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 SystemEdit

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

 

 

 

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

The DataEdit

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

The Optimization ProblemEdit

The problem is to find   such that:

 ,  , and   are satisfied and   is a scalar variable.

The LMI: LMI for Schur stabilizationEdit

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.

ImplementationEdit

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

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

Related LMIsEdit

LMI for Generalized Eigenvalue Problem

LMI for Matrix Norm Minimization

LMI for Maximum Singular Value of a Complex Matrix

External LinksEdit

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


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