# 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.

**The System**[edit]

Assume that we have a matrix function of variables :

where are symmetric matrices.

**The Data**[edit]

The symmetric matrices () are given.

**The Optimization Problem**[edit]

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[edit]

This optimization problem can be converted to an LMI problem.

The mathematical description of the LMI formulation can be written as follows:

**Conclusion:**[edit]

As a result, the variables after solving this LMI problem and we obtain that is the norm of matrix function .

**Implementation**[edit]

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

https://github.com/asalimil/LMI-for-Matrix-Norm-Minimization

**Related LMIs**[edit]

LMI for Matrix Norm Minimization

LMI for Generalized Eigenvalue Problem

LMI for Maximum Singular Value of a Complex Matrix

**External Links**[edit]

A list of references documenting and validating the LMI.

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