Về các ước lượng dựa trên chuẩn cho các miền thu hút trong hệ thống phi tuyến có độ trễ thời gian

Springer Science and Business Media LLC - Tập 100 - Trang 2027-2045 - 2020
Tessina H. Scholl1, Veit Hagenmeyer1, Lutz Gröll1
1Institute for Automation and Applied Informatics, Karlsruhe Institute of Technology (KIT), Eggenstein-Leopoldshafen, Germany

Tóm tắt

Đối với các hệ thống phi tuyến có độ trễ thời gian, các miền thu hút hiếm khi được nghiên cứu mặc dù chúng rất quan trọng cho các ứng dụng công nghệ. Bài báo hiện tại cung cấp những gợi ý phương pháp để xác định giới hạn trên về bán kính thu hút bằng phương pháp số. Do đó, không gian Banach tương ứng cho các hàm khởi đầu cần được lựa chọn và các hàm khởi đầu chính cần được chọn. Các hàm này sẽ được sử dụng trong các mô phỏng tiếp diễn theo thời gian để xác định một giới hạn trên ban đầu cho bán kính thu hút. Sau đó, giới hạn trên này sẽ được tinh chỉnh bởi các hàm khởi đầu thứ cấp, mà kết quả sẽ được rút ra từ các mô phỏng trước đó. Thêm vào đó, cần phải thực hiện một phân tích phân nhánh. Phân tích này cho phép cải thiện được ước lượng trước đó. Một ví dụ về phương trình dao động có độ trễ thời gian minh họa cho các khía cạnh khác nhau.

Từ khóa

#hệ thống phi tuyến #miền thu hút #độ trễ thời gian #phân tích phân nhánh #mô phỏng số

Tài liệu tham khảo

Agrawal, V., Zhang, C., Shapiro, A.D., Dhurjati, P.S.: A dynamic mathematical model to clarify signaling circuitry underlying programmed cell death control in arabidopsis disease resistance. Biotechnol. Prog. 20(2), 426–442 (2004) Aguirregabiria, J.M., Etxebarria, J.R.: Fractal basin boundaries of a delay-differential equation. Phys. Lett. A 122(5), 241–244 (1987) Balanov, A.G., Janson, N.B., Schöll, E.: Delayed feedback control of chaos: bifurcation analysis. Phys. Rev. E Stat. Nonlinear Soft. Matter. Phys. 71(1 Pt 2), 016222 (2005) Bellen, A., Zennaro, M.: Numerical Methods for Delay Differential Equations. Oxford Univ. Press, Oxford (2003) Bellman, R., Cooke, K.L.: Differential-Difference Equations. Rand Corporation, Santa Monica (1963) Bernier, C., Manitius, A.: On semigroups in \(\mathbb{R}^n \times L^p\) corresponding to differential equations with delays. Can. J. Math. 30(5), 897–914 (1978) Breda, D., Maset, S., Vermiglio, R.: Stability of Linear Delay Differential Equations: A Numerical Approach with MATLAB. Springer, New York (2015) Briat, C.: Robust Stability Analysis in the \(\ast \)-Norm and Lyapunov–Razumikhin Functions for the Stability Analysis of Time-Delay Systems: (CDC-ECC 2011); Orlando, Florida, USA, 12–15 December 2011. IEEE, Piscataway, NJ (2011) Briat, C.: Linear Parameter-Varying and Time-Delay Systems: Analysis, Observation, Filtering & Control. Springer, Berlin (2015) Broer, H.W., Takens, F.: Dynamical Systems and Chaos. Springer, New York (2011) Cao, Y.Y., Lin, Z., Hu, T.: Stability analysis of linear time-delay systems subject to input saturation. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 49(2), 233–240 (2002) Chiang, H.D.: Direct Methods for Stability Analysis of Electric Power Systems: Theoretical Foundation, BCU Methodologies, and Applications. Wiley, Hoboken (2011) Chiang, H.D., Fekih-Ahmed, L.: Quasi-stability regions of nonlinear dynamical systems: theory. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 43(8), 627–635 (1996) Chiang, H.D., Tada, Y.: Design and implementation of on-line dynamic security assessment. IEEJ Trans. Electr. Electron. Eng. 4(3), 313–321 (2009) Coutinho, D.F., de Souza, C.E.: Delay-dependent robust stability and \(L^2\)-gain analysis of a class of nonlinear time-delay systems. Automatica 44(8), 2006–2018 (2008) Curtain, R.F., Zwart, H.: An Introduction to Infinite-Dimensional Linear Systems Theory. Springer, New York (1995) Dambrine, M.: Contribution à l’étude de la stabilité des systèmes à retards. Ph.D. thesis, Université Lille1-Sciences et Technologies (1994). http://ori.univ-lille1.fr/notice/view/univ-lille1-ori-127166 Daza, A., Wagemakers, A., Sanjuán, M.A.F.: Wada property in systems with delay. Commun. Nonlinear Sci. Numer. Simul. 43, 220–226 (2017) Delfour, M., Mitter, S.: Hereditary differential systems with constant delays. I. General case. J. Differ. Equ. 12(2), 213–235 (1972) de Souza, C.E., Coutinho, D.: Delay-dependent regional stabilization of nonlinear quadratic time-delay systems. IFAC Proc. Vol. 47(3), 10084–10089 (2014) Diekmann, O., Verduyn Lunel, S.M., Gils, S.A., Walther, H.O.: Delay Equations: Functional-, Complex-, And Nonlinear Analysis. Springer, New York (1995) Dombovari, Z., Iglesias, A., Molnar, T.G., Habib, G., Munoa, J., Kuske, R., Stépán, G.: Experimental observations on unsafe zones in milling processes. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 377(2153), 20180125 (2019) Dombovari, Z., Wilson, R.E., Stepan, G.: Estimates of the bistable region in metal cutting. Proc. Math. Phys. Eng. Sci. 464(2100), 3255–3271 (2008) Dudkowski, D., Jafari, S., Kapitaniak, T., Kuznetsov, N.V., Leonov, G.A., Prasad, A.: Hidden attractors in dynamical systems. Phys. Rep. 637, 1–50 (2016) Efimov, D., Schiffer, J., Ortega, R.: Robustness of delayed multistable systems with application to droop-controlled inverter-based microgrids. Int. J. Control 89(5), 909–918 (2015) Engelborghs, K., Luzyanina, T., Roose, D.: Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL. ACM Trans. Math. Softw. 28(1), 1–21 (2002) Fridman, E.: Introduction to Time-Delay Systems: Analysis and Control. Springer, Cham (2014) Fridman, E.: Tutorial on Lyapunov-based methods for time-delay systems. Eur. J. Control 20(6), 271–283 (2014) Goldsztejn, A., Chabert, G.: Estimating the robust domain of attraction for non-smooth systems using an interval Lyapunov equation. Automatica 100, 371–377 (2019) Gu, K., Kharitonov, V.L., Chen, J.: Stability of Time-Delay Systems. Birkhäuser, Boston (2003) Haddock, J.R., Terjéki, J.: Liapunov-Razumikhin functions and an invariance principle for functional differential equations. J. Differ. Equ. 48(1), 95–122 (1983) Hahn, W.: Stability of Motion. Springer, Berlin (1967) Halanay, A.: Mathematics in Science and Engineering: Differential Equations: Stability, Oscillations, Time Lags. Elsevier, Amsterdam (1966) Hale, J.K.: Sufficient conditions for stability and instability of autonomous functional-differential equations. J. Differ. Equ. 1(4), 452–482 (1965) Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Springer, New York (1993) Hassard, B.D., Kazarinoff, N.D., Wan, Y.H.: Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge (1981) Hinrichsen, D., Pritchard, A.J.: Real and complex stability radii: a survey. In: Hinrichsen, D., Mårtensson, B. (eds.) Control of Uncertain Systems, Progress in Systems and Control Theory, pp. 119–162. Birkhäuser Boston, Boston (1990) Hu, G., Davison, E.J.: Real stability radii of linear time-invariant time-delay systems. Syst. Control Lett. 50(3), 209–219 (2003) Hunt, B.R., Sauer, T., Yorke, J.A.: Prevalence: a translation-invariant ’almost every’ on infinite-dimensional spaces. Bull. Am. Math. Soc. 27(2), 217–239 (1992) Ilyashenko, Y.: Centennial history of Hilbert’s 16th problem. Bull. Am. Math. Soc. 39(03), 301–355 (2002) Insperger, T., Ersal, T., Orosz, G. (eds.): Time Delay Systems: Theory, Numerics, Applications, and Experiments. Springer, Cham (2017) Janssens, S.G.: On a normalization technique for codimension two bifurcations of equilibria of delay differential equations. Master thesis, Utrecht University, Utrecht (2010) Jarlebring, E.: The spectrum of delay-differential equations: numerical methods, stability and perturbation. Dissertation, Technische Universität Carolo-Wilhelmina zu Braunschweig (2008) Kazarinoff, N.D., Wan, Y.H., van den Driessche, P.: Hopf bifurcation and stability of periodic solutions of differential-difference and integro-differential equations. IMA J. Appl. Math. 21(4), 461–477 (1978) Khalil, H.K.: Nonlinear Systems. Prentice Hall, Upper Saddle River (2002) Kharitonov, V.L.: Time-Delay Systems: Lyapunov Functionals and Matrices. Birkhäuser Springer, New York (2013) Kloeden, P.E., Rasmussen, M.: Nonautonomous Dynamical Systems. American Mathematical Society, Providence (2011) Krasovskii, N.N.: The approximation of a problem of analytic design of controls in a system with time-lag. J. Appl. Math. Mech. 28(4), 876–885 (1964) Krasovskii, N.N., Brenner, J.L.: Stability of Motion: Applications of Lyapunov’s Second Method to Differential Systems and Equations with Delay. Stanford University Press, Stanford (1963) Kuznetsov, N.V., Leonov, G.A.: Hidden attractors in dynamical systems: systems with no equilibria, multistability and coexisting attractors. IFAC Proc. Vol. 47(3), 5445–5454 (2014) Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory. Springer, New York (1998) Lakshmanan, M., Senthilkumar, D.V.: Dynamics of Nonlinear Time-Delay Systems. Springer, Berlin (2010) LaSalle, J.P., Artstein, Z.: The Stability of Dynamical Systems. Society for Industrial and Applied Mathematics, Philadelphia (1976) Lee, E., Neftci, S., Olbrot, A.: Canonical forms for time delay systems. IEEE Trans. Autom. Control 27(1), 128–132 (1982) Leng, S., Lin, W., Kurths, J.: Basin stability in delayed dynamics. Sci. Rep. 6, 21449 (2016) Liu, K., Fridman, E.: Delay-dependent methods and the first delay interval. Syst. Control Lett. 64, 57–63 (2014) Losson, J., Mackey, M.C., Longtin, A.: Solution multistability in first-order nonlinear differential delay equations. Chaos 3(2), 167–176 (1993) Melchor-Aguilar, D., Niculescu, S.I.: Estimates of the attraction region for a class of nonlinear time-delay systems. IMA J. Math. Control Inf. 24(4), 523–550 (2006) Menck, P.J., Heitzig, J., Marwan, N., Kurths, J.: How basin stability complements the linear-stability paradigm. Nat. Phys. 9(2), 89–92 (2013) Michiels, W., Niculescu, S.I.: Stability, Control, and Computation for Time-Delay Systems: An Eigenvalue-based Approach. SIAM Soc. for Indust. and Appl. Math, Philadelphia (2014) Minorsky, N.: Self-excited mechanical oscillations. J. Appl. Phys. 19(4), 332–338 (1948) Molnar, T.G., Dombovari, Z., Insperger, T., Stépán, G.: On the analysis of the double hopf bifurcation in machining processes via centre manifold reduction. Proc. Math. Phys. Eng. Sci. 473(2207), 20170502 (2017) Molnar, T.G., Dombovari, Z., Insperger, T., Stépán, G.: Bifurcation analysis of nonlinear time-periodic time-delay systems via semidiscretization. Int. J. Numer. Meth. Eng. 115(1), 57–74 (2018) Niculescu, S.I.: Delay Effects on Stability: A Robust Control Approach. Springer, Berlin (2001) Niculescu, S.I., Gu, K. (eds.): Advances in Time-Delay Systems. Springer, Berlin (2004) Oliva, W.: Functional differential equations on compact manifolds and an approximation theorem. J. Differ. Equ. 5(3), 483–496 (1969) Ott, W., Yorke, J.A.: Prevalence. Bull. Am. Math. Soc. 42(03), 263–291 (2005) Otto, A., Just, W., Radons, G.: Nonlinear dynamics of delay systems: an overview. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 377(2153), 20180389 (2019) Oxtoby, J.C.: Measure and Category: A Survey of the Analogies Between Topological and Measure Spaces. Springer, New York (1980) Rantzer, A.: A dual to Lyapunov’s stability theorem. Syst. Control Lett. 42(3), 161–168 (2001) Roose, D., Szalai, R.: Continuation and bifurcation analysis of delay differential equations. In: Krauskopf, B., Osinga, H.M., Galán-Vioque, J. (eds.) Numerical Continuation Methods for Dynamical Systems, pp. 359–399. Springer, Dordrecht (2007) Schäfer, B., Matthiae, M., Timme, M., Witthaut, D.: Decentral smart grid control. New J. Phys. 17(5), 059502 (2015) Scholl, T.H., Gröll, L.: Time delay in the swing equation: a variety of bifurcations. Chaos Interdiscip. J. Nonlinear Sci. 29(12), 123118 (2019) Seuret, A., Gouaisbaut, F., Baudouin, L.: D1.1–Overview of Lyapunov methods for time-delay systems: Rapport laas no. 16308. HAL archives-ouvertes.fr (hal-01369516) (2016) Shampine, L.F., Thompson, S.: Solving DDEs in matlab. Appl. Numer. Math. 37(4), 441–458 (2001) Shang, H., Xu, J.: Delayed feedbacks to control the fractal erosion of safe basins in a parametrically excited system. Chaos Solitons Fractals 41(4), 1880–1896 (2009) Sieber, J., Engelborghs, K., Luzyanina, T., Samaey, G., Roose, D.: DDE-BIFTOOL manual: Bifurcation analysis of delay differential equations. arXiv preprint (arXiv:1406.7144) (2014) Smith, H.: An Introduction to Delay Differential Equations with Applications to the Life Sciences. Springer, New York (2011) Sprott, J.C.: A simple chaotic delay differential equation. Phys. Lett. A 366(4–5), 397–402 (2007) Stépán, G.: Chaotic motion of wheels. Veh. Syst. Dyn. 20(6), 341–351 (1991) Sullivan, T.J.: Introduction to Uncertainty Quantification. Springer, Cham (2015) Taylor, S.R., Campbell, S.A.: Approximating chaotic saddles for delay differential equations. Phys. Rev. E 75(4), 046215 (2007) Villafuerte, R., Mondié, S.: On improving estimate of the region of attraction of a class of nonlinear time delay system. IFAC Proc. Vol. 40(23), 227–232 (2007) Wu, M., He, Y., She, J.H.: Stability Analysis and Robust Control of Time-Delay Systems. Springer, Berlin (2010) Yan, Y., Xu, J., Wiercigroch, M.: Estimation and improvement of cutting safety. Nonlinear Dyn. 53(2), 619 (2019) Zaborszky, J., Huang, G., Zheng, B., Leung, T.C.: On the phase portrait of a class of large nonlinear dynamic systems such as the power system. IEEE Trans. Autom. Control 33(1), 4–15 (1988)