A two‐time‐scale system involves both fast and slow dynamics. This article studies observer design for general nonlinear two‐time‐scale systems and presents two alternative nonlinear observer design approaches, one full‐order and one reduced‐order. The full‐order observer is designed by following a scheme to systematically select design parameters, so that the fast and slow observer dynamics are assigned to estimate the corresponding system modes. The reduced‐order observer is derived based on a lower dimensional model to reconstruct the slow states, along with the algebraic slow‐motion invariant manifold function to reconstruct the fast states. Through an error analysis, it is shown that the reduced‐order observer is capable of providing accurate estimation of the states for the detailed system with an exponentially decaying estimation error. In the last part of the article, the two proposed observers are designed for an anaerobic digestion process, as an illustrative example to evaluate their performance and convergence properties.

中文翻译：

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两尺度系统的非线性观测器设计

两级系统涉及快速和慢速动力学。本文研究了一般非线性二次系统的观测器设计，并提出了两种可供选择的非线性观测器设计方法：一种全阶和一种降阶。通过遵循一种方案来系统地选择设计参数来设计全序观测器，以便分配快速和慢速观测器动态来估计相应的系统模式。降阶观测器是基于低维模型来重建慢状态的，而代数慢动作不变流形函数则是用于重建快速状态的。通过误差分析表明，降阶观测器能够为详细系统提供精确的状态估计，且估计误差呈指数衰减。