lecture 3 - is it a stable?adapted from s. n. sreenach et al., modeling the dynamics of signaling...
TRANSCRIPT
Lecture 3 - is it a stable? -
1
Review
2
Modeling• A simplified, quantified representation • Dynamical system or process (driven by time) • Answer questions via formal analysis and simulation • Driven by the rise of big data • Good models resemble real-world data
3
Adapted from S. N. Sreenach et al., Modeling the dynamics of signaling pathways, Essays in Biochemestry, vol. 45, 2008.4
Dynamic modeling
deterministic, stochastic
hybrid systems
continuous time (differential equations),
discrete time (difference equation)
discrete event systems
time-driven event-driven
Central concept in modeling• The state of the system! • Dynamic model: state, input, outputs, dynamics • What it is:
• Independent quantities that determine future evolution • What it does:
• Captures effects of the past
5
Phase planes
CDS 110, 10 Oct 07 R. Murray, Caltech
ODEs can also be used to prove stability of a systems
• Try to reason about the long term behavior of all solutions
• Stability ! all solutions return to equilibrium point (more precise defn later)
Example: spring mass system
• Can we show that all solutions return to rest w/out explicitly solving ODE?
• Idea: look at how energy evolves in time
• Start by writing equations in state space form
• Compute energy and its derivative
• Energy is positive " x2 must eventually go to zero
• If x2 goes to zero, can show that x1 must also approach zero (Lasalle, W3)
8
Analyzing Models Using ODEs: Stability
mq + cq + kq = 0
dx
dt=
�x2
� kmx1 � c
mx2
⇥x1 = q
x2 = q
V (x) =12kx2
1 +12mx2
2dV
dt= kx1x1 + mx2x2
= kx1x2 + mx2(�c
mx2 �
k
mx1) = �cx2
2
CDS 110, 10 Oct 07 R. Murray, Caltech
ODEs can also be used to prove stability of a systems
• Try to reason about the long term behavior of all solutions
• Stability ! all solutions return to equilibrium point (more precise defn later)
Example: spring mass system
• Can we show that all solutions return to rest w/out explicitly solving ODE?
• Idea: look at how energy evolves in time
• Start by writing equations in state space form
• Compute energy and its derivative
• Energy is positive " x2 must eventually go to zero
• If x2 goes to zero, can show that x1 must also approach zero (Lasalle, W3)
8
Analyzing Models Using ODEs: Stability
mq + cq + kq = 0
dx
dt=
�x2
� kmx1 � c
mx2
⇥x1 = q
x2 = q
V (x) =12kx2
1 +12mx2
2dV
dt= kx1x1 + mx2x2
= kx1x2 + mx2(�c
mx2 �
k
mx1) = �cx2
2
9 Oct 06 R. M. Murray, Caltech CDS 3
Today: Dynamic Behavior (and Stability)
Goal #1: Stability
! Check if closed loop response is stable
Goal #2: Performance
! Look at how the closed loop system behaves, in a dynamic context
Goal #3: Robustness (later)
system input
control law
SenseVehicle Speed
ComputeControl “Law”
ActuateGas Pedal
Response
depends on
choice of control
(all are stable)
9 Oct 06 R. M. Murray, Caltech CDS 4
Phase Portraits (2D systems only)
Phase plane plots show 2D dynamics as vector fields & stream functions
! Plot f(x) as a vector on the plane; stream lines follow the flow of the arrows
-1 0 1-1
-0.5
0
0.5
1
x1
x2
-1 0 1-1
-0.5
0
0.5
1
x1
x2
phaseplot(‘dosc’, ... [-1 1 10], [-1 1 10], ... boxgrid([-1 1 10], [-1 1 10]));9 Oct 06 R. M. Murray, Caltech CDS 3
Today: Dynamic Behavior (and Stability)
Goal #1: Stability
! Check if closed loop response is stable
Goal #2: Performance
! Look at how the closed loop system behaves, in a dynamic context
Goal #3: Robustness (later)
system input
control law
SenseVehicle Speed
ComputeControl “Law”
ActuateGas Pedal
Response
depends on
choice of control
(all are stable)
9 Oct 06 R. M. Murray, Caltech CDS 4
Phase Portraits (2D systems only)
Phase plane plots show 2D dynamics as vector fields & stream functions
! Plot f(x) as a vector on the plane; stream lines follow the flow of the arrows
-1 0 1-1
-0.5
0
0.5
1
x1
x2
-1 0 1-1
-0.5
0
0.5
1
x1
x2
phaseplot(‘dosc’, ... [-1 1 10], [-1 1 10], ... boxgrid([-1 1 10], [-1 1 10]));
matlab >> phaseplot()
6
mathematica >> VectorPlot[]
k
Today• ODEs for (state space) modeling • Linear v. nonlinear systems • Equilibrium points • Types of stability • Local v. global properties • Why care about stability?
7
Linear v. nonlinear systems
8
Differential equations
General form
First order Linear systems
1 O
ct 0
7R
. M. M
urra
y, Calte
ch C
DS
9
Tw
o M
ain
Prin
cip
les o
f Feed
back
Ro
bu
stn
es
s to
Un
certa
inty
thro
ug
h
Fe
ed
bac
k
!F
ee
db
ack a
llow
s h
igh p
erfo
rma
nce
in th
e
pre
se
nce
of u
ncerta
inty
!E
xa
mp
le: re
pea
table
perfo
rma
nce
of
am
plifie
rs w
ith 5
X c
om
pon
ent v
aria
tion
!K
ey id
ea
: accura
te s
en
sin
g to
co
mp
are
a
ctu
al to
desire
d, c
orre
ctio
n th
rou
gh
com
pu
tatio
n a
nd a
ctu
atio
n
Des
ign
of D
yn
am
ics th
rou
gh
Fe
ed
ba
ck
!F
ee
db
ack a
llow
s th
e d
yn
am
ics (b
eh
avio
r) of
a s
yste
m to
be m
od
ified
!E
xa
mp
le: s
tab
ility a
ugm
enta
tion fo
r hig
hly
a
gile
, un
sta
ble
airc
raft
!K
ey id
ea
: inte
rco
nn
ectio
n g
ive
s c
losed
loop
tha
t mod
ifies n
atu
ral b
eh
avio
r
X-2
9 ex
perim
ental aircraft
1 O
ct 0
7R
. M. M
urra
y, Calte
ch C
DS
10
Exam
ple
#2: S
peed
Co
ntro
lCo
ntro
lS
ystem
++
-
distu
rban
ce
reference
Sta
bility
/perfo
rma
nce
!S
tead
y s
tate
velo
city
appro
aches
de
sire
d v
elo
city
as k
! "
!S
mo
oth
resp
on
se; n
o o
vers
ho
ot o
r oscilla
tions
Dis
turb
an
ce
reje
ctio
n
!E
ffect o
f dis
turb
ance
s (e
g, h
ills)
ap
pro
ach
es z
ero
as k
! "
Ro
bu
stn
es
s
!R
esu
lts d
on’t d
ep
end o
n th
e s
pe
cific
va
lue
s o
f b, m
or k
, for k
suffic
ien
tly
larg
e
time
velo
city
“Bob
”
�1
ask�⇥
�0
ask�⇥
1 O
ct 0
7R
. M. M
urra
y, Calte
ch C
DS
9
Tw
o M
ain
Prin
cip
les o
f Feed
back
Ro
bu
stn
es
s to
Un
certa
inty
thro
ug
h
Fe
ed
bac
k
!F
ee
db
ack a
llow
s h
igh p
erfo
rma
nce
in th
e
pre
se
nce
of u
ncerta
inty
!E
xa
mp
le: re
pe
ata
ble
perfo
rma
nce
of
am
plifie
rs w
ith 5
X c
om
pon
en
t va
riatio
n
!K
ey id
ea
: accu
rate
sen
sin
g to
co
mp
are
a
ctu
al to
de
sire
d, c
orre
ctio
n th
rou
gh
com
pu
tatio
n a
nd
actu
atio
n
Des
ign
of D
yn
am
ics th
rou
gh
Fe
ed
ba
ck
!F
ee
db
ack a
llow
s th
e d
yn
am
ics (b
eh
avio
r) of
a s
yste
m to
be
mo
difie
d
!E
xa
mp
le: s
tab
ility a
ugm
enta
tion fo
r hig
hly
a
gile
, un
sta
ble
airc
raft
!K
ey id
ea
: inte
rco
nne
ctio
n g
ives c
losed
loop
tha
t mod
ifies n
atu
ral b
eh
avio
r
X-2
9 ex
perim
ental aircraft
1 O
ct 0
7R
. M. M
urra
y, Calte
ch C
DS
10
Exam
ple
#2: S
peed
Co
ntro
lCon
trol
Sy
stem+
+-
distu
rban
ce
reference
Sta
bility
/pe
rform
an
ce
!S
tea
dy s
tate
ve
locity
app
roa
ch
es
de
sire
d v
elo
city
as k
! "
!S
mo
oth
respo
nse
; no
overs
hoo
t or
oscilla
tions
Dis
turb
an
ce
reje
ctio
n
!E
ffect o
f dis
turb
an
ce
s (e
g, h
ills)
ap
pro
ach
es z
ero
as k
! "
Ro
bu
stn
es
s
!R
esu
lts d
on
’t dep
en
d o
n th
e s
pe
cific
va
lue
s o
f b, m
or k
, for k
suffic
iently
larg
e
time
velo
city
“Bob
”
�1
ask�⇥
�0
ask�⇥
What about higher order linear differential equations?
9
Linear systems
1 O
ct 0
7R
. M. M
urra
y, Calte
ch C
DS
9
Tw
o M
ain
Prin
cip
les o
f Feed
back
Ro
bu
stn
ess
to U
nc
erta
inty
thro
ug
h
Fee
db
ac
k
!F
eed
ba
ck a
llow
s h
igh
pe
rform
ance
in th
e
pre
se
nce o
f unce
rtain
ty
!E
xam
ple
: rep
eata
ble
perfo
rma
nce o
f am
plifie
rs w
ith 5
X c
om
pon
en
t varia
tion
!K
ey id
ea
: accu
rate
sen
sin
g to
com
pare
actu
al to
de
sire
d, c
orre
ctio
n th
rou
gh
co
mp
uta
tion
an
d a
ctu
atio
n
De
sig
n o
f Dy
na
mic
s th
rou
gh
Fee
db
ack
!F
eed
ba
ck a
llow
s th
e d
yn
am
ics (b
eh
avio
r) of
a s
yste
m to
be m
od
ified
!E
xam
ple
: sta
bility
aug
me
nta
tion fo
r hig
hly
ag
ile, u
nsta
ble
airc
raft
!K
ey id
ea
: inte
rco
nn
ectio
n g
ive
s c
losed lo
op
tha
t mo
difie
s n
atu
ral b
eh
avio
r
X-2
9 ex
perim
ental aircraft
1 O
ct 0
7R
. M. M
urra
y, Calte
ch C
DS
10
Exam
ple
#2: S
peed
Co
ntro
lCo
ntro
lS
ystem
++
-
distu
rban
ce
reference
Sta
bility
/perfo
rman
ce
!S
tea
dy s
tate
ve
locity
appro
ach
es
de
sire
d v
elo
city
as k
! "
!S
moo
th re
spo
nse; n
o o
vers
hoo
t or
oscilla
tions
Dis
turb
an
ce
reje
ctio
n
!E
ffect o
f dis
turb
ance
s (e
g, h
ills)
ap
pro
ach
es z
ero
as k
! "
Ro
bu
stn
ess
!R
esu
lts d
on’t d
ep
end o
n th
e s
pe
cific
va
lue
s o
f b, m
or k
, for k
suffic
iently
larg
e
time
velo
city
“Bo
b”
�1
ask�⇥
�0
ask�⇥
10
General form
1 O
ct 0
7R
. M. M
urra
y, Calte
ch C
DS
9
Tw
o M
ain
Prin
cip
les o
f Feed
back
Ro
bu
stn
es
s to
Un
certa
inty
thro
ug
h
Fe
ed
bac
k
!F
ee
db
ack a
llow
s h
igh p
erfo
rma
nce
in th
e
pre
se
nce
of u
ncerta
inty
!E
xa
mp
le: re
pe
ata
ble
perfo
rma
nce
of
am
plifie
rs w
ith 5
X c
om
pon
en
t va
riatio
n
!K
ey id
ea
: accu
rate
sen
sin
g to
co
mp
are
a
ctu
al to
de
sire
d, c
orre
ctio
n th
rou
gh
com
pu
tatio
n a
nd
actu
atio
n
Des
ign
of D
yn
am
ics th
rou
gh
Fe
ed
ba
ck
!F
ee
db
ack a
llow
s th
e d
yn
am
ics (b
eh
avio
r) of
a s
yste
m to
be
mo
difie
d
!E
xa
mp
le: s
tab
ility a
ugm
enta
tion fo
r hig
hly
a
gile
, un
sta
ble
airc
raft
!K
ey id
ea
: inte
rco
nne
ctio
n g
ives c
losed
loop
tha
t mod
ifies n
atu
ral b
eh
avio
r
X-2
9 ex
perim
ental aircraft
1 O
ct 0
7R
. M. M
urra
y, Calte
ch C
DS
10
Exam
ple
#2: S
peed
Co
ntro
lCon
trol
Sy
stem+
+-
distu
rban
ce
reference
Sta
bility
/pe
rform
an
ce
!S
tea
dy s
tate
ve
locity
app
roa
ch
es
de
sire
d v
elo
city
as k
! "
!S
mo
oth
respo
nse
; no
overs
hoo
t or
oscilla
tions
Dis
turb
an
ce
reje
ctio
n
!E
ffect o
f dis
turb
an
ce
s (e
g, h
ills)
ap
pro
ach
es z
ero
as k
! "
Ro
bu
stn
es
s
!R
esu
lts d
on
’t dep
en
d o
n th
e s
pe
cific
va
lue
s o
f b, m
or k
, for k
suffic
iently
larg
e
time
velo
city
“Bob
”
�1
ask�⇥
�0
ask�⇥
Linear systems
1 O
ct 0
7R
. M. M
urra
y, Calte
ch C
DS
9
Tw
o M
ain
Prin
cip
les o
f Feed
back
Ro
bu
stn
es
s to
Un
certa
inty
thro
ug
h
Fe
ed
bac
k
!F
ee
db
ack a
llow
s h
igh p
erfo
rma
nce
in th
e
pre
se
nce
of u
ncerta
inty
!E
xa
mp
le: re
pea
table
perfo
rma
nce
of
am
plifie
rs w
ith 5
X c
om
pon
ent v
aria
tion
!K
ey id
ea
: accura
te s
en
sin
g to
co
mp
are
a
ctu
al to
desire
d, c
orre
ctio
n th
rou
gh
com
pu
tatio
n a
nd a
ctu
atio
n
Des
ign
of D
yn
am
ics th
rou
gh
Fe
ed
ba
ck
!F
ee
db
ack a
llow
s th
e d
yn
am
ics (b
eh
avio
r) of
a s
yste
m to
be m
od
ified
!E
xa
mp
le: s
tab
ility a
ugm
enta
tion fo
r hig
hly
a
gile
, un
sta
ble
airc
raft
!K
ey id
ea
: inte
rco
nn
ectio
n g
ive
s c
losed
loop
tha
t mod
ifies n
atu
ral b
eh
avio
r
X-2
9 ex
perim
ental aircraft
1 O
ct 0
7R
. M. M
urra
y, Calte
ch C
DS
10
Exam
ple
#2: S
peed
Co
ntro
lCo
ntro
lS
ystem
++
-
distu
rban
ce
reference
Sta
bility
/pe
rform
an
ce
!S
tea
dy s
tate
ve
locity
appro
ach
es
de
sire
d v
elo
city
as k
! "
!S
mo
oth
respo
nse
; no
overs
hoo
t or
oscilla
tions
Dis
turb
an
ce
reje
ctio
n
!E
ffect o
f dis
turb
ance
s (e
g, h
ills)
ap
pro
ach
es z
ero
as k
! "
Ro
bu
stn
es
s
!R
esu
lts d
on
’t dep
end o
n th
e s
pe
cific
va
lue
s o
f b, m
or k
, for k
suffic
iently
larg
e
time
velo
city
“Bo
b”
�1
ask�⇥
�0
ask�⇥
Difference equations
xk+1 = Adxk + Bduk
yk = Cdxk
Continuous differential equations into discrete difference equations!
11
ODE with derivative input
12
Considerw + 5w + 6w = r + r + 2r
Letu = r
x1 = w � �0ux2 = w � �1u � �0u
Thenx1 = w � �0u= x2 + �1u
x2 = w � �1u � �0u= �5w � 6w + u + u + 2u � �1u � �0u
Letting �0 = 1, �1 = �4
x2 = �5(w � u)� 6w + 2u= �5(x2 � u)� 6(x1 + u) + 2u= �6x1 � 5x2 + 26u= �5(x2 � 4u)� 6(x1 + u) + 2u= �6x1 � 5x2 + 16u
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• Stability: closed-loop stable?
• Performance: closed-loop behavior?
• Robustness (later)
13
Actuate
Compute
Sense Plant
xu
9 Oct 06 R. M. Murray, Caltech CDS 3
Today: Dynamic Behavior (and Stability)
Goal #1: Stability
! Check if closed loop response is stable
Goal #2: Performance
! Look at how the closed loop system behaves, in a dynamic context
Goal #3: Robustness (later)
system input
control law
SenseVehicle Speed
ComputeControl “Law”
ActuateGas Pedal
Response
depends on
choice of control
(all are stable)
9 Oct 06 R. M. Murray, Caltech CDS 4
Phase Portraits (2D systems only)
Phase plane plots show 2D dynamics as vector fields & stream functions
! Plot f(x) as a vector on the plane; stream lines follow the flow of the arrows
-1 0 1-1
-0.5
0
0.5
1
x1
x2
-1 0 1-1
-0.5
0
0.5
1
x1
x2
phaseplot(‘dosc’, ... [-1 1 10], [-1 1 10], ... boxgrid([-1 1 10], [-1 1 10]));
control lawplant input
CONTROL SCHEME
PROCESSSYS
TEM
9 Oct 06 R. M. Murray, Caltech CDS 3
Today: Dynamic Behavior (and Stability)
Goal #1: Stability
! Check if closed loop response is stable
Goal #2: Performance
! Look at how the closed loop system behaves, in a dynamic context
Goal #3: Robustness (later)
system input
control law
SenseVehicle Speed
ComputeControl “Law”
ActuateGas Pedal
Response
depends on
choice of control
(all are stable)
9 Oct 06 R. M. Murray, Caltech CDS 4
Phase Portraits (2D systems only)
Phase plane plots show 2D dynamics as vector fields & stream functions
! Plot f(x) as a vector on the plane; stream lines follow the flow of the arrows
-1 0 1-1
-0.5
0
0.5
1
x1
x2
-1 0 1-1
-0.5
0
0.5
1
x1
x2
phaseplot(‘dosc’, ... [-1 1 10], [-1 1 10], ... boxgrid([-1 1 10], [-1 1 10]));
Responses to stable controllers
Calling something a system does not make it stable, controllable, or even analyzable...
14
Equilibrium points
15
16
e.g.
17
18
Stationary conditions for the pendulum
19
-2 0 2 4 6
-2
-1
0
1
2
x1
x 2
find states such that
Givenx = f(x)
f(xe) = 0
No friction case
xe =
±n⇡0
�
d
dt
x1
x2
�=
x2
gl sinx1
�
• Consider with an equilibrium xe ≠ 0
• Note that because
• The equilibrium ze of the new system is ze = 0.
Can always assume xe = 0
20
g(0) = f(0 + xe) = f(xe) = 0
z = x� xe
z = x = f(x)
f(x) = f(z + xe) := g(z)
x = f(x), f(xe) = 0
z = g(z)
e.g.
21
22
Stability of equilibrium points
23
Stable (in the sense of Lyapunov)
24
9 Oct 06 R. M. Murray, Caltech CDS
Equilibrium points represent stationary conditions for the dynamics
The equilibria of the system x = f(x) are the points xe such that f(xe) = 0.
5
Equilibrium Points
!
-2" 0 2"
-2
0
2
x1
x2
9 Oct 06 R. M. Murray, Caltech CDS 6
Stability of Equilibrium Points
An equilibrium point is:
Asymptotically stable if all nearby initial conditions con-verge to the equilibrium point
! Equilibrium point is an attractor
or sink
Unstable if some initial conditions diverge from the equilibrium point
! Equilibrium point is a source
(or saddle)
Stable if initial conditions that start near the equilibrium point, stay near
! Equilibrium point is a center
-1 0 1
-1
-0.5
0
0.5
1
-1 0 1-1
-0.5
0
0.5
1
0 5 10
-1
0
1
0 5 10
-1
0
1
0 5 10
-1
0
1
-1 0 1
-1
-0.5
0
0.5
1
Stable (in the sense of Lyapunov)
9 Oct 06 R. M. Murray, Caltech CDS
Equilibrium points represent stationary conditions for the dynamics
The equilibria of the system x = f(x) are the points xe such that f(xe) = 0.
5
Equilibrium Points
!
-2" 0 2"
-2
0
2
x1
x2
9 Oct 06 R. M. Murray, Caltech CDS 6
Stability of Equilibrium Points
An equilibrium point is:
Asymptotically stable if all nearby initial conditions con-verge to the equilibrium point
! Equilibrium point is an attractor
or sink
Unstable if some initial conditions diverge from the equilibrium point
! Equilibrium point is a source
(or saddle)
Stable if initial conditions that start near the equilibrium point, stay near
! Equilibrium point is a center
-1 0 1
-1
-0.5
0
0.5
1
-1 0 1-1
-0.5
0
0.5
1
0 5 10
-1
0
1
0 5 10
-1
0
1
0 5 10
-1
0
1
-1 0 1
-1
-0.5
0
0.5
1
e.p. (equilibrium point) is a center
25
For every ✏ > 0, there exists a � = �(✏) > 0 such that,if kx(0)� xek < �, then kx(t)� xek < ✏, for every t � 0.
Asymptotically Stable
9 Oct 06 R. M. Murray, Caltech CDS
Equilibrium points represent stationary conditions for the dynamics
The equilibria of the system x = f(x) are the points xe such that f(xe) = 0.
5
Equilibrium Points
!
-2" 0 2"
-2
0
2
x1
x2
9 Oct 06 R. M. Murray, Caltech CDS 6
Stability of Equilibrium Points
An equilibrium point is:
Asymptotically stable if all nearby initial conditions con-verge to the equilibrium point
! Equilibrium point is an attractor
or sink
Unstable if some initial conditions diverge from the equilibrium point
! Equilibrium point is a source
(or saddle)
Stable if initial conditions that start near the equilibrium point, stay near
! Equilibrium point is a center
-1 0 1
-1
-0.5
0
0.5
1
-1 0 1-1
-0.5
0
0.5
1
0 5 10
-1
0
1
0 5 10
-1
0
1
0 5 10
-1
0
1
-1 0 1
-1
-0.5
0
0.5
1
26
e.p. is an attractor or sink
Lyapunov stable
There exists � > 0 such that if kx(0)� xek < �,then limt!1 kx(t)� xek = 0.
1.
2.
Exponentially stable
27
9 Oct 06 R. M. Murray, Caltech CDS
Equilibrium points represent stationary conditions for the dynamics
The equilibria of the system x = f(x) are the points xe such that f(xe) = 0.
5
Equilibrium Points
!
-2" 0 2"
-2
0
2
x1
x2
9 Oct 06 R. M. Murray, Caltech CDS 6
Stability of Equilibrium Points
An equilibrium point is:
Asymptotically stable if all nearby initial conditions con-verge to the equilibrium point
! Equilibrium point is an attractor
or sink
Unstable if some initial conditions diverge from the equilibrium point
! Equilibrium point is a source
(or saddle)
Stable if initial conditions that start near the equilibrium point, stay near
! Equilibrium point is a center
-1 0 1
-1
-0.5
0
0.5
1
-1 0 1-1
-0.5
0
0.5
1
0 5 10
-1
0
1
0 5 10
-1
0
1
0 5 10
-1
0
1
-1 0 1
-1
-0.5
0
0.5
1
Asymptotically stable1.
2. There exist ↵,�, � > 0 such that if kx(0)� xek < �,then kx(t)� xek ↵kx(0)� xeke��t, for t � 0.
Any e.p. that is exponentially stable systems is also asymptotically stable.
↵kx(0)� xeke��t
Unstable
9 Oct 06 R. M. Murray, Caltech CDS
Equilibrium points represent stationary conditions for the dynamics
The equilibria of the system x = f(x) are the points xe such that f(xe) = 0.
5
Equilibrium Points
!
-2" 0 2"
-2
0
2
x1
x2
9 Oct 06 R. M. Murray, Caltech CDS 6
Stability of Equilibrium Points
An equilibrium point is:
Asymptotically stable if all nearby initial conditions con-verge to the equilibrium point
! Equilibrium point is an attractor
or sink
Unstable if some initial conditions diverge from the equilibrium point
! Equilibrium point is a source
(or saddle)
Stable if initial conditions that start near the equilibrium point, stay near
! Equilibrium point is a center
-1 0 1
-1
-0.5
0
0.5
1
-1 0 1-1
-0.5
0
0.5
1
0 5 10
-1
0
1
0 5 10
-1
0
1
0 5 10
-1
0
1
-1 0 1
-1
-0.5
0
0.5
1
28
e.p. is a source or saddle
Pendulum with friction
29
d
dt
x1
x2
�=
x2
gl sinx1 � k
mx2
�
-2 0 2 4 6
-2
-1
0
1
2
x1
x 2
k = 0
-2 0 2 4 6
-2
-1
0
1
2
x1
x 2
k > 0
Chaotic attractor
30
-2 -1 0 1 2
-2
-1
0
1
2
x1
x 2
d
dt
x1
x2
�=
x21 � x2
2
2x1x2
�
Convergence by itself does not imply Stability
All trajectories converge to xe = 0 but xe is not stable
31
9 Oct 06 R. M. Murray, Caltech CDS 7
Example #1: Double Inverted Pendulum
Stability of equilibria
! Eq #1 is stable
! Eq #3 is unstable
! Eq #2 and #4 are unstable, but with some stable “modes”
Two series coupled pendula
! States: pendulum angles (2), velocities (2)
! Dynamics: F = ma (balance of forces)
! Dynamics are very nonlinear
Eq #1 Eq #2
Eq #3 Eq #4
9 Oct 06 R. M. Murray, Caltech CDS 8
Local versus Global Behavior
Stability is a local concept
! Equilibrium points define the local behavior of the dynamical system
! Single dynamical system can have stable and unstable equilibrium points
Region of attraction
! Set of initial conditions that converge to a given equilibrium point
-2! 0 2!
-2
0
2
x1
x2
9 Oct 06 R. M. Murray, Caltech CDS 7
Example #1: Double Inverted Pendulum
Stability of equilibria
! Eq #1 is stable
! Eq #3 is unstable
! Eq #2 and #4 are unstable, but with some stable “modes”
Two series coupled pendula
! States: pendulum angles (2), velocities (2)
! Dynamics: F = ma (balance of forces)
! Dynamics are very nonlinear
Eq #1 Eq #2
Eq #3 Eq #4
9 Oct 06 R. M. Murray, Caltech CDS 8
Local versus Global Behavior
Stability is a local concept
! Equilibrium points define the local behavior of the dynamical system
! Single dynamical system can have stable and unstable equilibrium points
Region of attraction
! Set of initial conditions that converge to a given equilibrium point
-2! 0 2!
-2
0
2
x1
x2
9 Oct 06 R. M. Murray, Caltech CDS 7
Example #1: Double Inverted Pendulum
Stability of equilibria
! Eq #1 is stable
! Eq #3 is unstable
! Eq #2 and #4 are unstable, but with some stable “modes”
Two series coupled pendula
! States: pendulum angles (2), velocities (2)
! Dynamics: F = ma (balance of forces)
! Dynamics are very nonlinear
Eq #1 Eq #2
Eq #3 Eq #4
9 Oct 06 R. M. Murray, Caltech CDS 8
Local versus Global Behavior
Stability is a local concept
! Equilibrium points define the local behavior of the dynamical system
! Single dynamical system can have stable and unstable equilibrium points
Region of attraction
! Set of initial conditions that converge to a given equilibrium point
-2! 0 2!
-2
0
2
x1
x2
9 Oct 06 R. M. Murray, Caltech CDS 7
Example #1: Double Inverted Pendulum
Stability of equilibria
! Eq #1 is stable
! Eq #3 is unstable
! Eq #2 and #4 are unstable, but with some stable “modes”
Two series coupled pendula
! States: pendulum angles (2), velocities (2)
! Dynamics: F = ma (balance of forces)
! Dynamics are very nonlinear
Eq #1 Eq #2
Eq #3 Eq #4
9 Oct 06 R. M. Murray, Caltech CDS 8
Local versus Global Behavior
Stability is a local concept
! Equilibrium points define the local behavior of the dynamical system
! Single dynamical system can have stable and unstable equilibrium points
Region of attraction
! Set of initial conditions that converge to a given equilibrium point
-2! 0 2!
-2
0
2
x1
x2
9 Oct 06 R. M. Murray, Caltech CDS 7
Example #1: Double Inverted Pendulum
Stability of equilibria
! Eq #1 is stable
! Eq #3 is unstable
! Eq #2 and #4 are unstable, but with some stable “modes”
Two series coupled pendula
! States: pendulum angles (2), velocities (2)
! Dynamics: F = ma (balance of forces)
! Dynamics are very nonlinear
Eq #1 Eq #2
Eq #3 Eq #4
9 Oct 06 R. M. Murray, Caltech CDS 8
Local versus Global Behavior
Stability is a local concept
! Equilibrium points define the local behavior of the dynamical system
! Single dynamical system can have stable and unstable equilibrium points
Region of attraction
! Set of initial conditions that converge to a given equilibrium point
-2! 0 2!
-2
0
2
x1
x2
32
Local v. global behavior• Stability is a local behavior
• define local behavior of the system • feature of each e.p. • system can have stable and unstable e.p.
• Region of attraction • initial conditions that converge to e.p. • local or global feature
9 Oct 06 R. M. Murray, Caltech CDS 7
Example #1: Double Inverted Pendulum
Stability of equilibria
! Eq #1 is stable
! Eq #3 is unstable
! Eq #2 and #4 are unstable, but with some stable “modes”
Two series coupled pendula
! States: pendulum angles (2), velocities (2)
! Dynamics: F = ma (balance of forces)
! Dynamics are very nonlinear
Eq #1 Eq #2
Eq #3 Eq #4
9 Oct 06 R. M. Murray, Caltech CDS 8
Local versus Global Behavior
Stability is a local concept
! Equilibrium points define the local behavior of the dynamical system
! Single dynamical system can have stable and unstable equilibrium points
Region of attraction
! Set of initial conditions that converge to a given equilibrium point
-2! 0 2!
-2
0
2
x1
x2
33
Key questions• Do equilibrium states exist in the data? • Are equilibrium states unique? • Are equilibrium states optimal? • Are equilibrium states stable?
34