Mode Transition Behavior in Hybrid Dynamic Systemsbdeschutter/disc_hs_summer_school_2003/...© 2002...

© 2001 The MathWorks, Inc.

Mod

e Tr

ansi

tion

Beha

vior

in H

ybrid

Dyn

amic

Sy

stem

s

Piet

er J

. Mos

term

anR

eal-t

ime

and

Sim

ulat

ion

Tech

nolo

gies

The

Mat

hWor

ks, I

nc.

Nat

ick,

MA

piet

er_j

_mos

term

an@

mat

hwor

ks.c

omht

tp://

ww

w.x

s4al

l.nl/~

mos

term

a


Intro

duct

ion

■M

ode

Tran

sitio

ns in

Hyb

rid M

odel

s of

Phy

sica

l Sys

tem

s●

hybr

id b

ecau

se◆

cont

inuo

us, d

iffer

entia

l equ

atio

ns◆

disc

rete

, fin

ite s

tate

mac

hine

●ov

ervi

ew o

f phe

nom

ena

invo

lved

■Ill

ustra

ted

by H

ydra

ulic

Act

uato

r Use

d fo

r Airc

raft

Attit

ude

Con

trol

Surfa

ces


Mod

elin

g of

Phy

sica

l Sys

tem

s

■Id

eal P

ictu

re M

odel

(Sch

emat

ic)

■Id

entif

y Be

havi

oral

Phe

nom

ena

■Fo

r Exa

mpl

e, A

Hyd

raul

ic A

ctua

tor


Equa

tion

Gen

erat

ion

■C

ompi

le C

onst

ituen

t Equ

atio

ns◆

Rin

◆R

oil

◆C

oil

◆m

p

◆R

rel

◆I re

l

◆0,

cyl

inde

r cha

mbe

r◆

1, re

lief f

low

pip

e◆

1, in

take

pip

e◆

1, o

il co

mpr

essi

on

fR

pin

inRi

n=

If

pre

lre

lre

l&

=

pp

fR

pre

lsm

pre

lre

lcy

l=

−+

vf

fp

inre

l=

−

mv

Ap

pp

pcy

l&=

fR

pR

oil

Roil

=C

pf

oil

CR

&=

fR

pre

lre

lre

l=

pp

pRi

nin

cyl

=−

pp

pRo

iloi

lC

=−


Equa

tion

Proc

essi

ng

■Be

fore

Sim

ulat

ion

●th

e nu

mbe

r of e

quat

ions

is re

duce

d◆

subs

titut

ion/

elim

inat

ion

●eq

uatio

ns a

re s

orte

d◆

each

equ

atio

n co

mpu

tes

one

varia

ble

●eq

uatio

ns a

re s

olve

d◆

high

inde

x pr

oble

ms

may

requ

ire d

iffer

entia

tion

of c

erta

in e

quat

ions


Hyb

rid B

ehav

ior

■In

trodu

ce V

alve

s●

mak

e hi

ghly

non

linea

r beh

avio

r pie

cew

ise

linea

r◆

inta

ke v

alve

◆re

lief v

alve

■Sw

itchi

ng B

etw

een

Mod

es o

f Con

tinuo

us B

ehav

ior

●in

take

val

ve,v

in, e

xter

nal s

witc

h (c

ontro

l law

)●

relie

f val

ve,v

rel,

auto

nom

ous

switc

h tri

gger

ed b

y ph

ysic

al q

uant

ities

●di

ffere

nt s

ets

of e

quat

ions

vp

pre

lcy

lth

=>

i fv

then

pp

pel

sef

inRi

nin

cyl

in=

−=0

ifv

then

pp

fR

pel

sef

rel

rel

smp

rel

rel

cyl

rel

=−

+=0


Com

puta

tiona

l Cau

salit

y

■W

hen

Switc

hing

Equ

atio

ns●

com

puta

tiona

l cau

salit

y m

ay c

hang

e◆

re-o

rder

ing

◆re

-sol

ving

■Ex

ampl

e●

whe

n th

e in

take

val

ve c

lose

s, e

quat

ions

cha

nge

◆Fr

om

◆To

●th

eref

ore,

in th

is e

quat

ion

◆p R

inbe

com

es u

nkno

wn

◆f in

beco

mes

kno

wn

f in=0

pp

pRi

nin

cyl

=−


Impl

icit

Mod

elin

g

■D

eal W

ith C

ausa

l Cha

nges

Num

eric

ally

■Va

lve

Beha

vior

●re

sidu

e on

f in

●re

sidu

e on

f rel

■Im

plic

it N

umer

ical

Sol

ver (

e.g.

, DAS

SL)

●de

sign

ed to

han

dle

this

form

ulat

ion

0=−

+−

+if

vth

enp

pf

Rp

else

fre

lre

lsm

pre

lre

lcy

lre

l

0=−

+−

ifv

then

pp

pel

sef

inRi

nin

cyl

in


Hyb

rid D

ynam

ic B

ehav

ior

■G

eom

etric

Vie

w●

mod

es o

f con

tinuo

us, s

moo

th, b

ehav

ior

●pa

tche

s of

adm

issi

ble

stat

e va

riabl

e va

lues


Spec

ifica

tion

Parts

■H

ybrid

Beh

avio

r Spe

cific

atio

n●

a fu

nctio

n, f,

that

def

ines

con

tinuo

us, s

moo

th, b

ehav

ior f

or e

ach

mod

e

●an

ineq

ualit

y, γ

, tha

t def

ines

adm

issi

ble

stat

e va

riabl

e va

lues

γ ααα

αii

ii

Cx

Du

++

≥1

0:

fE

xA

xB

ui

ii

iα

αα

α:

&+

+=0


Dyn

amic

s

■Be

havi

or C

hara

cter

istic

s●

C0 ,

i.e.,

no ju

mps

in s

tate

va

riabl

es●

stee

p gr

adie

nts

■Ex

ampl

e●

whe

n th

e in

take

val

ve

clos

es, p

isto

n ve

loci

ty

quic

kly

redu

ces

to 0


The

Nex

t Ste

p

■R

emov

e St

eep

Gra

dien

ts●

e.g.

, sin

gula

r per

turb

atio

n■

Alge

brai

c C

onst

rain

ts B

etw

een

Stat

e Va

riabl

es●

high

inde

x sy

stem

s●

subs

pace

with

adm

issi

ble

(con

tinuo

us) d

ynam

ic b

ehav

ior

●di

scon

tinui

ties

(jum

ps) i

n st

ate

beha

vior


Hyb

rid D

ynam

ic B

ehav

ior -

Ref

ined

■G

eom

etric

Vie

w●

mod

es o

f con

tinuo

us, s

moo

th, b

ehav

ior

●pa

tche

s of

adm

issi

ble

stat

e va

riabl

e va

lues

●m

anifo

ld o

f dyn

amic

beh

avio

r


Spec

ifica

tion

Parts

■H

ybrid

Beh

avio

r Spe

cific

atio

n●

a fu

nctio

n, f,

that

impl

icitl

y de

fines

for e

ach

mod

e◆

cont

inuo

us, s

moo

th, b

ehav

ior

◆st

ate

varia

ble

valu

e ju

mps

●an

ineq

ualit

y, g

, tha

t def

ines

adm

issi

ble

gene

raliz

ed s

tate

var

iabl

e va

lues

●fo

r exp

licit

rein

itial

izat

ion

(sem

antic

s of

x-)

γ ααα

αii

ii

Cx

Du

++

≥1

0:

0:

=+

+u

Bx

Ax

Ef

ii

ii

αα

αα

&

fE

xA

xB

uB

xi

ii

ii

ux

αα

αα

α:

&+

++

=−

0


Han

dlin

g of

Sys

tem

s W

ith H

igh

Inde

x

■D

ASSL

Han

dles

Inde

x 2

Syst

ems

●im

plic

it fo

rmul

atio

n fo

r con

tinuo

us b

ehav

ior

■R

equi

res

Con

sist

ent I

nitia

l Con

ditio

ns W

hen

Mod

e C

hang

es O

ccur

●co

mpu

te fr

om im

plic

it fo

rmul

atio

n to

mak

e ju

mp

spac

e (p

roje

ctio

n)

expl

icit

●fo

r exa

mpl

e, s

eque

nces

of s

ubsp

ace

itera

tion

◆sp

ace

of d

ynam

ic b

ehav

ior:

Vn+

1=

A-1

E V

n , V

0=

Rn

◆ju

mp

spac

e:Tn+

1=

E-1

A T

n ,T0

= {0

}●

or, d

ecom

posi

tion

in (p

seud

o) K

rone

cker

Nor

mal

For

m


Proj

ectio

ns

■Li

near

Tim

e In

varia

nt In

dex

2 Sy

stem

●de

rive

pseu

do K

rone

cker

Nor

mal

For

m (n

umer

ical

ly s

tabl

e)

●af

ter i

nteg

ratio

n (n

o im

puls

ive

inpu

t beh

avio

r), c

onsi

sten

t val

ues

are

EE

x x x

AA

AA

A A

x x x

B B Bu

f i i

f i i

11

2212

1 2

11121

122

2211

2212

2222

1 2

1 21

22

00

00

00

00 0

00

,, ,

,,

,, ,

, ,

, ,

& & &

+

+

=

xx

EA

AE

xx

xA

Bu

Ex

Ax

xA

Bu

ff

ii

ii

i

i

=−

−

=−

+−

=−−

−−

−

−

−

11112122111

2212

22

122111

21

2212

22212

2

222221

22,

,,

,,

,,

,,

,,

,

,,

,

()

(&)


The

Hyd

raul

ic A

ctua

tor

■G

ener

aliz

ed S

tate

Jum

ps fo

r Eac

h M

ode

Mode

Projection

α00

f rel =

0v p

= 0

α01

v p =

(mpv

p- – I r

elf re

l- )/(m

rel +

mp)

f rel =

(mpv

p- – I r

elf re

l- )/(m

rel +

mp)

α10

v p =

vp-

f rel =

0α

11v p

= v

p-

f rel =

f rel

-


A Sc

enar

io

■In

take

Val

ve Is

Ope

n●

pist

on s

tarts

to m

ove

■In

take

Val

ve C

lose

s●

pist

on in

ertia

cau

ses

pres

sure

bui

ld-u

p●

pres

sure

reac

hes

criti

cal v

alue

■R

elie

f Val

ve O

pens

●cy

linde

r pre

ssur

e de

crea

ses

⇒In

tera

ctio

n Be

twee

n M

ode

Tran

sitio

n Be

havi

or


Phas

e Sp

ace

of C

ylin

der S

cena

rio

■Pr

ojec

tion

Is A

borte

d●

imm

edia

tely

●af

ter p

artia

l com

plet

ion

(b)

(a)


Sequ

ence

s of

Mod

e C

hang

es

■(a

) Sta

te O

utsi

de o

f a P

atch

in th

e N

ew M

ode

■(b

) Dur

ing

Proj

ectio

n St

ate

Valu

es a

re R

each

ed O

utsi

de o

f a P

atch

in

the

New

Mod

e

(a)

(b)


Impu

lses

■H

igh

Inde

x Sy

stem

s M

ay C

onta

in Im

puls

ive

Beha

vior

●in

cas

e of

the

hydr

aulic

cyl

inde

r, p

> p th

, wou

ld a

lway

s ho

ld if

not

vp

= v p-

●un

know

n w

here

the

patc

h is

aba

ndon

ed■

In-D

epth

Ana

lysi

s of

Sw

itchi

ng C

ondi

tions

●so

lve

for r

equi

red

x(t)

●co

mpu

te e

arlie

st t

= t s

at w

hich

γ(x

(t), u

(t), t

) ≥0

●su

bstit

ute

t sto

com

pute

x(t s)

■C

ompl

ex S

witc

hing

Stru

ctur

e■

Addi

tiona

l Diff

icul

ty W

hen

Inte

ract

ing

Fast

Tra

nsie

nts

(e.g

., co

llisio

n)


Det

aile

d An

alys

is o

f the

Pro

ject

ion

■C

ylin

der E

xam

ple

(Imag

inar

y Ei

genv

alue

s, λ

= λ r

+ i λ

i)●

from

det

aile

d m

odel

◆so

lve

for p

◆su

bstit

ute

tat w

hich

p(t)

> p

th

pt

ep

tC

vp

trt

ii

pr

i()

(cos(

)(

)sin())

=−

+−

−−

λλ

λλ

λ11 1

ve

vt

R Iv

p Iv

tp

tp

ip

rp

is

ir

s=

−−

+−

−−

λλ

λλ λ

(cos(

)(

)sin(

) )2 1

1 1


Com

plex

Sw

itchi

ng S

truct

ure

■Ex

plic

it R

e-In

itial

izat

ion

ELSE

DAE

α 10

DAE

α 00

DAE

α 11

DAE

α 01

¬v in

/t s

= f v(p

- , v p- ,

p th)

t s>0

/v p

= g

p,a0

0(p- ,

v p- , t s)


Cha

tterin

g

■W

hat I

f the

New

Mod

e Sw

itche

s Ba

ck●

imm

edia

tely

⇒in

cons

iste

nt m

odel

, no

solu

tion

●af

ter i

nfin

itesi

mal

per

iod

of ti

me ⇒

chat

terin

g be

havi

or, s

olve

with

◆eq

uiva

lent

con

trol

◆eq

uiva

lent

dyn

amic

s

γ α12

αγ α

21α

αα 12

xx

α1

α2

γ α12

αγ α

21α

αα 12

xx

α1

α2


Equi

vale

nt D

ynam

ics

■C

hatte

ring

●fa

st c

ompo

nent

◆re

mov

e●

slow

com

pone

nt◆

wei

ghte

d m

ean

of in

stan

tane

ous

vect

or fi

elds

(Filip

pov

Con

stru

ctio

n)●

slid

ing

beha

vior


Ont

olog

y

■Ph

ase

Spac

e Tr

ansi

tion

Beha

vior

Cla

ssifi

catio

n●

myt

hica

l (st

ate

inva

riant

)●

pinn

acle

(sta

te p

roje

ctio

n ab

orte

d)●

cont

inuo

us◆

inte

rior (

cont

inuo

us b

ehav

ior)

◆bo

unda

ry (f

urth

er tr

ansi

tion

afte

r inf

inite

sim

al ti

me

adva

nce)

◆sl

idin

g (re

peat

ed tr

ansi

tions

afte

r eac

h in

finite

sim

al ti

me

adva

nce)

■C

ombi

natio

ns o

f Beh

avio

r Cla

sses


Con

clus

ions

■M

ode

Tran

sitio

n Be

havi

or●

Ric

h●

Com

plex

■R

equi

res

●sp

ecia

l alg

orith

ms/

com

puta

tions

●m

odel

ver

ifica

tion

anal

yses

■H

ow to

Effi

cien

tly G

ener

ate

Beha

vior

(e.g

., fo

r Rea

l-tim

e Ap

plic

atio

ns)?


Ref

eren

ces

■ht

tp://

ww

w.x

s4al

l.nl/~

mos

term

a/pu

blic

atio

ns.h

tml

■Pi

eter

J. M

oste

rman

, "M

ode

Tran

sitio

n Be

havi

or in

Hyb

rid D

ynam

icSy

stem

s," W

inte

r Si

mul

atio

n C

onfe

renc

e, D

ecem

ber 7

-10,

New

Orle

ans,

Lou

isia

na, 2

003,

invi

ted

pape

r.■

Piet

er J

. Mos

term

an, "

HYB

RSI

M-A

Mod

elin

g an

d Si

mul

atio

n En

viro

nmen

t for

Hyb

rid B

ond

Gra

phs,

" in

Jour

nal o

f Sys

tem

s an

d C

ontro

l Eng

inee

ring,

vol

. 216

, Par

t I, p

p. 3

5-46

, 200

2,

spec

ial i

ssue

pap

er.

■Pi

eter

J. M

oste

rman

, MAS

IM-A

Hyb

rid D

ynam

ic S

yste

ms

Sim

ulat

or, t

echn

ical

repo

rt D

LR-

IB-5

15-0

1-07

, Ins

titut

e of

Rob

otic

s an

d M

echa

troni

cs, D

LR O

berp

faffe

nhof

en, 2

001.

■

Piet

er J

. Mos

term

an, "

Impl

icit

Mod

elin

g an

d Si

mul

atio

n of

Dis

cont

inui

ties

in P

hysi

cal S

yste

m

Mod

els,

" in

The

4th

Inte

rnat

iona

l Con

fere

nce

on A

utom

atio

n of

Mix

ed P

roce

sses

: Hyb

rid

Dyn

amic

Sys

tem

s, p

p. 3

5-40

, Dor

tmun

d, G

erm

any,

Sep

tem

ber,

2000

, inv

ited

pape

r. ■

Piet

er J

. Mos

term

an, F

eng

Zhao

and

Gau

tam

Bis

was

, "An

Ont

olog

y fo

r Tra

nsiti

ons

in

Phys

ical

Dyn

amic

Sys

tem

s," i

n Pr

ocee

ding

s of

AAA

I-98,

pp.

219

-224

, Jul

y, M

adis

on, W

I, 19

98.

■Pi

eter

J. M

oste

rman

and

Gau

tam

Bis

was

, "A

Theo

ry o

f Dis

cont

inui

ties

in P

hysi

cal S

yste

m

Mod

els,

" Jou

rnal

of t

he F

rank

lin In

stitu

te, V

olum

e 33

5B, N

umbe

r 3, p

p. 4

01-4

39, 1

998

■Pi

eter

J. M

oste

rman

and

Gau

tam

Bis

was

, "A

Form

al H

ybrid

Mod

elin

g Sc

hem

e fo

r Han

dlin

g D

isco

ntin

uitie

s in

Phy

sica

l Sys

tem

Mod

els,

" Pro

c. o

f AAA

I-96,

pp.

985

-990

, Aug

ust 2

-4,

Portl

and,

Ore

gon,

199

6.

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