Delay systemsBatz-sur-Mer October 17-21 2011

Thomas ErneuxUniversité Libre de Bruxelles

Optique Nonlinéaire ThéoriqueT. Erneux, “Applied Delay Differential Equations”, Springer (2009)

1. Familiar problems

2. Emerging problems

3. Delay equation ?

4. Biology

5. Engineering

6. Optical feedback

7. Stabilization with delay

8. Delayed negative feedback (Leon Glass)

9. Square-waves (Laurent Larger)

10.Control theory (Jean-Pierre Richard)

PLAN

• The hot shower problem

( )( ( ) )

1

0.5 ( 0)

d

d

dT ta T t T

dta T

T t

τ

ττ

= − − −

= = == − < ≤

Delay: time for the increased (or decreased) hot/cold water to flow from the tap to the shower head

time t

T

Tdesired

Tcold

τ0

1. Familiar delay problems

( )( ( ) )

1

0.5 ( 0)

d

d

dT ta T t T

dta T

T t

τ

ττ

= − − −

= = == − < ≤

2

1

8( )

0.36 : damped oscillations

/ 2 1.57 : growing oscillations

large: high human sensitivity

large: low , large

l

p R

a e

a

a

p l

µτ

ττ π

τ

−

=∆

> ≈> ≈

∆

time t

T

Tdesired

Tcold

τ0

• Remote controlImages are sent to Earth and a signal is sent back

For the Moon, the time delay in the control loop is 2-10 sFor Mars, it is 40 minutes !

• Traffic flow

1 2

2 1 2a (t + τ) = α(v - v )

braking speed difference

τ: reaction time

↑

• Traffic flow

1 2

2 1 2

2 1 2

-1

1

1 2

a (t + τ) = α(v - v )

v '(t + τ) = α(v - v )

0.5 s , 1 s

v (km/h) 80 20( ) ( 0)

'

t t

d v v

α ττ τ

= == + − − ≤ <

= −

t/τ-1 0 1 2 3 4 5 6 7 8 9 10

d

4

5

6

7

8

9

10

11

-1 0 1 2 3 4 5 6 7 8 9 10

v

50

60

70

80

90

v1

v2

d

1 2

2 1 2a (t + τ) = α(v - v )

: reaction timeτ

t/τ-1 0 1 2 3 4 5 6 7 8 9 10

v

60

70

80

90

τ = 1.5 s

τ = 1 s

sober driver

τ 1s

0.5g/l in blood

1 shot

= 2 glasses of wine

= 2 cans of beer

τ 1.5s

=

=

=

• Wind instruments such as the clarinet

reed pressure waves

Researchers (J.-P. Dalmont Université du Maine) et J. Kergomard (Université d’Aix-Marseille) study how it works

Goal: synthesize the sound of a clarinet or a trumpet

Delay = round trip of the pressure wave

2. Emerging problems

• Neuron synchronization (Parkinson disease)

Researchers J. Henry (INRIA Bordeaux)A. Beuter, J. Modolo (Université Bordeaux 2) model networks of coupled neurons

Tremors result from a a too synchronized population of neurons

What is the role of the delayed communication between neurons ?

• Anticipation mechanismsThe reflex time is too long. Researchers think that the brain develop anticipation mechanisms to compensate for the delay.

For example, our hand is preparing itself to catch the ball

signal hand – brain = 0.1 s

signal round trip time = 0.2 s

Too long!

( )( ) exp( )

dy tky t y A kt

dt= − → = −

t

y

3. Delay Equation ?

( )( ) exp( )

dy tky t y A kt

dt= − → = −

( )( ) y sin( t)

if 2

dy tky t A

dtτ ω

πωτ

= − − → =

=

t

y

t

y

3. Delay Equation ?

( )( ) exp( )

dy tky t y A kt

dt= − → = −

( )( ) y sin( t)

if 2

dy tky t A

dtτ ω

πωτ

= − − → =

=

t

y

t

y

[ ]

Check

cos( ) sin( ( ))

cos( ) sin( ) cos( ) cos( ) sin( )

(1) sin( ) (2) cos( ) 0

/ 2 and

A t kA t

t k t t

k

k

ω ω ω τω ω ω ωτ ω ωτ

ω ωτ ωτωτ π ω

= − −= − −

= == =

3. Delay Equation ?

1. A time delay is taken into account when a mechanical or physiological feedback takes time

2. Oscillatory regimes possible

Summary

2009 – 2011

1. T. Erneux, “Applied Delay Differential Equations”, Springer (2009)2. A. Balachandran, T. Kamár-Nagy, D.E. Gilsinn, Eds. “Delay Differential

Equations, Recent Advances and New Directions”, Springer (2009)3. F.M. Atay, Ed. “Complex Time-Delay Systems” , Springer (2010)4. Theme Issue “Delay effects in brain dynamics” compiled by G. Stepan,

Phil. Trans. Roy. Soc A 367, 1059 (2009)5. Theme Issue “Delayed complex systems” compiled by W. Just, A. Pelster,

M. Schanz and E. Schöll, Phil. Trans. Roy. Soc A 368, 303 (2010)6. Special Issue on Time Delay Systems, T. Kalmár-Nagy, N. Olgac, and G.

Stépán, Eds., J. Vibration & Control, June/July 16 (2010)7. Hal Smith, An Introduction to Delay Differential Equations with

Applications to the Life Sciences, Springer (2010)8. M. Lakshmanan and D.V. Senthilkumar, Dynamics of Nonlinear Time-

Delay Systems , Springer Series in Synergetics (2011)9. T. Insperger and G. Stépán, Semi-discretization for time-delay systems –

Engineering applications, Springer (2011)

Population dynamics• Density of lemmings (number of individuals per hectare) in the Churchill area in

Canada (T.J. Case, Oxford 2002)

broken line: solution of the

delayed logistic equation

(1 ( ))

r 1/(0.3 years),

τ 0.72 years

dNrN N t

dtτ= − −

==Delay: the population growth depends on the

present food levels

Plant levels depends on the number of grazers that lived between now and τ time steps earlier

4. Biology

• The pupil eye reflex(Beuter et al. Eds., Nonlinear Dynamics in Physiolog y and Medicine, Springer 2003)

When a narrow pencil of light is placed at the iris margin, a cycle of contraction and dilatation of the pupil is possible

P = 0,9 s useful for clinical tests

τ = 0,3 s

Slit lamp beam

• Physiological diseases• Sustained periodic breathing (PB)

patients with cardiac problems sleep in altitude

( ( ))

dpM pV p t

dtτ= − −

↑ ↑

Model p = pressure of CO2

production ventilation

CO2 high in lungs Blood carried the information to the brain The brain commands ventilation of the lungs after a delay τ (0.3 – 0.5 minutes)

• Blood disorders showing oscillatory clinical data

Control air/fuel ratio (best 15:1 = 15 parts of air for one part of fuel)

sensors

5. Engineering

• Mechanical engineering

High speed machining (aeronautics, tool fabrication)

Danger: self-sustained vibrations = chatter

22

[ ]20

Tool equation:

'' ' ( ) y y y F y y tε ω τ+ + = − −

Stépan, Retarded Dynamical Systems, Longman, London, 1989

Machine-tool vibrations chatter instabilities

-1 -20

-1 -1 -2

ω 579 s T 10 s

14 turns s τ Ω 7 10 s

= → ≈

Ω = → = = ×

wave fromcurrent pass

y(t)

wave fromprevious pass

y(t-τ)

23

• Semiconductor laser-12

photons

-9round-trip

τ 10 s

τ 10 s

=

=

6. Optical feedback

24

• Semiconductor laser-12

photons

-9round-trip

τ 10 s

τ 10 s

=

=

Large delay = oscillations = noise

Undesirable in optical communication

6. Optical feedback

y'' y 0

'' ( )

delayed control

' ...

' ..

'' ' 0

damping

y y y t y

y y y

y

y y y

τ

ττ

τ

+ =

+ = − −

= − + −= − +

+ + =↑

t0 5 10 15 20

y

-1.0

-0.5

0.0

0.5

1.0

t0 5 10 15 20

y

-1.0

-0.5

0.0

0.5

1.0

τ = 0.5

7. Stabilization with delay

Container cranesGoal: move the container fast (24 tons)Danger: oscillations

Idea: use a delayed control

y

2

2( ) ( ( ) )

d yF y k y t y

dtτ= + − −

↑ ↑Newton delayed control

Erneux and Kalmár-Nagy, J. of Vib. and Contr. 13 603–6 16 (2007) Erneux, J. of Vib. and Contr. 16, 1141-1149 (2010)

Model scale 1:10for a 65 tons crane Masoud and DaqaqJJMIE 1, 57 (2007)

Results:

Oscillations are quickly damped when the control in ON

Summary

• Oscillatory instabilities caused by a delay occur in all scientific disciplines

• The way we explore them depends on our background

• A delayed feedback can be used to stabilize a system

1. Negative feedback

1

1 p

dyby

dt y= −

+

protein

A protein represses the transcription of its own gene [for example, PER in the circadian control system of fruit flies]

τ is the time delay that is required for transcription and translation

2. Delayed negative feedback

1

1 ( )p

dyby

dt y t τ= −

+ −

8. Delayed negative feedback (Leon)

Numerous applications in biology• Mackey 1996: Periodic crashes in circulating red blood cells (RBC).

p = 7.6 τ = 1.8• Pupil eye reflex

Other sciencesEl-Nino/Southern-Oscillations (ENSO) variabilityForced DDE for surface Temperature T (Ghil 2008, 2009)

Delayed negative feedback

1

1 ( )p

dyby

dt y t τ= −

+ −

tanh( ( )) cos(2 )dT

T t tdt

κ τ π= − − +

Delayed negative feedback atmosphere – ocean coupling Seasonal cycle in the trade winds

κ = 100, τ = 0.01 κ = 100, τ = 0.01 κ = 100, τ = 0.01 κ = 100, τ = 0.01 −−−− 1111

1

2

1

2

1

1 ( )

1 11. Steady state : 0

1 (1 )

2. Stability : ( | | 1)

( )(1 )

3. Try : exp( )

exp( ) exp( ((1 )

p

sp ps s s

s

ps

ps

ps

ps

dyby

dt y t

by by y y

y y u u

pyduu t bu

dt y

u c t

pyc t c

y

τ

τ

λ

λ λ λ

−

−

= −+ −

− = → =+ +

= + <<

→ = − − −+

=

= −+

1

2

1

)) exp( )

exp( )(1 )

exp( )

is the characteristic equation for

ps

ps

ps

t bc t

pyb

y

pby b

τ λ

λ λτ

λ λτλ

−

+

− −

= − − −+

→ = − − −

1

1

1

1

1

2 1 1

1

(1 )

exp( )

Hopf conditions

exp( )

Re : 0 cos( )

Im : sin( )

3 equations for b, y and

2

1

1 ln( ) ( )

ps s

ps

ps

ps

ps

s

H

H

by y

pby b

i

i pby i b

pby b

pby

p

bp

bp p O p

λ λτλ ω

ω ωτωτ

ω ωτω

πτ

+

+

+

+

− −

=+

= − − −=

= − − −

= − −

=

→ ∞

≈

≈− + p=20

τ0.0 0.5 1.0 1.5 2.0

b

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1

2

min

1 1max

1

1 ( )

0 ( ) 1by

1 ( ) 1

y exp( b )

y (1 b )exp( b ) b

p

dyby

dt y t

p

y tdy

y tdt

τ

ττ

ττ− −

= −+ −

→ ∞− >

= − +− <

= −

= − − +b = 0.4, p = 20, τ = 1.8

t-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

y

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

y(t - τ)

b0.0 0.5 1.0

0

1

2

3

0

1

2

3p = = = = 7.6

p = = = = 20

y

y

Bifurcation diagrams

min

1 1max

y exp( b )

y (1 b )exp( b ) b

ττ− −

= −

= − − +

35

0.0 0.2 0.4 0.6 0.8 1.0 1.2

x

0.0

0.5

1.0

1.5

2.0

2.5

3.0

b0.0 0.2 0.4 0.6 0.8 1.0 1.2

period/τ

2.5

3.0

3.5

4.0

4.5

5.0

min

1 1max

1 max min

min max

p 20

y exp( b )

y (1 b )exp( b ) b

1period ln

1

by yb

by y

ττ− −

−

=

= −

= − − +

−= − −

36

1

1

( ( 1))

0

( ( 1)) 0

( )n n

dxx f x t

dt

x f x t

x f x

ε

ε τ

ε

−

+

= − + −

≡

→− + − =

=

t2000 2002 2004 2006 2008 2010

x(t)

1.0

1.5

2.0

2.5

3.0

3.5

2τ – periodic square - waves of scalar DDEs

1983 - 1996: S.N. Chow, J. Mallet-Paret, R.D. Nussbaum, J.K. Hale and W. Huang

9. Square-waves (Laurent)

37

' - ( ( -1))

11 cos( ( 1))

2

x x f x t

f x t

ε

β

= +

= + −

Larger et al. JOSA B 18, 1063 (2001)

Experiments using an optoelectronic oscillator modeled by a scalar DDE

38

1

' - ( ( -1))

11 cos( ( 1))

2

( ( 1)) 0

( )n n

x x f x t

f x t

x f x t

x f x

ε

β

+

= +

= + −

− + − ==

Hopf1 Hopf2 Chaos

map 2.08 5.04 6.59

experiment 2.07 5.30 6.69

Larger et al. JOSA B 18, 1063 (2001)

Experiments using an optoelectronic oscillator modeled by a scalar DDE

β1 2 3 4 5 6 7 8

xn

0

2

4

6

8

10

12

39

Experiments in Besançon

Opto electronic oscillator

- periodic solutionτ

40

-3 -3

'

' ( ( 1))

ε 10 δ 8 10

y x

x x y f x tε δ== − − + −

= = ×

41

-3 -3

'

' ( ( 1))

ε 10 δ 8 10

y x

x x y f x tε δ== − − + −

= = ×

10007 10008 10009 s

x

-0.8

0.0

0.8

s0

0.5 1.0 1.5 2.0β β0.5 1.0 1.5 2.0

s0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

1 - s0(a) (b) (c)

Numerical solution Asymptotic analysis

10. Control theory (Jean-Pierre)

x' ax

x 0 stable if a0

Can we do better?

x' ax bx(t 1)

== <

= + −

control

x' ax bx(t 1) = + −

The characteristic equation is

a bexp(- ) 0

real: exp( )( )

complex:

exp( )cos( ) 0

exp( )sin( ) 0

b a

i

a b

b

σ σσ σ σσ σ γ ωγ γ ωω γ ω

− − == −

= +− − − =+ − =

a-3 -2 -1 0 1 2

b

-3

-2

-1

0

1

2

σ=0

σ=iω

stable

x' ax bx(t 1) = + −

The characteristic equation is

a bexp(- ) 0

real: exp( )( )

complex:

exp( )cos( ) 0

exp( )sin( ) 0

b a

i

a b

b

σ σσ σ σσ σ γ ωγ γ ωω γ ω

− − == −

= +− − − =+ − =

tan( )tan( )

xp( )sin( )

0 :

0 : , tan( ) sin( )

aa

b e

b a

a b

ω ωω γγ ω

ωγω

σω ωγ

ω ω

= − → = −−

= −

= = −

= = = −

Stable if a 1<

Act and wait

x' ax G(t)x(t 1)

0 (0 1)G(t)

(1 2)

( ) ( 2 ) 1,2,...

t

b t

G t G t k k

= + −

< <=

< <= + =

a-3 -2 -1 0 1 2

b

-5

-4

-3

-2

-1

0

1

2

µ=1µ=0

µ=−1

stable0 0

0

0

Act and wait

x' ax G(t)x(t 1)

0 (0 1)G(t)

(1 2)

( ) ( 2 ) 1,2,...

0 1: ' , (0) exp( )

1 2 : ' exp( ( 1)),

(1) exp( )

At

t

b t

G t G t k k

t x ax x x x x at

t x ax bx a t

x x a

= + −

< <=

< <= + =

< < = = → =

< < = + −=

0 0

t 2

x(2) exp(2 )(1 exp( ))

stable if | | 1, when t 2k increases

The stability domain is limited by the lines 1

x a b a xµµ

µ

== + − =

< == ±

stable if a 1>

a-3 -2 -1 0 1 2

b

-5

-4

-3

-2

-1

0

1

2

µ=1µ=0

µ=−1

stable

Full delayed feedback control

' ( 1)

stable if 1

Act and wait control

x' ax G(t)x(t 1)

0 (0 1)G(t)

(1 2)

( ) ( 2 ) 1,2,...

stable if 1

x ax bx t

a

t

b t

G t G t k k

a

= + −<

= + −

< <=

< <= + =

>

a-3 -2 -1 0 1 2

b

-3

-2

-1

0

1

2

σ=0

σ=iω

stable

SummaryAnalytical tools

1. Delayed negative feedback

• Linear stability and Hopf bifurcation

• Delay is moderate but feedback can be strong (limit of strong feedback)

2. Square-waves

• The large delay limit and maps

3. Control theory

• New forms of delayed feedback are designed