# LMIs in Control/pages/KBH theorem

WIP, Description in progress

Consider the set of matrices

${\displaystyle {\mathcal {A}}=\{{\textbf {A}}={\begin{bmatrix}{\textbf {0}}_{(n-1)\times 1}&&{\textbf {1}}_{(n-1)\times (n-1)}\\-a_{0}&\cdots &-a_{n-1}\end{bmatrix}}|{\underline {a_{j}}}\leq a_{j}\leq {\overline {a_{j}}},j=0,1,2,\cdots ,n-1\}}$

Every matrix in the set ${\displaystyle {\mathcal {A}}}$  is Hurwitz if and only if there exist ${\displaystyle {\textbf {P}}_{i}\in mathbb{S}^{n},i=1,2,3,4}$  where ${\displaystyle {\textbf {P}}_{i}>0,i=1,2,3,4,}$  such that

${\displaystyle {\textbf {P}}_{i}A_{i}+{\textbf {A}}_{i}^{T}{\textbf {P}}_{i}<0,\quad i=1,2,3,4,}$

where

${\displaystyle {\textbf {A}}_{i}={\begin{bmatrix}{\begin{bmatrix}{\textbf {0}}_{(n-1)\times 1}&{\textbf {1}}_{(n-1)\times (n-1)}\end{bmatrix}}\\{\textbf {a}}_{i}\end{bmatrix}},\quad i=1,2,3,4,}$

${\displaystyle {\textbf {a}}_{1}=-[{\underline {a}}_{0}\quad {\underline {a}}_{1}\quad {\overline {a}}_{2}\quad {\overline {a}}_{3}\quad \cdots \quad {\underline {a}}_{n-4}\quad {\underline {a}}_{n-3}\quad {\overline {a}}_{n-2}\quad {\overline {a}}_{n-1}],}$

${\displaystyle {\textbf {a}}_{2}=-[{\underline {a}}_{0}\quad {\overline {a}}_{1}\quad {\overline {a}}_{2}\quad {\underline {a}}_{3}\quad \cdots \quad {\underline {a}}_{n-4}\quad {\overline {a}}_{n-3}\quad {\overline {a}}_{n-2}\quad {\underline {a}}_{n-1}],}$

${\displaystyle {\textbf {a}}_{3}=-[{\overline {a}}_{0}\quad {\underline {a}}_{1}\quad {\underline {a}}_{2}\quad {\overline {a}}_{3}\quad \cdots \quad {\overline {a}}_{n-4}\quad {\underline {a}}_{n-3}\quad {\underline {a}}_{n-2}\quad {\overline {a}}_{n-1}],}$

${\displaystyle {\textbf {a}}_{3}=-[{\overline {a}}_{0}\quad {\underline {a}}_{1}\quad {\underline {a}}_{2}\quad {\overline {a}}_{3}\quad \cdots \quad {\overline {a}}_{n-4}\quad {\underline {a}}_{n-3}\quad {\underline {a}}_{n-2}\quad {\overline {a}}_{n-1}],}$

${\displaystyle {\textbf {a}}_{4}=-[{\overline {a}}_{0}\quad {\overline {a}}_{1}\quad {\underline {a}}_{2}\quad {\underline {a}}_{3}\quad \cdots \quad {\overline {a}}_{n-4}\quad {\overline {a}}_{n-3}\quad {\underline {a}}_{n-2}\quad {\underline {a}}_{n-1}].}$