Backstepping design for cascade systems with relaxed assumption on Lyapunov functions

This study proposes a novel backstepping technique for non-linear cascade systems whose driven subsystems have a feedforward structure and include higher order terms. A small control is first assigned to stabilise the driven subsystem, and a simple backstepping procedure is then followed. When dealing with the driven subsystem, we adopt an approach of using a small control to attenuate the higher order term, rather than the usual method in which the higher order term is tackled under the guidance of a single Lyapunov function. The stability analysis is carried out using some boundedness information to explicitly compute the higher order term, and the global asymptotical stability of the whole closed-loop system is obtained using the 'converging-input bounded-state' criterion. This is in sharp contrast with previous designs where (involved) Lyapunov functions are utilised in both the control design and stability analysis. As applications, global stabilisation designs are presented for several classical mechanical systems including the inertia wheel pendulum, the translational oscillator with rotating actuator and the cart-pole system.