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

LMI Condition

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

There exists $X$ such thatTemplate:NumBlk if and only if Template:NumBlkAny matrix $X\in {𝕊}^m$ satisfying

$X<$ $-P_{22}$ + $\left[\begin{array}{cc}{P}_{22}^T& P_{23}\end{array}\right]$${\left[\begin{array}{cc}{P}_{11}& P_{13}\\ *& P_{33}\end{array}\right]}^{-1}$$\left[\begin{array}{c}{P}_{12}\\ P_{23}^T\end{array}\right]$ is a solution to (1)

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

There exists $X$ such thatTemplate:NumBlk

if and only if

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

There exists $X$ such thatTemplate:NumBlk if and only ifTemplate:NumBlk

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

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

Consider $P_1\in {𝕊}^n$, $P_2$, $X\in {𝕊}^q$, $Q_1\in {ℝ}^{n\times m}$, $Q_2\in {ℝ}^{q\times p}$, $R_1\in {𝕊}^m$, and $R_2\in {𝕊}^p$

LMI givesTemplate:NumBlkif and only ifTemplate:NumBlk

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

LMI givesTemplate:NumBlkare satisfied if and only ifTemplate:NumBlk

Consider $P_1$, $Q\in {𝕊}^n$, $P_2$, $Q_2\in {ℝ}^{n\times m}$, and $P_3$, $Q_3\in {𝕊}^m$, 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 thatTemplate:NumBlkif and only ifTemplate:NumBlk

Proof

Proof for (3)

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

Sufficiency ((4) $⟹$(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 Template:NumBlkorTemplate:NumBlkwhich is equivalent to (3) using the Schur complement lemma.

Proof for (6)

the LMI in (5) can be written using the Schur complement lemma asTemplate:NumBlk

Proof for (7)

Using the Schur complement lemma on (7) for $X>H^T{G}^{-1}P{G}^{-T}H$

Using the property $G+G^T-P$$\le$$G^T{P}^{-1}G$ or equivalent $(G+G^T-P{)}^{-1}\ge G^{-1}P{G}^{-T}$ gives

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

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.