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 $H_{\mathrm{\infty }}$ 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 $x$:

$\begin{array}{c}A(x)=A_0+A_1{x}_1+...+A_n{x}_n\end{array}$

where $\begin{array}{c}{A}_i,i=1,2,...,n\end{array}$ are symmetric matrices.

The Data

The symmetric matrices $A_i$ ($\begin{array}{c}{A}_0,A_1,...,A_n\end{array}$) are given.

The Optimization Problem

The optimization problem is to find the variables $\begin{array}{c}x=[x_1{x}_2...x_n]\end{array}$ in order to minimize the following cost function:

$\begin{array}{c}J(x)=||A(x)|{|}_2\end{array}$

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:

$\begin{array}{c}{A}^{T}A-t^{2}I\le 0⟺\left[\begin{array}{cc}-tI& A\\ A^T& -tI\end{array}\right]\le 0\\ \end{array}$

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:

$\begin{array}{ccc}& \text{min}t& \\ & \text{s.t.}\left[\begin{array}{cc}-tI& A(x)\\ A(x{)}^T& -tI\end{array}\right]\le 0\\ \end{array}$

Conclusion:

As a result, the variables $x_i,i=1,2,...,n$ 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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