WIP, Description in progress
Consider the set of matrices
A={A=[0(n−1)×11(n−1)×(n−1)−a0⋯−an−1]|aj_≤aj≤aj¯,j=0,1,2,⋯,n−1}{\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 A{\displaystyle {\mathcal {A}}} is Hurwitz if and only if there exist Pi∈mathbbSn,i=1,2,3,4{\displaystyle {\textbf {P}}_{i}\in mathbb{S}^{n},i=1,2,3,4} where Pi>0,i=1,2,3,4,{\displaystyle {\textbf {P}}_{i}>0,i=1,2,3,4,} such that
PiAi+AiTPi<0,i=1,2,3,4,{\displaystyle {\textbf {P}}_{i}A_{i}+{\textbf {A}}_{i}^{T}{\textbf {P}}_{i}<0,\quad i=1,2,3,4,}
where
Ai=[[0(n−1)×11(n−1)×(n−1)]ai],i=1,2,3,4,{\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,}
a1=−[a_0a_1a¯2a¯3⋯a_n−4a_n−3a¯n−2a¯n−1],{\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}],}
a2=−[a_0a¯1a¯2a_3⋯a_n−4a¯n−3a¯n−2a_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}],}
a3=−[a¯0a_1a_2a¯3⋯a¯n−4a_n−3a_n−2a¯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}],}
a4=−[a¯0a¯1a_2a_3⋯a¯n−4a¯n−3a_n−2a_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}].}
WIP, additional references to be added
A list of references documenting and validating the LMI.