User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/LMI for Generalized Eigenvalue Problem

LMI for Generalized Eigenvalue Problem

Technically, the generalized eigenvalue problem considers two matrices, like $$A$$ and $$B$$, to find the generalized eigenvector, $$x$$, and eigenvalues, $$\lambda $$, that satisfies $$ Ax = \lambda Bx $$. If the matrix $$ B $$ 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 $$ x = [x_{1} \quad x_{2} \quad ... \quad x_{n}]^{\text{T}} \in \mathbb{R}^{n} $$ as follows:

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

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

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

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

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

The Optimization Problem
The problem is to find $$\begin{align} x = [x_{1} \quad x_{2} ... x_{n}] \end{align}$$ such that:

$$A(x) < \lambda B(x) $$, $$ B(x) >0 $$, and $$ C(x) <0 $$ are satisfied and $$\lambda$$ 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:

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

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

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

Related LMIs
LMI for Generalized Eigenvalue Problem

LMI for Matrix Norm Minimization

LMI for Maximum Singular Value of a Complex Matrix

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