Robust generation of almost‐periodic oscillations in a class of nonlinear systems

This paper deals with the problem of generating, by state feedback, stable oscillations in high order nonlinear systems. The desired oscillations are robust in the presence of disturbances, such as unmodelled dynamics and bounded noise signals, which result in bounded deviations from the nominal target orbit. The method consists of two steps. First, a globally attractive oscillation is generated in a nominal second‐order subsystem. Based on a partition of the state space and solving the Lyapunov equation on each part, a strict Lyapunov function is obtained that ensures exponential convergence, even in the presence of disturbances, to a ring‐shaped region containing the target limit cycle. Then, the oscillation stabilizing controller and the strict Lyapunov function are extended to arbitrary order systems, via backstepping. Notwithstanding backstepping is intended for cascade systems, the acquired ability to deal with unmodelled dynamics permits the analytical treatment of non‐triangular structures, as is illustrated with the Ball and Beam example.