# 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 $M$  we need matrices $A$ , $P$  & $Q$ .

## The Optimization Problem

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

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

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

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

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