Adaptive RBFNN finite-time control of normal forms for underactuated mechanical systems

This paper presents a constructive design of a continuous finite-time controller for a class of mechanical systems known as underactuated systems that satisfy the symmetry properties. An adaptive radial basis function neural network (RBFNN) finite-time control scheme is proposed to stabilize the underactuated system at a given equilibrium, regardless of the various uncertainties and disturbances that the system contains. First, a coordinate transformation is introduced to decouple the control input so that an n-th order underactuated system can be represented into a special cascade form. Next, an adaptive robust finite-time controller is derived from adding a power integrator technique and the RBFNN to approximate the nonlinear unknown dynamics in the new space, whose bounds are supposedly unknown. The stability and finite-time convergence of the closed-loop system are established by using Lyapunov theory. To show the effectiveness of the proposed method, simulations are carried out on the rotary inverted pendulum, a typical example of an underactuated mechanical system.