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 t racking objectives common in classic control, in some applications it is de sirable to achieve an oscillatory behavior. Towards this end, bifurcation c ontrol aims at designing a controller to modify the bifurcative dynamical b ehavior 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 a pplied to obtain oscillations of prescribed amplitude in a simple mechanica l system: an underactuated pendulum. Two different nonlinear control laws a re described and analyzed. Both are designed to modify the coefficients of the linearization matrix of tile system via feedback. The first law modifie s those coefficients that correspond to the physical parameters, whereas ti le second one changes some null coefficients of the linearization matrix. T he 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 approximat ions, and over a larger range by numerical continuation of tile periodic so lutions. Theoretical results are contrasted by both computer simulations an d experimental results.