Oleg Kiselev

The dynamics of an inverted wheeled pendulum controlled by a proportional plus integral plus derivative action controller in various cases is investigated. The properties of trajectories are studied for a pendulum stabilized on a horizontal line, an inclined straight line and on a soft horizontal line. Oscillation regions on phase portraits of dynamical systems are shown. In particular, an analysis is made of the stabilization of the pendulum on a soft surface, modeled by a differential inclusion. It is shown that there exist trajectories tending to a semistable equilibrium position in the adopted mathematical model. However, in numerical simulations, as well as in the case of real robotic devices, such trajectories turn into a limit cycle due to round-off errors and perturbations not taken into account in the model.

We study the asymptotic behavior of nonlinear oscillators under an external driver with slowly changing frequency and amplitude. As a result, we obtain formulas for properties of the amplitude and frequency of the driver when the autoresonant behavior of the nonlinear oscillator is observed. Also, we find the measure of autoresonant asymptotic behaviors for such a driven nonlinear oscillator.

We obtain criteria for the stability of fast straight-line motion of a wheeled robot using proportional or proportional derivative feedback control. The motion of fast robots with discrete feedback control is defined by the discrete dynamical system. The stability criteria are obtained for the discrete system for proportional and proportional-derivative feedback control.