LMIs in Control/Matrix and LMI Properties and Tools/Variable Reduction Lemma
Introduction
editThe 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
editIn order to find the unknown matrix we need matrices , & .
The Optimization Problem
editGiven 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
editThis can be implemented in any LMI solver such as YALMIP, using an algorithmic solver like Gurobi.
Conclusion
editUsing this technique we can get the value of unknown matrix .
External Links
editA 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