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

Matrix Inequality

Definition-1

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

$\begin{array}{c}G(x)=G_0+\sum _{i=1}^p{f}_i(x)G_i\le 0\end{array}$,

where $x^T=[x_1\cdots x_m],G_0\in {𝕊}^n$ and $G_i\in {ℝ}^{n\times n}$, $i=1,\dots ,p.$

Linear Matrix Inequality

Definition-2

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

$\begin{array}{c}F(x)=F_0+\sum _{i=1}^m{x}_i{F}_i\le 0\end{array}$,

where $x^T=[x_1\dots x_m]$ and $F_i\in {𝕊}^n$, $i=0\dots ,m.$

Bilinear Matrix Inequality

Definition-3

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

$\begin{array}{c}H(x)=H_0+\sum _{i=1}^m{x}_i{H}_i+\sum _{i=1}^m\sum _{j=1}^m{x}_i{x}_j{H}_{i,j}\le 0,\end{array}$

where $x^T=[x_1\cdots x_m]$, and $H_i$, $H_{i,j}\in {𝕊}^n,$ $i=0,\dots ,m$, $j=0\dots ,m.$

Example

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

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

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 $x=[p_1{p}_2{p}_3{]}^T.$ Writing the matrix $P$ in terms of a basis $E_i\in {𝕊}^2,$ $i=1,2,3$, yields

$\begin{array}{c}P=\left[\begin{array}{cc}{p}_1& p_2\\ p_2& p_3\end{array}\right]=p_1\underset{E_1}{\underbrace{\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]}}+p_2\underset{E_2}{\underbrace{\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]}}+p_3\underset{E_3}{\underbrace{\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]}}\end{array}$

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

$\begin{array}{c}{p}_1(E_{1}A+A^T{E}_1)+p_2(E_{2}A+A^T{E}_2)+p_3(E_{3}A+A^T{E}_3).\end{array}$

Defining $F_0=Q$ and $F_i=E_{i}A+A^T{E}_i,$ $i=1,2,3,$ yields

$\begin{array}{c}{F}_0+\sum {i=1}^3{p}_i{F}_i<0,\end{array}$

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.