Generating self-excited oscillation in a class of mechanical systems by relay-feedback

Self-generated limit cycle tracking of the underactuated inertia wheel inverted pendulum under IDA-PBC

“…In this Appendix, we show generation of a self-sustained oscillation (limit cycle) in the nonlinear dynamics of the underactuated IWIP under the IDA-PBC. Thus, we present first stability conditions of the equilibrium point xeq, given by expression (9), of the nonlinear dynamics (8). Moreover, we derive conditions proving existence of the Hopf bifurcation and hence the periodic solutions in the nonlinear dynamics (8).…”

“…The return time, say τr, represents the time between two successive intersections with the Poincaré section and it defines the period of the stable period-1 limit cycle. We note that the return time is calculated when integrating the nonlinear dynamics (8)…”

“…Thus, we present first stability conditions of the equilibrium point xeq, given by expression (9), of the nonlinear dynamics (8). Moreover, we derive conditions proving existence of the Hopf bifurcation and hence the periodic solutions in the nonlinear dynamics (8). Finally, we discuss briefly stability of the periodic solutions using the fist Lyapunov value [4].…”

“…In order to characterize the stability of the periodic solutions generated through each Hopf bifurcation, the so-called fist Lyapunov value (coefficient) [4], say ℓ 1 , must be computed and analyzed for the system (8).…”

“…Some examples of dynamical systems that exhibit a self-excited oscillation are beating of a heart, rhythms in body temperature, hormone secretion, chemical reactions, vibrations in bridges and airplane wings, inverted pendulums, the Van der Pol oscillator, the Duffing oscillator, biped robots, etc. [4][5][6][7] (see also [8,9] for further examples). In general, selfoscillators give a regular output robustly: they approach the same limit cycle regardless of initial conditions and transient perturbations [2].…”