Definition-1
A Matrix Inequality,
, in the variable
is an expression of the form
,
where
and
,
Definition-2
A Linear Matrix Inequality,
, in the variable
is an expression of the form
,
where
and
,
Definition-3
A Bilinear Matrix Inequality (BMI),
, in the variable
is an expression of the form
![{\displaystyle {\begin{aligned}H(x)=H_{0}+\sum _{i=1}^{m}x_{i}H_{i}+\sum _{i=1}^{m}\sum _{j=1}^{m}x_{i}x_{j}H_{i,j}\leq 0,\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56671ad2f43d49ecf74e3b94e13534b37f7b9239)
where
, and
,
,
Consider the matrices
and
, where
. It is desired to find a symmetric matrix
satisfying the inequality
![{\displaystyle {\begin{aligned}PA+A^{T}P+Q<0,\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (1)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93da2529ef8a632e8bd451317d8391b6dd23c3eb)
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
![{\displaystyle {\begin{aligned}P={\begin{bmatrix}p_{1}&p_{2}\\p_{2}&p_{3}\end{bmatrix}}=p_{1}\underbrace {\begin{bmatrix}1&0\\0&0\end{bmatrix}} _{E_{1}}+p_{2}\underbrace {\begin{bmatrix}0&1\\1&0\end{bmatrix}} _{E_{2}}+p_{3}\underbrace {\begin{bmatrix}0&0\\0&1\end{bmatrix}} _{E_{3}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f296c9d59ca4f862d250b55b918b5644b1cd7758)
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
![{\displaystyle {\begin{aligned}p_{1}(E_{1}A+A^{T}E_{1})+p_{2}(E_{2}A+A^{T}E_{2})+p_{3}(E_{3}A+A^{T}E_{3}).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8171db32b5c8943dbd85240400002a93f98ea220)
Defining
and
yields
![{\displaystyle {\begin{aligned}F_{0}+\sum _{i=1}^{3}p_{i}F_{i}<0,\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48b4b25b947a9e23ecb2e09e50f76a83183b684c)
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.