This paper presents a nonlinear observer-based actuator fault diagnosis framework for a class of Lipschitz nonlinear systems in the presence of actuator faults, including both multiplicative and additive types. The proposed diagnosis algorithm leverages a nonlinear observer designed using a modified linear quadratic Riccati equation that ensures exponential convergence of the state estimation error, independent of the Lipschitz constant. A deterministic threshold-based detection mechanism is then developed to detect fault occurrences. The framework further distinguishes between multiplicative and additive faults by leveraging specific control strategies and dynamic analysis. Theoretical results are validated with rigorous stability proofs, ensuring robustness and fault isolation capabilities. This framework provides a systematic solution for fault detection, isolation, and diagnosis in complex nonlinear systems.

In the field of solid-state luminescence, Cu²⁺ has long been widely acknowledged for its capacity to emit infrared light. However, the occurrence of visible emission from Cu²⁺ ions had been infrequently observed and reported. In this study, we made an intriguing discovery by examining the behavior of Cu²⁺ within an irregular coordination environment of Ba in BaGa₂O₄. When excited by UV light, Cu²⁺ unexpectedly gave a vibrant yellow–red emission, covering a wavelength range spanning from 500 to 750 nm. More noteworthy, by simply manipulating the excitation wavelength or adjusting the temperature, the peak wavelength of the emission could be effectively tuned from approximately 600 to 660 nm, which could be attributed to the luminescence nature of the charge transfer (CT) between O²⁻ and Cu²⁺. Moreover, the phosphor material displayed a remarkable persistent luminescence (PerL) lasting up to 12 h after UV light excitation. Through thermoluminescence (TL) measurements and first-principle calculations, we found that the intrinsic defects, such as vacancies of oxygen and gallium (V_O and V_Ga^″), played important roles for the PerL phenomena. These findings highlighted the exceptional tunability and PerL properties of BaGa₂O₄:Cu²⁺. Our study provided a new potential guideline for the design of Cu²⁺-activated phosphors in visible region, and opened up new avenues for the research in related functional luminescence materials.