# LMIs in Control/Template

This methods uses LMI techniques iteratively to obtain the result.

## The System

Given a state-space representation of a system $G(s)$  and an initial estimate of reduced order model ${\hat {G}}(s)$ .

{\begin{aligned}\ G(s)&=C(sI-A)B+D,\\\ {\hat {G}}(s)&={\hat {C}}(sI-{\hat {A}}){\hat {B}}+{\hat {D}},\\\end{aligned}}

Where $A\in \mathbb {R} ^{n\times n},B\in \mathbb {R} ^{n\times m},C\in \mathbb {R} ^{p\times n},D\in \mathbb {R} ^{p\times m},{\hat {A}}\in \mathbb {R} ^{k\times k},{\hat {B}}\in \mathbb {R} ^{k\times m},{\hat {C}}\in \mathbb {R} ^{p\times k}$  and ${\hat {D}}\in \mathbb {R} ^{p\times m}$ .

## The Data

The full order state matrices $A,B,C,D$ .

## The Optimization Problem

The objective of the optimization is to reduce the $H_{\infty }$  norm .

## The LMI: The Lyapunov Inequality

Objective: $\min \gamma$ .

Subject to:: {\begin{aligned}\ P&={\begin{bmatrix}\ P11&P12\\\ P21&P22\\\end{bmatrix}}\ >0,\end{aligned}}

{\begin{aligned}{\begin{bmatrix}\ A^{T}P11+P11A&A^{T}P12+P12{\hat {A}}&P11B-P12{\hat {B}}&C^{T}\\\ {\hat {A}}^{T}P12^{T}+P12^{T}A&{\hat {A}}^{T}P22+P22{\hat {A}}&P12^{T}B-P22{\hat {B}}&{\hat {C}}^{T}\\\ B^{T}P11-{\hat {B}}^{T}P12^{T}&B^{T}P12-{\hat {B}}^{T}P22&-\gamma {I}&D^{T}-{\hat {D}}^{T}\\\ C&{\hat {C}}&D-{\hat {D}}&-\gamma {I}\\\end{bmatrix}}\ >0\end{aligned}}

## Conclusion:

The LMI techniques results in model reduction close to the theoretical bounds.