# LMIs in Control/pages/LMI for Generalized eigenvalue problem

**LMI for Generalized Eigenvalue Problem**

Technically, the generalized eigenvalue problem considers two matrices, like and , to find the generalized eigenvector, , and eigenvalues, , that satisfies . If the matrix is an identity matrix with the proper dimension, the generalized eigenvalue problem is reduced to the eigenvalue problem.

**The System**[edit]

Assume that we have three matrice functions which are functions of variables as follows:

where are , , and () are the coefficient matrices.

**The Data**[edit]

The , , and are matrix functions of appropriate dimensions which are all linear in the variable and , , are given matrix coefficients.

**The Optimization Problem**[edit]

The problem is to find such that:

, , and are satisfied and is a scalar variable.

**The LMI:** LMI for Schur stabilization[edit]

A mathematical description of the LMI formulation for the generalized eigenvalue problem can be written as follows:

**Conclusion:**[edit]

The solution for this LMI problem is the values of variables such that the scalar parameter, , is minimized. In practical applications, many problems involving LMIs can be expressed in the aforementioned form. In those cases, the objective is to minimize a scalar parameter that is involved in the constraints of the problem.

**Implementation**[edit]

A link to Matlab codes for this problem in the Github repository:

https://github.com/asalimil/LMI-for-Schur-Stability

**Related LMIs**[edit]

LMI for Generalized Eigenvalue Problem

LMI for Matrix Norm Minimization

LMI for Maximum Singular Value of a Complex Matrix

**External Links**[edit]

- [1] - LMI in Control Systems Analysis, Design and Applications