Part II: Noise sensitivies in systems
with delays and multiple time scales
Acknowle
The auth
NSF Grant
sional Meas
sity of Mich
Nomencla
0.8
0.6
0.4
0.2
0
!0.2
!0.4
!0.6
!0.8
700
Wednesday, June 6, 2012
750
800
850
900
Ak ! p
am ! a
#
% ! s
d ! d
& ! t
f ! s
'(t) ! t
( ! L
Kc ! c
k ! i
M (t;z) ! u
) ! o
p max ! m
d
p(i) ! d
so ! n
s(t) ! s
t ! t
u(m) ! p
* bw ! o
*n ! n
x(t) ! r
p
z ! h
+ ! s
Models of Balance
• Applications: Human Postural Sway, Stick
Balancing, Robotics
• Mathematical Challenges: Nonlinear dynamics
with delayed feedback, discontinuous control,
noise
• What are the contributing factors to stability,
instability, balance, sway, other behaviors?
Ideas from models in mechanics/
optics, bring to biological
applications: balance, blood
diseases, epidemiology
Wednesday, June 6, 2012
• Applications: Human Postural Sway, Stick
Balancing, Robotics
461
cm) attached at either end to a metal stand (see Holden et al.
1994). The stand rested on a rigid wooden platform (155 cm!
70 cm) that overlay the force plate and extended beyond its lateral edges. The touch device apparatus on one side of the platform
was balanced by a comparable mass on the other side (see Fig. 1).
This arrangement ensured that the Kistler force platform detected
all forces applied to the touch device as well as all forces generated by the subject’s feet and that it did not interpret forces at the
fingertip as changes in CFPx or CFPy. In this circumstance, the
force applied by the finger to the touchbar and transmitted to the
platform by the touchbar’s base will be exactly balanced by the
resulting reaction force generated at the subject’s feet; consequently, there will be no spurious change in CFPx or CFPy as a result of the force at the fingertip. The horizontal bar was adjusted
in height to allow individual subjects to assume a comfortable lateral arm position while touching the contact plate with their index
finger. Two, dual-element, temperature-compensated strain gauges (Kulite Semiconductor, Type M(12) DGP-350-500) mounted
on the metal bar transduced the lateral (FingerL) and vertical (FingerV) forces applied by the finger. The strain gauge signals were
amplified and calibrated in units of force (newtons). A comparator could trigger an auditory tone when a specified threshold
force was reached.
Procedure
Fig. 1 Schematic illustration of the experimental apparatus with a
subject in the tandem Romberg stance
signed an informed consent statement that had been approved by
the Brandeis University Committee for the Protection of Human
Subjects.
Derivative of
the Angle
from vertical
Apparatus and Measures
Figure 1 depicts the test situation with a subject standing in the
tandem Romberg position (heel-to-toe) on a force platform, touching a device used to measure the forces applied by the fingertip.
The force platform (Kistler Model 9261A) measured the ground
reaction forces at the feet.
Center of foot pressure
Medial-lateral (CFPx) and anterior-posterior (CFPy) coordinates of
foot pressure were computed from the force components registered by piezoelectric crystals in the corners of the force platform.
Wednesday, June
6, 2012
Head motion
The subject stood with right foot behind left along the center of the
anterior-posterior axis of the force platform. Adhesive tape was
used to mark the position of the subject’s feet on the platform so
that the same foot position could be repeated for the different conditions of the experiment. The touch bar was adjusted to a comfortable height (approximately waist level) and lateral distance for the
subject to make contact with the right INDEX fingertip.
The experimental conditions varied in terms of allowing vision
(Vision) or having eyes closed (Dark) and type of fingertip contact: no contact (NoTouch) in which the subject’s arms hung passively, touch contact (Touch) in which the subject was limited to
1 N of applied force on the touch apparatus, and force contact
(Force) during which the alarm was turned off and subjects could
apply as much force as desired. One newton of applied force at the
fingertip is mechanically inadequate to attenuate sway amplitude
by more than 2.3% (Holden et al. 1994). The six experimental
conditions included: Vision-NoTouch, Vision-Touch, VisionForce, Dark-NoTouch, Dark-Touch, and Dark-Force.
One practice trial was given for each condition before the experiment began. Subjects started a trial by getting on the platform
and looking straight ahead at a fixation target on a wall 2 m away
that was covered with a black cloth. The subject’s peripheral visual field beyond 30° provided a rich, complex visual environment
with many horizontal and vertical features. Subjects were told to
let go of the safety railing surrounding them and take as much
Asai, etal,
2009,
PLOS1
Human
postural sway
Angle from vertical
"
−
!
sec
""
.
ẍ
=
g
tan
3
ystem. Finally, in Section 5 we conclude and point to some open problems.
Simple model: inverted pendulum
hus when the pendulum is at the inverted position, " = 0 and
called “labyrinthus” in the inner ear which helps in the self-balancing
¨" = 0, equations
.= Model
the cart is not atouraeyesfixed
position but moves with
are closed.
ome
constant speed.
The dynamics of the setup in Fig. 1 are governed
by the second-order
2. MECHANICAL
MODEL differential
OF BALAN
An alternate
approach
isConsider
to
the
penisplacement
θ of the pendulum,
which
canassume
be
as (see
alsoinverted
Refs.
the written
simplest that
planar
mechanical
model[2,9,17]):
of the
inverted pen
Its lowest point slides smoothly along the horizontal line. This mechan
!
"
ulum is 3m
stabilized not
by
the
application
of
forces
at
the
possible
model
describing
the
“man-machine”
system
when
somebody
3m 2
3g
3F
2
his fingertip,
tries
to
point ofcos
the stick
in0.
a way t
1 − by the
θ̈ + application
− and
cos θdirect
θ̇ onsin(2θ)
sin
θ move
+ thisatlowest
θ=
ase but
of
torque
the
pivot.
In
4
8 its upper position.
2 LThe system has
2L(M
+ Mofc )freedom described by
two p
degrees
z. The angle cp is detected together with its derivatives and the ho
is case the model is veryandsimple,
M. LANDRY, S. CAMPBELL, K. MORRIS, AND C. AGUILAR
y
l
m
θ
O
j
i
l
g = -g j
P
F(t) i
G. STP.~N AND L. KOLL~R
200
M
x
x(t)
Figure 1. Inverted pendulum system.
Table 1
Notation.
x(t)
θ(t)
M
m
L
l
P
F (t)
Displacement of the center of mass of the cart from point O
Angle the pendulum makes with the top vertical
Mass of the cart
Mass of the pendulum
Length of the pendulum
Distance from the pivot to the center of mass of the pendulum (l = L/2)
Pivot point of the pendulum
Force applied to the cart
experimental setup, the force F (t) on the cart is
determined
by them
in acart,
way that
s 0 position
be
where F is the horizontal driving force
applied
to the
andthe
m upper
= Mcpp /(M
p + Mshould
c ) is th
niform pendulum Mp with respect to the sum of the mass of the cart Mc and the mass of
2¨
f the4 m!
pendulum
is L and
g is""the
gravitational
acceleration. We adjust the time
scale t√
"
=
T
,
!10"
−
mg!
sin!
control
3
f the pendulum by rescaling time, introducing a new dimensionless time tnew = told ·
F (t) = αV (t) − β ẋ(t),
V (t) is the voltage in the motor driving the cart, and the second term represents
al resistance in the motor. The voltage V (t) can be varied and is used to control the
The values of the parameters for our experimental setup [21] are given in Table 2.
ecise value of the viscous friction # is difficult to measure; however, we expect that it
of a similar order of magnitude as β. Our work will thus consider # in the range [0, 15].
more convenient form of the equations is found by solving for ẍ and θ̈ from equations
nd (1.2) and introducing the variables
θ
y = (y1 , y2 , y3 , y4 )T
T
= (x, θ, ẋ, θ̇) .
here
the more
linear feedback
control torque is
Even
simple model:
�
inverted pendulum w/
˙.
=
q
"
+
q
"
T
control control
2
4 (torque at
pivot
T
control
the
pivot)
he linearization of Eq. !10" about " = 0 is very similar to
Figure 1. Mechanical model of stick balancing.
Wednesday, June 6, 2012
also explain the center manifold reduction and present the bifurcation diagram of the redu
erted
or“Reduced
field model near the
triple-zero
eigenvalue
bifurcation.
In Section
3, we show that there a
4
¨
to
the
essentials”
balance
model:
""give
. examples of complex balancing mo
ẍ =model,
g tan
3 ! sec
amics in the reduced
and"in−Section
4 we
em. Finally, in Section 5 we conclude and point to someor
open
as problems.
a system
Thus
when
the
pendulum
is at the inverted position, " =
θ̈
−
sin
θ
+
F
(θ,
θ̇)
cos
θ
=
0
!4"
θ̇ = φ
F ="˙ aθ(t
+ bcart
θ̇(t −isτ )not at a fixed position but move
= "¨ =−0,τ )the
φ̇ = sin θ − F (θ, θ̇) cos θ
Model equations
istic
some constant speed.
F
=
aθ(t
−
τ
)
+
b
θ̇(t
−
τ
)
nThe!3"
dynamics of the
in Fig. 1approach
are governedisby
second-order
equ
Ansetup
alternate
tothe
assume
that differential
the inverte
lacement
the pendulum,
which can be
written
(see application
also Refs. [2,9,17]):
po- θ ofdulum
is stabilized
not
by asthe
of forces
!
"
of torque at the pi
3m base
3m 2the direct3 gapplication 3F
2 but by
1−
cos θ θ̈ +
θ̇ sin(2θ) −
sin θ +
cos θ = 0.
4 this case the
8 model is2 very
L
2L(Mp + Mc )
simple,
re F is the horizontal driving force applied to the cart, and m = Mp /(Mp + Mc ) is the r
2¨
to the sum of the mass of the cart Mc and the mass of the
orm pendulum Mp4with respect
m!
"
"
"
=
T
,
−
mg!
sin!
control
3 g is the gravitational acceleration. We adjust the time scale to t
he pendulum is L and
√
he!5"
pendulum by rescaling time, introducing a new dimensionless time tnew = told · 3g
conn the
and
•
where the linear feedback control torque is
Vertical Pendulum: need for control/
external force applied
for
stabilization
Tcontrol = q2" + q4"˙ .
Wednesday, June 6, 2012
cart becomes
of
Model equations
ted
4
¨(PD)
Proportional
/differential
control:
"
−
!
sec
""
.
ẍ
=
g
tan
The dynamics of the setup in Fig.3 1 are governed by the second-order differential e
placement θ of the pendulum, which can be written as (see also Refs. [2,9,17]):
Thus when the pendulum is at the inverted position, " =
!4"!
"
3m
3g
3m 2
3F
˙
¨
2
" =θ0,θ̈ the
is not
a θfixed
positioncos
but
1 − " =cos
+ cart
− at sin
θ̇ sin(2θ)
+
θ =moves
0.
4
8
2
L
2L(M
+
M
)
p
c
stic
some constant speed.
!3"
Mc ) is the
ere F is the horizontal
driving force
applied to the
cart,
and m = M
p /(M
p + inverted
An alternate
approach
is
to
assume
that
the
F
=
aθ
+
b
θ̇
iform pendulum Mp with respect to the sum of the mass of the cart Mc and the mass of th
podulum
is
stabilized
not
by
the
application
of
forces
a
the pendulum is L and g is the gravitational acceleration. We adjust the time scale to
√
base
by time,
the introducing
direct application
of torque
at =the
the pendulum
by but
rescaling
a new dimensionless
time tnew
toldpiv
·
Derivative
Proportional
this
case the model is very
simple,
or
4
2¨
m! " − mg! sin!"" = T
,
!5"
onthe
control
3
Aθ + B θ̇control torque is
where the-Tlinear
feedback
control =
Wednesday, June 6, 2012
˙
Biological considerations:
• Delay in the application of the control: neural
transmission
• On-off control: not active control all the time
• Noise: Small random fluctuations can result in
large changes in certain circumstances
F = aθ(t − τ ) + bθ̇(t − τ )
Wednesday, June 6, 2012
2. The model. A nondimensional model of the hematopoietic production system is
by
Hematopoietic
Stem cells
Neutrophils
(2)
Platelets
Erythrocytes
dq
dt
dn
dt
dp
dt
dr
dt
(blood production) system:
= −qb1 hq (q) + b1 µ1 q1 hq (q1 ) − q {b2 hn (n) + b3 hp (p) + b4 hr (r)} ,
= −γn n + an b2 qτnm hn (nτnm ),
!
"
= −γp p + ap b3 qτpm hp (pτpm ) − µ3 qτpsum hp (pτpsum ) ,
= −γr r + ar b4 {qτrm hr (rτrm ) − µ4 qτrsum hr (rτrsum )} ,
where q, n, r, and
are nondimensional stem cells, neutrophils,
Colijn,erythrocytes,
Mackey, 2007and pla
Fluctuations
inpproliferation
respectively. Subscripts indicate delays: q1 = q(t − 1) and so on.
times
in lower
Theand
functions
hq , hn , hcell
r , and hp are Hill functions given by
population levels
can drive transitions to
large oscillations (disease):
Weak attraction to healthy
(steady) state (slow
damping)
10
8
6
4
2
0
40
60
80
100
120
140
160
180
120
140
160
180
10
8
6
4
2
0
Wednesday, June 6, 2012
200
220
240
260
Mathematical Challenges:
• Nonlinearity: non-uniqueness, complex dynamics
• Delays: Delay Differential Equations (DDEs)
instead of Ordinary Differential Equations (ODEs),
functional/history dependence instead of initial
conditions at a single point
• On/off control: discontinuous equations rather
than continuous or smooth behavior.
• Noise: Stochastic DDEs (SDDEs), minimal theory,
limited computational methods, additional
complexities with nonlinearities
References: Campbell, Milton, Ohira, Sieber, Krauskopf, Stepan,
others
Wednesday, June 6, 2012
Chatter in
machine tool
dynamics
chip
tool
previous cut
nonlinear force
present cut
dx = ydt
dy
=
(−2κy − x +
cj [x(t − τ ) − x(t)]j )dt
1
damped linear oscillator
+
3
X
η(1 +
3
X
cj [x(t − τ ) − x(t)]j )
1
Key parameters:
variation in material
x = fraction of variation from desired chip thickness
Oscillations of tool position (about equilibrium)
c1 = =
material
parameter,
κ =cut
damping
(fixed)
Is x(t)
0 stable?
i.e. is a straight
stable?
τ = inversely proportional to rotation velocity
Noise sources:
δ = percentage variation of material parameter, η = δdw
Material parameters (additive + multiplicative noise)
Speed of rotation (delay)
Wednesday, June 6, 2012
What about delays?
Brief crash course in Delay Differential Equations
(DDEs)
�
y (t) = −y(t − τ ),
τ = 1 is the delay
Need an initial history function: solution for the time
interval [0, τ [ depends on initial history function
Applied Delay Differential Equations
y = 1 for the time interval [−1, 0[
Discontinuity in derivative at t=0,
propagates as discontinuity in the
second derivative at t=1, etc.
Wednesday, June 6, 2012
http://site.ebrary.c
Delays and cyclic/period behaviors
Logistic equation: Populations
�
y(t − τ )
y (t) = ry(t) 1 −
K
�
�
Growth in birth rate,
Decay due to overpopulation
Delayed feedback in deaths from
overpopulation
Longer delays yield oscillations Lower graph has longer delay
Wednesday, June 6, 2012
Carrying capacity = K:
Bifurcation diagram for periodic behavior
Logistic equation:
Populations
ations
Bifurcation: fixed point stable
for smaller delays,
http://site.ebrary.com/lib/ubc/docPrint.action?encrypted=41e5c...
For larger delays, branch
indicates amplitude of
Applied Delay Differential Equations
oscillations
http://site.ebrary.com/lib/ubc/docP
Carrying capacity: K
y
Hopf bifurcation
S
U
τ
Wednesday, June 6, 2012
S
which
yieldsstability
Linear
2x
analysis: machine tool example
dx
�
x
(t)
+
κx
(t)
+
x(t)
=
c
(x(t
−
τ
)
−
x(t))
(x(s
−
τ
)
−
x(s)).
+
x
=
c
−
2κ
1
1
s2
ds
��
and linearized about x = 0, which yields
(3.10)
x(t) = e
e Look
characteristic
equation
for eigenvalues
λt
dx
d2 x
+ x = c1 (x(s − τ ) − x(s)).
− 2κ
2
ds
ds
−λτ
+ 1 = c1x(e
− 1).
λ2 − 2κλSubstituting
the characteristic equation
= eλt yields
(3.11)
λ2 − 2κλ + 1 = c1 (e−λτ − 1).
termined
by setting
λ=
iωdecay?
in (3.11),
dividing
this
equation
into its
λ
=
σ
+
iω
Oscillations:
grow
or
Look
for
The stability boundary is determined by setting λ = iω in (3.11), dividing this equation
τ . These curves are shown
d solving
two
equations
for c 1 and
σ = 0 these
pure
imaginary
eigenvalues
real and
imaginary parts,
and solving
these two equations for c 1 and τ . These curves a
in Figureparameters
3, in terms of theand
non-dimensional
parameters and
the dimensional paramete
non-dimensional
the
dimensional
parameters.
x=0 unstable (oscillations grow)The
c1
c1
0.115
0.11
0.105
0.1
x=0 stable (oscillations decay)
40
6
Wednesday, June 6, 2012
3.7
x 10
45
50
55
ττ
60
65
70
75
ds2
− 2κ
ds
+ x = c1 (x(s − τ ) − x(s)).
Full nonlinear model:
Substituting x = eλt yields the characteristic equation
��
�
λ2 − 2κλ + 1 = c1 (e−λτ − 1).
The stability
boundary is−
determined
setting λ = iω in (3.11), dividing this equati
x (t) + κx (t) + x(t)
= F(x(t
τ ) −byx(t))
real and imaginary parts, and solving these two equations for c 1 and τ . These curves
in Figure 3, in terms of the non-dimensional parameters and the dimensional parame
x=0 unstable (oscillations grow)
material c
c
Noise induced chatter
parameter 1
1
0.115
0.11
Bifurcation scenario: Near stability boundaries
x=0 stable (oscillations decay)
0.105
0.1
maximum
amplitude
40
45
50
55
60
65
τ
x
delay
70
75
6
x 10
Hopf bifurcation: oscillatory
large amplitude
solutions
for certain delay
K
nonlinear mode
Smaller nonlinear oscillatory
Hopf bifurcation
solutions unstable
Ω
Figure 3:
small amplitude
Larger nonlinear oscillatory
nonlinear mode
(stable)
solutions
bi-stable
with
x=0
zero solution
solution
shape
of
the
stability
boundary implies that for certain
control parameter (c )
t values of τ there is a larger rang
1.53.7
3.6
3.5
1
3.4
0
3.3
0.53.2
3.1
3
0
700
800
900
1000
1100
1200
rpm
The stability !0.5
boundary for the zero solution of the linearized deterministic equation
δ = 0; that is, there is a stable equilibrium at x = 0 for parameter values below these curve
a series of curves, due to the periodicity of the functions in (3.11) with λ = iω. The top figu
!1 of the non-dimensional parameters c and τ . The bottom figure sho
stability boundary in terms
1
1.75
boundary for the dimensional parameters K0 (N/m ) and Ωrpm (rpm). Here Ωrpm = 60 · Ω/
for consistency with graphs given in previous references, for example [12].
!1.5
2200
1
Wednesday, June 6, 2012
2250
2300
2350
2400
2450
2500
which the steady state x = 0 is linearly stable. Since the delay τ is inversely proport
the angular velocity of the rotating workpiece, then there may be certain preferred valu
Variation in material parameters:
noise source amplifies oscillations through CR
x
0.8
0.6
0.4
0.2
0
Chatter sustained by noise in subthre
!0.2
Computation of probability density for am
!0.4
p(x)
!0.6
!0.8
700
750
800
850
2.5
900
2
t
p(x)
1.5
1
Probability density for x
(x=0 is fixed point)
0.5
0
!1.5
!1
!0.5
0
x
0.5
1
x
Wednesday, June 6, 2012
1.5
FIGURES
SDDE: (additive noise)
dx = y dt, dx = f (x(t), x(t − τ ); c1 ) dt + δdw
τ ∼ τc + �2 τ2 or c1 = c1 c + �2 c1 2
T is a slow time T = �2 t,
0.13
material c
parameter
1
Look for solution of the form
0.125
0.12
0.115
0.11
x = 0 stable
0.105
0.1
35
x̂(t) = A(T )cos bt + B(T )sin bt
40
45
50
55
τ
6
3.8
60
delay
65
x 10
Information from the deterministic problem: on t time scale
3.6
Derive equations for noisy amplitudes on T time scaleK1
0
@
dA
dB
1
A
=
0
@
ψA
ψB
1
0
A dT + Σ @
Σ = constant matrix for added noise
3.4
3.2
dβA (T )
3
700
A
800
dβB (T )
x = 0 stable
900
1000
1100
1200
Ωrpm
Figure 1: The stability boundary for the zero solution for the linearized deterministic equatio
Klosek, K. 2005
δ = 0. The equilibrium at x = 0 is stable for parameter values below these curves. The series o
result of the periodicity of eλt in (2.10) for λ = iω.
Find ψA , ψB and Σ by equating two equations for dx:
Ito’s formula: change of variables from x to A, B
Substitution in the SDDE (involves A(T ), A(T
Wednesday, June 6, 2012
1
− �2 τ ) , etc.)
Envelope equations for oscillations:
ditional terms in (3.12) related to the multiplicative noise, but they are not included here, since
K, 2006
he
form of the equations for A and B is

 arean order of magnitude




ey describe effects
that
smaller
than those
due to the additive
noise
δ " 1.
 dA   ψA (A, B)

 

=

 

δ

 dT +

$
 σ11



 dξ1 (T ) 
σ12 




dB
B)in detail in [19]. The
dξB)
σ21 form
σ22 of ψA (A,
B (A,
2 (Tis)
are
given
The coefficients
ψA and ψB ψ
,

(3.12)
amplification
(
'
ξj (t), j = 1, 2 independent standard
the deterministic
B(TFor
− $parameters
τ ) − B(T ) infactor
A(T − $ τBrownian
) − A(T ) motions.
+ g (ω, κ)
ψ(A, B) = c g (ω, κ)
c
1
2
2
1
2
$2
$2
q = 0) stability region for x = 0, the amplitude (A, B) of the vibrations decay on the T
+ c12 [h1 (ω, κ)A(T ) + h2 (ω, κ)B(T )]
0.12
(3.13)
Compare probability density p(x)
0.1
scale. However, for δ != 0 the envelope of the vibrationsforhas
standard deviation on the
x(t)aand
noisec1
2 τ ), B(T −$2 τ ),
d similarly for ψB . Note that these coefficients depend on A(T ), B(T ) and A(T −$subthreshold
x̂(t) = A(T )cos bt + B(T )sin bt
driven
oscillations
of δ/$ (σ11 and σ22 are O(κ) while σ12 and σ21 are O(1)), indicating
that2 the
amplification
c1 =time
c1 c +
� cT1 2 .this
at is, they also depend on the delayed value of the envelope. On Here
the slow
scale
Time scale of coherence resonance (CR):
T (slow)
0.8
versely
proportional
to the square root of the proximity to the stability boundary. Figure 2
rresponds
to a small delay.
x(t)
5
0.10
4
0
0.4
graphs of the invariant density p(x) (for large time) for different values of c1 , showing the
3.5
0.2
3
p(x)
ase in the variance as the parameter values approach the stability boundary. One can include
0
2.5
2
!0.2
Time
scale
of
ional terms in (3.12) related to the multiplicative
noise, but they are not included here, since
8
oscillations: t
1.5
!0.4
1
!0.6
0.5
t than those due to the additive
describe effects thatBuckwar,K.
are an order
of
magnitude
smaller
noise
x
, L’esperance, Soo, 2006
Wednesday, June 6, 2012
0.1
4.5
0.6
!0.8
0.11
700
750
800
850
900
0
!0.5
!0.4
!0.3
!0.2
!0.1
0
0.1
0.2
0.3
0.4
0.5
chip
tool
previous cut
nonlinear force
present cut
dx = ydt
dy
=
(−2κy − x +
cj [x(t − τ ) − x(t)]j )dt
1
damped linear oscillator
+
3
X
η(1 +
3
X
cj [x(t − τ ) − x(t)]j )
1
Key parameters:
variation in material
x = fraction of variation from desired chip thickness
Oscillations of tool position (about equilibrium)
c1 = material
parameter,
κ = damping
(fixed)
Is x(t)
= 0 stable?
i.e. is a straight
cut stable?
τ = inversely proportional to rotation velocity
Noise sources:
δ = percentage variation of material parameter, η = δdw
Material parameters (additive + multiplicative noise)
Speed of rotation (delay)
Wednesday, June 6, 2012
Results for variation of delay w/o variation
of material parameters
Ackno
The
NSF G
sional
sity of
Nome
Without CR: eigenvalues of
discretized problem
(Yilmaz, 2003)
Ak
am
%
d
&
f
'(t)
(
Kc
k
M (t;z)
)
p max
Sinusoidal variation of delay:
Must be tuned precisely for
damping
(Namachchivaya, et al, 2003)
p(i)
so
s(t)
t
u(m)
* bw
*n
x(t)
z
+
Refer
Wednesday, June 6, 2012
Fig. 12 MRSSV hold duration, z, effect in chatter suppression
#1$ Sl
in
x
0.4
Random switching of
speed (delay) damps
coherent oscillations
−1
τ
(w/o variation in material
parameters)
0.2
x
0
!0.2
!0.4
0
1
2
3
4
5
6
7
8
9
10
5
6
7
8
9
10
s
t
900
800
Ωrpm
700
600
500
0
1
2
3
4
t
s
• Noise-induced order via transients
• Switching between exponential
Figure 3: Top: The time series for (2.6) with δ = 0, τ = 66, c1 = .115 and q = 0 (dotted line), q = .25
(solid line). Bottom: Variation in Ωrpm corresponding to the time series in the top figure for q = .25.
FIGURES
damping and slow growth near a
stability boundary
0.13
• Apparent controlled
oscillations/quiescence is a
sequence of transients
Wednesday, June 6, 2012
0.125
0.12
c1
15
0.115
0.11
x = 0 stable
0.105
0.1
35
40
45
50
τ
6
3.8
3.6
x 10
55
60
65
70
ditional terms in (3.12) related to the multiplicative noise, but they are not included here, since
Both types of noise: Envelope equations for oscillations
he form of the equations for A and B is
y describe effects that are an order of magnitude smaller than those due to the additive noise

 




 dA   ψA (A, B) 
 σ11 σ12   dξ1 (T ) 
δ

 





=
 dT + 

,

 



$
(A,
The coefficients
dBψA and ψB
ψBare
(A,given
B) in detail in [19].
σ21 The
σ22 form ofdξψ2A(T
) B) is
δ " 1.
'
ξj (t), j = 1, 2 independent
c
standard
A(T −Brownian
$2 τ ) − A(Tmotions.
)
ψ(A, B) = c1 g1 (ω, κ)
$2
(3.12)
(
For
parameters
in the
B(T
− $2 τ ) − B(T
)
+ g2 (ω, κ)
$2
deterministic
q = 0) stability region for x = 0, the amplitude (A, B) of the vibrations decay on the T
+ c12 [h1 (ω, κ)A(T ) + h2 (ω, κ)B(T )]
(3.13)
scale. However, for δ != 0 the envelope of the vibrations has a standard
on the
K. deviation
2006
Time scale of coherence resonance (CR): T
at is,
they also
dependof
on oscillations:
the delayed value of
the envelope. On the slow time scale T this
Time
scale
t
versely proportional to the square root of the proximity to the stability boundary. Figure 2
Timeto ainterval
responds
small delay. for varying delay: L (discrete r.v.)
d similarly for ψB . Note that these coefficients depend on A(T ), B(T ) and A(T −$2 τ ), B(T −$2 τ ),
of δ/$ (σ11 and σ22 are O(κ) while σ12 and σ21 are O(1)), indicating that the amplification
graphs of the invariant density p(x) (for large time) for different values of c1 , showing the
ase in the variance as the parameter values approach the stability boundary. One can include
Variance of resonant mode on intermediate scale:
8
Validthatfor
large
describe effects
are L
an sufficiently
order of magnitude
smaller than those due to the additive noise
ional terms in (3.12) related to the multiplicative noise, but they are not included here, since
Wednesday, June 6, 2012
19
CROSSING BOUNDARIES OF DELAY SYSTEMS IN LEUKEMIA
Complicated stability regions not uncommon for DDE’s:
Controlled container crane, Erneux, Kalmar-Nagy, 2008
FIGURES
1
−5
0.8
0.2
0.6
0.19
0.185
0.18
0.175
Immune response to
leukemia treatment:
(a)
delays for cell
division/interaction
q0(ρ, jω) + q1(ρ,jω)
0.195
ρ(ω)
x 10
0.4
0.13
0.2
0.125
0
0.12
−0.2
c1
−0.4
0.115
0.17
−0.6
0.11
0.165
−0.8
0.105
0.16
0
0.2
0.4
0.6
0.8
−1
0
1
ω
x = 0 stable
0.002 0.10.004
35
0.006
6
3.8
x 10
0.008
40
ρ
(b)
0.01
45
0.012
0.014
50
55
60
65
70
τ
BIFURCATION ANALYSIS OF A MOD
Delayed response for therapies:
Fig. 4.1. (a) Graph of the stability crossing point ω(ρ) as a function of the time delay ρ based
as a function of ρ.
on the implicit relation (4.3). (b) Graph of q0 (ρ, jω(ρ)) + q1 (ρ, jω(ρ))e−jω(ρ)ρ
3.6
We are interested in when the expression equals 0, thus satisfying (4.2).
100
K1
0.015
3.4
0.01
3.2
90
80
x = 0 stable
70
0.005
3
60
700
800
900
1000
1100
1200
1300
Ωrpm
0
∼
τ
∼
τ
50
stable region
40
−0.005
Figure
1: The stability boundary for the zero solution for the linearized deterministic equation (2.6) with
30
δ = 0. The equilibrium at x = 0 is stable for parameter values below these curves. The series of curves is a
−0.01of the periodicity of eλt in (2.10) for λ = iω.
result
20
10
0
0
20
40
60
80
σ
(a)
Niculescu, et al, 2008
100
−0.015
−0.02
Colijn,
Mackey,
(b)
2007
−0.01
0
0.01
σ
Fig. 4.2. Crossing curves with respect to σ and τ̃ when υ̃ = 0. (a) The zoomed-out plot shows
the overall structure of the crossing curves. (b) The zoomed-in plot shows the small region of stability
around the origin. Note that only nonnegative delays make sense.
Wednesday, June 6, 2012
0.02
Hematopoietic system:
dq
dt
dn
Neutrophil
dt
)
dp
Platelets
dt
dr
dt
Stem cells
= −qb1 hq (q) + b1 µ1 q1 hq (q1 ) − q {b2 hn (n) + b3 hp (p) + b4 hr (r)} ,
= −γn n + an b2 qτnm hn (nτnm ),
!
"
= −γp p + ap b3 qτpm hp (pτpm ) − µ3 qτpsum hp (pτpsum ) ,
= −γr r + ar b4 {qτrm hr (rτrm ) − µ4 qτrsum hr (rτrsum )} ,
N ANALYSIS OF A MODEL OF HEMATOPOIESIS
383
here q, n, r, and p are nondimensional stem cells, neutrophils, erythrocytes, an
15
spectively. Subscripts indicate
delays: q1 = q(t − 1) and so on.
The functions hq , hn , hr , and hp are Hill functions given by
10
5
0
20
30
40
50
60
70
9
8
Steady-state and periodic oscillations in the decoupled
stem cell compartment. Where the steadyshown
as aJune
thick
line (left side), it is stable, and7where it is thin (right side), it is unstable. The
Wednesday,
6, 2012
80
90
100
110
120
2. The model. A nondimensional model of the hematopoietic production system is
by
Hematopoietic
Stem cells
Neutrophils
(2)
Platelets
Erythrocytes
dq
dt
dn
dt
dp
dt
dr
dt
system:
= −qb1 hq (q) + b1 µ1 q1 hq (q1 ) − q {b2 hn (n) + b3 hp (p) + b4 hr (r)} ,
= −γn n + an b2 qτnm hn (nτnm ),
!
"
= −γp p + ap b3 qτpm hp (pτpm ) − µ3 qτpsum hp (pτpsum ) ,
= −γr r + ar b4 {qτrm hr (rτrm ) − µ4 qτrsum hr (rτrsum )} ,
where q, n, r, and
are nondimensional stem cells, neutrophils,
Colijn,erythrocytes,
Mackey, 2007and pla
Fluctuations
inpproliferation
respectively. Subscripts indicate delays: q1 = q(t − 1) and so on.
times
in lower
Theand
functions
hq , hn , hcell
r , and hp are Hill functions given by
population levels
can drive transitions to
large oscillations (disease):
Weak attraction to healthy
(steady) state (slow
damping)
10
8
6
4
2
0
40
60
80
100
120
140
160
180
120
140
160
180
10
8
6
4
2
0
Wednesday, June 6, 2012
200
220
240
260
2. The model. A nondimensional model of the hematopoietic production system is given
by
Hematopoietic system:
dq
Stem cells dt
dn
dt
(2)
Platelets dp
dt
dr
dt
= −qb1 hq (q) + b1 µ1 q1 hq (q1 ) − q {b2 hn (n) + b3 hp (p) + b4 hr (r)} ,
= −γn n + an b2 qτnm hn (nτnm ),
15
!
"
10
= −γp p + ap b3 qτpm hp (pτpm ) − µ3 qτpsum hp (pτpsum ) ,
= −γr r + ar b4 {qτrm hr (rτrm ) − µ4 qτrsum hr (rτrsum )} ,
5
0
20
30
40
50
60
70
80
90
100
110
120
9
where q, n, r, and p are nondimensional stem cells, neutrophils, erythrocytes, and platelets,
respectively.
Subscripts
indicate
q1 =OF
q(tHEMATOPOIESIS
− 1) and so on.
BIFURCATION
ANALYSIS
OF delays:
A MODEL
The functions hq , hn , hr , and hp are Hill functions given by
8
7
Variable delay (production)
39
6
Platelet amplification
5
4
3
15
cell
( #):
0
50
100
150
10
5
0
30
delay
2
40
50
60
70
80
90
100
110
120
130
40
50
60
70
80
90
100
110
120
130
2.5
2
1.5
1
0.5
Colijn, Mackey, 2007
Wednesday, June 6, 2012
0
30
t
Balance model:
θ̇
φ̇
Stability diagram for PD
control (w/delay), control
always on
202
(b)
=
=
φ
sin θ − F (θ, θ̇) cos θ
F = aθ(t − τ ) + bθ̇(t − τ )
G. ST&PAN AND L. KOLL~
Hopf
bifurcation
Pitchfork
bifurcation
(a)
”
mg
Figure 2. Stability
chart.D-shaped region
θ = 0 is stable in shaded
area,
Pitchfork bifurcation: stability of non-zero fixed point
Hopf
bifurcation,
stability
ofof oscillatory
The corresponding
stability chart
in the plane
the gain parameters behavior
P, D for constant delays
can be proved by checking the limit condition Pmin = Pm=(w) for the existence of any stability
domain for P in (6).
71 < 72 < * ** is presented qualitatively in Figure 2. The encircled numbers show the number of
Wednesday, June 6, 2012
se of the controllerp
(solid
ffiffiffiffiffiffiffiffiffiffiline),
ffiffiffiffiffiffiffiffiffiffiffiand
ffiffiffi c) plots the position of the target (dashed line)
d c), DðxðtÞ,yðtÞÞ~ x2 ðtÞzy2 ðtÞ is the length of the position vector measured a
ively, by the Qualisys motion capture system and the computer program. No ambig
ment angle is small (see Figure 7d) and the movements of the fingertip and tip o
What about other effects?
On/off control, together with delays:
Below the threshold: Control is off (growth)
Above the threshold: Control is on (decay)
bserved only when vertical
with active control (‘‘act’’) taken only on
System
moves
back
and
forth
across
the
threshold
certain thresholds [7,10,13,22,23,25–27
rtip, is associated with a small
Balancing with Vibration
generic model with ‘‘drift–and–act’’ con
produces no changes in the
F (x(t − τ ))
with parametric excitation takes the form
made by the fingertip. Taken
that the skill enhancement is
Toy model
ertip and not to the effects of
dx
~F ðxðt{tÞÞxðtÞzkxðtÞsin
o–skeletal system. We suggest
dt
unexpected observation is to
where k is a constant, f is the forcing f
ced position is not a simple
delay, xðMilton,
tÞ,xðt{tÞetare,
the
lex bounded time–dependent
al, respectively,
2009
variable at times t and t{t, and jðtÞ d
of attraction whose size is of
uently, for sufficiently large
noise with variance g2 . The feedback funct
If fixed
point is and
not the
reached,
periodic
behavior
he basin
of attraction,
step–like
shape
shown incontinues
Figure 9a. Mode
Calculate
conditions
to periodic
solution
g, any
mechanism
that biasesthat lead
successfully
employed,
for example, to
Figure 9. Effects of parametric excitation on the dynamics of a simple ‘‘drift and act’’ controller. a) Graphical representation of a simple
realization of the feedback function that produces a limit cycle oscillation in (2) in the absence of parametric excitation and noisy perturbations,
where F ðxðt{tÞÞ~ðazbÞ{b=ð1zexpðQðxðt{tÞ{ThÞ and a~0:18, b~{0:20, Q~500, and Th~1. The displacement from the upright position,
xðt{tÞ, grows when xðt{tÞvTh and decreases when xðt{tÞwTh. b) Periodic parametric excitation is turned on at the ;. The effect is to decrease
the amplitude of the limit cycle oscillation. Parameters are f ~2 and k~0:14.
doi:10.1371/journal.pone.0007427.g009
Wednesday,
June
h
i 6, 2012
What about other effects?
On/off control: Observations indicate control on only for
large enough variation from upright position
Asai, etal,
2009,
PLOS1
Figure 8. Sample of human postural sway, collected from a subject in quiet standing for 120 s. Left upper pane
Left lower panel: trajectory in the phase plane; Right panel: power spectral density of the angular sway.
doi:10.1371/journal.pone.0006169.g008
Effects of delays: Delays can naturally introduce oscillations
Noise-free DDE simulations of Model 4 for various sets of P-D-a
parameters show that the stable regions in the P-D-a parameter
space of Model 4 are the same as those of Model 3, with the
difference that the attractor of Model 3 is an asymptotically stable
equilibrium point and that of Model 4 is a limit cycle. This fact
provides a common basis for understanding the noisy dynamics of
Wednesday, June 6, 2012
both Models 3 and 4. As shown in Fig. 7, the power spectra of
How do we analyze these?
mgh<0.79 for the right panel) and commo
(D = 10 Nms/rad, a = 20.4 s21, s = 0.2
the switching parameter a, the optimal va
of Model 3 was about 60% of mgh regard
the dark band was located at P=mgh&0
values of P used for Fig. 9 left and right a
and larger than the optimal value of P. A
we also explain the center manifold reduction and present the bifurcation diagram of the
vector field model near the triple-zero eigenvalue bifurcation. In Section 3, we show that th
dynamics in the reduced model, and in Section 4 we give examples of complex balancin
system. Finally, in Section 5 we conclude and point to some open problems.
Feedback control in
balance models:
2. Model equations
PD control with delay
or
θ̇
φ̇
=
=
φ
sin θ − F (θ, θ̇) cos θ
F = aθ(t − τ ) + bθ̇(t − τ )
The dynamics of the setup in Fig. 1 are governed by the second-order differentia
displacement θ of the pendulum, which can be written as (see also Refs. [2,9,17]):
!
"
3m
3m 2
3g
3F
1−
cos2 θ θ̈ +
θ̇ sin(2θ) −
sin θ +
cos θ = 0.
4
8
2L
2L(Mp + Mc )
where F is the horizontal driving force applied to the cart, and m = Mp /(Mp + Mc ) is
uniform pendulum Mp with respect to the sum of the mass of the cart Mc and the mass o
of the pendulum is L and g is the gravitational acceleration. We adjust the time scale
Asai,
etal, 2009,
PLOS1
of the pendulum by rescaling time, introducing a new
dimensionless
time
tnew = told ·
State-dependent on/off control: control is
on in state drifting away from origin
q
q
OFF
s
e
ON
e
Ws
W
Fig. 1. Sketch of the inverted pendulum of length L and mass Mp on a cart of mass Mc at horizontal position δ.
such that it is positive in the position shown in the sketch.
ON
Wednesday, June 6, 2012
OFF
or field model near the triple-zero eigenvalue bifurcation. In Section 3, we show that there are complex symmetric
amics in the reduced model, and in Section 4 we give examples of complex balancing motion in the full control
em. Finally, in Section 5 we conclude and point to some open problems.
Back to the balance models:
PD control with delay
θ̈ − sin θ + F (θ, θ̇) cos θ = 0
On/Off Control :
F =
aθ(t
τ )1 +
bθ̇(t −
τ )second-order differential equation
he dynamics
of the
setup −
in Fig.
are governed
by the
for the angular
Note delay in
Model equations
or
lacement θ of the pendulum, which can be written as (see also Refs. [2,9,17]):
!
"
3m
3m 2
3g
3F
1−
cos2 θ θ̈ +
θ̇ sin(2θ) −
sin θ +
cos θ = 0.
4
8
2L
2L(Mp + Mc )
switching from on/off,
(1)
system enters off
of the massbefore
of the
re F is the horizontal driving force applied to the cart, and m = M /(M + M ) is the ratio
region
control
orm pendulum M with respect to the sum of the mass of the cart M and the mass of the pendulum. The length
Less control:
he pendulum
is L and g is zig-zag
the gravitational acceleration. We adjust the time scale to
intrinsic
time scale off
switched
√
√ theis
he pendulum by rescaling time, introducing a new dimensionless time t
= t · 3g/ 2L. This converts
p
p
p
c
behavior
Intermittent Postural Control
new
old
ON
ON
OFF
c
. Sketch of the inverted pendulum of length L and mass Mp
that it is positive in the position shown in the sketch.
Stronger control:
OFF
on a cart of mass M at horizontal position δ. The inclination angle θ is defined
yields spiral
c
on/off switch
Wednesday,
6, 2012
gure
3. TypicalJune
solutions
of Model 3 in the phase plane. In each plane, the initial state at t = 0 is represented by the thick curve segment
Asai, etal,
2009,
PLOS1
Periodic behavior:
θ̈ − sin θ + F (θ, θ̇) cos θ = 0
F = aθ(t − τ ) + bθ̇(t − τ )
PD control with delay
Zig-zag orbits - periodic solutions away from origin,
moving back and forth from on/off
θ̇
"
(!0,"0)
t=0
$
t = !#
ON
!
t = Tint
t = Tint + #
(!1,"1)
OFF
%
1
Figure 6: The forward orbit, Γ, of a point on Σ1 . The primary purpose of this figure
is to illustrate some labels used in expansions given the text. Unlike elsewhere in this
Note, t = 0 corresponds to a switching point instead of a point on Σ1 .
Wednesday, June 6, 2012
Periodic behavior: zig-zag orbits
θ̈ − sin θ + F (θ, θ̇) cos θ = 0
F = aθ(t − τ ) + bθ̇(t − τ )
Calculating Zig-zag orbits:
•Calculate solution in OFF
region (no delays in equation)
•Calculate solution in ON
region, using off solution as
(Asymptotic analysis)
initial history,
ON
" θ̇ = φ
•Approximate with small
$
delays (next page)
•Does the value at end of
%
on/off switch match with
OFF
initial condition in off position?
Figure 6: The forward orbit, Γ, of a point on Σ . The primary purpose o
is to illustrate some labels used in expansions given the text. Unlike elsew
If yes, then periodic
(!0,"0)
t=0
t = !#
t = Tint
t = Tint + #
(!1,"1)
1
Note, t = 0 corresponds to a switching point instead of a point on Σ1 .
Wednesday, June 6, 2012
and assuming, without loss of generality, that this point corresponds to t =
Some calculations:
F = aθ(t − τ ) + bθ̇(t − τ )
θ̈ − sin θ + F (θ, θ̇) cos θ = 0
"
ON
(!0,"0)
t=0
$
t = !#
OFF
!
t = Tint
%
t = Tint + #
(!1,"1)
1
where
The forward orbit, Γ, of a point on Σ1 . The primary purpose of this figure
strate some labels used in expansions given the text. Unlike elsewhere in this
3.2 Series
in s instead
and τ of a point on Σ .
= 0 corresponds
to a expansions
switching point
1
β4 = − bθ0 cos(θ0 ) ,
2
1
0 2<(θ0θ)0, < π2 ,
β̂5 = − abθ0 cos
#
2
2
1
−
<0 )s. ≤ 0 ,
β6 = − b sin(θ0 ) cos(θ
π
6
#
$
1
(θ20 ) < 0 . 1
β̂6 = β6 + abθF
0 cos (θ0 ) = − b cos(θ0 ) sin(θ0 ) − aθ0 cos(θ0 ) .
6
6
(63)
(64)(29)
21 and ensures that the control is sufficiently strong.
e last inequality
refers
(21)
It may be
showntothat
n that φ(−τ ) = sθ(−τ )% for the orbit 2Γ, the value
φ0 is completely determined
ξ1 τ + ξ2 sτ + ξ3 τ + O(|s, τ |3) , φ(τ ) − sθ(τ ) ≤ 0
T
=
,
(67)it is
int
alue of θ0 , so we first derive
relation
between
φ0 and θ0 . From
(4),
3
ˆ 2
ξ1 τ + ξthe
2 sτ + ξ3 τ + O(|s, τ | ) , φ(τ ) − sθ(τ ) ≥ 0
rified that21 for t ∈ [−τ, 0],
6
ully this isn’t too confusing in that we’ve been thinking of initial conditions as points on Σ 1 .
hat it is somewhat superfluous to also write φ(t) because φ(t) = θ̇(t).
Wednesday, June 6, 2012
13
"3 .
2
a2 θ02
"2
∆H =
#
.
ζ2
ζ3
ζ5
ζ̂4
1
b sin3 (θ0 ) cos(θ0 )
2
"3 .
sin(θ0 ) − aθ0 cos(θ0 )
1
sin2 (θ0 ) − 3aθ0 sin(θ0 ) cos(θ0 ) + a2 θ02 cos2 (θ0 )
b cos(θ0 )
.
ξˆ3 =
!
"2
2
sin(θ0 ) − aθ0 cos(θ0 )
Also, it is useful to introduce
(72)
ξ˜1 = ξ1 + 1 =
−aθ0 cos(θ0 )
.
sin(θ0 ) − aθ0 cos(θ0 )
Then
2
2
2
3
4
ζ1 sτ + ζ2 τ + ζ3 s τ + ζ4 sτ + ζ5 τ + O(|s, τ | ) , Tint ≤ τ
ζ1 sτ + ζ2 τ 2 + ζ3 s2 τ + ζ̂4 sτ 2 + ζ̂5 τ 3 + O(|s, τ |4) , Tint ≥ τ
,
(73)
∆H =
#
ζ1 sτ + ζ2 τ 2 + ζ3 s2 τ + ζ4 sτ 2 + ζ5 τ 3 + O(|s, τ |4) , Tint ≤ τ
ζ1 sτ + ζ2 τ 2 + ζ3 s2 τ + ζ̂4 sτ 2 + ζ̂5 τ 3 + O(|s, τ |4) , Tint ≥ τ
,
where
a2 θ03 cos2 (θ0 )
,
sin(θ0 ) − aθ0 cos(θ0 )
sin(θ0 ) − 21 aθ0 cos(θ0 )
,
= a2 θ02 cos2 (θ0 )
sin(θ0 ) − aθ0 cos(θ0 )
sin(θ0 ) − 21 aθ0 cos(θ0 )
= 2abθ03 cos2 (θ0 ) !
"2 ,
sin(θ0 ) − aθ0 cos(θ0 )
ζ4 =
ξ3 = !
(71)
where
ζ1 =
ξ2
2
cos (θ0 )
− sin(θ0 )
,
sin(θ0 ) − aθ0 cos(θ0 )
−bθ0 sin(θ0 ) cos(θ0 )
= !
"2 ,
sin(θ0 ) − aθ0 cos(θ0 )
ξ1 =
(70)
sin(θ0 ) − aθ0 cos(θ0 )
Then
Continuity of Tint may be verified at φ(τ ) − sθ(τ ) = 0 as follows: When φ(τ ) − sθ(τ ) = 0, by (59),
0)
0)
2
a = 2 tan(θ
−bs− b sin(θ
). With this value of a, ξ1 +ξ2 s+ξ3 τ = ξ1 +ξ2 s+ ξ̂3 τ = 1+O(|s, τ |2 )
θ0
2θ0 τ +O(|s,
ζ̂5
1= τ byτ | definition.
1
as
expected,
since
here
T
int
θ(t) = θ0 + φ0 t + sin(θ0 )t2 + φ0 cos(θ0 )t3 + O(t4 )16 .
(32)
2
1
b sin3 (θ0 ) cos(θ0 )
2
−aθ0 cos(θ0 )
.
ξ˜1 = ξ1 + 1 =
sin(θ0 ) − aθ0 cos(θ0 )
(65)(30)
(66)(31)
(69)
Also, it is useful to introduce
1
2
mary goal motivating
ourabθanalysis
in the
β̂0 = −
0 cos (θ0 ) , is the determination of the change (60)
6
nian, (5), of Γ between1 t = 02 and t = Tint + τ ; this will indicate whether the
abθ0 cos (θ0 ) ,
(61)
β̂2 =
2away from the origin.
pproaching or moving
"
1!
is point we makeβ3the
assumptions
sin(θ0 ) −
= following
aθ0 cos(θ0 ) , on a, b, s and θ0 and as we progress
(62)
2
the analysis the role of each
assumption
should
become
clear:
1
ξ2
1
sin (θ0 ) − 3aθ0 sin(θ0 ) cos(θ0 ) +
b cos(θ0 )
ξˆ3 =
!
2
sin(θ0 ) − aθ0 cos(θ0 )
15
%of generality,
& that this point corresponds to t = 0 . We let
ming, without loss
θ0 + %sθ0 + sin(θ0 )τ &t + %12 sin(θ0 )t2& + O(|s, τ, t|4 ) ,
t ∈ [−τ, 0]



2
3
te the smallest positive
intersects
Σ1 .τ,Then
sin(θ0 )τatt which
+ β3 + βΓ
t|4 ) , thet next
∈ [0, τ ]switching
θ0 + sθ0 +time
4 s &t +%β6 t + O(|s,
&
%
θ(t) =
3 + sθ + sin(θ )τ + β̂ τ 2 t + β + β s + β̂ τ t2 + β̂ t3 + O(|s, τ, t|4 ) ,
+
β̂
τ
θ
3
4
5
6
0
0
0
0
2
r at the point


t ∈ [τ, Tint + τ ]
!
"
(59)
(28)
(θ1 , φ1 ) = θ(Tint + τ ), φ(Tint + τ ) .
where
(68)
ξ3 = !
Below we will need higher order terms of (47). To prevent calculations and statements
becoming too messy we find it useful to perform series expansions in both s and τ . Thus
we rewrite (46) as:
where
− sin(θ0 )
,
sin(θ0 ) − aθ0 cos(θ0 )
−bθ0 sin(θ0 ) cos(θ0 )
= !
"2 ,
sin(θ0 ) − aθ0 cos(θ0 )
ξ1 =
sin
2abθ02 cos2 (θ0 )
3
(θ0 ) −
11
2
4 aθ0 sin (θ0 ) cos(θ0 ) +
2a2 θ02 sin(θ0 ) cos2 (θ0 ) − 12 a3 θ03 cos3 (θ0 )
,
"3
!
sin(θ0 ) − aθ0 cos(θ0 )
a2 θ03 cos2 (θ0 )
ζ1 =
,
(74)
sin(θ0 ) − aθ0 cos(θ0 )
sin(θ0 ) − 21 aθ0 cos(θ0 )
,
ζ2 = a2 θ02 cos2 (θ0 )
(75)
sin(θ0 ) − aθ0 cos(θ0 )
sin(θ0 ) − 21 aθ0 cos(θ0 )
ζ3 = 2abθ03 cos2 (θ0 ) !
"2 ,
(76)
sin(θ0 ) − aθ0 cos(θ0 )
ζ4 = 2abθ02 cos2 (θ0 )
(77)
sin3 (θ0 ) −
11
2
4 aθ0 sin (θ0 ) cos(θ0 ) +
!
2a2 θ02 sin(θ0 ) cos2 (θ0 ) − 12 a3 θ03
"3
sin(θ0 ) − aθ0 cos(θ0 )
1
sin3 (θ0 ) − 6aθ0 sin2 (θ0 ) cos(θ0 ) + 8a2 θ02 sin(θ0 ) cos2 (θ0 ) −
2
"3
!
1
sin3 (θ0 ) − 6aθ0 sin2 (θ0 ) cos(θ0 ) + 8a2 θ02 sin(θ0 ) cos2 (θ0 ) − 3a3 θ03 cos3 (θ0ζ)5 = 6 abθ0 sin(θ0 ) cos (θ0 )
2
=
(78),
abθ0 sin(θ0 ) cos (θ0 )
!
"3
sin(θ0 ) − aθ0 cos(θ0 )
6
sin(θ0 ) − aθ0 cos(θ0 )
sin2 (θ0 ) − 43 aθ0 sin(θ0 ) cos(θ0 ) + 14 a2 θ02 cos2 (θ0 )
3
1 2 2
2
2
,
ζ̂4 = 2abθ02 cos2 (θ0 )
!
"2
sin
(θ
)
−
aθ
sin(θ
)
cos(θ
)
+
0
0
0
0
2
2
4
4 a θ0 cos (θ0 )
= 2abθ0 cos (θ0 )
,
(79)
sin(θ0 ) − aθ0 cos(θ0 )
"2
!
sin(θ0 ) − aθ0 cos(θ0 )
1
sin3 (θ0 ) + 7aθ0 sin2 (θ0 ) cos(θ0 ) − 9a2 θ02 sin(θ0 ) cos2 (θ0 ) + 3a3 θ03 c
abθ0 cos2 (θ0 )
ζ̂5 =
"2
!
3
2
2 θ 2 sin(θ ) cos 2 (θ ) + 3a3 θ 3 cos3 (θ )
(θ
)
+
7aθ
sin
(θ
)
cos(θ
)
−
9a
1
sin
6
0
0
0
0
0
0
0
0
0
sin(θ0 ) − aθ0 cos(θ0 )
=
. (80)
abθ0 cos2 (θ0 )
"2
!
6
sin(θ0 ) − aθ0 cos(θ0 )
18
18
Back to the balance models: (w/ cart)
5
4
PD
control
with
delay
!!/s
super
DIB
F =3 aθ(t − τ ) + bθ̇(t − τ )
SN
(!0,"0)
"
2
ON
t=0
$
t = !#
PF
1
OFF
0
0.5
1
1.5
a
2
!
t = Tint
%
t = Tint + #
(!1,"1)
1
zig-zag orbits
2.5
Figure 6: The forward orbit, Γ, of a point on Σ1 . The primary purpose of this figure
is to illustrate some labels used in expansions given the text. Unlike elsewhere in this
Note, t = 0 corresponds to a switching point instead of a point on Σ1 .
B
HC
1
U
"
0.5
0
0.5
super
DIB
PF
1
and assuming, without loss of generality, that this point corresponds to t = 015 . We let
Tint denote the smallest positive time at which Γ intersects Σ1 . Then the next switching
will occur at the point
!
"
(28)
(θ1 , φ1 ) = θ(Tint + τ ), φ(Tint + τ ) .
U
S
sub
DIB
1.5
a
The primary goal motivating our analysis is the determination of the change in the
Hamiltonian, (5), of Γ between t = 0 and t = Tint + τ ; this will indicate whether the
orbit is approaching or moving away from the origin.
At this point we make the following assumptions on a, b, s and θ0 and as we progress
through the analysis the role of each assumption should become clear:
Zigzag orbits
S
2
0 < θ0 < π2 ,
#
− π2 < s ≤ 0 ,
2.5
Bifurcation diagram: control a vs. θ
F (θ0 ) < 0 .
Wednesday, June 6, 2012
(29)
(30)
(31)
where the last inequality refers to (21) and ensures that the control is sufficiently strong.
Given that φ(−τ ) = sθ(−τ ) for the orbit Γ, the value φ0 is completely determined
Simpson, K., Li, J. Nonl Sci, to appear
A A
7
7
HC
delay
6
5
HC
6
sub
DIB
sub
DIB
State
diagram:
control vs.
delay
5
4
!!/s !!/s4
3
super
super
DIB
DIB
3
SN
2
2
1
0
0.5
B B
a
$
1
"
0.5
1.5
1
2
a
HC
!
HC
t = Tint
U S
t = Tint + #
(!1,"1)
0.5
2
1.5
control
2.5
2.5
t=0
t = !#
1
PF
1
(!0,"0)
"
zig-zag orbits
PF
1
0
0.5
"
SN
%
S
U
1
ure 6: The forward orbit, Γ, of a point on Σ1 . The
primary purpose ofsub
this figure
super
super
sub
o illustrate some labels used in expansionsPF
given the
text.
Unlike
elsewhere
DIB
DIBin this
DIB
0 to a switching point instead ofPF
e, t = 0 corresponds
a pointDIB
on Σ1 .
0
0.5
1
1.5
0.5
1
1.5
a
a
assuming, without loss of generality, that this point corresponds to t = 015 . We let
denote the smallest positive time at which Γ intersects Σ1 . Then the next switching
occur at the point
!
"
(28)
(θ1 , φ1 ) = θ(Tint + τ ), φ(Tint + τ ) .
Wednesday, June 6, 2012
S
2
2
2.5
2.5
θ
Bifurcation
diagram:
control
a
vs.
Panel A is a bifurcation set of (4)-(6) when s = −0.01, b = 2 and G(θ) =
4:
ure 4: Panel A is a bifurcation set of (4)-(6) when s = −0.01, b = 2 and G(θ) =
Mathematical Challenges:
A
7
6
5
• Nonlinearity: non-uniqueness, complex dynamics
• Delays: Delay Differential Equations (DDEs)
HC
sub
DIB
instead of Ordinary Differential Equations (ODEs),
functional/history dependence instead of initial
conditions at a single point
4
!!/s
super
DIB
3
SN
2
1
• On/off control: discontinuous equations rather
0
0.5
B
PF
than continuous
or smooth behavior.
a
Biophysical model:
impact of noise?
U
U S
S
a
1
1.5
2
2.5
HC
1
"
0.5
0
0.5
super
DIB
PF
1
sub
DIB
1.5
2
2.5
Bifurcation diagram: control a vs.
: Wednesday,
Panel June
A 6,is2012
a bifurcation set of (4)-(6) when s = −0.01, b = 2 and G(θ) =
θ
Balance model w/ noise: small random fluctuation
A
For larger control:
oscillations near the origin:
weak attraction to the
origin
7
0.2
HC
6
0.15
OFF
0.1
5
ON
0.05
4
"
sub
DIB
!!/s 0
super
DIB
!0.05
3
ON
!0.1
SN
OFF
2
!0.15
!0.2
1
!0.5
!0.4
!0.3
PF
!0.1
!0.2
0
0.5
0
!
0.1
1
0.2
0.3
0.4
0.5
1.5
2
a
B
For smaller control, noise
can cause transition to zigzag orbit
2.5
HC
U
1
"
0.5
0
0.5
super
DIB
PF
1
U
S
sub
DIB
1.5
S
2
2.5
a
Bifurcation diagram: control a vs.
Wednesday, June 6, 2012
θ
Model 3.
In this m
intermittently switched
mechanism, which div
four regions separated
and the ordinate axis:
Balance model w/ noise:
Asai, 2009, PLOS1
Experimental observations
(
(
fP (hD )~P hD
fD (h_ D )~D h_ D
fP (hD )~0
fD (h_ D )~0
,
, i:e:, PD c
i:e:, PD co
Note that the phase
D.0 is infinite dimens
system at time t is
Therefore the h{h_ pl
Nevertheless, with kee
refer to the h{h_ plane
PD-on regions corresp
phase plane, augment
second quadrants, resp
the switching paramet
areas of the phase plan
between 50% to 100
Figure 1. Characterization of the 4 control models in the phase
a?{?, the PD-off
plane (h_ vs: h). In Models 1 and 2 the control is active in the whole
becomes
to
ample of human postural sway, collected from a subject in quiet
for
1203 s.
upper
panel:
swayidentical
sequen
plane.standing
The shaded areas
in Models
andLeft
4 identify
the areas
where Angular
the
condition for the cont
control
switched
off.
anel: trajectory in the phase plane; Right panel: power spectral density
ofis the
angular
sway.
Fig. 3: it describes a
doi:10.1371/journal.pone.0006169.g001
1.2
journal.pone.0006169.g008
breaking the stability
0.2
equilibrium would be
Noise drives Control off near origin
A
C
transitions
!0.028
OFF
to
larger
ON values of D, a, and
U
mgh<0.79 for the right panel) and common
e DDE simulations of Model
4 for various sets of P-D-a
0.8
zig-zag(D = 10 Nms/rad, a = 20.4 s21, s = 0.2 Nm). For this value
show
that the stable regions in the P-D-a parameter
!0.03
!
Model
4 are the sameSas those of Model" 3,
with
the even
the switching parameter a, the optimal value of P for the stabi
orbits,
hat the attractor of Model 3 is an asymptotically stable
of Model 3 was about 60% of mgh regardless the value of D, i
for
0.4
point
and that of Model 4 is a limit cycle. This
factlarger
the dark band was located at P=mgh&0:6 in Fig. 6C. Thus t
!0.032
U the noisy dynamicscontrol
common basis for understanding
of
values of P used for Fig. 9 left and right are, respectively, smal
ls 3 and 4. As shown in Fig. 7, the power spectra of
and larger thanON
the optimal valueOFF
of P. As in these two exampl
values
S the two power
nd 04, if affected by noise, exhibit
law
the
showed
0.3
0.31
0.32 PSD
0.33
0.34the two power law scaling for smaller values of
0.15
PLoS ONE | www.plosone.org
3
0.1
0.05
"
0
!0.05
U
a physiological sway movements.
mes that are typical of
1
1.2
1.4
1.6
B power law scaling factor at the low
ar, 0.08
the first
egime of Model 4 changes depending on the values of
" 0.04
Wednesday,
6, 2012 s as these parameters determine
the
noise June
intensity
!0.1
and! it was more like a second order system and similar to the PS
of Model 2 with or without a resonance for large values of P.
Figure 10 shows, for Model 4, the dependence of the scal
factor a of the power spectrum of the noisy sway at the l
!0.15
!0.2
!0.5
!0.4
!0.3
!0.2
!0.1
0
!
0.1
0.2
0.3
0.4
0.5
duced bifurcation was analyzed in [24] (see also [28, 29]) for a general time-delaye
ecewise-smooth system comprised of ordinary differential equations on each side
Bursting-like
Experimental
e switching
manifold.observations
In that paper itoscillations
was proved that in a neighbourhood of t
A
1
SN
0.8
! 0.6
0.4
sub
DIB
0.2
PF
0
1
1.2
B
a
Text
1.4
1.6
C
0.32
0.04
"
!
0.28
0
0.24
400
420
440
460
time
480
500
!0.04
0.24
0.28
0.32
!
gure 8: Panel A is a bifurcation diagram of (4)-(6) when τ = 0.3, s = −0.1, b =
nd G(θ) = cos(θ). The meanings of the abbreviations and colours are the same as
6, 2012
g.Wednesday,
4-B.JunePanel
B shows a partial time series of an orbit with transients decayed wh
Biophysical models of balance:
• Human postural sway, stick balancing
• Nonlinearity: non-uniqueness, complex dynamics
• Bifurcations in context of delays and on/off control
• Identify sensitivies to control conditions and noise
spiral, along the dotted curve the orbit falls onto W s and limits upon the origin without
again crossing either of the switching manifolds. The dash-dot curve is a Hopf-like
bifurcation of (4) at which a symmetric spiral periodic orbit is born.
Note that one may choose the control parameters such that zigzag orbits approach
the origin and spiral orbits head away from the origin (as in Fig. 1) and vice-versa. We
have not been able to identify an intersection between the dashed and dotted curves
of Fig. 9 for any τ and s, nor have been able to show that such an intersection cannot
occur. Such an intersection could permit for the existence of an orbit that repeatedly
switches between zigzag and spiral motion.
4
Discts control:
regions of
zigzags/spirals
depend on
control coeffs
spiral out
or zigzag in
spiral out
or zigzag out
spiral in
or zigzag out
b2
1
spiral in
or zigzag in
zigzag to
spiral in
spiral to
zigzag out
D-shaped
region
(cts control)
spiral to
zigzag in
1
Wednesday, June 6, 2012
zigzag to
spiral out
3
1.5
2
a
2.5
3
3.5
Figure 9: A bifurcation set of the linearization, (56), with (5)-(6), τ = 0.5 and s = 0.
Noise driven ZZ-spiral, amplified spirals,
stabilized transients
0.4
0.4
0.2
0.2
ON
0
0
ï0.2
ï0.2
ï0.4
OFF
ï0.4
ï0.2
0
0.2
0.4
0.6
ï0.4
0.4
0.4
0.2
0.2
0
0
ï0.2
ï0.2
ï0.4
ï0.4
Wednesday, June 6, 2012
ï0.4
ï0.2
0
0.2
0.4
0.6
ï0.4
ï0.2
0
0.2
0.4
0.6
ï0.4
ï0.2
0
0.2
0.4
0.6
Lots of mathematical challenges:
• New analytical and computational methods needed
• Nonlinear models with delay: complex behaviors
• Piecewise continuous nonlinear systems: recently
receiving more attention
• Stochastic modeling for systems with delay and
discontinuities: open problems analytically and
computationally
• Act and wait: apply control for times just beyond
the delay time for stabilization, some suggestions of
sensitivies to these optimal control strategies
Wednesday, June 6, 2012