Post on 20-Aug-2019
transcript
© 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
© 2002 The MathWorks, Inc.
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
© 2002 The MathWorks, Inc.
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
© 2002 The MathWorks, Inc.
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
=−
© 2002 The MathWorks, Inc.
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
© 2002 The MathWorks, Inc.
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
© 2002 The MathWorks, Inc.
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
=−
© 2002 The MathWorks, Inc.
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
© 2002 The MathWorks, Inc.
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
© 2002 The MathWorks, Inc.
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
© 2002 The MathWorks, Inc.
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
© 2002 The MathWorks, Inc.
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
© 2002 The MathWorks, Inc.
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
© 2002 The MathWorks, Inc.
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
© 2002 The MathWorks, Inc.
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
© 2002 The MathWorks, Inc.
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,
,,
,,
,,
,,
,,
,
,,
,
()
(&)
© 2002 The MathWorks, Inc.
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
-
© 2002 The MathWorks, Inc.
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
© 2002 The MathWorks, Inc.
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)
© 2002 The MathWorks, Inc.
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)
© 2002 The MathWorks, Inc.
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)
© 2002 The MathWorks, Inc.
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
© 2002 The MathWorks, Inc.
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)
© 2002 The MathWorks, Inc.
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
© 2002 The MathWorks, Inc.
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
© 2002 The MathWorks, Inc.
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
© 2002 The MathWorks, Inc.
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)?
© 2002 The MathWorks, Inc.
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
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