A MATLAB LMI (Linear Matrix Inequality) Implementation Example for Control Systems

The LMI (Linear Matrix Inequality) Toolbox in MATLAB serves as an indispensable tool for control system design and analysis. It effectively handles various optimization problems involving linear matrix inequalities, with particularly extensive applications in robust control, stability analysis, and performance optimization.

A typical implementation example involves designing a robust controller that maintains system stability under parameter uncertainties. Consider a linear system with a state-space model containing uncertainties. The objective is to find a controller that ensures closed-loop stability against all possible uncertainty variations.

Using MATLAB's LMI Toolbox, engineers first define system matrices and uncertainty parameter bounds. Subsequently, appropriate LMI conditions (such as Lyapunov inequalities) are constructed to express stability requirements. Through LMI solvers like feasp (feasibility problem solver) or mincx (linear objective minimization), the toolbox computes controller gain matrices satisfying these conditions. Key implementation steps include:

- Defining matrix variables using lmivar function - Setting LMI constraints with lmiterm commands - Solving feasibility problems with feasp or optimization with mincx - Extracting solutions using dec2mat function

For instance, LMIs can verify quadratic stability or solve H-infinity control problems to guarantee system performance under disturbances. The complete workflow encompasses matrix variable declaration, LMI constraint establishment, and optimization problem resolution, ultimately yielding qualified controller parameters.

The methodology's strength lies in its rigorous mathematical formulation for complex uncertainty problems, combined with MATLAB's efficient solving algorithms that rapidly deliver feasible solutions. For control system engineers, the LMI Toolbox represents a powerful instrument for achieving high-performance robust control designs through systematic matrix inequality computations.