Adaptive observer design and control for nonlinear systems with Hölderian dynamics

This paper introduces a Generalized Adaptive Nonlinear Observer (GANO) for state estimation in nonlinear systems exhibiting Hölder-like and non-differentiable dynamics. Traditional observer designs often fail under such conditions due to their reliance on strict Lipschitz continuity. By employing adaptive gains and generalized nonlinear corrections, the proposed method achieves robust performance in the presence of both stochastic disturbances and structural uncertainties. Theoretical guarantees are provided for both deterministic and stochastic scenarios, ensuring convergence of the estimation error. The observer is further integrated into adaptive control schemes, demonstrating effective stabilization of uncertain nonlinear systems. Numerical experiments involving nonlinear oscillators, electromechanical systems, and chaotic models with random perturbations confirm the method’s accuracy, robustness, and broad applicability. This contribution advances the state of the art in observer design by addressing the estimation problem for systems with weak regularity and unpredictable behaviors.