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 | edit source]
Assume that we have three matrice functions which are functions of variables as follows:
where are , , and () are the coefficient matrices.
The Data[edit | edit source]
The , , and are matrix functions of appropriate dimensions which are all linear in the variable and , , are given matrix coefficients.
The Optimization Problem[edit | edit source]
The problem is to find such that:
, , and are satisfied and is a scalar variable.
The LMI: LMI for Schur stabilization[edit | edit source]
A mathematical description of the LMI formulation for the generalized eigenvalue problem can be written as follows:
Conclusion:[edit | edit source]
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 | edit source]
A link to Matlab codes for this problem in the Github repository:
Related LMIs[edit | edit source]
External Links[edit | edit source]
-  - LMI in Control Systems Analysis, Design and Applications