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{align} \ A + P^{T}M^{T}Q + Q^{T}M^{T}P < 0\\ \end{align}$$

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



\begin{align} \ W_P^{T}AW_P < 0\\ \ W_Q^{T}AW_Q < 0\\ \end{align}$$

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 $$M$$.

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