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

## Matrix Inequality

Definition-1

A Matrix Inequality, ${\displaystyle G:\mathbb {R} ^{m}\to \mathbb {S} ^{n}}$, in the variable ${\displaystyle x\in \mathbb {R} ^{m}}$ is an expression of the form

{\displaystyle {\begin{aligned}G(x)=G_{0}+\sum _{i=1}^{p}f_{i}(x)G_{i}\leq 0\end{aligned}}},

where ${\displaystyle x^{T}=[x_{1}\cdots x_{m}],G_{0}\in \mathbb {S} ^{n}}$ and ${\displaystyle G_{i}\in \mathbb {R} ^{n\times n}}$, ${\displaystyle i=1,\ldots ,p.}$

## Linear Matrix Inequality

Definition-2

A Linear Matrix Inequality, ${\displaystyle F:\mathbb {R} ^{m}\to \mathbb {S} ^{n}}$, in the variable ${\displaystyle x\in \mathbb {R} ^{m}}$ is an expression of the form

{\displaystyle {\begin{aligned}F(x)=F_{0}+\sum _{i=1}^{m}x_{i}F_{i}\leq 0\end{aligned}}},

where ${\displaystyle x^{T}=[x_{1}\ldots x_{m}]}$ and ${\displaystyle F_{i}\in \mathbb {S} ^{n}}$, ${\displaystyle i=0\ldots ,m.}$

## Bilinear Matrix Inequality

Definition-3

A Bilinear Matrix Inequality (BMI), ${\displaystyle H:\mathbb {R} ^{m}\to \mathbb {S} ^{n}}$, in the variable ${\displaystyle x\in \mathbb {R} ^{m}}$ 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}}}

where ${\displaystyle x^{T}=[x_{1}\cdots x_{m}]}$, and ${\displaystyle H_{i}}$, ${\displaystyle H_{i,j}\in \mathbb {S} ^{n},}$ ${\displaystyle i=0,\ldots ,m}$, ${\displaystyle j=0\ldots ,m.}$

## Example

Consider the matrices ${\displaystyle A\in \mathbb {R} ^{n\times n}}$ and ${\displaystyle Q\in \mathbb {S} ^{n}}$, where ${\displaystyle Q>0}$. It is desired to find a symmetric matrix ${\displaystyle P\in \mathbb {S} ^{n}}$ satisfying the inequality

{\displaystyle {\begin{aligned}PA+A^{T}P+Q<0,\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (1)\end{aligned}}}

where ${\displaystyle P>0}$. The elements of ${\displaystyle P}$ are the design variables in this problem, and although equation ${\displaystyle (1)}$ is indeed an LMI in the matrix ${\displaystyle P}$, it does not look like the LMI in definition 3. For simplicity, let us consider the case of ${\displaystyle n=2}$ so that each matrix is of dimension ${\displaystyle 2\times 2}$, and ${\displaystyle x=[p_{1}\quad p_{2}\quad p_{3}]^{T}.}$ Writing the matrix ${\displaystyle P}$ in terms of a basis ${\displaystyle E_{i}\in \mathbb {S} ^{2},}$ ${\displaystyle i=1,2,3}$, 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}}}

Note that the matrices ${\displaystyle E_{i}}$ are linearly independent and symmetric, thus forming a basis for the symmetric matrix ${\displaystyle P}$. The matrix inequality in equation ${\displaystyle (1)}$ 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}}}

Defining ${\displaystyle F_{0}=Q}$ and ${\displaystyle F_{i}=E_{i}A+A^{T}E_{i},}$ ${\displaystyle i=1,2,3,}$ yields

{\displaystyle {\begin{aligned}F_{0}+\sum {i=1}^{3}p_{i}F_{i}<0,\end{aligned}}}

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 ${\displaystyle (1)}$ rather than the scalar form of definition 2.