LMIs in Control/pages/LMIforMinimizingMatrixNorm

The System


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

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

The Data


\begin{align} A_{0}, A_{1}, ..., A_{n} \quad \text{are given matrices.} \end{align}$$

The Optimization Problem
Find

$$\begin{align} x = [x_{1} \quad x_{2} ... x_{n}] \end{align}$$

to minimize,

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

According to Lemma 1.2 in [1] page 11, 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
Mathematical description of the LMI formulation:



\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:
This problem is a slight generalization of the eigenvalue minimization problem for a matrix.

$$\begin{align} x_{i}, \quad i =1, 2, ... , n \quad \text{and} \quad t > 0 \end{align}$$ are parameters to be optimized

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 Eigenvalue Minimization