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 $$x$$:

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

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

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

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

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

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:

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

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:

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

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

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