Backstepping designs based on small differentiable controls

This paper suggests a backstepping technique which is based on small differentiable controls. The design procedure is divided into two stages. At the first stage, small differentiable controls are assigned to stabilize an upper subsystem that includes some higher order terms. The second stage is a simple backstepping procedure. The design and analysis at the first stage are crucial: use boundedness information in finite time to calculate the higher order terms, and the associated estimate again guides the tuning of small control amplitude. Through elaborately assigning small controls, the higher order terms are guaranteed to have a minor effect on the stability, and the convergence can be verified in a simple manner. As for the global asymptotical stability of the whole closed-loop system, it is followed from the “converging-input bounded-state” theorem. In this way, the proposed algorithm does not heavily depend on Lyapunov functions. The algorithm is then applied to several classical nonlinear systems including the well-known inertia wheel pendulum and the translational oscillator with rotating actuator.