User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/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{align}

G(x)=G_0+\sum_{i=1}^p f_i(x)G_i\leq0 \end{align}$$,

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{align}

F(x)=F_0+\sum_{i=1}^mx_iF_i\leq0 \end{align}$$,

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{align}

H(x)=H_0+\sum_{i=1}^mx_iH_i+\sum_{i=1}^m\sum_{j=1}^mx_ix_jH_{i,j}\leq0, \end{align}$$

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{align}

PA+A^TP+Q<0,   \qquad \qquad \qquad \qquad \qquad  \qquad \qquad \qquad \qquad(1) \end{align}$$

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{align}

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{align}$$

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{align}

p_1(E_1A+A^TE_1)+p_2(E_2A+A^TE_2)+p_3(E_3A+A^TE_3). \end{align}$$

Defining $$F_0=Q$$ and $$F_i=E_iA+A^TE_i,$$ $$i=1,2,3,$$ yields


 * $$\begin{align}

F_0+\sum{i=1}^3p_iF_i<0, \end{align}$$

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.