LMIs in Control/Matrix and LMI Properties and Tools/Matrix Inequalities and LMIs

Matrix InequalityEdit

Definition-1

A Matrix Inequality,  , in the variable   is an expression of the form

 ,

where   and  ,  

Linear Matrix InequalityEdit

Definition-2

A Linear Matrix Inequality,  , in the variable   is an expression of the form

 ,

where   and  ,  

Bilinear Matrix InequalityEdit

Definition-3

A Bilinear Matrix Inequality (BMI),  , in the variable   is an expression of the form

 

where  , and  ,    ,  

ExampleEdit

Consider the matrices   and  , where  . It is desired to find a symmetric matrix   satisfying the inequality

 

where  . The elements of   are the design variables in this problem, and although equation   is indeed an LMI in the matrix  , it does not look like the LMI in definition 3. For simplicity, let us consider the case of   so that each matrix is of dimension  , and   Writing the matrix   in terms of a basis    , yields


 

Note that the matrices   are linearly independent and symmetric, thus forming a basis for the symmetric matrix  . The matrix inequality in equation   can be written as

 

Defining   and     yields

 

which now resembles the definition of LMI given in definition 2. Through out this wiki book, LMIs are typically written in the matrix form of equation   rather than the scalar form of definition 2.

External LinksEdit