Sergey A. Reshmin

Properties of the Time-Optimal Feedback Control for a Pendulum-Like System

A time-optimal control problem for a pendulum-like system is considered. The system describes the dynamics of an inertial object under the action of a bounded control force and an external force which is periodic in coordinate. The terminal set consists of points on the abscissa axis of the phase plane, and the distance between two neighboring points is equal to the period of the external force. In the general case, the solution can be obtained only numerically. An estimate is found for the amplitude of the control for which the time-optimal feedback control has the simplest structure: the number of switchings is not greater than one for any initial conditions. For the estimated interval of the control constraints, we analyze the feedback control pattern.

Dispersal Curve Properties in the Time-Optimal Control Problem for a Second-Order Nonlinear System* *

Abstract We consider the time-optimal control problem for a nonlinear mechanical system with one degree of freedom. The system describes the dynamics of an inertial object under the action of a bounded control force and a given disturbance force that is periodic in the coordinate. The terminal set consists of points located on the abscissa axis of the phase plane; here, the distance between two adjacent points is equal to the period of the disturbance force in coordinate. We investigate properties of the dispersal curve on the phase cylinder when the magnitude of the control force is sufficiently large.

METHOD OF DECOMPOSITION AND ITS APPLICATIONS TO UNCERTAIN DYNAMICAL SYSTEMS

Abstract A nonlinear dynamical system described by Lagranges equations is considered. The system is subjected to uncertain external forces and controls bounded by geometric constraints. A feedback control that satisfies the imposed conditions and brings the system to a prescribed terminal state in a finite time is proposed. The control is based on the decomposition of the system and uses ideas of optimal control and differential games. Explicit formulas for the feedback control are presented. Applications to control of robots and underactuated systems are discussed. Computer simulation of motions of a double pendulum controlled by one torque is presented.

Decomposition of Control for Nonlinear Lagrangian Systems

Abstract A feedback control for nonlinear dynamical systems described by the Lagrangian equations of motion is proposed. The system is subjected to bounded control forces and uncertain but bounded disturbances. The design of feedback control is based on the decomposition of the original system with many degrees of freedom into subsystems with one degree of freedom each. Under certain assumptions, the nonlinearities are estimated, and the feedback control is designed which counteracts the influence of nonlinearities and disturbances for each subsystem. Thus, the decomposition of the original system is accomplished, and explicit formulas for the feedback control in the original system are obtained. This control brings the nonlinear system from a given initial state to the prescribed terminal state in finite time. The time of control is estimated from above. Other versions of the decomposition approach are discussed. Applications of the described approach to control of robots are presented.