LMIs in Control/Matrix and LMI Properties and Tools/Dualization Lemma

Dualization Lemma

Consider $P_i\in {\text{S}}^n$ and the subspaces $U,V$, where $P$ is invertible and $U+V={\text{R}}^n$. The following are equivalent.

$X^{T}PX<0$ for all $X\in \text{U}$\$\left\{0\right\}$ and $X^{T}PX\ge 0$ for all $X\in \text{V}$.

$X^T{P}^{-1}X>0$ for all $X\in {\text{U}}^{\mathrm{\perp }}$\$\left\{0\right\}$ and $X^T{P}^{-1}X\le 0$ for all $X\in {\text{V}}^{\mathrm{\perp }}$.

Example

Consider the matrices $Q\in {\text{S}}^n,S\in {\text{R}}^{n\times m},R\in {\text{S}}^m,M\in {\text{R}}^{m\times n}$ where $R\ge 0,$ which define the quadratic matrix inequality

$\begin{array}{c}\left[\begin{array}{cc}1& M\end{array}\right]\left[\begin{array}{cc}Q& S\\ S^T& R\end{array}\right]\left[\begin{array}{c}1\\ M\end{array}\right]<0.(1)\end{array}$

Define $\begin{array}{c}P=\left[\begin{array}{cc}Q& S\\ S^T& R\end{array}\right],U=R(\left[\begin{array}{c}0\\ 1\end{array}\right])\end{array}$ where $U+V=R^{n+m}$. Notice that $(1)$ is equivalent to $X^{T}PX<0$ for all $X\in \text{U}$\$\left\{0\right\}$.Additionally, $X^{T}PX<0\ge$ for all $X\in \text{V}$ is euaivalent to

$\begin{array}{c}\left[\begin{array}{cc}0& 1\end{array}\right]\left[\begin{array}{cc}Q& S\\ S^T& R\end{array}\right]\left[\begin{array}{c}0\\ 1\end{array}\right]=R\ge 0,\end{array}$

which is satisfied based on the definition of $R$ . By the dualization lemma, $(1)$ is satisfied with $R\ge 0$ if and only if

$\begin{array}{c}\left[\begin{array}{cc}-M^T& 1\end{array}\right]\left[\begin{array}{cc}\tilde{Q}& \tilde{S}\\ {\tilde{S}}^T& \tilde{R}\end{array}\right]\left[\begin{array}{c}-M^T\\ 1\end{array}\right]>0,\tilde{Q}\le 0,\end{array}$

where $\begin{array}{c}\left[\begin{array}{cc}\tilde{Q}& \tilde{S}\\ {\tilde{S}}^T& \tilde{R}\end{array}\right]={\left[\begin{array}{cc}Q& S\\ S^T& R\end{array}\right]}^{-1},U^{\mathrm{\perp }}=N([1M^T])=R(\left[\begin{array}{c}-M^T\\ 1\end{array}\right])\end{array}$ , and $\begin{array}{c}{V}^{\mathrm{\perp }}=N([01])=R(\left[\begin{array}{c}1\\ 0\end{array}\right])\end{array}$.

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).