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

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

if and only if,

$\begin{array}{c}{N}_G^T\mathrm{\Psi }{N}_G<0\end{array}$,

$\begin{array}{c}{N}_H^T\mathrm{\Psi }{N}_H<0\end{array}$,

where $ℛ(N_G)=𝒩(G^T)$ and $ℛ(N_H)=𝒩(H^T)$.

External Links

A list of references documenting and validating the LMI.