02[ ]小林(責).indd

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1 43 Delayed Feedbac Control Delayed Feedbac Control Prediction-based Feedbac Control 1, 2 GDP

2 Ott, Grebogi Yore OGY 4 OGY OGY OGY OGY OGY 3, 5, 6 OGY Pyragas Delayed Feedbac Control DFC 7 DFC 2 DFC OGY 3, 8, 9, 1 DFC DFC 11, 12

3 45 2 Delayed Feedbac Control Delayed Feedbac Control 3 Delayed Feedbac Control Prediction-based Feedbac 4 5 Delayed Feedbac Control DFC 7 f R n R n n x 1 f x K x τ x, 1 K R n n Z τ Z x R n 1 K x τ x Delayed Feedbac Control K τ K τ DFC x τ x x τ x τ DFC DFC

4 x 1 ax 1 x, 2 a a a 3.7 x 1 ax 1 x K x τ x. 3 K 2 2 x f 1, x f 2 x f 1 a 1 a, x f δx i 1 a 2ax fi δx i, 4 δx x i x fi, i 1, 2 2 x fi, i 1, 2 1 τ 1 K x 1 x x 1 ax 1 x K x 1 x, 5 w x 1 x 1 ax 1 x K w x, 6 w 1 x

5 x fi, i 1, 2 δx i 1 a 2ax fi K K δx i δw i 1 1 δw i 8 δx x i x fi, δw w i x fi, i 1, 2 x f 1 DFC x f a 2ax f 1 2 a a a 1 x f 1 x f 1 DFC 5 x f K 3 a 2 K x f 1 Jury 13 x f 2 DFC x f 2 x f 2 DFC 11, 12 1 DFC x f 2 x f 2 4 a a 3.7 x f 2 1

6 x f 2 DFC 6 7 x f 2 a K K 1 a Jury K x f 2 DFC 1 Prediction-based Feedbac Control PFC 14, 15, 16 DFC τ PFC τ u K x τ x, x R n K x τ τ PFC x 1 f x K x 1 x 9 DFC PFC DFC PFC

7 49 PFC 17 PFC 1 x f 1, x f 2 PFC x 1 ax 1 x K x 1 x, 1 a 2 x fi, i 1, 2 δx i 1 2a 1 K x fi a ak K δx i, 11 δx i x x fi. DFC 1 K 2 2a 1 K x fi a ak K DFC x f 2 PFC x f 2 12 a 1 1 a K 1 K PFC DFC PFC x τ 1

8 u K x p τ x, x p τ τ PFC x τ x p τ PFC 18, 19 2 PFC 1 16 u x 1 f x, u. 13 τ u K x p τ x, K x p τ 13 x 1 f x, τ PFC PFC u K x τ x

9 51 DFC 7 u K x τ x PFC 14, 15 PFC 2 x system x system f x system, x data x data x data f x data, x system τ x p τ x data f x data, t p x data t p x data x system 1 x data, t p x system f x system, x system τ x system 13 x x data τ x system τ x * τ x p τ x * τ u K x p τ x K x * τ x x 1 3.7x 1 x x data, x system τ x system 1 τ data t p 1 x data x system x p x system t p τ * data τ 1 1 data

10 x p x τ system t p x data x system t p τ t p data x system x τ τ τ radial basis function 19 PFC a 3.7 x 1 3.7x 1 x. 14 x 1 PFC 14 x 1 3.7x 1 x K x p τ x,

11 53 x x x distribution function of x, P(x) x p τ 14 τ K u K x p τ x, if x τ x.1,, otherwise. 15 x fp a 1 a K.5 12 K x τ x K x p τ x 12 K x p τ 12 K 2 x p t p 1 2 x x fp.7297

12 x distribution function of x, P(x) x K.5 t p 1 x x distribution function of x, P(x) x K.5 t p 5 x 3 t p 5 3 x x fp t p 5 4 t p 1

13 55 x K.5 t p 1 x distribution function of x, P(x) x t p.1 t p x fp 2 t p t p 1

14 Delayed Feedbac Control DFC Prediction-based Feedbac Control PFC DFC PFC 21, 22 DFC 23 Delayed Feedbac Control DFC PFC PFC DFC x 1 f x, x τ1, x τ2 x τ2

15 57 1 Khalil K. H.: Nonlinear Systems, Prentice Hall edited by Schuster, H. F.: Handboo of Chaos Control. Wiley-VCH, Weinheim Ott, E., Grebogi C., and Yore J. A.: Controlling chaos, Phys. Rev. Lett., 64, Romeiras, F. J., Grebogi, C., Ott, E. and Dayawansa, W. P.: Controlling chaotic dynamical systems, Physica D, 58, Shinbrot, T., Grebogi, C., Ott, E., and Yore, J. A.: Using small perturbation to control chaos, Nature, 363, Pyragas, K.: Continuous control of chaos by self-controlling feedbac, Phys. Lett. A, 17, Just, W.: Delayed feedbac control of periodic orbits in autonomous systems, Phys. Rev. Lett., 81, Bielawsi, S., Derozier, D., and Glorieux, P.: Controlling unstable periodic orbits by a delayed continuous feedbac, Phys. Rev. E, 49, R Kobayashi, U. M. and Aihara K.: Delayed feedbac control method for dynamical systems with chaotic saddles, AIP Conf. Proc., 1468, Just, W., Bernard, T., Ostheimer, M., Reibold, E., and Benner H.: Mechanism of time delayed feedbac control, Phys. Rev. Lett., 78, Naajima, H.: On analytical properties of delayed feedbac control of chaos, Phys. Lett. A, 232, Phillips, L. C., and Nagle, T. H., Digital Control System Analysis and Design, PrenticeHall Vieira, M. de S. and Lichtenberg, A. J.: Controlling chaos using nonlinear feedbac with delay, Phys. Rev. E, 54, Ushio, T. and Yamamoto, S.: Prediction-based control of chaos, Phys. Lett. A, 264, Kobayashi, U. M., Ueta, T., and Aihara, K.: Feedbac Control Method

16 based on Predicted Future States for Controlling Chaos, Springer, Analysis and Control of Complex Dynamical Systems, Bouabou, A., Chebbah A. and Mansouri N.: Predictive control of continuous chaotic systems, Int. J. Bifurcation and Chaos, 18, Sugihara, G. and May, R. M., Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series, Nature, 344, Casdagli, M., Nonlinear Prediction of Chaotic Time Series, Physica D, 35, Lorenz, E. N.: Atmospheric Predictability as Revealed by Naturally Occurring Analogues, Jour. Atmos. Scie., 26, Naajima, H. and Ushio, Y., Half-period delayed feedbac control for dynamical systems with symmetries, Phys. Rev. E, 58, Morita, Y., Fujiwara, N., Kobayashi, U. M., and Mizuguchi, T., Scytale decodes chaos: A method for estimating unstable sysmmetric solutions, CHAOS 2, Fiedler, B., Flunert, V., Georgi, M., Hövel, P., and Schöll, E., Refuting the odd-number limitation of time-delayed feedbac control, Phys. Rev. Lett. 98,

ohp_06nov_tohoku.dvi

ohp_06nov_tohoku.dvi 2006 11 28 1. (1) ẋ = ax = x(t) =Ce at C C>0 a0 x(t) 0(t )!! 1 0.8 0.6 0.4 0.2 2 4 6 8 10-0.2 (1) a =2 C =1 1. (1) τ>0 (2) ẋ(t) = ax(t τ) 4 2 2 4 6 8 10-2 -4 (2) a =2 τ =1!! 1. (2) A. (2)