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

**Introduction**Edit

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**Edit

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

**The Optimization Problem**Edit

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**Edit

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

**Conclusion**Edit

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

**External Links**Edit

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