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


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 DataEdit

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

The Optimization ProblemEdit

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.


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


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

External LinksEdit

A list of references documenting and validating the LMI.

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