LMIs in Control/pages/MatrixNormMinimization
LMI for Matrix Norm Minimization
This problem is a slight generalization of the eigenvalue minimization problem for a matrix. Calculating norm of a matrix is necessary in designing an or an optimal controller for linear time-invariant systems. In those cases, we need to compute the norm of the matrix of the closed-loop system. Moreover, we desire to design the controller so as to minimize the closed-loop matrix norm.
Assume that we have a matrix function of variables :
where are symmetric matrices.
The symmetric matrices () are given.
The Optimization Problem
The optimization problem is to find the variables in order to minimize the following cost function:
where is the cost function and indicates the norm of the matrix function .
According to Lemma 1.1 in LMI in Control Systems Analysis, Design and Applications (page 10), the following statements are equivalent:
The LMI: LMI for matrix norm minimization
This optimization problem can be converted to an LMI problem.
The mathematical description of the LMI formulation can be written as follows:
As a result, the variables after solving this LMI problem and we obtain that is the norm of matrix function .
A link to Matlab codes for this problem in the Github repository:
A list of references documenting and validating the LMI.
-  - LMI in Control Systems Analysis, Design and Applications