MATLAB Linear Matrix Inequalities (LMI) Detailed Technical Handbook

Linear Matrix Inequalities (LMI) play a crucial role in control theory and optimization problems, with MATLAB providing powerful toolbox capabilities for solving such problems. This handbook details MATLAB's LMI-related functions and typical application scenarios with implementation insights. The LMI Toolbox serves as the core component in MATLAB's control system for handling convex optimization problems, allowing users to describe matrix inequality constraints in a natural manner. Built upon semidefinite programming theory, this toolbox efficiently solves various control system design problems through functions like `lmivar` for variable declaration and `lmiterm` for constraint specification. In typical applications, LMI is commonly used to address robust control, stability analysis, and performance optimization challenges. By defining appropriate decision variables and constraint conditions using MATLAB's object-oriented approach, engineers can transform complex control problems into solvable LMI formulations. The toolbox provides a complete workflow from problem description to solution, significantly simplifying traditional control design complexity through functions such as `feasp` for feasibility problems and `mincx` for optimization. In practical implementation, users first establish the LMI framework using `setlmis` to initialize the system, then define various constraints through structured term definitions. MATLAB employs an object-oriented methodology where each inequality is treated as an independent object, enhancing code readability and maintainability through systematic matrix variable management. The solving process is highly automated with functions like `getlmis` retrieving the complete system description, allowing users to focus on problem formulation rather than algorithmic details. For result analysis, MATLAB offers comprehensive visualization tools and performance metrics through commands like `dec2mat` for retrieving solution variables and `showlmi` for constraint display, helping engineers evaluate solution quality. Notably, while LMI methods theoretically guarantee global optimal solutions, computational efficiency must be considered when handling large-scale problems through proper problem structuring and solver selection.