Anti-control of Hopf bifurcations

Bifurcation control generally means to design a controller that is capable of modifying the bifurcation characteristics of a bifurcating nonlinear system, thereby achieving some desirable dynamical behaviors. A typical objective is to delay and/or stabilize an existing bifurcation. In this paper, we consider the problem of anti-controlling bifurcations, that is, a certain bifurcation is created at a desired location with preferred properties by appropriate control. Washout-filter-aided dynamic feedback control laws are developed for the creation of Hopf bifurcations. As Hopf bifurcations give rise to limit cycles, anti-control of Hopf bifurcations suggests a new approach for designing limit cycles with specified oscillatory behaviors into a system via feedback control when such dynamical behaviors are desirable,.

[1]  H. O. Wang,et al.  Active Stabilization of Rotating Stall in Axial-Flow Gas Compressors , 1993, Proceedings. The First IEEE Regional Conference on Aerospace Control Systems,.

[2]  E. Abed,et al.  Local feedback stabilization and bifurcation control, II. Stationary bifurcation , 1987 .

[3]  John Baillieul,et al.  Nonlinear control designs for systems with bifurcations with applications to stabilization and control of compressors , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[4]  B. Hassard,et al.  Theory and applications of Hopf bifurcation , 1981 .

[5]  Alberto Tesi,et al.  Harmonic balance analysis of period-doubling bifurcations with implications for control of nonlinear dynamics , 1996, Autom..

[6]  J. Levine,et al.  Saddle-node bifurcation control with application to thermal runaway of continuous stirred tank reactors , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

[7]  Guanrong Chen,et al.  Bifurcation control of two nonlinear models of cardiac activity , 1997 .

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

[9]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[10]  Guanrong Chen,et al.  Hopf Bifurcation Analysis: A Frequency Domain Approach , 1996 .

[11]  Guanrong Chen,et al.  Feedback control of Hopf bifurcations , 1998, ISCAS '98. Proceedings of the 1998 IEEE International Symposium on Circuits and Systems (Cat. No.98CH36187).

[12]  E. Abed,et al.  Local feedback stabilization and bifurcation control, I. Hopf bifurcation , 1986 .

[13]  Hua O. Wang,et al.  Bifurcation control of a chaotic system , 1995, Autom..

[14]  Eyad H. Abed,et al.  Control of Nonlinear Phenomena at the Inception of Voltage Collapse , 1993, 1993 American Control Conference.

[15]  Eyad H. Abed,et al.  Stabilization of tethered satellites during station keeping , 1990 .

[16]  Eyad H. Abed,et al.  Bifurcations, chaos, and crises in voltage collapse of a model power system , 1994 .

[17]  Guanrong Chen,et al.  Anti-control of Hopf bifurcations through washout filters , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[18]  Alberto Tesi,et al.  Control of period doubling bifurcations using harmonic balance , 1993, Proceedings of 32nd IEEE Conference on Decision and Control.

[19]  Guanrong Chen,et al.  Bifurcation control of pathological heart rhythms , 1998, Proceedings of the 1998 IEEE International Conference on Control Applications (Cat. No.98CH36104).

[20]  Guanrong Chen,et al.  Bifurcation Control: Theories, Methods, and Applications , 2000, Int. J. Bifurc. Chaos.

[21]  Dong Chen,et al.  Amplitude control of bifurcations and application to Rayleigh-Benard convection , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[22]  J. Baillieul,et al.  Oscillatory control of bifurcations in rotating chains , 1997, Proceedings of the 1997 American Control Conference (Cat. No.97CH36041).

[23]  Xiang Chen,et al.  Bifurcation Stabilization with Local Output Feedback , 1999 .

[24]  R.J. Thomas,et al.  On voltage collapse in electric power systems , 1989, Conference Papers Power Industry Computer Application Conference.

[25]  E. H. Abed,et al.  Robust control of period doubling bifurcations and implications for control of chaos , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[26]  Hong Wang,et al.  Control of bifurcations and chaos in heart rhythms , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.

[27]  Eyad H. Abed,et al.  Stabilization of period doubling bifurcations and implications for control of chaos , 1994 .