Stabilization of inverted pendulum on a two-wheeled vehicle

A mechanical system consisting of a two-wheeled vehicle with a fixed inverted pendulum is considered. The objective lies in forming such control action which, on the one hand, would provide movement of the system on the specified trajectory, and on the other hand would provide stabilization of the inverted pendulum in a neighborhood of unstable equilibrium position. Characteristic feature of the set objective is the fact that the control action is performed with the use of a hysteretic connection formalized by equations of the Bouc–Wen model. Equations of the researched system’s model are divided into two independent subsystems of the second and the fourth order, which describe rotational and longitudinal modes of motion correspondingly. Control action on each of the subsystems, formed on the basis of the feedback principle, is constructed according to the relay law; at that in the phase space of each subsystem linear manifolds which determine switching surfaces are constructed. Theorems that ascertain the presence of errors in the assumptions made, which determine the discrepancy between the desired and simulated laws of the vehicle motion, as well as the deviation of the pendulum from the unstable equilibrium position asymptotically tend to zero, are formulated and proved in the article. In proving the theorems, the apparatus of Lyapunov functions was used. As it is shown in the article, the presence of the hysteresis component in the feedback loop makes manageability of the system more difficult, and as a result, in this case we can only talk about limitation of errors of the discrepancy and the dissipative motion of the pendulum. Results of computational simulation of dynamics of the system under study both under conditions of hysteresis connections and without them, which illustrate the proved theorems, are given in the article.