# LMIs in Control/Matrix and LMI Properties and Tools/Variable Reduction Lemma

## Introduction

The variable reduction lemma allows the solution of algebraic Riccati inequality that involve a matrix of unknown dimension. This often arises when finding the controller that minimizes the H norm.

## The Data

In order to find the unknown matrix ${\displaystyle M}$  we need matrices ${\displaystyle A}$ , ${\displaystyle P}$  & ${\displaystyle Q}$ .

## The Optimization Problem

Given a symmetric matrix ${\displaystyle A\in \mathbb {R} ^{n\times n}}$  and two matrices ${\displaystyle P}$  & ${\displaystyle Q}$  of column dimension n, consider the problem of finding matrix ${\displaystyle M}$  of compatible dimensions such that

{\displaystyle {\begin{aligned}\ A+P^{T}M^{T}Q+Q^{T}M^{T}P<0\\\end{aligned}}}

The above equation is solvable for some ${\displaystyle M}$  if and only if the following two conditions hold

{\displaystyle {\begin{aligned}\ W_{P}^{T}AW_{P}<0\\\ W_{Q}^{T}AW_{Q}<0\\\end{aligned}}}

Where ${\displaystyle W_{P}}$  and ${\displaystyle W_{Q}}$  are matrices whose columns are bases for the null spaces of ${\displaystyle P}$  & ${\displaystyle Q}$ , respectively.

## Implementation

This can be implemented in any LMI solver such as YALMIP, using an algorithmic solver like Gurobi.

## Conclusion

Using this technique we can get the value of unknown matrix ${\displaystyle M}$ .