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

Matrix Inequality

Definition-1

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

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

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

Linear Matrix Inequality

Definition-2

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

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

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

Bilinear Matrix Inequality

Definition-3

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

${\begin{aligned}H(x)=H_{0}+\sum _{i=1}^{m}x_{i}H_{i}+\sum _{i=1}^{m}\sum _{j=1}^{m}x_{i}x_{j}H_{i,j}\leq 0,\end{aligned}}$

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

Example

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

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

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

${\begin{aligned}P={\begin{bmatrix}p_{1}&p_{2}\\p_{2}&p_{3}\end{bmatrix}}=p_{1}\underbrace {\begin{bmatrix}1&0\\0&0\end{bmatrix}} _{E_{1}}+p_{2}\underbrace {\begin{bmatrix}0&1\\1&0\end{bmatrix}} _{E_{2}}+p_{3}\underbrace {\begin{bmatrix}0&0\\0&1\end{bmatrix}} _{E_{3}}\end{aligned}}$

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

${\begin{aligned}p_{1}(E_{1}A+A^{T}E_{1})+p_{2}(E_{2}A+A^{T}E_{2})+p_{3}(E_{3}A+A^{T}E_{3}).\end{aligned}}$

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

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

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 $(1)$ rather than the scalar form of definition 2.

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A downloadable book on LMIs by Ryan James Caverly and James Richard Forbes.