A Unified Multiscale Observer Framework for Complex Dynamical Systems: Fractal Geometry, Scaling Laws, and Artificial Intelligence

We develop a unified mathematical and computational framework for observer design in modern dynamical systems, with emphasis on multiscale dynamics, fractal geometry, and scaling laws arising in complex systems. The class of systems under consideration is described by [Formula: see text] and includes nonlinear, time-delay, distributed-parameter, and data-driven models commonly encountered in engineering and applied sciences. The proposed framework integrates high-gain, adaptive, sliding-mode, distributed, and learningbased observers within a single structure [Formula: see text] and is designed to capture estimation processes evolving across multiple temporal and spatial scales. In particular, we show that the estimation error dynamics have intrinsic scaling properties depending on observer gains, nonlinearities, and network interactions. We use Lyapunov theory, input-to-state stability, and incremental stability to derive rigorous conditions for convergence of the estimation errors to ensure exponential and nite-time convergence [Formula: see text] even in the presence of disturbances, parametric uncertainties, and actuator or sensor faults. In particular, we demonstrate that the error estimation dynamics exhibit scaling properties depending on observer gains, nonlinearities, and network interactions. Moreover, we observe from the analysis that convergence rates and robustness properties exhibit scaling laws depending on the multiscale or fractal nature of the underlying system dynamics. To tackle incomplete models and irregularities in the behavior of complex systems, arti cial intelligence tools such as neural networks and Gaussian processes are employed to learn unknown dynamics and capture potential self-similar or fractal structures in the underlying models. Fault detection and isolation are rigorously developed based on residual-based approaches and invariant sets, ensuring reliable performance in a wide range of applications. Numerical simulations on nonlinear and networked systems are presented to validate the theoretical analysis, achieving fast convergence rates, robustness over a wide range of scales, and accurate fault detection. The developed framework is a rigorous and versatile approach to observer-based estimation for a wide range of applications in robotics, aerospace, power networks, biomedical engineering, and multiscale physics.