Oleg Kiselev

The article is devoted to a comprehensive study of linear equations of the second order with an almost periodic coefficient. Using an asymptotic approach, the system of equations for parametric subresonant growth of the amplitude of oscillations is obtained. Moreover, the time of a turning point from the growth of the amplitude to the bounded oscillations in the slow variable is found. Also, a comparison between the asymptotic approximation for the turning time and the numerical one is shown.

This paper contains an analysis of the problems concerning the control of underactuated systems. As an underactuated system an inverted pendulum on a wheel system is chosen since it has one motor that affects both the motion of the wheel and the angular position of the pendulum. The first objective of this work is to develop a motion algorithm bringing the system from the initial to the final point while both points are connected with a straight line and the system starts motion from the equilibrium position. The second objective is to study applicability of the modal control technique to the system and observe the range of its applicability in the case of parametric uncertainties in the system.
The proposed motion trajectory is built on the basis of the maximum allowed angular velocity of the motor and linear optimization techniques. The modal controller is applied to the initial parametric configuration of the system and to the system with a significant degree of parametric uncertainty. The controller demonstrates high robustness to constant parametric uncertainty expressed by the stability of the trajectory tracking process and a wide range of applicability.