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

LMI for Generalized Eigenvalue Problem

Technically, the generalized eigenvalue problem considers two matrices, like ${\displaystyle A}$ and ${\displaystyle B}$, to find the generalized eigenvector, ${\displaystyle x}$, and eigenvalues, ${\displaystyle \lambda }$, that satisfies ${\displaystyle Ax=\lambda Bx}$. If the matrix ${\displaystyle B}$ is an identity matrix with the proper dimension, the generalized eigenvalue problem is reduced to the eigenvalue problem.

## The System

Assume that we have three matrice functions which are functions of variables ${\displaystyle x=[x_{1}\quad x_{2}\quad ...\quad x_{n}]^{\text{T}}\in \mathbb {R} ^{n}}$ as follows:

${\displaystyle A(x)=A_{0}+A_{1}x_{1}+...+A_{n}x_{n}}$

${\displaystyle B(x)=B_{0}+B_{1}x_{1}+...+B_{n}x_{n}}$

${\displaystyle C(x)=C_{0}+C_{1}x_{1}+...+C_{n}x_{n}}$

where are ${\displaystyle A_{i}}$, ${\displaystyle B_{i}}$, and ${\displaystyle C_{i}}$ (${\displaystyle i=1,2,...,n}$) are the coefficient matrices.

## The Data

The ${\displaystyle A(x)}$, ${\displaystyle B(x)}$, and ${\displaystyle C(x)}$ are matrix functions of appropriate dimensions which are all linear in the variable ${\displaystyle x}$ and ${\displaystyle A_{i}}$, ${\displaystyle B_{i}}$, ${\displaystyle C_{i}}$ are given matrix coefficients.

## The Optimization Problem

The problem is to find {\displaystyle {\begin{aligned}x=[x_{1}\quad x_{2}...x_{n}]\end{aligned}}} such that:

${\displaystyle A(x)<\lambda B(x)}$, ${\displaystyle B(x)>0}$, and ${\displaystyle C(x)<0}$ are satisfied and ${\displaystyle \lambda }$ is a scalar variable.

## The LMI: LMI for Schur stabilization

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

{\displaystyle {\begin{aligned}&{\text{min}}\quad \lambda \\&{\text{s.t.}}\quad A(x)<\lambda B(x)\\&\quad \quad B(x)>0\\&\quad \quad C(x)<0\end{aligned}}}

## Conclusion:

The solution for this LMI problem is the values of variables ${\displaystyle x}$ such that the scalar parameter, ${\displaystyle \lambda }$, 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

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