Optimal multi-dimensional OGY controller

Abstract A technique which generalizes the standard OGY control scheme and may be applied to higher dimensional systems to stabilize both saddle nodes and fully unstable fixed points is presented. The proposed method is designed for nonlinear flows rather than maps and also provides a powerful technique to assess the controllability of nonlinear systems with OGY type controllers. An optimal OGY control scheme designed for larger dimensional systems having improved convergence properties compared to the standard OGY controller is also presented. Numerical examples for the well known Van der Pol–Duffing oscillator are presented to illustrate the proposed control schemes.

[1]  Pei Yu,et al.  Parametrically excited non-linear systems : A comparison of certain methods , 1998 .

[2]  Dressler,et al.  Controlling chaos using time delay coordinates. , 1992, Physical review letters.

[3]  V. Belykh,et al.  Bifurcations for heteroclinic orbits of a periodic motion and a saddle-focus and dynamical chaos , 1998 .

[4]  M. Lakshmanan,et al.  Bifurcation and chaos in the double-well Duffing–van der Pol oscillator: Numerical and analytical studies , 1997, chao-dyn/9709013.

[5]  Glorieux,et al.  Experimental characterization of unstable periodic orbits by controlling chaos. , 1993, Physical review. A, Atomic, molecular, and optical physics.

[6]  Steven R. Bishop,et al.  Self-locating control of chaotic systems using Newton algorithm , 1996 .

[7]  Ding,et al.  Controlling chaos in high dimensions: Theory and experiment. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[8]  Michael F. Shlesinger,et al.  Chaotic and Fractal Dynamics: An Introduction for Applied Scientists and Engineers , 1993 .

[9]  Zoltán Toroczkai,et al.  Geometric method for stabilizing unstable periodic orbits , 1994 .

[10]  Jun Jiang,et al.  The global bifurcation characteristics of the forced van der Pol oscillator , 1996 .

[11]  Chyun-Chau Fuh,et al.  Robust control for a class of nonlinear oscillators with chaotic attractors , 1996 .

[12]  E. Ott,et al.  Controlling Chaotic Dynamical Systems , 1991, 1991 American Control Conference.

[13]  Earl H. Dowell,et al.  On the optimality of the Ott-Grebogi-Yorke control scheme , 1998 .

[14]  Steven H. Strogatz,et al.  Nonlinear Dynamics and Chaos , 2024 .

[15]  S. Wiggins.  Introduction to Applied Nonlinear Dynamical Systems and Chaos , 1989 .

[16]  Bogdan I. Epureanu,et al.  Stabilization of Unstable Limit Cycles in Systems with Limited Controllability: Expanding the Basin of Convergence of OGY-Type Controllers , 1998 .

[17]  Grebogi,et al.  Higher-dimensional targeting. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[18]  L. A. Belyakov,et al.  On Bifurcations of Periodic Orbits in the van der Pol-Duffing Equation , 1997 .

[19]  Ying-Cheng Lai,et al.  Controlling chaos , 1994 .

[20]  Wanda Szemplińska-Stupnicka,et al.  THE COEXISTENCE OF PERIODIC, ALMOST-PERIODIC AND CHAOTIC ATTRACTORS IN THE VAN DER POL-DUFFING OSCILLATOR , 1997 .

[21]  Earl H. Dowell,et al.  System identification for the Ott-Grebogi-Yorke controller design , 1997 .

[22]  Scott,et al.  Stabilizing and characterizing unstable states in high-dimensional systems from time series. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[23]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[24]  Guangrong Qin,et al.  Rich bifurcation behaviors of the driven van der Pol oscillator , 1989 .

[25]  Ditto,et al.  Experimental control of chaos. , 1990, Physical review letters.

[26]  E. Ott.  Chaos in Dynamical Systems: Contents , 1993 .

[27]  Tomasz Kapitaniak,et al.  Analytical conditions for strange chaotic and nonchaotic attractors of the quasiperiodically forced van der pol equation , 1991 .