LMIs in Control/Matrix and LMI Properties and Tools/Strict Projection Lemma

Introduction

Projection Lemma is also known as Matrix Elimination Lemma. Strict Projection Lemma is one of the characteristics of the Projection Lemma.

Strict Projection Lemma

Consider $\Psi \in \mathbb {S} ^{n}$, $G\in \mathbb {R} ^{n\times m}$, $\Lambda \in \mathbb {R} ^{m\times p}$, and $H\in \mathbb {R} ^{n\times p}.$ There exists $\Lambda$ such that:

${\begin{aligned}\ \Psi +G\Lambda H^{T}+H\Lambda ^{T}G^{T}<0,\end{aligned}}$

if and only if,

${\begin{aligned}\ N_{G}^{T}\Psi N_{G}<0\end{aligned}}$,

${\begin{aligned}\ N_{H}^{T}\Psi N_{H}<0\end{aligned}}$,

where ${\mathcal {R}}(N_{G})={\mathcal {N}}(G^{T})$ and ${\mathcal {R}}(N_{H})={\mathcal {N}}(H^{T})$.

External Links

A list of references documenting and validating the LMI.