A MATLAB LMI Implementation Example for Matrix Stability Analysis

The following presents a MATLAB Linear Matrix Inequality (LMI) related example:

We consider the following problem: Given a matrix $A$, does there exist a symmetric positive definite matrix $P$ such that the following matrix inequality holds?

$$AP + PA^T < 0$$

This problem can be represented in the form of a Linear Matrix Inequality (LMI). We can solve this using MATLAB's Robust Control Toolbox. First, we need to construct an LMI such that $AP + PA^T < 0$. Our LMI formulation is as follows:

$$\begin{bmatrix}A^TP + PA & A^T \ A & -I\end{bmatrix} < 0$$

We can then solve this problem using the CVX toolbox with semidefinite programming (SDP) formulation. The implementation approach involves:

cvx_begin sdp variable P(n,n) symmetric minimize( 0 ) A'*P + P*A + Q'*P*Q < 0 P > 0 cvx_end

Key implementation details: - The 'sdp' specification indicates semidefinite programming mode - The 'variable P(n,n) symmetric' declaration defines a symmetric matrix variable - The inequality constraints represent the LMI conditions for stability - The minimize(0) function indicates we're solving a feasibility problem rather than optimization

This example demonstrates how to use MATLAB's LMI capabilities to solve practical control system stability problems, showcasing the integration of matrix operations with convex optimization techniques.