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

Introduction

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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

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In order to find the unknown matrix   we need matrices  ,   &  .

The Optimization Problem

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Given a symmetric matrix   and two matrices   &   of column dimension n, consider the problem of finding matrix   of compatible dimensions such that

 

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

 

Where   and   are matrices whose columns are bases for the null spaces of   &  , respectively.

Implementation

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

Conclusion

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Using this technique we can get the value of unknown matrix  .

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A list of references documenting and validating the LMI.


Return to Main Page

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