Peaking Attenuation in High-Gain Observers Using Adaptive Saturation: Application to a Ball and Wheel System

Despite providing robustness, high-gain observers impose a peaking phenomenon, which may cause instability, on the system states. In this paper, an adaptive saturation is proposed to attenuate the undesirable mentioned phenomenon in high-gain observers. A real-valued and differentiable sigmoid function is considered as the saturating element whose parameters (height and slope) are adaptively tuned. The corresponding feedback and adaptation laws are derived based on the Lyapunov and LaSalle theorems to guarantee the asymptotic stability property for the closed-loop system’s equilibrium point. Compared to the conventional high-gain observers which suffer from states’ peaking, it is possible to increase the observer’s gain, up to a higher level, under which not only all system states and the adaptive saturation elements remain stable, but also robustness is reinforced in the presence of uncertainties and/or non-similarities in the system and observer’s dynamics, respectively. Both theoretical analysis and simulation results confirm the efficiency of the proposed scheme.