LMIs in Control/pages/Projection Lemma

WIP, Description in progress

A condition for eliminating a variable in an LMI using orthogonal complements is presented.

Definition 1: Orthogonal Complements

Let $A\in \mathbb {R} ^{m\times n}$, Then, $M_{a}$ is called a left orthogonal complement of $A$ if it satisfies

$M_{a}A=0,\quad {\text{rank}}(M_{a})=m-{\text{rank}}(A)$;

and $N_{a}$ is called a right orthogonal complement of $A$ if it satisfies

$AN_{a}=0,\quad {\text{rank}}(N_{a})=n-{\text{rank}}(A)$.

Using the definition of orthogonal complements, we have the following projection lemma:

Projection Lemma

Let $P$, $Q$ and $H=H^{T}$ be given matrices of appropriate dimensions, $N_{p}$ and $N_{q}$ be the right orthogonal complement of $P$ and $Q$, respectively. Then, there exists $X$ such that

$H+P^{T}X^{T}Q+Q^{T}XP<0$

if and only if

$N_{p}^{T}P^{T}=0,\quad N_{q}^{T}Q^{T}=0$.

WIP, additional references to be added

External Links

A list of references documenting and validating the LMI.