Control of an inverted pendulum on a wheel

We consider a mechanical system consisting of a wheel and a pendulum suspended on the wheel axis. The wheel rolls on a horizontal surface. The problem of simultaneous stabilization of the vertical position of the pendulum and a given position of the wheel is considered. A well-known difficulty associated with this problem is that the use of a single control serves two purposes — to stabilize the pendulum angle and the wheel rotation angle. The application of the output feedback linearization method, with the pendulum angle chosen as the output, leads to the appearance of unstable zero dynamics in the closed-loop system.
It is shown that if we take the sum of the pendulum angle and the wheel rotation angle as the output of the system, then the zero dynamics of the closed-loop system turns out to be stable, albeit not asymptotically. A method for asymptotic stabilization of the equilibrium of the closed-loop system is proposed, and an estimate for the attraction basin is constructed. The construction of the estimate is reduced to a problem on the solvability of linear matrix inequalities.