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

Dilation

Matrix inequalities can be dilated in order to obtain a larger matrix inequality. This can be a useful technique to separate design variables in a BMI (bi-linear matrix inequality), as the dilation often introduces additional design variables.

A common technique of LMI dilation involves using the projection lemma in reverse, or the "reciprocal projection lemma." For instance, consider the matrix inequality

$\left[\begin{array}{cc}𝐏𝐀+{𝐀}^{𝐓}𝐏-𝐏& 𝐏\\ *& \mathbf{-}𝐏\end{array}\right]<0,$

where $𝐏\in {\mathrm{\S }}^{n\times m}$, $𝐀\in {ℝ}^{n\times n}$, with $𝐏>0.$This can be rewritten as

$\left[\begin{array}{ccc}{𝐀}^{𝐓}& \mathrm{𝟏}& \mathrm{𝟎}\\ \mathrm{𝟏}& \mathrm{𝟎}& \mathrm{𝟏}\end{array}\right]\left[\begin{array}{ccc}\mathrm{𝟎}& 𝐏& \mathrm{𝟎}\\ *& \mathbf{-}𝐏& \mathrm{𝟎}\\ *& *& \mathbf{-}𝐏\end{array}\right]\left[\begin{array}{cc}𝐀& \mathrm{𝟏}\\ \mathrm{𝟏}& \mathrm{𝟎}\\ \mathrm{𝟎}& \mathrm{𝟏}\end{array}\right]<0.$ (1)

Then since $𝐏>0,$

$\left[\begin{array}{cc}\mathbf{-}𝐏& \mathrm{𝟎}\\ *& \mathbf{-}𝐏\end{array}\right]<0,$

which is equivalent to

$\left[\begin{array}{ccc}\mathrm{𝟎}& \mathrm{𝟏}& \mathrm{𝟎}\\ \mathrm{𝟎}& \mathrm{𝟎}& \mathrm{𝟏}\end{array}\right]\left[\begin{array}{ccc}\mathrm{𝟎}& 𝐏& \mathrm{𝟎}\\ *& \mathbf{-}𝐏& \mathrm{𝟎}\\ *& *& \mathbf{-}𝐏\end{array}\right]\left[\begin{array}{cc}\mathrm{𝟎}& \mathrm{𝟎}\\ \mathrm{𝟏}& \mathrm{𝟎}\\ \mathrm{𝟎}& \mathrm{𝟏}\end{array}\right]<0.$ (2)

These expanded inequalities (1) and (2) are now in the form of the strict projection lemma, meaning they are equivalent to

$\mathrm{\Phi }(𝐏)+𝐆(𝐀)𝐕{𝐇}^{𝐓}+𝐇{𝐕}^{𝐓}{𝐆}^{𝐓}(𝐀),$ (3)

where $N({𝐆}^{𝐓}(𝐀))=R({𝐍}_G(𝐀)),N({𝐇}^{𝐓})=R({𝐍}_H),$and $V\in {ℝ}^{n\times n}.$ By choosing

$𝐆(𝐀)=\left[\begin{array}{c}\mathbf{-}\mathrm{𝟏}\\ {𝐀}^{𝐓}\\ \mathrm{𝟏}\end{array}\right],𝐇=\left[\begin{array}{c}\mathrm{𝟏}\\ \mathrm{𝟎}\\ \mathrm{𝟎}\end{array}\right],$

we can now rewrite the inequality (3) as

$\left[\begin{array}{ccc}-(𝐕+{𝐕}^{𝐓})& {𝐕}^{𝐓}𝐀+𝐏& {𝐕}^{𝐓}\\ *& \mathbf{-}𝐏& \mathrm{𝟎}\\ *& *& \mathbf{-}𝐏\end{array}\right]<0,$

which is the new dilated inequality.

Examples

Some useful examples of dilated matrix inequalities are presented here.

Example 1

Consider matrices $𝐀,𝐆\in {ℝ}^{n\times n},\mathrm{\Delta }\in {ℝ}^{m\times m},𝐏\in {\mathrm{\S }}^n,{\delta }_1,{\delta }_2,a,b\in {ℝ}_{>0},$where $𝐏>0$and $b=a^{-1}.$ The following matrix inequalities are equivalent:

$𝐀𝐏+𝐏{𝐀}^{𝐓}+{\delta }_1𝐏+{\delta }_2𝐀𝐏{𝐀}^{𝐓}+𝐏{\mathrm{\Delta }}^{𝐓}\mathrm{\Delta }𝐏<0;$

$\left[\begin{array}{ccccc}\mathrm{𝟎}& \mathbf{-}𝐏& 𝐏& \mathrm{𝟎}& 𝐏{\mathrm{\Delta }}^{𝐓}\\ *& \mathrm{𝟎}& \mathrm{𝟎}& \mathbf{-}𝐏& \mathrm{𝟎}\\ *& *& -{\delta }_1^{-1}𝐏& \mathrm{𝟎}& \mathrm{𝟎}\\ *& *& *& -{\delta }_2^{-1}𝐏& \mathrm{𝟎}\\ *& *& *& *& \mathbf{-}\mathrm{𝟏}\end{array}\right]+He(\left[\begin{array}{c}𝐀\\ \mathrm{𝟏}\\ \mathrm{𝟎}\\ \mathrm{𝟎}\\ \mathrm{𝟎}\end{array}\right]𝐆\left[\begin{array}{ccccc}\mathrm{𝟏}& -b\mathrm{𝟏}& b\mathrm{𝟏}& \mathrm{𝟏}& b{\mathrm{\Delta }}^T\end{array}\right])<0.$

Example 2

Consider matrices $𝐀,𝐕\in {ℝ}^{n\times n},𝐏,𝐗\in {\mathrm{\S }}^n,𝐁\in {ℝ}^{n\times m},C\in {ℝ}^{p\times n},𝐃\in {ℝ}^{p\times m},𝐑\in {\mathrm{\S }}^m,$and $𝐒\in {\mathrm{\S }}^p,$where $𝐏,𝐑,𝐒,𝐗>0.$The matrix inequality

$\left[\begin{array}{ccccc}-𝐕-{𝐕}^{𝐓}& 𝐕𝐀+𝐏& 𝐕𝐁& \mathrm{𝟎}& 𝐕\\ *& -2𝐏+𝐗& \mathrm{𝟎}& {𝐂}^{𝐓}& \mathrm{𝟎}\\ *& *& -𝐑& {𝐃}^{𝐓}& \mathrm{𝟎}\\ *& *& *& -𝐒& \mathrm{𝟎}\\ *& *& *& *& -𝐗\end{array}\right]<0$

implies the inequality

$\left[\begin{array}{ccc}𝐏𝐀+{𝐀}^{𝐓}𝐏& 𝐏𝐁& {𝐂}^{𝐓}\\ *& -𝐑& {𝐃}^{𝐓}\\ *& *& -𝐒\end{array}\right]$

Example 3

Consider matrices $𝐀,𝐕\in {ℝ}^{n\times n},𝐐,𝐗\in {\mathrm{\S }}^n,𝐁\in {ℝ}^{n\times m},C\in {ℝ}^{p\times n},𝐃\in {ℝ}^{p\times m},𝐑\in {\mathrm{\S }}^m,$and $𝐒\in {\mathrm{\S }}^p,$where $𝐐,𝐑,𝐒,𝐗>0.$The matrix inequality

$\left[\begin{array}{ccccc}-𝐕-{𝐕}^{𝐓}& {𝐕}^{𝐓}{𝐀}^{𝐓}+𝐐& \mathrm{𝟎}& {𝐕}^{𝐓}𝐂& {𝐕}^{𝐓}\\ *& -2𝐐+𝐗& 𝐁& \mathrm{𝟎}& \mathrm{𝟎}\\ *& *& -𝐑& {𝐃}^{𝐓}& \mathrm{𝟎}\\ *& *& *& -𝐒& \mathrm{𝟎}\\ *& *& *& *& -𝐗\end{array}\right]<0$

implies the inequality

$\left[\begin{array}{ccc}𝐀𝐐+𝐐{𝐀}^{𝐓}& 𝐁& 𝐐{𝐂}^{𝐓}\\ *& -𝐑& {𝐃}^{𝐓}\\ *& *& -𝐒\end{array}\right]$

Related Pages

• Projection Lemma - The projection lemma.
• Reciprocal Projection Lemma - The reciprocal projection lemma.

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
• Norm-Preserving Dilations - Norm-preserving dilations and their applications to optimal error bounds (Davis, Kahan, Weinberger).