An Experimental Application of the Anticontrol of Hopf bifurcations

The control of nonlinear systems exhibiting bifurcation phenomena has been the subject of active research in recent years. Contrary to regulation or tracking objectives common in classic control, in some applications it is desirable to achieve an oscillatory behavior. Towards this end, bifurcation control aims at designing a controller to modify the bifurcative dynamical behavior of a complex nonlinear system. Among the available methods, the so-called "anti-control" of Hopf bifurcations is one approach to design limit cycles in a system via feedback control. In this paper, this technique is applied to obtain oscillations of prescribed amplitude in a simple mechanical system: an underactuated pendulum. Two different nonlinear control laws are described and analyzed. Both are designed to modify the coefficients of the linearization matrix of the system via feedback. The first law modifies those coefficients that correspond to the physical parameters, whereas the second one changes some null coefficients of the linearization matrix. The latter results in a simpler controller that requires the measurement of only one state of the system. The dependence of the amplitudes as function of the feedback gains is obtained analytically by means of local approximations, and over a larger range by numerical continuation of the periodic solutions. Theoretical results are contrasted by both computer simulations and experimental results.

[1]  Ali H. Nayfeh,et al.  BIFURCATIONS IN A POWER SYSTEM MODEL , 1996 .

[2]  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).

[3]  Abdelfatah M. Mohamed,et al.  Nonlinear oscillations in magnetic bearing systems , 1993 .

[4]  Guanrong Chen,et al.  From Chaos to Order - Perspectives and Methodologies in Controlling Chaotic Nonlinear Dynamical Systems , 1993 .

[5]  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).

[6]  Leon O. Chua,et al.  The Hopf bifurcation theorem and its applications to nonlinear oscillations in circuits and systems , 1979 .

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

[8]  Po Ki Yuen,et al.  Rendering a subcritical Hopf bifurcation supercritical , 1996, Journal of Fluid Mechanics.

[9]  Y. Kuznetsov,et al.  Continuation techniques and interactive software for bifurcation analysis of ODEs and iterated maps: physics , 1993 .

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

[11]  Guanrong Chen,et al.  Feedback Control of Limit Cycle Amplitudes from A Frequency Domain Approach , 1998, Autom..

[12]  Jorge L. Moiola,et al.  On anticontrol of Hopf bifurcations , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[13]  Guanrong Chen,et al.  An overview of bifurcation, chaos and nonlinear dynamics in control systems , 1994 .

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