LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Minimizing Norm by Scaling

Minimizing Norm by Scaling
There are many cases in which a norm should be minimized, such as in applications of the $$ H_2 $$ or $$ H_{\infty} $$ norm optimal control.

The System
$$ M $$ is a matrix $$M \in \mathbb{C}^{nxn}$$. $$ D $$ is some diagonal, nonsingular $$ D $$.

The Data
The optimal diagonally scaled norm of a matrix $$M \in \mathbb{C}^{nxn}$$ is defined as $$ \nu(M) \overset{\underset{\mathrm{def}}{}}{=} inf \left \lbrace \lVert DMD^{-1} \rVert \quad | D \in \mathbb{C}^{n \times n} \right \rbrace $$,  where $$ D $$ is diagonal and nonsingular.

The LMI:Minimizing Norm by Scaling
Therefore, $$ \nu(M) $$ is the optimal value of the generalized eigenvalue problem

minimize $$ \quad \gamma $$

subject to $$ \quad P>0 $$ and diagonal, $$ \quad M^TPM-\gamma^2P<0 $$

Conclusion:
This result can be extended in many ways, such as in applications of $$ H_2 $$ or $$ H_{\infty} $$ optimal control.

Implementation
This implementation requires Yalmip and Sedumi.

Minimizing Norm by Scaling

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