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 $H_{2}$ or an $H_{\infty }$ 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

Assume that we have a matrix function of variables $x$:

${\begin{aligned}A(x)=A_{0}+A_{1}x_{1}+...+A_{n}x_{n}\end{aligned}}$

where ${\begin{aligned}A_{i},\quad i=1,2,...,n\end{aligned}}$ are symmetric matrices.

The Data

The symmetric matrices $A_{i}$ (${\begin{aligned}A_{0},A_{1},...,A_{n}\end{aligned}}$) are given.

The Optimization Problem

The optimization problem is to find the variables ${\begin{aligned}x=[x_{1}\quad x_{2}...x_{n}]\end{aligned}}$ in order to minimize the following cost function:

${\begin{aligned}J(x)=||A(x)||_{2}\end{aligned}}$

where $J(x)$ is the cost function and $||.||_{2}$ indicates the norm of the matrix function $A$.

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

${\begin{aligned}A^{T}A-t^{2}I\leq 0\iff {\begin{bmatrix}-tI&A\\A^{T}&-tI\end{bmatrix}}\leq 0\\\end{aligned}}$

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:

${\begin{aligned}&{\text{min}}\quad t&\\&{\text{s.t.}}\quad {\begin{bmatrix}-tI&A(x)\\A(x)^{T}&-tI\end{bmatrix}}\leq 0\\\end{aligned}}$

Conclusion:

As a result, the variables $x_{i},\quad i=1,2,...,n$ after solving this LMI problem and we obtain $t$ that is the norm of matrix function $A(x)$.

Implementation

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

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

Related LMIs

LMI for Matrix Norm Minimization

LMI for Generalized Eigenvalue Problem

LMI for Maximum Singular Value of a Complex Matrix

LMI for Matrix Positivity

External Links

A list of references documenting and validating the LMI.

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

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