Delayed state feedback and chaos control for time-periodic systems via a symbolic approach. (English) Zbl 1082.93048
The monodromy matrix of linear delay systems is approximated by Chebyshev polynomials. Based on this discretization tracking of a desired orbit is achieved by stabilizing the linearized error dynamics. The resulting feedback is used for the original nonlinear system. Two explicit examples are treated where the method yields good results. The computations are based on Matematica. Convergence problems are not discussed.
Reviewer: Fritz Colonius (Augsburg)
MSC:
| 93D15 | Stabilization of systems by feedback |
| 93C30 | Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems) |
Software:
dde23References:
| [1] | Averina Victoria. Symbolic stability of delay differential equations, M.S. thesis, Department of Mathematical Science, University of Alaska Fairbanks, AK, August 2002; Averina Victoria. Symbolic stability of delay differential equations, M.S. thesis, Department of Mathematical Science, University of Alaska Fairbanks, AK, August 2002 |
| [2] | Barnett, S., Polynomials and linear control systems (1983), Madison Dekker Inc: Madison Dekker Inc New York · Zbl 0528.93003 |
| [3] | Butcher, E. A.; Ma, H.; Bueler, E.; Averina, V.; Szabo, Z., Stability of linear time-periodic delay-differential equations via Chebyshev polynomials, Int. J. Num. Meth. Engr, 59, 895-922 (2004) · Zbl 1065.70002 |
| [4] | Curtain, R. F.; Zwart, H. J., An introduction to infinite-dimensional linear systems theory (1995), Springer-Verlag: Springer-Verlag New York · Zbl 0839.93001 |
| [5] | Fox, L.; Parker, I. B., Chebyshev polynomials in numerical analysis (1968), Oxford Univ. Press: Oxford Univ. Press London · Zbl 0153.17502 |
| [6] | Ma, H-. T.; Butcher, E. A.; Bueler, E., Chebyshev expansion of linear dynamic systems with time delay and periodic coefficients under control excitations, J. Dyn. Syst. Meas. and Control, 125, 236-243 (2003) |
| [7] | Hale, J.; Verduyn Lunel, S. M., Introduction to functional differential equations (1993), Spring · Zbl 0787.34002 |
| [8] | Sinha, S. C.; Butcher, E. A., Symbolic computation of fundamental solution matrices for linear time-periodic dynamical systems, J. Sound Vib, 206, 1, 61-85 (1997) · Zbl 1235.15003 |
| [9] | Sinha, S. C.; Wu, D. H., An efficient computational scheme for the analysis of periodic systems, J. Sound Vib, 151, 1, 91-117 (1991) |
| [10] | Yakubovich, V. A.; Starzhinski, V. M., Linear differential equations with periodic coefficients, Part (1975), Wiley: Wiley New York · Zbl 0308.34001 |
| [11] | Sinha, S. C.; Henrichs, J. T.; Ravindra, B., A general approach in the design of active controllers for nonlinear systems exhibiting chaos, Int. J. Bifurcat. Chaos, 10, 1, 165-178 (2000) · Zbl 1090.37528 |
| [12] | Sinha, S. C.; Joseph, P., Control of general dynamic systems with periodically varying parameters via Liapunov-Floquet transformation, ASME J. Dyn. Syst. Meas. Control, 116, 650-658 (1994) · Zbl 0843.93032 |
| [13] | Shampine LF, Thompson S. Solving DDEs in Matlab, Available from: http://www.radford.edu/ · Zbl 0983.65079 |
| [14] | Malkus, M. V.R., Non-periodic convection at high and low Prandtl number, Mem. Soc. Roy. Sci. Liege, Ser. 6, 4, 125-128 (1972) · Zbl 0269.76025 |
| [15] | Yorke, E. D.; Yorke, J. A., Metastable chaos: transition to sustained chaotic behavior in the Lorenz model, J. Stat. Phys, 21, 263-277 (1979) |
| [16] | Knobloch, E., Chaos in a segmented disc dynamo, Phys. Lett, 82A, 439-440 (1981) |
| [17] | Pedlosky, J., Limit cycles and unstable baroclinic waves, J. Atmos. Sci, 29, 53 (1972) |
| [18] | Pedlosky, J.; Frenzen, C., Chaotic and periodic behavior of finite amplitude baroclinic waves, J. Atmos. Sci, 37, 1177-1196 (1980) |
| [19] | Chen, G.; Dong, X., From chaos to order, perspectives and methodologies in controlling chaotic nonlinear dynamical systems, Int. J. Bifurcat. Chaos, 3, 1363-1409 (1993) · Zbl 0886.58076 |
| [20] | Jackson, E. A., Controls of dynamic flows with attractors, Phys. Rev. A, 44, 4839-4853 (1991) |
| [21] | Ott, E.; Grebogi, C.; Yorke, J. A., Controlling chaos, Phys. Lett. A, 64, 1169-1199 (1990) · Zbl 0964.37501 |
| [22] | Kapitaniak, T., Controlling chaos. Theoretical and practical methods in nonlinear dynamics (1996), Academic Press · Zbl 0883.58021 |
| [23] | David A, Sinha SC. Control of chaos in nonlinear systems with time-periodic coefficients. In: Proceedings of the 2000 American Control Conference, Chicago, Illinois, 28-30 June 2000; David A, Sinha SC. Control of chaos in nonlinear systems with time-periodic coefficients. In: Proceedings of the 2000 American Control Conference, Chicago, Illinois, 28-30 June 2000 |
| [24] | Ma H, Deshmukh V, Butcher E, Averina V. Controller design for linear time-periodic delay systems via a symbolic approach. In: Proceedings of 2003 American Control Conference, Denver, CO, 4-6 June 2003; Ma H, Deshmukh V, Butcher E, Averina V. Controller design for linear time-periodic delay systems via a symbolic approach. In: Proceedings of 2003 American Control Conference, Denver, CO, 4-6 June 2003 |
| [25] | Guan, X.-P.; Chen, C.-L.; Peng, H.-P.; Fan, Z.-P., Time-delayed feedback control of time-delay chaotic systems, Int. J. Bifurcat. Chaos, 13, 1, 193-205 (2003) · Zbl 1129.93399 |
| [26] | Liao, X.-X.; Chen, G.-R., Chaos synchronization of general Lur’e systems via time-delay feedback control, Int. J. Bifurcat. Chaos, 13, 1, 207-213 (2003) · Zbl 1129.93509 |
| [27] | Butcher, E. A.; Sinha, S. C., A hybrid formulation for the analysis of time-periodic linear systems via Chebyshev polynomials, J. Sound Vib, 195, 3, 518-527 (1996) |
| [28] | Phat; Vu Ngoc, Stabilization of linear continuous time-varying systems with state delays in Hilbert spaces, Electron. J. Differ. Equat, 67, 1-13 (2001) · Zbl 1020.93017 |
| [29] | Gowrdon G, Sinha SC. Design of controllers with symbolic computation for nonlinear system exhibiting chaos, Technical Report, Department of Mechanical Engineering, Auburn University, 2003; Gowrdon G, Sinha SC. Design of controllers with symbolic computation for nonlinear system exhibiting chaos, Technical Report, Department of Mechanical Engineering, Auburn University, 2003 |
| [30] | Bleich ME, Socolar JES. Stability of periodic orbits controlled by time-delay feedback, Available from: http://arxiv.org/abs/chao-dyn/9510019; Bleich ME, Socolar JES. Stability of periodic orbits controlled by time-delay feedback, Available from: http://arxiv.org/abs/chao-dyn/9510019 |
| [31] | Maccari, A., The response of a parametrically excited van der Pol oscillator to a time delay state feedback, Nonlinear Dyn, 26, 105-119 (2001) · Zbl 1018.93015 |
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