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

$$ \begin{bmatrix} \bold{PA}+\bold{A^TP}-\bold{P} & \bold{P}\\ * & \bold{-P}\end{bmatrix} <0, $$

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

$$ \begin{bmatrix} \bold{A^T} & \bold{1} & \bold{0} \\ \bold{1} & \bold{0} & \bold{1}\end{bmatrix} \begin{bmatrix} \bold{0} & \bold{P} & \bold{0} \\ \begin{bmatrix} \bold{A} & \bold{1}\\ \bold{1} & \bold{0} \\ \bold{0} & \bold{1} \end{bmatrix}<0. $$                  (1)
 * & \bold{-P} & \bold{0}\\
 * &*&\bold{-P}\end{bmatrix}

Then since $$\bold{P}>0,$$

$$ \begin{bmatrix} \bold{-P} & \bold{0}\\ * & \bold{-P}\end{bmatrix}<0, $$

which is equivalent to

$$                           \begin{bmatrix} \bold{0} & \bold{1} & \bold{0} \\ \bold{0} & \bold{0} & \bold{1}\end{bmatrix} \begin{bmatrix} \bold{0} & \bold{P} & \bold{0} \\ \begin{bmatrix} \bold{0} & \bold{0}\\ \bold{1} & \bold{0} \\ \bold{0} & \bold{1} \end{bmatrix}<0. $$                       (2)
 * & \bold{-P} & \bold{0}\\
 * &*&\bold{-P}\end{bmatrix}

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

$$\bold{\Phi}(\bold{P})+\bold{G}(\bold{A})\bold{VH^T}+\bold{HV^TG^T}(\bold{A}),$$                                    (3)

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

$$                           \bold{G}(\bold{A})=\begin{bmatrix}\bold{-1}\\ \bold{A^T}\\ \bold{1} \end{bmatrix}, \bold{H}=\begin{bmatrix} \bold{1} \\ \bold{0} \\ \bold{0} \end{bmatrix}, $$

we can now rewrite the inequality (3) as

$$ \begin{bmatrix} -(\bold{V}+\bold{V^T}) & \bold{V^TA}+\bold{P} & \bold{V^T}\\ \end{bmatrix}<0, $$
 * & \bold{-P} & \bold{0}\\
 * & * & \bold{-P}

which is the new dilated inequality.

Examples
Some useful examples of dilated matrix inequalities are presented here.

Example 1

Consider matrices $$\bold{A,G}\in \R^{n\times n}, \bold{\Delta} \in \R^{m\times m}, \bold{P}\in \S^n, \delta_1, \delta_2, a, b \in \R_{>0},$$where $$\bold{P}>0$$and $$b=a^{-1}.$$ The following matrix inequalities are equivalent:

$$\bold{AP}+\bold{PA^T}+\delta_1\bold{P}+\delta_2\bold{APA^T}+\bold{P\Delta^T\Delta P}<0;$$

$$ \begin{bmatrix} \bold{0}&\bold{-P}&\bold{P}&\bold{0}&\bold{P\Delta^T}\\ \end{bmatrix}+He( \begin{bmatrix} \bold{A}\\ \bold{1}\\ \bold{0}\\ \bold{0}\\ \bold{0}\\ \end{bmatrix} \bold{G} \begin{bmatrix} \bold{1} & -b\bold{1} & b\bold{1} & \bold{1} & b\bold{\Delta}^T \end{bmatrix})<0. $$
 * &\bold{0}&\bold{0}&\bold{-P}&\bold{0}\\
 * &*&-\delta_1^{-1}\bold{P}&\bold{0}&\bold{0}\\
 * &*&*&-\delta_2^{-1}\bold{P}&\bold{0}\\
 * &*&*&*&\bold{-1}\\

Example 2

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

$$ \begin{bmatrix} -\bold{V}-\bold{V^T} & \bold{VA}+\bold{P} & \bold{VB} & \bold{0} & \bold{V}\\ \end{bmatrix}<0 $$
 * & -2\bold{P}+\bold{X} & \bold{0} & \bold{C^T} & \bold{0}\\
 * & * & -\bold{R} & \bold{D^T} & \bold{0}\\
 * & * & * & -\bold{S} & \bold{0}\\
 * & * & * & * & -\bold{X}\\

implies the inequality

$$ \begin{bmatrix} \bold{PA}+\bold{A^TP} & \bold{PB} & \bold{C^T}\\ \end{bmatrix} $$
 * & -\bold{R} & \bold{D^T}\\
 * & * & -\bold{S}

Example 3

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

$$ \begin{bmatrix} -\bold{V}-\bold{V^T} & \bold{V^TA^T}+\bold{Q} & \bold{0} & \bold{V^TC} & \bold{V^T}\\ \end{bmatrix}<0 $$
 * & -2\bold{Q}+\bold{X} & \bold{B} & \bold{0} & \bold{0}\\
 * & * & -\bold{R} & \bold{D^T} & \bold{0}\\
 * & * & * & -\bold{S} & \bold{0}\\
 * & * & * & * & -\bold{X}\\

implies the inequality

$$ \begin{bmatrix} \bold{AQ}+\bold{QA^T} & \bold{B} & \bold{QC^T}\\ \end{bmatrix} $$
 * & -\bold{R} & \bold{D^T}\\
 * & * & -\bold{S}

Related Pages

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