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 $M$.

External Links

A list of references documenting and validating the LMI.

• https://web.mit.edu/braatzgroup/33 A tutorial on linear and bilinear matrix inequalities.pdf - A journal paper on the said LMI
• https://onlinelibrary.wiley.com/doi/abs/10.1002/rnc.4590040403 - Research paper on the said LMI and its proof

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