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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, $\quad M^{T}PM-\gamma ^{2}P<0$

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

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
• LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013.

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