Tian, Yu-Chu; Tadé, Moses O.; Tang, Jinyu Nonlinear open-plus-closed-loop (NOPCL) control of dynamic systems. (English) Zbl 0958.93041 Chaos Solitons Fractals 11, No. 7, 1029-1035 (2000). The authors consider the entrainment (or asymptotic tracking) control problem, where the goal is to find a control \(u\) such that the solution \(x(t)\) of the system \(\dot{x}=F(x,t)+u(t)\) satisfies \(\lim_{t\to\infty}|x(t)-g(t)|=0\) for some prescribed function \(g(t)\). For the solution to this problem the authors propose a time-varying feedback controller, denoted as nonlinear-open-plus-closed-loop or NOPLC control. This method extends the OPCL control proposed by E. A. Jackson and I. Grosu [Physica D 85, No. 1–2, 1–9 (1995; Zbl 0888.93034)] by taking into account higher order derivatives of \(F\). It is shown that the resulting control law solves the problem, at least for a nonempty set of initial conditions called the basin of entrainment. For special cases like, e.g., the Lorenz, Roessler, Duffing and other systems, the basin of entrainment is shown to be global. Reviewer: Lars Grüne (Frankfurt) Cited in 5 Documents MSC: 93C10 Nonlinear systems in control theory 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior 34C28 Complex behavior and chaotic systems of ordinary differential equations Keywords:entrainment control; asymptotic tracking; time-varying feedback; chaotic system Citations:Zbl 0888.93034 PDFBibTeX XMLCite \textit{Y.-C. Tian} et al., Chaos Solitons Fractals 11, No. 7, 1029--1035 (2000; Zbl 0958.93041) Full Text: DOI References: [1] Chen, G.; Dong, X., From chaos to order: perspectives and methodologies in controlling chaotic nonlinear dynamic systems, Int. J. Bifurc and Chaos, 3, 1343 (1993) [2] Ott, E.; Grebogi, C.; Yorke, J. A., Controlling chaos, Phys. Rev. Lett., 64, 1196-1199 (1990) · Zbl 0964.37501 [3] Pyragas, K., Continuous control of chaos by self-controlling feedback, Phys. Lett. A, 170, 421-428 (1992) [4] Chen, G.; Dong, X., On feedback control of chaotic nonlinear dynamic systems, Int. J. Bifurc and Chaos, 2, 407-411 (1992) · Zbl 0875.93176 [5] Chen, G.; Dong, X., On feedback control of chaotic continuous-time systems, IEEE Trans. Circuit and Syst. I, 40, 591-601 (1993) · Zbl 0800.93758 [6] Fradkov, A. L.; Pogromsky, A. Yu., Speed gradient control of chaotic continuous-time systems, IEEE Trans. Circuit and Syst. I, 43, 907-913 (1996) [7] Hubler, A. W.; Luscher, E., Resonant stimulation and control of nonlinear oscillations, Naturwissenschaft, 76, 67 (1989) [8] Jackson, E. A.; Grosu, I., An OPCL control of complex dynamic systems, Phys. D, 85, 1-9 (1995) · Zbl 0888.93034 [9] Tian, Y.-C., Adaptive control of a chaotic system with delay, Chin Phys. Lett., 15, 477-479 (1998) [10] Jackson, E. A., The entrainment and migration controls of multiple-attractor systems, Phys. Lett. A, 151, 474-478 (1990) [11] Jackson, E. A., Control of dynamic flows with attractor, Phys. Rev. A, 44, 4839-4853 (1991) [12] Jackson, E. A., On the control of complex dynamic systems, Phys. D, 50, 341-366 (1991) · Zbl 0746.93041 [13] J.-J.E. Slotine, W. Li, Applied nonlinear control, Englewood Cliffs, NJ: Prentice-Hall, 1991; J.-J.E. Slotine, W. Li, Applied nonlinear control, Englewood Cliffs, NJ: Prentice-Hall, 1991 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.