LMIs in Control/Matrix and LMI Properties and Tools/Schur Complement Lemma-Based Properties

LMI Condition
Consider $$ P_{11} \in \mathbb{S}^{n} $$, $$ P_{12} \in \mathbb{R}^{n\times m} $$, $$ P_{22} $$, $$ X \in \mathbb{S}^{m} $$, $$ P_{13} \in \mathbb{R}^{n\times p}, P_{23} \in \mathbb{R}^{m\times p} $$, and $$ P_{33} \in \mathbb{S}^{p} $$.

There exists $$ X $$ such that if and only if Any matrix $$ X \in \mathbb{S}^{m} $$ satisfying

$$ X< $$ $$ -P_{22} $$ + $$\begin{bmatrix} P_{22}^{T} & P_{23} \end{bmatrix}$$$$\begin{bmatrix} P_{11} & P_{13} \\ * & P_{33} \end{bmatrix}^{-1}$$$$\begin{bmatrix} P_{12} \\ P_{23}^T \end{bmatrix}$$ is a solution to ($$)

Consider $$ P_{11} \in \mathbb{S}^{n} $$, $$ P_{12} $$, $$ X \in \mathbb{R}^{n \times m} $$, $$ P_{22} \in \mathbb{S}^{m} $$,$$P_{13} \in \mathbb{R}^{n \times p}$$, $$P_{23} \in \mathbb{R}^{m \times p}$$ and $$ P_{33} \in \mathbb{S}^{p} $$.

There exists $$ X $$ such that

if and only if

If two inequalities in ($$) hold, then a solution to ($$) is given by

$$X = P_{23}P_{33}^{-1}P_{13}^{T}-P_{12}^{T} $$

Consider $$ P_{11} $$, $$ X \in \mathbb{S}^{n} $$, $$ P_{12} \in \mathbb{R}^{n \times m} $$, and $$ P_{22} \in \mathbb{S}^{m} $$, where $$ X>0 $$.

There exists $$ X $$ such that if and only if

Consider $$ X \in \mathbb{S}^{n} $$, $$ H \in \mathbb{R}^{m \times n} $$,$$ G \in \mathbb{R}^{m \times m} $$, and $$ P \in \mathbb{S}^{m} $$, where $$ P>0 $$.implies

$$X>H^{T}G^{-1}PG^{-T}H $$

Consider $$ P_1 \in \mathbb{S}^{n} $$, $$ P_2 $$, $$ X \in \mathbb{S}^{q} $$, $$ Q_1 \in \mathbb{R}^{n \times m} $$, $$ Q_2 \in \mathbb{R}^{q \times p} $$, $$ R_1 \in \mathbb{S}^{m} $$, and $$ R_{2} \in \mathbb{S}^{p} $$

LMI givesif and only if

Consider $$ P \in \mathbb{S}^{n} $$, $$ R \in \mathbb{S}^{m} $$, $$ S \in \mathbb{S}^{p} $$, $$ Q \in \mathbb{R}^{n \times m} $$, $$ X \in \mathbb{R}^{n \times p} $$, $$ V \in \mathbb{R}^{m \times p} $$, and $$ E \in \mathbb{R}^{p \times m} $$.

LMI givesare satisfied if and only if

Consider $$ P_1 $$, $$ Q \in \mathbb{S}^{n} $$, $$ P_2 $$, $$ Q_{2} \in \mathbb{R}^{n \times m} $$, and $$ P_3 $$, $$ Q_{3} \in \mathbb{S}^{m} $$, where $$ P_1>0 $$, $$ P_3>0 $$, $$ Q_1>0 $$, and $$ Q_3>0 $$.

There exists $$ P_2 $$, $$ P_3 $$, $$ Q_2 $$, and $$ Q_3 $$ such thatif and only if

Proof for ($$)
Necessity (($$) $$\Longrightarrow$$($$)) comes from the requirement that the submatrices corresponding to the principle minors of ($$) are negative definite

Sufficiency (($$) $$\Longrightarrow$$($$)) is shown by rewriting the matrix inequalities of ($$) in the equivalent form

$$P_{11}-P_{13}^{T}P_{33}^{-1}P_{13}<0 $$, and $$P_{22}-P_{23}^{T}P_{33}^{-1}P_{23}<0 $$

Concatenating the two matrices and choosing $$X=P_{23}^{T}P_{33}^{-1}P_{23}-P_{12}^{T} $$ gives the equivalent matrix inequality orwhich is equivalent to ($$) using the Schur complement lemma.

Proof for ($$)
the LMI in ($$) can be written using the Schur complement lemma as

Proof for ($$)
Using the Schur complement lemma on ($$) for $$X>H^{T}G^{-1}PG^{-T}H $$

Using the property $$G+G^{T}-P $$$$\leq $$$$G^{T}P^{-1}G $$ or equivalent $$(G+G^{T}-P)^{-1}\geq G^{-1}PG^{-T} $$ gives

$$X>H^{T}(G+G^{T}-P)^{-1}H \leq H^{T}G^{-1}PG^{-T}H $$