LMIs in Control/Stability Analysis/Kharitonov-Bernstein-Haddad

Kharitonov-Bernstein-Haddad

Consider the set of matrices A = { $\begin{array}{c}A=\left[\begin{array}{cc}{0}_{(n-1)\times 1}& 1_{(n-1)\times (n-1)}\\ -a_0& -a_{n-1}\end{array}\right]|{\underset{\_}{a}}_j\le {\underset{\_}{a}}_j\le {\bar{a}}_j,j=1,2,3,4,...,n-1\end{array}$ }

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

$\begin{array}{c}{P}_i{A}_i+A_i^T{P}_i<0i=1,2,3,4,\end{array}$

where

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

$\begin{array}{c}{a}_1=-\left[\begin{array}{ccccccccc}{\underset{\_}{a}}_0& {\underset{\_}{a}}_1& {\bar{a}}_2& {\bar{a}}_2& ...& {\underset{\_}{a}}_{n-4}& {\underset{\_}{a}}_{n-3}& {\bar{a}}_{n-2}& {\bar{a}}_{n-1}\end{array}\right],\end{array}$

$\begin{array}{c}{a}_2=-\left[\begin{array}{ccccccccc}{\underset{\_}{a}}_0& {\bar{a}}_1& {\bar{a}}_2& {\underset{\_}{a}}_2& ...& {\underset{\_}{a}}_{n-4}& {\bar{a}}_{n-3}& {\bar{a}}_{n-2}& {\underset{\_}{a}}_{n-1}\end{array}\right],\end{array}$

$\begin{array}{c}{a}_3=-\left[\begin{array}{ccccccccc}{\bar{a}}_0& {\underset{\_}{a}}_1& {\underset{\_}{a}}_2& {\bar{a}}_2& ...& {\bar{a}}_{n-4}& {\underset{\_}{a}}_{n-3}& {\underset{\_}{a}}_{n-2}& {\bar{a}}_{n-1}\end{array}\right],\end{array}$

$\begin{array}{c}{a}_4=-\left[\begin{array}{ccccccccc}{\bar{a}}_0& {\bar{a}}_1& {\underset{\_}{a}}_2& {\underset{\_}{a}}_2& ...& {\bar{a}}_{n-4}& {\bar{a}}_{n-3}& {\underset{\_}{a}}_{n-2}& {\underset{\_}{a}}_{n-1}\end{array}\right],\end{array}$

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

$\begin{array}{c}{A}_i{Q}_i+Q_i{A}_i^T<0i=1,2,3,4,\end{array}$

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.