Adaptive control for nonlinear systems with unknown actuator dynamics based on a novel extended Nussbaum function

In this article, we introduced a novel definition for a class of extended Nussbaum functions that have looser requirements than the traditional Nussbaum function. The extended Nussbaum function achieves comparable performance to the traditional Nussbaum function while addressing the challenges of unknown control directions and time-varying actuator faults. We then provide a specific example function based on the extended Nussbaum function definition. Unlike the traditional Nussbaum function, which oscillates with fixed frequency and constantly increasing amplitude, this example function constrains the amplitude and oscillates with decreasing frequency. Consequently, its amplitude and derivative remain bounded. The oscillation of the system caused by the too large amplitude of the traditional Nussbaum function can be avoided. Furthermore, this example function is constructed using portions of elliptic curve. As a result, it is continuous and differentiable, in contrast to the normal decreasing-frequency Nussbaum function using trigonometric function, which is only continuous. The overall closed-loop system's global stability and asymptotic convergence of system states are successfully achieved. Simulation results demonstrate the effectiveness of the proposed function.