Observer Design for Nonlinear Systems: LMI Approach with Algorithm Implementation

This paper investigates the design of observers for nonlinear systems and proposes a solution based on Linear Matrix Inequality (LMI) methods. The approach decomposes nonlinear systems into finite linear subsystems and designs an observer to monitor these subsystems, thereby achieving full-system observation. Compared to traditional methods, the LMI-based design approach enables more efficient resolution of nonlinear system observer design challenges. The implementation involves key algorithmic steps: system linearization through sector nonlinearity approximation, formulation of observer gain conditions as LMIs, and solving convex optimization problems using MATLAB's LMI Control Toolbox. The core function lmivar() defines matrix variables while feasp() solves the feasibility problem to determine observer gains. We provide detailed algorithmic workflows and corresponding MATLAB code implementations. Experimental validation demonstrates the method's effectiveness and practical applicability through numerical simulations involving Lyapunov stability analysis and error dynamics evaluation. The code includes modular functions for system decomposition, LMI construction, and observer performance verification.