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.
Assume that we have three matrice functions which are functions of variables as follows:
where are , , and () are the coefficient matrices.
The , , and are matrix functions of appropriate dimensions which are all linear in the variable and , , are given matrix coefficients.
The Optimization Problem
The problem is to find such that:
, , and are satisfied and 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:
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.
A link to Matlab codes for this problem in the Github repository:
-  - LMI in Control Systems Analysis, Design and Applications