Delay systems Batz-sur-Mer October

Transcription

1 Delay systems Batz-sur-Mer October Thomas Erneux Université Libre de Bruxelles Optique Nonlinéaire Théorique T. Erneux, Applied Delay Differential Equations, Springer (2009)

2 PLAN 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)

3 1. Familiar delay problems The hot shower problem T T desired T cold dt ( t) = a( T ( t τ ) Td ) dt a = T = τ = 1 d T = 0.5 ( τ < t 0) τ 0 time t Delay: time for the increased (or decreased) hot/cold water to flow from the tap to the shower head

4 T desired T cold T dt ( t) = a( T ( t τ ) Td ) dt a = T = τ = 1 d T = 0.5 ( τ < t 0) τ 0 time t 8 µ l τ = ( ) p R aτ e 1 > : damped oscillations aτ > π / : growing oscillations a large: high human sensitivity τ large: low p, large l

5 Remote control Images are sent to Earth and a signal is sent back For the Moon, the time delay in the control loop is 2-10 s For Mars, it is 40 minutes!

6 Traffic flow 1 2 a 2(t + τ) = α(v 1 - v 2) braking speed difference τ: reaction time

7 Traffic flow v v 1 v 2 d 1 2 a (t + τ) = α(v - v ) v '(t + τ) = α(v - v ) d t/τ α 1 = = s, τ 1 s v (km/h) = ( t τ ) ( τ t < 0) d ' = v v 1 2

8 1 2 v a 2(t + τ) = α(v 1 - v 2) τ: reaction time τ = 1 s τ = 1.5 s sober driver τ = 1s 0.5g/l in blood = 1 shot = 2 glasses of wine = 2 cans of beer τ = 1.5s t/τ

9 2. Emerging problems 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

10 Neuron synchronization (Parkinson disease) Tremors result from a a too synchronized population of neurons What is the role of the delayed communication between neurons? Researchers J. Henry (INRIA Bordeaux) A. Beuter, J. Modolo (Université Bordeaux 2) model networks of coupled neurons

11 Anticipation mechanisms The reflex time is too long. Researchers think that the brain develop anticipation mechanisms to compensate for the delay. signal hand brain = 0.1 s signal round trip time = 0.2 s Too long! For example, our hand is preparing itself to catch the ball

12 3. Delay Equation? dy( t) dt = ky( t) y = A exp( kt) y t

13 3. Delay Equation? dy( t) dt = ky( t) y = A exp( kt) y dy( t) dt if ωτ = = ky( t τ ) y = Asin( ω t) π 2 y t t

14 3. Delay Equation? dy( t) dt = ky( t) y = A exp( kt) y dy( t) dt if ωτ = Check = ky( t τ ) y = Asin( ω t) π 2 Aω cos( ωt) = ka sin( ω ( t τ )) y t t [ ] ω cos( ωt) = k sin( ωt) cos( ωτ ) cos( ωt) sin( ωτ ) (1) ω = k sin( ωτ ) (2) cos( ωτ ) = 0 ωτ = π / 2 and k = ω

15 Summary 1. A time delay is taken into account when a mechanical or physiological feedback takes time 2. Oscillatory regimes possible

16 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)

17 Population dynamics 4. Biology Density of lemmings (number of individuals per hectare) in the Churchill area in Canada (T.J. Case, Oxford 2002) 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 broken line: solution of the delayed logistic equation dn = rn (1 N ( t τ )) dt r = 1/(0.3 years), τ = 0.72 years

18 The pupil eye reflex (Beuter et al. Eds., Nonlinear Dynamics in Physiology and Medicine, Springer 2003) τ = 0,3 s 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 Slit lamp beam

19 Physiological diseases Sustained periodic breathing (PB) patients with cardiac problems sleep in altitude Model p = pressure of CO 2 dp = M pv ( p ( t τ)) dt production ventilation CO 2 high in lungs Blood carried the information to the brain The brain commands ventilation of the lungs after a delay τ ( minutes) Blood disorders showing oscillatory clinical data

20 5. Engineering Control air/fuel ratio (best 15:1 = 15 parts of air for one part of fuel) sensors

21 Mechanical engineering High speed machining (aeronautics, tool fabrication) Danger: self-sustained vibrations = chatter

22 Stépan, Retarded Dynamical Systems, Longman, London, 1989 Machine-tool vibrations chatter instabilities wave from current pass y(t) wave from previous pass y(t-τ) Tool equation: 2 0 [ ] y '' + ε y ' + ω y = F y y( t τ ) -1-2 ω0 = 579 s T 10 s Ω = = = turns s τ Ω 7 10 s 22

23 Semiconductor laser -12 τ = 10 s τ 6. Optical feedback photons round-trip = s 23

24 Semiconductor laser -12 τ = 10 s τ 6. Optical feedback photons round-trip = s Large delay = oscillations = noise Undesirable in optical communication 24

25 y'' + y = 0 7. Stabilization with delay 1.0 y '' + y = y( t τ ) y delayed control = y τ y ' +... y = τ y ' +.. y '' + y + τ y ' = 0 damping y y t τ = t

26 Container cranes Goal: move the container fast (24 tons) Danger: oscillations Idea: use a delayed control y 2 d y = F ( y) + k( y( t τ ) y) 2 dt Newton delayed control Erneux and Kalmár-Nagy, J. of Vib. and Contr (2007) Erneux, J. of Vib. and Contr. 16, (2010)

27 Results: Oscillations are quickly damped when the control in ON Model scale 1:10 for a 65 tons crane Masoud and Daqaq JJMIE 1, 57 (2007)

28 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

29 8. Delayed negative feedback (Leon) 1. Negative feedback protein dy 1 by p dt = 1+ y A protein represses the transcription of its own gene [for example, PER in the circadian control system of fruit flies] 2. Delayed negative feedback dy 1 = by p dt 1 + y ( t τ ) τ is the time delay that is required for transcription and translation

30 Delayed negative feedback dy 1 = by p dt 1 + y ( t τ ) Numerous applications in biology Mackey 1996: Periodic crashes in circulating red blood cells (RBC). p = 7.6 τ = 1.8 Pupil eye reflex Other sciences El-Nino/Southern-Oscillations (ENSO) variability Forced DDE for surface Temperature T (Ghil 2008, 2009) dt dt = tanh( κt ( t τ )) + cos(2 π t) Delayed negative feedback atmosphere ocean coupling Seasonal cycle in the trade winds κ = 100, τ =

31 dy 1 = by p dt 1 + y ( t τ ) Steady state: bys = 0 b = p p 1 + y y (1 + y ) s s s 2. Stability: y = y + u ( u << 1) 3. Try: u = c exp( λt) s du py = u( t τ) bu dt (1 + ) p 1 s p 2 ys p 1 s p 2 ys py λcexp( λt) = cexp( λ( t τ)) bc exp( λt) (1 + ) py λ = exp( λτ ) b (1 + ) p 1 s p 2 ys λ = pby exp( λτ ) b p+ 1 s is the characteristic equation for λ

32 b = p+ 1 s p+ 1 i = pbys exp( i ) b p+ 1 Re : 0 pbys cos( ) p+ 1 Im : = pbys sin( ) 1H y s 1 p (1 + y ) λ = pby exp( λτ ) b 3 equations for b, y and ω π 2 pτ s Hopf conditions λ = iω ω p b b ω ωτ = ωτ b ωτ 1 p=20 1 p ln( p) + O( p ) 2H 1 1 s b τ

33 dy 1 = by p dt 1 + y ( t τ ) p dy 0 y( t τ ) > 1 = by + dt 1 y( t τ ) < 1 y y(t - τ) 0.6 y = exp( b τ ) min y = (1 b )exp( b τ ) + b max b = 0.4, p = 20, τ = 1.8 t

34 Bifurcation diagrams y 3 2 p = 7.6 y = exp( b τ ) min y = (1 b )exp( b τ ) + b max 1 1 y p = b

35 p = 20 x y = exp( b τ ) min y = (1 b )exp( b τ ) + b max period 1 1 by 1 1 max min = b ln bymin 1 ymax y period/τ b

36 9. Square-waves (Laurent) 2τ periodic square - waves of scalar DDEs dx ε = x + f ( x( t 1)) dt 1 ε τ 3.5 x(t) ε 0 x + f ( x( t 1)) = 0 x n+ 1 = f ( x ) n t : S.N. Chow, J. Mallet-Paret, R.D. Nussbaum, J.K. Hale and W. Huang 36

37 Experiments using an optoelectronic oscillator modeled by a scalar DDE Larger et al. JOSA B 18, 1063 (2001) ε x ' = - x + f ( x( t -1)) 1 f = β 1+ cos( x( t 1)) 2 37

38 Experiments using an optoelectronic oscillator modeled by a scalar DDE Larger et al. JOSA B 18, 1063 (2001) ε x ' = - x + f ( x( t -1)) 1 f = β 1+ cos( x( t 1)) 2 x + f ( x( t 1)) = 0 x n+ 1 = f ( x ) n Hopf1 Hopf2 Chaos map experiment x n β 38

39 Experiments in Besançon Opto electronic oscillator τ - periodic solution 39

40 y ' = x ε x ' = x δ y + f ( x( t 1)) ε = 10 δ =

41 y ' = x ε x ' = x δ y + f ( x( t 1)) ε = 10 δ = x (a) (b) s 0 (c) 1 - s s s β β 2.0 Numerical solution Asymptotic analysis 41

42 10. Control theory (Jean-Pierre) x' = ax x = 0 stable if a < 0 Can we do better? x' = ax + bx(t 1) control

43 x' = ax + bx(t 1) The characteristic equation is σ a bexp(- σ ) = 0 σ real: b = exp( σ )( σ a) σ complex: σ = γ + iω γ a b exp( γ ) cos( ω) = 0 ω + b exp( γ )sin( ω) = 0

44 x' = ax + bx(t 1) The characteristic equation is σ a bexp(- σ ) = 0 σ real: b = exp( σ )( σ a) σ complex: σ = γ + iω γ a b exp( γ ) cos( ω) = 0 ω + b exp( γ )sin( ω) = 0 ω ω = tan( ω) γ = a γ a tan( ω) ω b = e xp( γ ) sin( ω ) σ = 0 : b = a ω ω γ = 0 : a =, b = tan( ω) sin( ω) b stable σ=0 σ=iω a Stable if a < 1

45 Act and wait x' = ax + G(t)x(t 1) 0 (0 < t < 1) G(t) = b (1 < t < 2) G( t) = G( t + 2 k) k = 1, 2,...

46 Act and wait x' = ax + G(t)x(t 1) 2 G(t) G( t) = G( t + 2 k) k = 1, 2,... 0 < t < 1: x ' = ax, x(0) = x x = x exp( at) < t < 2 : x ' = ax + bx exp( a( t 1)), At 0 (0 < t < 1) = b (1 < t < 2) t = 2 x(1) = x exp( a) x(2) = x exp(2 a)(1 + b exp( a)) = µ x 0 0 stable if µ < 1, when t = 2k increases The stability domain is limited by the lines µ = ± 1 b µ=0 stable µ= 1 µ= a stable if a > 1

47 2 Full delayed feedback control 1 σ=0 x ' = ax + bx( t 1) stable if a < 1 b 0 stable -1-2 σ=iω Act and wait control x' = ax + G(t)x(t 1) a µ=0 µ=1 0 (0 < t < 1) G(t) = b (1 < t < 2) G( t) = G( t + 2 k) k = 1, 2,... b stable -4 µ= 1 stable if a > a

48 Summary Analytical 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