LMIs in Control/pages/MatrixEigenValueMinimization

LMI for Minimizing Eigenvalue of a Matrix

Synthesizing the eigenvalues of a matrix plays an important role in designing controllers for linear systems. The eigenvalues of the state matrix of a linear time-invariant system determine if the system is stable or not. The system is stable if all the eigenvalues of the state matrix are located in the left half of the complex plane. Thus, we may desire to minimize the maximal eigenvalue of the state matrix such that the minimized eigenvalue is placed in the left half-plane, which guarantees that the system is stable.

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   to minimize the following cost function:

 

where   is the cost function and   indicates the maximim eigenvalue of a matrix.

According to Lemma 1.1 in LMI in Control Systems Analysis, Design and Applications (page 10), the following statements are equivalent

 

where   is defined as the maximim eigenvalue of the matrix  .

The LMI: LMI for eigenvalue 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.

Moreover, we obtain the maximum eigenvalue,  , of the matrix  .

ImplementationEdit

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

https://github.com/asalimil/LMI-for-Minimizing-the-Maximum-Eigenvalue-of-Matrix

Related LMIsEdit

LMI for Generalized Eigenvalue Problem

LMI for Matrix Norm Minimization

LMI for Maximum Singular Value of a Complex Matrix

LMI for Matrix Positivity

External LinksEdit

  • [1] - LMI in Control Systems Analysis, Design and Applications
  • Eigenvalues and Eigenvectors of a Matrix


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