Suppression of Oscillations of a Trolley with a Double Pendulum by Controlling Its Acceleration

The problem of transferring a mechanical system from one phase state to another is one of the most important problems of the control theory. In this case, a model problem is to find the optimal control force that transfers the trolley with pendulums moving horizontally, for instance, from its state of rest to a new state of rest over the given distance during the fixed time. In their previous works, the authors showed that when using the Pontryagin maximum principle with the functional of the squared control force being minimized to solve such a problem, a high-order constraint is implemented automatically (for instance, an eighth-order constraint for the motion of a trolley with two pendulums). This is the reason the generalized Gauss principle was used to solve the same problem, making it possible to find the control force in the form of a polynomial. In this work, the problem of suppression of oscillations of a trolley with a double pendulum is solved using the same principle. It is proposed to find the acceleration of the trolley as a control instead of the force first and then to search immediately for the control force by the obtained law of variation of the optimal acceleration of the trolley and the motion of the entire mechanical system.