LMIs in Control/pages/KBH theorem

WIP, Description in progress

Kharitonov-Bernstein-Haddad (KBH) Theorem

Consider the set of matrices

$𝒜=\{\mathbf{\text{A}}=\left[\begin{array}{ccc}{\mathbf{\text{0}}}_{(n-1)\times 1}& & {\mathbf{\text{1}}}_{(n-1)\times (n-1)}\\ -a_0& \cdots & -a_{n-1}\end{array}\right]|\underset{\_}{a_j}\le a_j\le \overline{a_j},j=0,1,2,\cdots ,n-1\}$

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

${\mathbf{\text{P}}}_i{A}_i+{\mathbf{\text{A}}}_i^T{\mathbf{\text{P}}}_i<0,i=1,2,3,4,$

where

${\mathbf{\text{A}}}_i=\left[\begin{array}{c}\left[\begin{array}{cc}{\mathbf{\text{0}}}_{(n-1)\times 1}& {\mathbf{\text{1}}}_{(n-1)\times (n-1)}\end{array}\right]\\ {\mathbf{\text{a}}}_i\end{array}\right],i=1,2,3,4,$

${\mathbf{\text{a}}}_1=-[{\underset{\_}{a}}_0{\underset{\_}{a}}_1{\bar{a}}_2{\bar{a}}_3\cdots {\underset{\_}{a}}_{n-4}{\underset{\_}{a}}_{n-3}{\bar{a}}_{n-2}{\bar{a}}_{n-1}],$

${\mathbf{\text{a}}}_2=-[{\underset{\_}{a}}_0{\bar{a}}_1{\bar{a}}_2{\underset{\_}{a}}_3\cdots {\underset{\_}{a}}_{n-4}{\bar{a}}_{n-3}{\bar{a}}_{n-2}{\underset{\_}{a}}_{n-1}],$

${\mathbf{\text{a}}}_3=-[{\bar{a}}_0{\underset{\_}{a}}_1{\underset{\_}{a}}_2{\bar{a}}_3\cdots {\bar{a}}_{n-4}{\underset{\_}{a}}_{n-3}{\underset{\_}{a}}_{n-2}{\bar{a}}_{n-1}],$

${\mathbf{\text{a}}}_3=-[{\bar{a}}_0{\underset{\_}{a}}_1{\underset{\_}{a}}_2{\bar{a}}_3\cdots {\bar{a}}_{n-4}{\underset{\_}{a}}_{n-3}{\underset{\_}{a}}_{n-2}{\bar{a}}_{n-1}],$

${\mathbf{\text{a}}}_4=-[{\bar{a}}_0{\bar{a}}_1{\underset{\_}{a}}_2{\underset{\_}{a}}_3\cdots {\bar{a}}_{n-4}{\bar{a}}_{n-3}{\underset{\_}{a}}_{n-2}{\underset{\_}{a}}_{n-1}].$

WIP, additional references to be added

External Links

A list of references documenting and validating the LMI.