Adaptive state observer for nonlinear systems with time-varying gain and output restrictions

This paper proposes a novel adaptive state observer for nonlinear systems with uncertain mathematical models and output restrictions. The main contribution is the development of an observer architecture that does not require an accurate system model for its implementation. Instead, only a nominal reference model (A^∗,B^∗) defined around the operating conditions of the system is required, while the observer matrices and gain are adaptively adjusted online to compensate for modeling uncertainties and time-varying dynamics. This feature enables state estimation even when the available model differs from the actual system, overcoming a common limitation of conventional model-based observers. The proposed observer preserves a simple affine structure with a Luenberger-like gain while incorporating adaptation mechanisms for both the observer dynamics and the gain. A Lyapunov-based analysis is developed to establish sufficient conditions for boundedness, asymptotic convergence of the estimation error, and stability of the adaptive laws. The effect of output restrictions on the estimation process is also analyzed. The proposed methodology is validated through two complementary case studies. First, a time-varying linear benchmark system is employed to systematically analyze the adaptation capabilities, transient behavior, and robustness of the observer under controlled parameter variations. Second, the observer is implemented on a robotic manipulator, which constitutes a nonlinear system and provides an experimental assessment of the proposed approach. The results show that the proposed observer accurately tracks the system states despite model uncertainties and achieves faster convergence, smaller transient peaks, and greater robustness to measurement noise than a high-gain observer designed using the same reference model. These results confirm the effectiveness and practical applicability of the proposed adaptive observer for nonlinear systems under output restrictions.