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

${\begin{aligned}\ {\begin{bmatrix}P_{11}&P_{12}&P_{13}\\*&P_{22}+X&P_{23}\\*&*&P_{33}\end{bmatrix}}<0\\\end{aligned}}$

(1)

if and only if

${\begin{aligned}\ {\begin{bmatrix}P_{11}&P_{13}\\*&P_{33}\end{bmatrix}}<0\\\end{aligned}}$

(2)

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 (1)

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

${\begin{aligned}\ {\begin{bmatrix}P_{11}&P_{12}+X^{T}&P_{13}\\*&P_{22}&P_{23}\\*&*&P_{33}\end{bmatrix}}<0\\\end{aligned}}$

(3)

if and only if

${\begin{aligned}\ {\begin{bmatrix}P_{11}&P_{13}\\*&P_{33}\end{bmatrix}}<0\\\end{aligned}}$, and ${\begin{aligned}\ {\begin{bmatrix}P_{22}&P_{23}\\*&P_{33}\end{bmatrix}}<0\\\end{aligned}}$

(4)

If two inequalities in (4) hold, then a solution to (3) 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

${\begin{aligned}\ {\begin{bmatrix}P_{11}-X&P_{12}&X\\*&P_{22}&0\\*&*&-X\end{bmatrix}}<0\\\end{aligned}}$

(5)

if and only if

${\begin{aligned}\ {\begin{bmatrix}P_{11}&P_{12}\\*&P_{22}\end{bmatrix}}<0\\\end{aligned}}$

(6)

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 $P>0$.

${\begin{aligned}\ {\begin{bmatrix}X&H^{T}\\*&G+G^{T}-P\end{bmatrix}}>0\\\end{aligned}}$

(7)

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 gives

${\begin{aligned}\ {\begin{bmatrix}P_{1}-LXL^{T}&Q_{1}\\*&R_{1}\end{bmatrix}}>0\\\end{aligned}}$, and ${\begin{aligned}\ {\begin{bmatrix}P_{2}+X&Q_{2}\\*&R_{2}\end{bmatrix}}>0\\\end{aligned}}$

(8)

if and only if

${\begin{aligned}\ {\begin{bmatrix}P_{1}+LP_{2}L^{T}&Q_{1}&LQ_{2}\\*&R_{1}&0\\*&*&R_{2}\end{bmatrix}}>0\\\end{aligned}}$

(9)

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 gives

${\begin{aligned}\ {\begin{bmatrix}P&Q\\*&R-VE-E^{T}V^{T}+E^{T}SE\end{bmatrix}}>0\\\end{aligned}}$, and ${\begin{aligned}\ {\begin{bmatrix}R&V\\*&S\end{bmatrix}}>0\\\end{aligned}}$

(10)

are satisfied if and only if

${\begin{aligned}\ {\begin{bmatrix}P&Q+XE&X\\*&R&V\\*&*&S\end{bmatrix}}>0\\\end{aligned}}$

(11)

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 that

${\begin{aligned}\ {\begin{bmatrix}P_{1}&P_{2}\\*&P_{3}\end{bmatrix}}>0\\\end{aligned}}$, and ${\begin{aligned}\ {\begin{bmatrix}P_{1}&P_{2}\\*&P_{3}\end{bmatrix}}^{-1}\\\end{aligned}}$=${\begin{aligned}\ {\begin{bmatrix}Q_{2}&Q_{2}\\*&Q_{3}\end{bmatrix}}\\\end{aligned}}$

(12)

if and only if

${\begin{aligned}\ {\begin{bmatrix}P_{1}&1\\*&Q_{1}\end{bmatrix}}>0\\\end{aligned}}$, and $rank($${\begin{aligned}\ {\begin{bmatrix}P_{1}&1\\*&Q_{1}\end{bmatrix}}\\\end{aligned}}$$)$$<=n+m$

(13)

Proof

Proof for (3)

Necessity ((3) $\Longrightarrow$(4)) comes from the requirement that the submatrices corresponding to the principle minors of (3) are negative definite

Sufficiency ((4) $\Longrightarrow$(3)) is shown by rewriting the matrix inequalities of (4) 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

${\begin{aligned}\ {\begin{bmatrix}P_{11}-P_{13}^{T}P_{33}^{-1}P_{13}&P_{12}-P_{13}^{T}P_{33}^{-1}P_{23}+X^{T}\\*&P_{22}-P_{23}^{T}P_{33}^{-1}P_{23}\end{bmatrix}}>0\\\end{aligned}}$

(3-1)

or

${\begin{aligned}\ {\begin{bmatrix}P_{11}&P_{12}+X^{T}\\*&P_{22}\end{bmatrix}}\\\end{aligned}}-$${\begin{aligned}\ {\begin{bmatrix}P_{13}^{T}\\P_{23}^{T}\end{bmatrix}}\\\end{aligned}}$$P_{33}^{-1}$${\begin{aligned}\ {\begin{bmatrix}P_{13}&P_{23}\end{bmatrix}}<0\\\end{aligned}}$

(3-2)

which is equivalent to (3) using the Schur complement lemma.

Proof for (6)

the LMI in (5) can be written using the Schur complement lemma as

${\begin{aligned}\ {\begin{bmatrix}P_{11}-X&P_{12}\\*&P_{22}\end{bmatrix}}\\\end{aligned}}-$${\begin{aligned}\ {\begin{bmatrix}X\\0\end{bmatrix}}\\\end{aligned}}$$X^{-1}$${\begin{aligned}\ {\begin{bmatrix}X&0\end{bmatrix}}<0\\\end{aligned}}$

( 5-1)

${\begin{aligned}\ {\begin{bmatrix}P_{11}-X&P_{12}\\*&P_{22}\end{bmatrix}}\\\end{aligned}}-$${\begin{aligned}\ {\begin{bmatrix}X&0\\*&0\end{bmatrix}}\\\end{aligned}}<0$

(5-2)

${\begin{aligned}\ {\begin{bmatrix}P_{11}&P_{12}\\*&P_{22}\end{bmatrix}}\\\end{aligned}}<0$

(5-3)

Proof for (7)

Using the Schur complement lemma on (7) 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

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

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.