x1

∫

Non

linea

r St

abili

ty: E

xam

ples

Example: Mass-Spring-Dashpot System

)(yk

y )(dtdyf

V (x) = 12mx2

2 + k(ς )dς0

x1

∫⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

)(1)(121

2

2

1

xfm

xkm

x

dtdxdtdx

dV (x )dt

=mx2

dx2dt

+ k (x1)dx1dt

=

!0)()())(1)(1( 2221212 ≤−=+−− xxfxxkxfm

xkm

mx

f(.), k(.) – 1st and 3rd quadrants f(0)=k(0)=0

V(x)>0!

Non

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abili

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xam

ples

Example: Mass-Spring-Dashpot System

)(yk

y )(dtdyf

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

)(1)(121

2

2

1

xfm

xkm

x

dtdxdtdx

!0)()(22 ≤− xxf

dtxdV

2x

1x

!00 2 == xfordtdV

Examine dynamics here!

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

)(10

12

1

xkmdt

dxdtdx

Trajectory leaves W unless x1=0!

W

M is the origin. The origin is asymptotically stable!

Non

linea

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abili

ty: E

xam

ples

Example: Mass-Spring-Dashpot System

dvdt

= −v |v |+u

dxdt

=v

v -v|v|

AUVmovinginthewaterwithspeedvundertheactionofanappliedforceu.

Objective:drivethepositionxoftheAUVtox*(byproperchoiceofu)

x 0 x*

Example: AUV position control N

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mpl

es

u = −k1(x − x*)− k2v ; k1, k2 > 0

Suggestedcontrollaw

ControllawexhibitsProporcional+DerivativeactionsTheplantitselfhasapureintegrator(todrivethestaticerrorto0)

dvdt

= −v |v |+u

dxdt

=v

dvdt

= −v |v |+u dxdt

=vk1

k2

x * xv

Non

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Example: AUV position control

Showasymptoticstabilityofthe(equilibriumpointofthe)system

Step1.Startbyre-writingtheequationsintermsofthevariablesthatmustbedriventoo.Objective:

v (t )→ 0e (t ) = x (t )− x *(t )→ 0

dvdt

= −v |v |+u

dedt

=dxdt

−dx *dt

=dxdt

=v

dvdt

= −v |v |−k1e − k2v

dedt

=v

u = −k1(x − x*)− k2v ; k1, k2 > 0

Non

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Example: AUV position control

Step2.Proveglobalasymptoticstabilityoftheorigin

[v ,e ]T = [0,0]T

isanequilibriumpointofthesystem!

Seekinspirationfromthespring-mass-dashpotsystem

dvdt

= −v |v |−k1e − k2v

dedt

=v

EnergyPotentialEnergyKineticxV +=)(

V (x) = 12mx2

2 + k(ς )dς0

x1

∫)(yk

y )(dtdyf

Non

linea

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ty: E

xam

ples

Example: AUV position control

EnergyPotentialEnergyKineticxV +=)( V (x ) = 12v 2 + k1ς d ς

0

e

∫

V(x)>0!

dVdt

=∂V∂vdvdt

+∂V∂ededt

=v dvdt

+ k1ededt

=v[−v |v |−k1e − k2v ]+ k1ev

=v 2 |v |−k2v2 ≤ o!

Using La Salle´s theorem it follows that the origin is globally asymptotically stable (GAS)

Non

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Example: AUV position control

33

Func

tiona

l Spe

cifi

catio

ns /

Sys

tem

s Th

eory

xB

yBxy

{B}

V

0

AUV trajectory

AUV

ψ

AUV Path Following P

ath

to b

e fo

llow

ed

Non

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34

xB

yBxy

{B}

V

0

AUV trajectory ψ

Set the speed of the AUV equal to V>0 (constant)

Recruit the heading angle so that

the lateral distance to the vertical path

will converge to 0!

AUV Path Following N

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xB

yBxy

{B}

V

0

AUV trajectory ψ

Plant Model I:

AUV Path Following

dy(t)

dt= V sin (t)

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y<latexit sha1_base64="YBnOz4OfA0qe2MXK9oswnbSi1XA=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx6rGFtoQ9lsN+3SzW7Y3Qgh9B948aDi1Z/kzX/jps1BWx8MPN6bYWZemHCmjet+O5WV1bX1jepmbWt7Z3evvn/wqGWqCPWJ5FJ1Q6wpZ4L6hhlOu4miOA457YSTm8LvPFGlmRQPJktoEOORYBEj2FjpPqsN6g236c6AlolXkgaUaA/qX/2hJGlMhSEca93z3MQEOVaGEU6ntX6qaYLJBI9oz1KBY6qDfHbpFJ1YZYgiqWwJg2bq74kcx1pncWg7Y2zGetErxP+8XmqiyyBnIkkNFWS+KEo5MhIVb6MhU5QYnlmCiWL2VkTGWGFibDhFCN7iy8vEP2teNd2780brukyjCkdwDKfgwQW04Bba4AOBCJ7hFd6cifPivDsf89aKU84cwh84nz+KSozk</latexit><latexit sha1_base64="YBnOz4OfA0qe2MXK9oswnbSi1XA=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx6rGFtoQ9lsN+3SzW7Y3Qgh9B948aDi1Z/kzX/jps1BWx8MPN6bYWZemHCmjet+O5WV1bX1jepmbWt7Z3evvn/wqGWqCPWJ5FJ1Q6wpZ4L6hhlOu4miOA457YSTm8LvPFGlmRQPJktoEOORYBEj2FjpPqsN6g236c6AlolXkgaUaA/qX/2hJGlMhSEca93z3MQEOVaGEU6ntX6qaYLJBI9oz1KBY6qDfHbpFJ1YZYgiqWwJg2bq74kcx1pncWg7Y2zGetErxP+8XmqiyyBnIkkNFWS+KEo5MhIVb6MhU5QYnlmCiWL2VkTGWGFibDhFCN7iy8vEP2teNd2780brukyjCkdwDKfgwQW04Bba4AOBCJ7hFd6cifPivDsf89aKU84cwh84nz+KSozk</latexit><latexit sha1_base64="YBnOz4OfA0qe2MXK9oswnbSi1XA=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx6rGFtoQ9lsN+3SzW7Y3Qgh9B948aDi1Z/kzX/jps1BWx8MPN6bYWZemHCmjet+O5WV1bX1jepmbWt7Z3evvn/wqGWqCPWJ5FJ1Q6wpZ4L6hhlOu4miOA457YSTm8LvPFGlmRQPJktoEOORYBEj2FjpPqsN6g236c6AlolXkgaUaA/qX/2hJGlMhSEca93z3MQEOVaGEU6ntX6qaYLJBI9oz1KBY6qDfHbpFJ1YZYgiqWwJg2bq74kcx1pncWg7Y2zGetErxP+8XmqiyyBnIkkNFWS+KEo5MhIVb6MhU5QYnlmCiWL2VkTGWGFibDhFCN7iy8vEP2teNd2780brukyjCkdwDKfgwQW04Bba4AOBCJ7hFd6cifPivDsf89aKU84cwh84nz+KSozk</latexit>Plant Model

<latexit sha1_base64="HdkFIKDvlU4NCBi6pyctNR8SRU8=">AAAB93icbVBNS8NAFHypX7V+NOrRy2IRPJVEBBUvRS9ehArGFtpQNptNu3STDbsboYb+Ei8eVLz6V7z5b9y0OWh1YGGYeY83O0HKmdKO82VVlpZXVteq67WNza3tur2ze69EJgn1iOBCdgOsKGcJ9TTTnHZTSXEccNoJxleF33mgUjGR3OlJSv0YDxMWMYK1kQZ2vc1xolH/At2IkPLawG44TWcG9Je4JWlAifbA/uyHgmQxTTThWKme66Taz7HUjHA6rfUzRVNMxnhIe4YmOKbKz2fBp+jQKCGKhDTPpJipPzdyHCs1iQMzGWM9UoteIf7n9TIdnfk5S9JM04TMD0UZR1qgogUUMkmJ5hNDMJHMZEVkhCUm2nRVlOAufvkv8Y6b503n9qTRuizbqMI+HMARuHAKLbiGNnhAIIMneIFX69F6tt6s9/loxSp39uAXrI9vh7uR5Q==</latexit><latexit sha1_base64="HdkFIKDvlU4NCBi6pyctNR8SRU8=">AAAB93icbVBNS8NAFHypX7V+NOrRy2IRPJVEBBUvRS9ehArGFtpQNptNu3STDbsboYb+Ei8eVLz6V7z5b9y0OWh1YGGYeY83O0HKmdKO82VVlpZXVteq67WNza3tur2ze69EJgn1iOBCdgOsKGcJ9TTTnHZTSXEccNoJxleF33mgUjGR3OlJSv0YDxMWMYK1kQZ2vc1xolH/At2IkPLawG44TWcG9Je4JWlAifbA/uyHgmQxTTThWKme66Taz7HUjHA6rfUzRVNMxnhIe4YmOKbKz2fBp+jQKCGKhDTPpJipPzdyHCs1iQMzGWM9UoteIf7n9TIdnfk5S9JM04TMD0UZR1qgogUUMkmJ5hNDMJHMZEVkhCUm2nRVlOAufvkv8Y6b503n9qTRuizbqMI+HMARuHAKLbiGNnhAIIMneIFX69F6tt6s9/loxSp39uAXrI9vh7uR5Q==</latexit><latexit sha1_base64="HdkFIKDvlU4NCBi6pyctNR8SRU8=">AAAB93icbVBNS8NAFHypX7V+NOrRy2IRPJVEBBUvRS9ehArGFtpQNptNu3STDbsboYb+Ei8eVLz6V7z5b9y0OWh1YGGYeY83O0HKmdKO82VVlpZXVteq67WNza3tur2ze69EJgn1iOBCdgOsKGcJ9TTTnHZTSXEccNoJxleF33mgUjGR3OlJSv0YDxMWMYK1kQZ2vc1xolH/At2IkPLawG44TWcG9Je4JWlAifbA/uyHgmQxTTThWKme66Taz7HUjHA6rfUzRVNMxnhIe4YmOKbKz2fBp+jQKCGKhDTPpJipPzdyHCs1iQMzGWM9UoteIf7n9TIdnfk5S9JM04TMD0UZR1qgogUUMkmJ5hNDMJHMZEVkhCUm2nRVlOAufvkv8Y6b503n9qTRuizbqMI+HMARuHAKLbiGNnhAIIMneIFX69F6tt6s9/loxSp39uAXrI9vh7uR5Q==</latexit>

input<latexit sha1_base64="LKDUr8nARCFfy/y8jbjzLyMGHTQ=">AAAB7HicbVBNSwMxFHzrZ61fVY9egkXwVHZFUG9FLx4ruG2hXUo2zbax2WRJskJZ+h+8eFDx6g/y5r8xu92Dtg4Ehpn3yJsJE860cd1vZ2V1bX1js7JV3d7Z3duvHRy2tUwVoT6RXKpuiDXlTFDfMMNpN1EUxyGnnXBym/udJ6o0k+LBTBMaxHgkWMQINlZqM5Gkpjqo1d2GWwAtE68kdSjRGtS++kNJ0pgKQzjWuue5iQkyrAwjnM6q/VTTBJMJHtGepQLHVAdZce0MnVpliCKp7BMGFervjQzHWk/j0E7G2Iz1opeL/3m91ERXQVZEooLMP4pSjoxEeXQ0ZIoSw6eWYKKYvRWRMVaYGFtQXoK3GHmZ+OeN64Z7f1Fv3pRtVOAYTuAMPLiEJtxBC3wg8AjP8ApvjnRenHfnYz664pQ7R/AHzucPu7aOww==</latexit><latexit sha1_base64="LKDUr8nARCFfy/y8jbjzLyMGHTQ=">AAAB7HicbVBNSwMxFHzrZ61fVY9egkXwVHZFUG9FLx4ruG2hXUo2zbax2WRJskJZ+h+8eFDx6g/y5r8xu92Dtg4Ehpn3yJsJE860cd1vZ2V1bX1js7JV3d7Z3duvHRy2tUwVoT6RXKpuiDXlTFDfMMNpN1EUxyGnnXBym/udJ6o0k+LBTBMaxHgkWMQINlZqM5Gkpjqo1d2GWwAtE68kdSjRGtS++kNJ0pgKQzjWuue5iQkyrAwjnM6q/VTTBJMJHtGepQLHVAdZce0MnVpliCKp7BMGFervjQzHWk/j0E7G2Iz1opeL/3m91ERXQVZEooLMP4pSjoxEeXQ0ZIoSw6eWYKKYvRWRMVaYGFtQXoK3GHmZ+OeN64Z7f1Fv3pRtVOAYTuAMPLiEJtxBC3wg8AjP8ApvjnRenHfnYz664pQ7R/AHzucPu7aOww==</latexit><latexit sha1_base64="LKDUr8nARCFfy/y8jbjzLyMGHTQ=">AAAB7HicbVBNSwMxFHzrZ61fVY9egkXwVHZFUG9FLx4ruG2hXUo2zbax2WRJskJZ+h+8eFDx6g/y5r8xu92Dtg4Ehpn3yJsJE860cd1vZ2V1bX1js7JV3d7Z3duvHRy2tUwVoT6RXKpuiDXlTFDfMMNpN1EUxyGnnXBym/udJ6o0k+LBTBMaxHgkWMQINlZqM5Gkpjqo1d2GWwAtE68kdSjRGtS++kNJ0pgKQzjWuue5iQkyrAwjnM6q/VTTBJMJHtGepQLHVAdZce0MnVpliCKp7BMGFervjQzHWk/j0E7G2Iz1opeL/3m91ERXQVZEooLMP4pSjoxEeXQ0ZIoSw6eWYKKYvRWRMVaYGFtQXoK3GHmZ+OeN64Z7f1Fv3pRtVOAYTuAMPLiEJtxBC3wg8AjP8ApvjnRenHfnYz664pQ7R/AHzucPu7aOww==</latexit>

output<latexit sha1_base64="fLL7CYRKqua5Cng0Dxa8RPrk964=">AAAB7XicbVBNS8NAFHypX7V+VT16CRbBU0lEUG9FLx4rGFtoQ9lsN+3SzW7YfRFK6I/w4kHFq//Hm//GTZuDVgcWhpn32DcTpYIb9Lwvp7Kyura+Ud2sbW3v7O7V9w8ejMo0ZQFVQuluRAwTXLIAOQrWTTUjSSRYJ5rcFH7nkWnDlbzHacrChIwkjzklaKWOyjDNsDaoN7ymN4f7l/glaUCJ9qD+2R8qmiVMIhXEmJ7vpRjmRCOngs1q/cywlNAJGbGepZIkzIT5/NyZe2KVoRsrbZ9Ed67+3MhJYsw0iexkQnBslr1C/M/rZRhfhjmXNhKTdPFRnAkXlVtkd4dcM4piagmhmttbXTommlC0DRUl+MuR/5LgrHnV9O7OG63rso0qHMExnIIPF9CCW2hDABQm8AQv8OqkzrPz5rwvRitOuXMIv+B8fAOmo49O</latexit><latexit sha1_base64="fLL7CYRKqua5Cng0Dxa8RPrk964=">AAAB7XicbVBNS8NAFHypX7V+VT16CRbBU0lEUG9FLx4rGFtoQ9lsN+3SzW7YfRFK6I/w4kHFq//Hm//GTZuDVgcWhpn32DcTpYIb9Lwvp7Kyura+Ud2sbW3v7O7V9w8ejMo0ZQFVQuluRAwTXLIAOQrWTTUjSSRYJ5rcFH7nkWnDlbzHacrChIwkjzklaKWOyjDNsDaoN7ymN4f7l/glaUCJ9qD+2R8qmiVMIhXEmJ7vpRjmRCOngs1q/cywlNAJGbGepZIkzIT5/NyZe2KVoRsrbZ9Ed67+3MhJYsw0iexkQnBslr1C/M/rZRhfhjmXNhKTdPFRnAkXlVtkd4dcM4piagmhmttbXTommlC0DRUl+MuR/5LgrHnV9O7OG63rso0qHMExnIIPF9CCW2hDABQm8AQv8OqkzrPz5rwvRitOuXMIv+B8fAOmo49O</latexit><latexit sha1_base64="fLL7CYRKqua5Cng0Dxa8RPrk964=">AAAB7XicbVBNS8NAFHypX7V+VT16CRbBU0lEUG9FLx4rGFtoQ9lsN+3SzW7YfRFK6I/w4kHFq//Hm//GTZuDVgcWhpn32DcTpYIb9Lwvp7Kyura+Ud2sbW3v7O7V9w8ejMo0ZQFVQuluRAwTXLIAOQrWTTUjSSRYJ5rcFH7nkWnDlbzHacrChIwkjzklaKWOyjDNsDaoN7ymN4f7l/glaUCJ9qD+2R8qmiVMIhXEmJ7vpRjmRCOngs1q/cywlNAJGbGepZIkzIT5/NyZe2KVoRsrbZ9Ed67+3MhJYsw0iexkQnBslr1C/M/rZRhfhjmXNhKTdPFRnAkXlVtkd4dcM4piagmhmttbXTommlC0DRUl+MuR/5LgrHnV9O7OG63rso0qHMExnIIPF9CCW2hDABQm8AQv8OqkzrPz5rwvRitOuXMIv+B8fAOmo49O</latexit>

Objective

Compute (t) so that lim t!1 y(t) = 0<latexit sha1_base64="RwRA7KEufflgpmHdGqd8uOoyZGk=">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</latexit><latexit sha1_base64="RwRA7KEufflgpmHdGqd8uOoyZGk=">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</latexit><latexit sha1_base64="RwRA7KEufflgpmHdGqd8uOoyZGk=">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</latexit>

Non

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0

AUV trajectory ψ

AUV Path Following

y(t) : deviation from the path (path following ERROR)<latexit sha1_base64="U7ZULyhlIbeGxEwa+LN0kOFhVkc=">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</latexit><latexit sha1_base64="U7ZULyhlIbeGxEwa+LN0kOFhVkc=">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</latexit><latexit sha1_base64="U7ZULyhlIbeGxEwa+LN0kOFhVkc=">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</latexit>

dy(t)

dt= V sin (t)

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Objective: reduce the error to o!

Simplified (linearized) model

dy(t)

dt= V (t)

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small (t)<latexit sha1_base64="EP/8fQslvmB9suyCYNmsel//OKo=">AAAB+HicbVBNS8NAEJ34WetX1KOXxSLUS0lFUPFS9OKxgrGFJpTNdtMu3d2E3U2hhP4TLx5UvPpTvPlv3LY5aOuDgcd7M8zMi1LOtPG8b2dldW19Y7O0Vd7e2d3bdw8On3SSKUJ9kvBEtSOsKWeS+oYZTtupolhEnLai4d3Ub42o0iyRj2ac0lDgvmQxI9hYqeu6WmDOUXCDglSzqjnruhWv5s2Alkm9IBUo0Oy6X0EvIZmg0hCOte7UvdSEOVaGEU4n5SDTNMVkiPu0Y6nEguown10+QadW6aE4UbakQTP190SOhdZjEdlOgc1AL3pT8T+vk5n4KsyZTDNDJZkvijOOTIKmMaAeU5QYPrYEE8XsrYgMsMLE2LDKNoT64svLxD+vXde8h4tK47ZIowTHcAJVqMMlNOAemuADgRE8wyu8Obnz4rw7H/PWFaeYOYI/cD5/AIeRkns=</latexit><latexit sha1_base64="EP/8fQslvmB9suyCYNmsel//OKo=">AAAB+HicbVBNS8NAEJ34WetX1KOXxSLUS0lFUPFS9OKxgrGFJpTNdtMu3d2E3U2hhP4TLx5UvPpTvPlv3LY5aOuDgcd7M8zMi1LOtPG8b2dldW19Y7O0Vd7e2d3bdw8On3SSKUJ9kvBEtSOsKWeS+oYZTtupolhEnLai4d3Ub42o0iyRj2ac0lDgvmQxI9hYqeu6WmDOUXCDglSzqjnruhWv5s2Alkm9IBUo0Oy6X0EvIZmg0hCOte7UvdSEOVaGEU4n5SDTNMVkiPu0Y6nEguown10+QadW6aE4UbakQTP190SOhdZjEdlOgc1AL3pT8T+vk5n4KsyZTDNDJZkvijOOTIKmMaAeU5QYPrYEE8XsrYgMsMLE2LDKNoT64svLxD+vXde8h4tK47ZIowTHcAJVqMMlNOAemuADgRE8wyu8Obnz4rw7H/PWFaeYOYI/cD5/AIeRkns=</latexit><latexit sha1_base64="EP/8fQslvmB9suyCYNmsel//OKo=">AAAB+HicbVBNS8NAEJ34WetX1KOXxSLUS0lFUPFS9OKxgrGFJpTNdtMu3d2E3U2hhP4TLx5UvPpTvPlv3LY5aOuDgcd7M8zMi1LOtPG8b2dldW19Y7O0Vd7e2d3bdw8On3SSKUJ9kvBEtSOsKWeS+oYZTtupolhEnLai4d3Ub42o0iyRj2ac0lDgvmQxI9hYqeu6WmDOUXCDglSzqjnruhWv5s2Alkm9IBUo0Oy6X0EvIZmg0hCOte7UvdSEOVaGEU4n5SDTNMVkiPu0Y6nEguown10+QadW6aE4UbakQTP190SOhdZjEdlOgc1AL3pT8T+vk5n4KsyZTDNDJZkvijOOTIKmMaAeU5QYPrYEE8XsrYgMsMLE2LDKNoT64svLxD+vXde8h4tK47ZIowTHcAJVqMMlNOAemuADgRE8wyu8Obnz4rw7H/PWFaeYOYI/cD5/AIeRkns=</latexit>

Non

linea

r St

abili

ty: E

xam

ples

37

xB

yBxy

{B}

V

0

AUV Path Following Objective: reduce the error to o!

dy(t)

dt= V (t)

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Non

linea

r St

abili

ty: E

xam

ples

38

xB

yBxy

{B}

V

0

AUV Path Following Objective: reduce the error to o!

Non

linea

r St

abili

ty: E

xam

ples

1. Lyapunov function candidate

2. Check that V is positive definite

3. Check that dV(t)/dt is neg. def.

Because V(.) is radially unbounded, 0 is GAS!

3-5 Julho Ciência 2017 , Lisbon

Nonlinear Systems: An Introduction to Lyapunov Stability Theory

Antonio Pascoal

Modelação e Simulação, 2017-2018