Modification via averaging of partial-energy-shaping control for creating oscillations: cart-pendulum example

We consider the challenging problem of creating oscillations in underactuated mechanical systems. Target oscillatory motions of the indirectly actuated degree of freedom of a mechanical system can often be achieved via a straightforward to design feedback transformation. Moreover, the corresponding part of the dynamics can be forced to match a desired second-order system possessing the target periodic solution (Aracil, J., Gordillo, F., and Acosta, J.A. (2002), ‘Stabilization of Oscillations in the Inverted Pendulum’, in Proceedings of the 15th IFAC World Congress, Barcelona, Spain; Canudas-de-Wit, C., Espiau, B., and Urrea, C. (2002), ‘Orbital Stabilisation of Underactuated Mechanical Systems’, in Proceedings of the 15th IFAC World Congress, Barcelona, Spain). Sometimes, it is possible to establish the presence of periodic or bounded motions for the remaining degrees of freedom in the transformed system. However, typically this motion planning procedure leads to an open-loop unstable orbit and by necessity should be followed by a feedback control design. We propose a new approach for synthesis of a (practically) stabilising feedback controller, which ensures convergence of the solutions of the closed-loop system into a narrow tube around the preplanned orbit. The method is illustrated in detail by shaping oscillations in the inverted pendulum on a cart around its upright equilibrium. The complete analysis is based on application of a non-standard higher-order averaging technique assuming sufficiently high frequency of oscillations and is presented for this particular example.