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

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

and $N_a$ is called a right orthogonal complement of $A$ if it satisfies

$A{N}_a=0,\text{rank}(N_a)=n-\text{rank}(A)$.

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

$N_p^T{P}^T=0,N_q^T{Q}^T=0$.

WIP, additional references to be added

External Links

A list of references documenting and validating the LMI.