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 SystemEdit

Assume that we have a matrix function of variables  :

 

where   are symmetric matrices.

The DataEdit

The symmetric matrices   ( ) are given.

The Optimization ProblemEdit

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 minimizationEdit

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  .

ImplementationEdit

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

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

Related LMIsEdit

LMI for Matrix Norm Minimization

LMI for Generalized Eigenvalue Problem

LMI for Maximum Singular Value of a Complex Matrix

LMI for Matrix Positivity

External LinksEdit

A list of references documenting and validating the LMI.

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


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