Kęstutis Pyragas Semiconductor Physics Institute, Vilnius, Lithuania

Transcription

1 Kęstutis Pragas Semiconductor Phsics Institute, Vilnius, Lithuania Introduction Delaed feedback control (DFC) method Applications and Modifications Limitation of the DFC method Unstable delaed feedback controller Using unstable controller for stabilizing stead states Conclusions

2 Control of Chaos (dnamics of publications) Pioneering paper: E. Ott, C. Grebogi, J.A. Yorke, Phs.Rev. Lett., 64, 1196 (199)

3 Wh chaotic sstems are interesting objects for control theor and applications? Chaotic sstems are extremel sensitive to small perturbations (the butterfl effect) A tpical strange attractor has embedded within it an infinit number of unstable periodic orbits. One can choose an of them b desire and stabilize it with onl tin perturbation.

4 Unstable Periodic Orbits (Lorenz attractor) x& & z& = σ ( z) = rx xz = x bz x

5 Unstable Periodic Orbits (Rössler attractor) x& = z & = x a z& = b z ( x c) x

6 Delaed Feedback Control (DFC) Method K. Pragas, Phs. Lett. A 17, 421 (1992) d d r t d x d t P r = Q r ( x,) = F(t) F(t) = K{(t-τ) -(t)} r ( x,) τ = T - period of unstable periodic orbit F(t) CHAOTIC SISTEM (t) K{(t-τ) -(t)} - (t) (t-τ) τ

7 Demonstration of the DFC method (Rössler sstem) x& = z & = x a z& = b z K{(t-τ) -(t)} ( x c)

8 Explanation of the stabilization F= Simple example: x n1 =4x n (1-x n )- KF n, F n = x n - x n-1 x F= x n1 F n = x n 3/4 x n x F n = Stabilization is achieved through additional degrees of freedom introduced with the feedback perturbation! F n x n Stabilit condition: -1 <K< 1/2

9 Experimental Implementations Electronic chaos oscillators Pragas, Tamaševičius (1993) Gauthier et al. (1994) Kittel et. al. (1994) Celka (1994) Socolar et al. (1994) Mechanic pendulums Hikihara, Kawagoshi (1996) Christini et. al. (1997) Lasers Belawski et al. (1994) Lu, Yu, Harrison (1998) Gas discharge sstems Mausbach et al. (1997) Pierre et al. (1996) Chemical sstems Parmananda et al. (1999) Tsui, Jones (2) Helikopter roter blades Krokiewski, Faragher (2) Ferromagnetic resonance Benner et al. (1997) Tailor-Couette flow chaos Lüthje et al. (21) Cardiac sstems Hall et al. (1997) Rappel et al. (1999) Skučas et al. (21)

10 Controlling Chaos in a fast Diode Resonator D.J. Gauthier et al. (1994)

11 Controlling Magneto-Elastic Beam Sstem T. Hikihara and T. Kawagoshi (1995) V cosωt A M P em shaker LD sensor - τ

12 Control of Chaotic Talor-Couette Flow O.Lüthje, S. Wolf, and G. Pfister (21) Re = 2π f (r o - r i ) r i /ν v z (x,t) - output (axial velocit) f - control parameter

13 Control of Ionization Wave Chaos Th. Mausbach et al. (1997) The wave character of dnamics in some sstems allows us to D 1 λ D 2 simplif the DFC algorithm b replacing the dela line with the spatiall distributed detectors No dela line is needed! Due to dispersion relations the dela in time can be replaced with the spatial displacement, e.g., for i kx ωt the plane wave u x, t = Ae : λ =ωτ/k ( ) ( ) ( x, t τ) = u( x λ t) u, -

14 Control of isolated auricles of a rabbit heart M. Skučas, I. Grigaliūnienė, V. Dzenkauskas, R. Labrencas Cardiac institute, Kaunas, Lithuania V T n T n1 T n1 [sek] T t n 2 T n 3 1 Control algorithm: T T K( T T ) n = n 1 n 1 n K=,5; Stimulo uюvлlinimas= ms D Tn, sa B 3. C Ex A D Periodo Nr

15 Extended Delaed Feedback Controller: EDFC J. Socolar, D. Sukov, and D. Gauthier (1994) () ( ) ( ) ( ) ( ) [ ] ( ) ( ) [ ] () ( ) ( ) = = = 1 1 2, k k k t R R t t t R t t R t t t F τ τ τ τ τ τ L R<1 - convergence condition CHAOTIC SISTEM (t) F(t) R 1-R K τ -

16 Advantages of the EDFC Illustration for the Rössler sstem: x& = z & = x a z& = b z -KF(t) ( x c) F(t)= (t)-(t-t) R F(t-T) ReΛ.2. R= R=.2 R=.4 R=.6 R= K

17 Adaptive time-delaed feedback A. Kittel, J. Parisi, K. Pragas (1995) τ n-1 τ n τ n1 d dt.2 = 1 ( t τ ) ( t τ ) M. 1 1 M F n (t) (t) x(t-τ) (n) t max 4 2 F n1 =K[(t)-(t-τ n )] x (t) (n) t max n F n =K[(t)-(t-τ n-1 )] x(t) τ n n T F(t) t

18 Limitation of the DFC method Ushio (1996), Just et al. (1997), Nakajima and Ueda (1997, 1998) Characteristics of orbits Floquet exponent Λ =λ iω or e λ T ωτ Floquet multiplier µ = e ΛΤ The method works for ω ω The method fails for ω = ω Λ Λ * Λ The method fails for an periodic orbits with an odd number of real Floquet multipliers >1 λ λ

19 Unstable Delaed Feedback Controller K. Pragas, Phs. Rev. Lett. 86, 2265 (21) The ke idea: Artificiall enlarge a set of real Floquet multipliers greater than unit to an even number b introducing into a feedback loop an unstable degree of freedom ω Additional exponent Λ λ

20 Extended Delaed Feedback Controller for R>1 (Simple Discrete Time Sstem) Unstable sstem: n1 =µ s n -KF n, µ s >1 Wh do not tr R > 1??? Extended controller: F n = n - n-1 R( n-1 - n-2 ) R 2 ( n-2 - n-3 )... Im µ 1-1 µ =1 R µ s Re µ R < 1 convergence condition It works!

21 Extended Delaed Feedback Controller for R>1 (Simple Continuous Time Sstem) Unstable sstem: d/dt =λ s -KF(t), λ s > Extended controller: F(t)= (t)-(t-τ) R F(t-τ) Im λ 1 4π 2π λ s R > π -4π It does not work! -1 lnr 1 Re λ

22 Unstable Extended Delaed Feedback Controller: UEDFC (EDFC supplemented b an unstable degree of freedom) Dnamic sstem: x&r = UEDFC: r f r ( x, p ) p - control parameter r = g - output variable ( x) p= p K F u (t) F u (t) = F(t) w(t) F(t) = (t)-(t-τ) R F(t-τ) - usual EDFC (R<1) w& () ( t λ λ ) = λcw c c F(t) - additional unstable degree of freedom ( > ) λ c

23 Simple Continuous Time Sstem under UEDFC d/dt =λ s -KF u (t), λ s > 4π 2 1 2π Re λ 1 K 1 K 2 Im λ λ c λ s K -2π -1-4π -1 lnr Re λ 1 2 Im N(ω).2-1/K 1-1/K Re N(ω)

24 The Lorenz Sstem under UEDFC ( z) x& = σ & = rx xz z& = x bz -KF u (t) UEDFC: F u (t) = F(t) w(t) F(t)= (t)-(t-τ) R F(t-τ) w& () ( t λ λ ) = λcw c c R < 1, λc > F(t) Re λ K F u K 1 K K t (c) t

25 Block Diagram of the UEDFC p(t) CHAOTIC SISTEM (t) K - 1-R R τ ULPF -R C ULPF

26 Stabilizing and tracking unknown stead states of dnamical sstems K.Pragas, V.Pragas, I.Kiss, J.Hudson, Phs.Rev.Lett., (22) Dnamic sstem: r f r x&r = r f ( x, ) = r ( x, p ) p p - control parameter r = g - output variable ( x) x r is a fixed point for p = p Adaptive controller: p = p K(w-) dw/dt = λ(w-) λ> is a necessar condition to stabilize the stead states with an odd number of real positive eigenvalues!

27 Controlling a chemical reaction p(t) DINAMICAL SISTEM (t) K - LPF R or -R C LPF

28 Conclusions Time-delaed feedback control technique is a convenient and universal tool for various experimental applications. An unstable degree of freedom introduced into a feedback loop can overcome the well known limitation of the delaed feedback schema. The modified controller can stabilize unstable periodic orbits with a finite torsion and can be successfull used for a wider class of chaotic sstems. The idea of using an unstable controller for stabilizing unstable states of dnamical sstems seems to be new in control theor. It is surprising that the idea works since common sense dictates that the instabilit introduced into a feedback loop should increase the instabilit of the closed loop sstem. Nevertheless, two coupled unstable sstems can stabilize each other and operate in a stable regime.

29 Acknowledgments Prof. Alexander Fradkov for inviting me for this talk M daughter Viktorija Pragaitė for preparing the demonstration program of two uncoupled double-pendulums